Datasets:
Answer audit
Every answer in this dataset was checked against the problem it belongs to by a symbolic engine (SymPy), and a share of the answers was checked a second time by an independent web calculator. The machine-made exercises were checked by a second, per-topic verifier that re-derives the answer from the published statement without using the generator. This file reports what was checked, what held, and what the source material got wrong.
Totals
- problems in the dataset: 12,592
- rows in the scored
verifiedconfig: 2,755 - answers refuted by the engine (concrete counterexample): 68
- answers refuted and replaced by a re-verified recomputation: 50
- answers only suspected to be wrong (the cause is not settled): 271
- rows set aside as
disputed(a third read of the source disagreed): 85 - rows with no answer at all (
no-gold): 5,185
By collection
| collection | rows | verified | corrected | refuted | suspected | disputed | unverifiable | no-gold |
|---|---|---|---|---|---|---|---|---|
algebra |
2,327 | 537 | 1 | 6 | 58 | 6 | 1,348 | 371 |
circles |
853 | 0 | 0 | 0 | 0 | 0 | 9 | 844 |
core |
3,617 | 1,328 | 18 | 52 | 141 | 78 | 1,550 | 450 |
exercises |
520 | 520 | 0 | 0 | 0 | 0 | 0 | 0 |
parameters |
1,287 | 117 | 23 | 6 | 66 | 0 | 913 | 162 |
planimetry |
2,000 | 2 | 0 | 0 | 3 | 0 | 172 | 1,823 |
syllabus |
1,751 | 201 | 8 | 3 | 3 | 1 | 232 | 1,303 |
trigonometry |
237 | 0 | 0 | 1 | 0 | 0 | 4 | 232 |
What each status means
| status | meaning |
|---|---|
verified |
the answer was substituted back into the problem and the equation held |
corrected |
the original answer was refuted; it was replaced by a recomputed answer that the engine then re-verified |
refuted |
the engine produced a concrete counterexample: the root was substituted back and failed, the tuple satisfies no equation of the system, or the interval misses a point that satisfies the inequality. The answer is demonstrably wrong. |
suspected |
the engine found a difference but it is not settled whether the source or the transcription is at fault: a difference in a radical's scope or in a notational convention can produce this, and so can a mistake in the source. These are not counted as errors in the source. |
disputed |
a third independent read of the source disagreed with the bulk transcription, so the row is not trusted |
unverifiable |
an answer exists but is outside the reach of automatic checking - word problems, open-ended proofs, answers that depend on a figure |
no-gold |
the source carries no answer for this problem |
Neither unverifiable, suspected, disputed nor no-gold means the answer is
wrong. They mean nobody checked it, or that the check was inconclusive.
The verified config is the honest scored split: every item in it has been
checked by a symbolic engine.
How the checking works
| problem type | check |
|---|---|
| equation | every root is substituted back into the equation; a solution family is checked at several integer values of its parameter |
| system of equations | every ordering of the tuple is substituted into every equation |
| inequality | the endpoints of the interval are evaluated exactly, and a grid of sample points is masked against the inequality |
identity (prove that A = B) |
both sides are compared numerically at a grid of sample points |
| simplify | the expression is evaluated numerically and compared with the stated result |
| limit / integral | the expression is evaluated symbolically with SymPy |
extremum (the greatest value is ...) |
a grid over the stated domain is searched for a counterexample to the claimed bound |
Answers given in degrees are converted to radians before checking, and
degrees-minutes-seconds (3^\circ45') are parsed as 3.75 degrees.
Answers that were refuted and replaced
| id | original answer | recomputed answer |
|---|---|---|
dgn.core.2.343 |
see the source | x_1 = \sqrt{3}; x_2 = \frac{\sqrt{3}}{3}; x_3 = -2\sqrt{3} |
dgn.core.2.356 |
see the source | x = - \sqrt{2}; x = \sqrt{2} |
dgn.core.5.037 |
see the source | x = - \frac{5 \pi}{6}; x = - \frac{4 \pi}{5}; x = - \frac{3 \pi}{5}; x = - \frac{\pi}{2}; x = - \frac{2 \pi}{5}; x = - \frac{\pi}{5}; x = - \frac{\pi}{6}; x = 0; x = \frac{\pi}{6}; x = \frac{\pi}{5}; x = \frac{2 \pi}{5}; x = \frac{\pi}{2}; x = \frac{3 \pi}{5}; x = \frac{4 \pi}{5}; x = \frac{5 \pi}{6}; x = \pi |
dgn.core.5.107 |
see the source | x = - \frac{\pi}{4}; x = \frac{3 \pi}{4} |
dgn.core.5.123 |
see the source | x = \operatorname{atan}{\left(\frac{1}{5} \right)}; x = \frac{\pi}{4} |
dgn.core.5.162 |
see the source | x = 0; x = \frac{2 \pi}{3} |
dgn.core.5.217 |
see the source | x = - \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}; x = - \frac{\pi}{8}; x = - \operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)}; x = \frac{3 \pi}{8} |
dgn.core.5.232 |
see the source | x = - \operatorname{atan}{\left(\frac{1}{5} + \frac{\sqrt{26}}{5} \right)}; x = - \frac{\pi}{8}; x = - \operatorname{atan}{\left(\frac{1}{5} - \frac{\sqrt{26}}{5} \right)}; x = \frac{3 \pi}{8} |
dgn.core.5.413 |
see the source | x = 0; x = \pi |
dgn.core.5.421 |
see the source | x = - \frac{\pi}{4} |
dgn.core.7.038 |
see the source | x = \frac{1}{100}; x = 100 |
dgn.core.7.039 |
see the source | x = \frac{1}{10}; x = 1000 |
dgn.core.7.096 |
see the source | x = \left(\frac{3^{\frac{2}{3}}}{3}\right)^{\frac{\sqrt[3]{3} \left(- \left(9 + \sqrt{105}\right)^{\frac{2}{3}} + 2 \sqrt[3]{3}\right)}{\sqrt[3]{9 + \sqrt{105}}}} |
dgn.core.7.242 |
see the source | x = \frac{1}{10}; x = \sqrt{10}; x = 100 |
dgn.core.7.243 |
see the source | x = \frac{1}{1000}; x = 1; x = 10 |
dgn.core.7.250 |
see the source | x = \frac{2^{\frac{2}{5}}}{8}; x = 1; x = 4 |
dgn.core.8.276 |
see the source | (\sqrt[5]{5},5) |
dgn.core.10.122 |
see the source | x = \frac{9}{2}; x = 6 |
dgn.algebra.454 |
see the source | x_1 = \frac{1}{2\sqrt{3}(\log_3 5 - 1)}; \; x_2 = \frac{4\log_3 2 + 1}{2\sqrt{3}(\log_3 5 - 1)} |
dgn.syllabus.p148.30 |
see the source | x = -\frac{\pi}{4} + \pi n, \ x = -\arctan(2 + \sqrt{3}) + \pi n, \ x = -\arctan(2 - \sqrt{3}) + \pi n, \ n \in \mathbf{Z} |
dgn.syllabus.p149.52 |
see the source | x = \frac{\pi n}{2}, \ n \in \mathbf{Z} |
dgn.syllabus.p149.61 |
see the source | x = \pi n, \ x = \frac{\pi}{6} + \pi n, \ x = \frac{5\pi}{6} + \pi n, \ x = \frac{7\pi}{12} + \pi n, \ x = \frac{11\pi}{12} + \pi n, \ n \in \mathbf{Z} |
dgn.syllabus.p149.65 |
see the source | x = \pm\frac{\pi}{4} + 2\pi n, \ x = \arctan\frac{1}{2} + (-1)^n \arcsin\frac{1}{\sqrt{10}} + \pi n, \ n \in \mathbf{Z} |
dgn.syllabus.p149.76 |
see the source | x = \frac{\pi}{2} + 2\pi n, \ n \in \mathbf{Z} |
dgn.syllabus.p149.77 |
see the source | x = \frac{\pi}{8} + \frac{\pi n}{4}, \ x = \pm\frac{\pi}{3} + 2\pi n, \ n \in \mathbf{Z} |
dgn.syllabus.p149.89 |
see the source | x = \frac{2\pi n}{7}, \ x = \frac{\pi}{9} + \frac{2\pi k}{9}, \ n \in \mathbf{Z}, \ k \in \mathbf{Z}, \ n \neq 7p, \ (p \in \mathbf{Z}), \ 2k + 1 \neq 9m, \ (m \in \mathbf{Z}) |
dgn.syllabus.p151.143 |
see the source | x = \pi n, \ x = \pm 0.205560277 + 2\pi n, \ x = \pm 2.019445231 + 2\pi n, \ x = \pm 2.765378489 + 2\pi n, \ n \in \mathbf{Z} |
dgn.parameters.p111.216 |
see the source | 72 |
dgn.parameters.243 |
see the source | x_1 = -1/\sqrt{2},\quad x_2 = (\sqrt{6}-\sqrt{2})/4 |
dgn.parameters.p157.3 |
see the source | {-\pi} \cup \{3\pi\} \cup [\pi/2; 3\pi/4) \cup [5\pi/2; 11\pi/4 - \arcsin((1-\sqrt{2})/\sqrt{2})] |
dgn.parameters.p160.1 |
see the source | (-\infty; -8/7] \cup [0; +\infty) |
dgn.parameters.p160.2 |
see the source | (-1; 0] |
dgn.parameters.p161.322 |
see the source | (-\infty;\, -\frac{53}{3}] \cup [\frac{73}{7};\, +\infty) |
dgn.parameters.p161.323 |
see the source | (8/3; 6] |
dgn.parameters.p184.3 |
see the source | (-\infty; -2/3] \cup \{0\} \cup [2/3; +\infty) |
dgn.parameters.p188.1 |
see the source | (-\infty;\, 1] \cup [5;\, +\infty) |
dgn.parameters.p228.2 |
see the source | x \in (-1; 1) |
dgn.parameters.p237.7 |
see the source | \{ (-\pi/4 + 2\pi m;\ \pi/4 + 2\pi k);\ (3\pi/4 + 2\pi m;\ \pi/4 + 2\pi k);\ k, m \in \mathbb{Z} \} |
dgn.parameters.p238.7 |
see the source | \varnothing \text{ (the system has no solutions)} |
dgn.parameters.Variant 2004 (iyul), psychology Faculty, 3. Task 1 |
see the source | x = \log_{4}(4 + 3\sqrt{2}) |
dgn.parameters.Variant 2004 (iyul), psychology Faculty, 3. Task 2 |
see the source | [-5;\, -1) \cup (3;\, 4] \cup (7;\, +\infty) |
dgn.parameters.p263.3 |
see the source | x \in [1; 2) \cup (2; \sqrt{5}] |
dgn.parameters.p267.2 |
see the source | \varnothing |
dgn.parameters.p268.Task 6 |
see the source | a \in [0;\, 2\pi - 6] \cup (2\pi - 4;\, \pi] |
dgn.parameters.p271.2 |
see the source | [1/4; +\infty) |
dgn.parameters.p274.6 |
see the source | 3 |
dgn.parameters.p280.4 |
see the source | (-\infty;\, 1-\sqrt{14}] \cup [-\sqrt{\frac{17}{3}};\, -2) \cup (1;\, \sqrt{\frac{17}{3}}] |
dgn.parameters.p289.2 |
see the source | (-\infty; -2] |
dgn.parameters.p290.2 |
see the source | (-3/2; -\sqrt{3/2}) \cup [-1; 0) \cup (0; \sqrt{3/2}) |
dgn.parameters.Task 1 (Variant 2006, aprel) |
see the source | [-2; (7+\sqrt{21})/2) |
Answers the engine refuted
These are answers that do not hold: the engine produced a
concrete counterexample. They are excluded from the verified config
because no model can be scored correct on them.
| id | collection | answer | what the engine found |
|---|---|---|---|
dgn.core.1.202 |
core |
\sqrt{mn} |
the source was re-read at high resolution and its answer was refuted exactly - third read of the page disagree … |
dgn.core.1.206 |
core |
\frac{a(a+1)}{2a+1} |
the source was re-read at high resolution and its answer was refuted exactly - third read of the page disagree … |
dgn.core.2.059 |
core |
x=2 |
the source was re-read at high resolution and its answer was refuted exactly - three independent reads agree o … |
dgn.core.2.068 |
core |
(-4;24);\ (-6;-2) |
three independent reads agree on the statement and the printed answer |
dgn.core.2.169 |
core |
x_1=0;\ x_2=4. |
claimed solution 'x=4' fails the equation |
dgn.core.2.205 |
core |
(2;1),\ (6;-3),\ \left(6+2\sqrt{3};-2-2\sqrt{3}\right),\ … |
three independent reads agree on the statement and the printed answer - source error: the source answer does n … |
dgn.core.2.223 |
core |
(49; 49) |
no assignment of the source tuple (49; 49) satisfies the system |
dgn.core.2.240 |
core |
(10;1),\ \left(-21/2;53/12\right) |
three independent reads agree on the statement and the printed answer |
dgn.core.2.243 |
core |
(1;4),(4;1),(-1;-4),(-4;-1) |
three independent reads agree on the statement and the printed answer |
dgn.core.2.295 |
core |
x=65 |
the source was re-read at high resolution and its answer was refuted exactly - third read of the page disagree … |
dgn.core.2.298 |
core |
x=4 |
the source was re-read at high resolution and its answer was refuted exactly - third read of the page disagree … |
dgn.core.5.002 |
core |
x_1=35^\circ+120^\circ\cdot k,\quad x_2=55^\circ+120^\circ … |
three independent reads agree on the statement and the printed answer |
dgn.core.5.039 |
core |
x_1=\frac{\pi k}{5},\ x_2=\frac{\pi}{8}(4k+3). |
three independent reads agree on the statement and the printed answer |
dgn.core.5.121 |
core |
z_1=2\arctan3+2\pi k,\quad z_2=-2\arctan7+2\pi k |
three independent reads agree on the statement and the printed answer |
dgn.core.5.147 |
core |
x=\frac{\pi}{12}(6k+1) |
three independent reads agree on the statement and the printed answer |
dgn.core.5.152 |
core |
x_1=\pi k,\quad x_2=(-1)^k\frac{k\pi}{6}+\frac{\pi k}{2} |
claimed solution 'x=(-1)^k\frac{k\pi}{6}+\frac{\pi k}{2}' fails the equation |
dgn.core.5.174 |
core |
t=\frac{\pi}{2}(4k+1) |
three independent reads agree on the statement and the printed answer |
dgn.core.5.176 |
core |
x_1=\frac{\pi}{6}+\pi k;\quad x_2=(-1)^k\frac{\pi}{8}+\frac … |
three independent reads agree on the statement and the printed answer |
dgn.core.5.177 |
core |
x=(-1)^k\frac{\pi}{3}+\pi k |
claimed solution 'x=(-1)^k\frac{\pi}{3}+\pi k' fails the equation |
dgn.core.5.221 |
core |
x_1=-\frac{\pi}{16}(4k+1),\quad x_2=-\frac{1}{4}\arctan … |
three independent reads agree on the statement and the printed answer |
dgn.core.5.227 |
core |
t=\arctan\frac{4}{5}+2\pi k |
three independent reads agree on the statement and the printed answer |
dgn.core.5.259 |
core |
x=-\frac{\pi}{8}(4k+1) |
three independent reads agree on the statement and the printed answer |
dgn.core.5.269 |
core |
x=-\frac{\pi}{4}(4k+1) |
three independent reads agree on the statement and the printed answer |
dgn.core.5.334 |
core |
t=-\frac{\pi}{4}(4k+1) |
three independent reads agree on the statement and the printed answer |
dgn.core.5.344 |
core |
t=-\frac{\pi}{4}(4k-1) |
claimed solution 't=-\frac{\pi}{4}(4k-1)' fails the equation |
dgn.core.5.348 |
core |
t=(-1)^k\frac{k\pi}{6}+\pi k |
three independent reads agree on the statement and the printed answer |
dgn.core.5.445 |
core |
t=-\frac{\pi}{4}(4k+1) |
three independent reads agree on the statement and the printed answer |
dgn.core.5.472 |
core |
t_1=\frac{\pi}{4}(8k+1),\quad t_2=-\arctan3+\pi(2k+1) |
three independent reads agree on the statement and the printed answer |
dgn.core.7.001 |
core |
x_1=-\frac{1}{2},\ x_2=6 |
the source was re-read at high resolution and its answer was refuted exactly - claimed solution 'x=-\frac{1} … |
dgn.core.7.006 |
core |
x=\frac{5}{3} |
the source was re-read at high resolution and its answer was refuted exactly - three independent reads agree o … |
dgn.core.7.013 |
core |
x=10 |
the source was re-read at high resolution and its answer was refuted exactly - claimed solution 'x=10' fails t … |
dgn.core.7.026 |
core |
x_1=3,\quad x_2=\frac{3}{\log_2 6}=3\log_6 2 |
the source was re-read at high resolution and its answer was refuted exactly - third read of the page disagree … |
dgn.core.7.029 |
core |
x_1=-\frac13,\quad x_2=0 |
the source was re-read at high resolution and its answer was refuted exactly - third read of the page disagree … |
dgn.core.7.035 |
core |
x_1=1\sqrt{10},\quad x_2=100 |
the source was re-read at high resolution and its answer was refuted exactly - third read of the page disagree … |
dgn.core.7.037 |
core |
x_1=1/\sqrt3,\quad x_2=3 |
three independent reads agree on the statement and the printed answer |
dgn.core.7.110 |
core |
x = 5 |
the source was re-read at high resolution and its answer was refuted exactly - third read of the page disagree … |
dgn.core.7.133 |
core |
x=\sqrt[3]{2} |
the source was re-read at high resolution and its answer was refuted exactly - third read of the page disagree … |
dgn.core.7.144 |
core |
x=-2 |
the source was re-read at high resolution and its answer was refuted exactly - three independent reads agree o … |
dgn.core.7.151 |
core |
(9; 16) |
no assignment of the source tuple (9; 16) satisfies the system |
dgn.core.7.160 |
core |
(3; 27) |
three independent reads agree on the statement and the printed answer |
dgn.core.7.244 |
core |
x = \pi n/2, n \in \mathbb{Z} |
claimed solution 'x=\pi n/2' fails the equation |
dgn.core.7.251 |
core |
x = 1/3. |
the source was re-read at high resolution and its answer was refuted exactly - third read of the page disagree … |
dgn.core.7.252 |
core |
x=4 |
the source was re-read at high resolution and its answer was refuted exactly - three independent reads agree o … |
dgn.core.7.258 |
core |
x = 1. |
the source was re-read at high resolution and its answer was refuted exactly - third read of the page disagree … |
dgn.core.7.263 |
core |
x_1=1/8,\ x_2=8. |
claimed solution 'x=1/8' fails the equation |
dgn.core.7.265 |
core |
x=256. |
the source was re-read at high resolution and its answer was refuted exactly - claimed solution 'x=256' fails … |
dgn.core.7.348 |
core |
x=100 |
the source was re-read at high resolution and its answer was refuted exactly - claimed solution 'x=100' fails … |
dgn.core.8.026 |
core |
(-1,\infty) |
three independent reads agree on the statement and the printed answer |
dgn.core.8.283 |
core |
x=3 |
claimed integer 3 gives F=-1.2910247, which violates the inequality |
dgn.core.10.012 |
core |
(-3,2)\cup(2,3) |
three independent reads agree on the statement and the printed answer |
dgn.core.10.166 |
core |
(-4,0)\cup(0,4) |
three independent reads agree on the statement and the printed answer |
dgn.core.11.246 |
core |
-\frac{1}{3} |
third read of the page corrected the row |
dgn.algebra.282 |
algebra |
x = \frac{n+\sqrt{n^2+4}}{2}, n \in \mathbb{N} |
claimed solution 'x=\frac{n+\sqrt{n^2+4}}{2}' fails the equation |
dgn.algebra.777 |
algebra |
-\frac{19}{4}<x<-4 |
the source was re-read at high resolution and its answer was refuted exactly - x=-2.375 satisfies the inequali … |
dgn.algebra.788 |
algebra |
-1 < x < -\frac{5}{6}. |
the source was re-read at high resolution and its answer was refuted exactly - the source answer excludes the … |
dgn.algebra.926 |
algebra |
[-18;-3] \cup (0;2). |
the source was re-read at high resolution and its answer was refuted exactly - the source answer includes the … |
dgn.algebra.p536.2123 |
algebra |
[0;\ \sqrt{5}] |
x=-0.65 satisfies the inequality but is missing from the source answer |
dgn.algebra.p544.2176 |
algebra |
2 \leq a \leq 2 + 2\sqrt{2}. |
claimed endpoint 4.82843 gives F=1.8284271, which violates the inequality |
dgn.syllabus.p190.Example 2 |
syllabus |
(1;2;3),\ (-1;-2;-3) |
no assignment of the source tuple (-1; -2; -3) satisfies the system |
dgn.syllabus.p194.Example 8 |
syllabus |
(2;\sqrt[3]{3}; \sqrt[3]{9}),\ \left(\frac{1}{\sqrt[3]{2}}; … |
no assignment of the source tuple (\frac{1}{\sqrt[3]{2}}; -\sqrt[3]{\frac{9}{2}}; \sqrt[3]{\frac{3}{2}} … |
dgn.syllabus.p222.Example 2 |
syllabus |
(1; 3); \left(\frac{9}{4}; 4\right) |
no assignment of the source tuple (\frac{9}{4}; 4\right) satisfies the system |
dgn.parameters.p121.45 |
parameters |
(x; y; z) = (\tg \alpha; \tg 2\alpha; \tg 4\alpha), where … |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.249 |
parameters |
(\frac{\tan\alpha}{3}; \frac{\tan2\alpha}{3}; \tan4 … |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.252 |
parameters |
(\tan\alpha; \frac{\tan2\alpha}{2}; \frac{\tan4\alpha}{2}), … |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p158.2003.1 |
parameters |
(1; -1) \cup [2; 3) |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p159.1 |
parameters |
(1/4; 1/3) \cup [2; +\infty) |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.Variant 2005 (iyul), geography Faculty, 1 - Task 1 |
parameters |
(x;y) = (-\pi/2 \pm \pi/3 + 2\pi k;\ 2\pi m),\ k,m \in … |
no assignment of the source tuple (-\pi/2 \pm \pi/3 + 2\pi k; \ 2\pi m) satisfies the system |
dgn.trigonometry.p158.26.6 |
trigonometry |
(a) \left[-1; \dfrac{1}{2}\right]. (b) (\cos 2; 1]. (c) … |
the source was re-read at high resolution and its answer was refuted exactly - multi-block question - source e … |
Answers only suspected to be wrong
For these rows a checker found a difference - either the symbolic engine or the independent web calculator - but it is not settled whether the source or the transcription is at fault. A difference in the radical scope or in a notational convention can produce this, so these are not counted as errors in the source and are kept out of the scored split as well.
| id | collection | answer | what was found |
|---|---|---|---|
dgn.core.1.004 |
core |
0.04 |
numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from … |
dgn.core.1.023 |
core |
0.5a^3(a-1) |
the source answer is contradicted at a=3: the expression is exactly 27/4, the source answer gives 27 - reporte … |
dgn.core.1.027 |
core |
-4 |
the source answer is contradicted at a=1, b=4: the expression is exactly (-27649 + 242(1/3) + 5762**(2/3)) … |
dgn.core.1.028 |
core |
\frac{2}{\sqrt[3]{r^2}}\left(r^2-\sqrt{r^4-16}\right) |
the source answer is contradicted at r=2: the expression is exactly 2**(1/3), the source answer gives 42*(1/ … |
dgn.core.1.039 |
core |
y^2 |
the source answer is contradicted at y=1: the expression is exactly (30**(1/10)(sqrt(2) + 3) + 6sqrt(2) + 81 … |
dgn.core.1.041 |
core |
-4 |
the source answer is contradicted at t=1: the expression is exactly 152/25, the source answer gives -4 - repor … |
dgn.core.1.050 |
core |
2\sqrt[6]{a^5/a} |
the source answer is contradicted at a=1: the expression is exactly 2**(11/12)/2, the source answer gives 2 - … |
dgn.core.1.053 |
core |
\sqrt{6x} |
the source answer is contradicted at x=3: the expression is exactly 0, the source answer gives 3*sqrt(2) - rep … |
dgn.core.1.054 |
core |
\sqrt[3]{20x} |
the source answer is contradicted at x=1: the expression is exactly -2sqrt(3) + 2sqrt(2)(4sqrt(6) + 11)**( … |
dgn.core.1.056 |
core |
1/\sqrt[12]{a^2b} |
the source answer is contradicted at a=1, b=2: the expression is exactly 1/2, the source answer gives 2**(11/1 … |
dgn.core.1.059 |
core |
-1 |
the source answer is contradicted at a=2: the expression is exactly -2**(1/4) - 1 - 2**(3/4)/2 + sqrt(sqrt(2) … |
dgn.core.1.061 |
core |
-(\sqrt[4]{x}-\sqrt[4]{y}) |
the source answer is contradicted at x=1, y=2: the expression is exactly 3*(1 - sqrt(2))/(-3 + 2**(1/4) + 2**( … |
dgn.core.1.063 |
core |
a^{1/3}+b^{1/3} |
the source answer is contradicted at a=1, b=2: the expression is exactly (-22(2/3) + 1 + 22**(1/3))/(1 - 2 … |
dgn.core.1.065 |
core |
(\sqrt{m}-\sqrt{n})/m |
the source answer is contradicted at m=1, n=2: the expression is exactly -sqrt(sqrt(5)/4 + 3/4), the source an … |
dgn.core.1.067 |
core |
1 |
the source answer is contradicted at : the expression is exactly (-sqrt(5) + 2*sqrt(6))/(-sqrt(3) + sqrt(2)), … |
dgn.core.1.068 |
core |
\frac{1}{a\left(\sqrt[m]{a}-\sqrt[n]{a}\right)} |
the source answer is contradicted at a=2, m=1, n=2: the expression is exactly 5*sqrt(2)/14 + 4/7, the source a … |
dgn.core.1.069 |
core |
6 |
the source answer is contradicted at : the expression is exactly (-18sqrt(5) - 12(1/3) + 30sqrt(3))/(-3*sq … |
dgn.core.1.073 |
core |
-\sqrt{ac} |
the source answer is contradicted at a=1, b=1, c=2: the expression is exactly 2**(190/213)3(308/639)5(40 … |
dgn.core.1.077 |
core |
\frac{x^{1/3}+y^{1/3}}{\sqrt[6]{x^3y^2}} |
the source answer is contradicted at x=2, y=1: the expression is exactly (-2**(1/6) - 1 + 2**(2/3) + 2**(5/6)) … |
dgn.core.1.080 |
core |
0 |
the source answer is contradicted at m=1, n=1: the expression is exactly -2, the source answer gives 0 - repor … |
dgn.core.1.083 |
core |
5 |
the source answer is contradicted at b=1, c=1, a=1/25: the expression is exactly 5sqrt(51)(sqrt(101) + 20)/5 … |
dgn.core.1.086 |
core |
\frac{1}{\sqrt{a}+\sqrt{2}} |
the source answer is contradicted at a=1: the expression is exactly -1/2 + 2*sqrt(2)/3, the source answer give … |
dgn.core.1.087 |
core |
-3n(m+p) |
the source answer is contradicted at m=1, n=1, p=1: the expression is exactly (sqrt(6) + 23(1/3))(-2**(1/3 … |
dgn.core.1.089 |
core |
(1-a)/\sqrt{a} |
the source answer is contradicted at a=2: the expression is exactly sqrt(3)*(-4 + sqrt(2))/24, the source answ … |
dgn.core.1.098 |
core |
\sqrt[3]{x+y}-\sqrt[3]{x-y} |
the source answer is contradicted at x=2, y=1: the expression is exactly 2**(1/3), the source answer gives -1 … |
dgn.core.1.101 |
core |
2 |
the source answer is contradicted at x=1: the expression is exactly 2/3 - 2*sqrt(2)/3, the source answer gives … |
dgn.core.1.107 |
core |
a/2 |
the source answer is contradicted at a=1, b=2, p=3: the expression is exactly sqrt(2)/(-22(5/6) - 22**(1/3 … |
dgn.core.1.116 |
core |
1/m^2 |
the source answer is contradicted at m=1, n=2: the expression is exactly sqrt(2)/(-62(3/4) - 82**(1/4) + 7 … |
dgn.core.1.117 |
core |
-a^2 |
the source answer is contradicted at a=1, b=2: the expression is exactly -2**(3/4)/2, the source answer gives … |
dgn.core.1.120 |
core |
3/5 |
the source answer is contradicted at : the expression is exactly (32(1/3)7(1/4) + 602(1/3))/(22**(11 … |
dgn.core.1.121 |
core |
31/3 |
the source answer is contradicted at : the expression is exactly 2**(1/3)(206**(1/3) + 63)/(2*(6**(2/3) + 15 … |
dgn.core.1.122 |
core |
\sqrt[12]{32} |
the source answer is contradicted at : the expression is exactly -32(7/12) + 22**(1/6) + 22*(11/12), the … |
dgn.core.1.123 |
core |
0.1 |
the source answer is contradicted at : the expression is exactly -36sqrt(5) - 23**(1/4)sqrt(5) + 8sqrt(30) … |
dgn.core.1.124 |
core |
0 |
the source answer is contradicted at : the expression is exactly 3**(1/3)(-27sqrt(2)3(1/3) - 112**(1/6) … |
dgn.core.1.125 |
core |
2 |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.1.142 |
core |
-\frac{\sqrt{6}}{2} |
the source answer is contradicted at x=sqrt(6): the expression is exactly 0, the source answer gives -sqrt(6)/ … |
dgn.core.1.160 |
core |
\frac{1}{z+2} |
the source answer is contradicted at x=1, z=2: the expression is exactly 1/3, the source answer gives 1/4 - re … |
dgn.core.1.173 |
core |
\frac{x+y}{2} |
the source answer is contradicted at x=1, y=2: the expression is exactly 38/17 + 61*sqrt(2)/34, the source ans … |
dgn.core.1.176 |
core |
\frac{m-8}{2} |
the source answer is contradicted at m=9: the expression is exactly 144*sqrt(5) + 322, the source answer gives … |
dgn.core.1.180 |
core |
\frac{1}{a(3a+b)} |
the source answer is contradicted at a=1, b=2: the expression is exactly 2/5 - 22*(1/6)/5, the source answer … |
dgn.core.1.184 |
core |
\frac{1}{x-\sqrt{2x+1}} |
the source answer is contradicted at x=2: the expression is exactly sqrt(6)/5 + 6/5, the source answer gives - … |
dgn.core.1.186 |
core |
\sqrt{x^3}-\sqrt{y^3} |
the source answer is contradicted at x=1, y=1: the expression is exactly 1/2, the source answer gives 0 - repo … |
dgn.core.1.194 |
core |
x^2\sqrt[3]{y^1} |
the source answer is contradicted at x=1, y=1: the expression is exactly -sqrt(3)(4sqrt(3) + 7)(1/6) + 2 … |
dgn.core.1.197 |
core |
\frac{{}^{4}\!\sqrt a}{6} |
the source answer is contradicted at a=2: the expression is exactly 32(1/12)(-(2 - sqrt(2))(1/6) + 2(1 … |
dgn.core.1.201 |
core |
1/(ab) |
the source answer is contradicted at a=1, b=2: the expression is exactly (2**(1/3)/16 + 1/2)/(1 + 2**(1/4)), t … |
dgn.core.1.222 |
core |
-\frac{\sqrt{x-2}}{x+2} |
the source answer is contradicted at x=3: the expression is exactly -sqrt(5)/5, the source answer gives -1/5 - … |
dgn.core.1.224 |
core |
\sqrt{a}\left(\sqrt{2a+1}-\sqrt{a}\right) |
the source answer is contradicted at a=1: the expression is exactly 1/2 + sqrt(3)/2, the source answer gives - … |
dgn.core.1.226 |
core |
2.4 |
the source answer is contradicted at c=1, a=24/5, b=6/5: the expression is exactly 12*sqrt(163112586)/298741 + … |
dgn.core.1.231 |
core |
\sqrt{p^2-q^2}\sqrt{p} |
the source answer is contradicted at p=2, q=1: the expression is exactly 15 - 2*sqrt(6), the source answer giv … |
dgn.core.1.233 |
core |
\frac{4}{9} |
the source answer is contradicted at b=1, a=9/4: the expression is exactly 2sqrt(-3960sqrt(2) + 3718*sqrt(3 … |
dgn.core.1.236 |
core |
\sqrt[3]{p-q} |
the source answer is contradicted at p=3, q=1: the expression is exactly 2**(7/8)/2, the source answer gives 2 … |
dgn.core.1.237 |
core |
2 |
the source answer is contradicted at m=3: the expression is exactly (-2*(4 - 3**(2/3))(15(2/3) + 10) + 55* … |
dgn.core.1.238 |
core |
1 |
the source answer is contradicted at a=1, b=2: the expression is exactly (1 + sqrt(2))(4/3)/(2*2(5/6) + 3* … |
dgn.core.1.240 |
core |
\sqrt{a}(a+4b) |
the source answer is contradicted at a=1, b=1: the expression is exactly sqrt(5)/5, the source answer gives 5 … |
dgn.core.1.241 |
core |
1 |
the source answer is contradicted at p=1, q=2: the expression is exactly 4, the source answer gives 1 - report … |
dgn.core.1.293 |
core |
1) \sqrt[3]{8}-3\sqrt[3]{5}; 2) \sqrt[6]{8}-6\sqrt[6]{5}; 3) … |
the source answer is an enumerated list - reported as suspected: the independent web calculator contradicts th … |
dgn.core.1.294 |
core |
Equality holds for all n\in[0,3m] |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.1.324 |
core |
b-4a if a\in[0,\sqrt{2}); 2(a-1)^2 if a\in[\sqrt{2},+\infty) |
the source value '2(a-1)^2' does not hold at a=3, where it is the only case that applies - reported as suspect … |
dgn.core.1.329 |
core |
-x if x\in(0,1); x if x\in(1,\infty) |
the source value '-x' does not hold at x=1/2, where it is the only case that applies - reported as suspected, … |
dgn.core.1.330 |
core |
\frac{x-1}{x} if x\in(-\infty,0)\cup[1,\infty); \frac{1-x} … |
the source value '\frac{x-1}{x}' does not hold at x=2, where it is the only case that applies - reported as s … |
dgn.core.2.122 |
core |
a^2x^2+(ab-ac)x-bc=0 |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.2.123 |
core |
ax^2+(b-2a)x+(c-b+a)=0 |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.2.133 |
core |
\frac{215}{27} |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.2.135 |
core |
c=1/3 |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.2.246 |
core |
ax^2+bx+\left(c+\sqrt{b^2-4ac}-a\right)=0 |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.2.342 |
core |
x_1=1.5,\,x_2=0.5+\sqrt{3},\,x_3=0.5-\sqrt{3} |
no candidate solution parsed - reported as suspected: the independent web calculator contradicts the printed a … |
dgn.core.2.345 |
core |
ab. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.2.368 |
core |
x=2 |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
dgn.core.3.326 |
core |
421 |
numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from … |
dgn.core.4.080 |
core |
\tan\frac{m+n}{2}\alpha |
the source answer is contradicted at alpha=1, m=1, n=2: the expression is exactly (-cos(1) + cos(2))/(-sin(2) … |
dgn.core.4.117 |
core |
\tan^6\alpha |
numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from … |
dgn.core.4.145 |
core |
2\sqrt2\sin^2\alpha\cos\left(\frac\pi4+2\alpha\right) |
the source answer is contradicted at alpha=1: the expression is exactly -sqrt(-sin(1) + tan(1)) + sqrt(sin(1) … |
dgn.core.4.241 |
core |
\displaystyle -\frac12\sin2\alpha |
the source answer is contradicted at alpha=1: the expression is exactly sqrt(-cos(2))/sqrt(-cot(1)**2 + tan(1) … |
dgn.core.4.248 |
core |
\displaystyle \sin4\alpha\cos^{-2}4\alpha |
the source answer is contradicted at alpha=1, x=2: the expression is exactly (sin(24) + sin(8))/(cos(16) + 1)* … |
dgn.core.4.264 |
core |
\tan5\alpha |
the source answer is contradicted at alpha=1, x=2: the expression is exactly tan(10), the source answer gives … |
dgn.core.4.265 |
core |
\sin3\alpha |
the source answer is contradicted at alpha=1, x=2: the expression is exactly sin(6), the source answer gives s … |
dgn.core.4.266 |
core |
\cos3\alpha |
the source answer is contradicted at alpha=1, x=2: the expression is exactly cos(6), the source answer gives c … |
dgn.core.4.270 |
core |
2\sin2\alpha |
the source answer is contradicted at alpha=1, x=1: the expression is exactly (-sin(pi/4 + 2) + sin(pi/4 + 6))/ … |
dgn.core.4.272 |
core |
\cot\left(\alpha-\frac{\pi}{4}\right) |
numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from … |
dgn.core.4.291 |
core |
-2\sin\frac{\alpha}{4} |
the source answer is contradicted at alpha=1: the expression is exactly -sqrt(1 - sin(1/2)) + sqrt(sin(1/2) + … |
dgn.core.4.300 |
core |
\frac{4\sin^2\left(\frac{\pi}{4}-4\alpha\right)} … |
the source answer is contradicted at alpha=1: the expression is exactly -tan(4) - cot(4) + 2, the source answe … |
dgn.core.4.375 |
core |
\tan \frac{x}{2}=2 or \tan \frac{x}{2}=-\frac{1}{3} |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.4.381 |
core |
a) (\cos 2\alpha)_1=-3/5; b) (\cos 2\alpha)_2=4/5 |
sides differ numerically - reported as suspected, not as a refutation: the difference can come from the readin … |
dgn.core.4.382 |
core |
a) 4/5; b) 3/5 |
sides differ numerically - reported as suspected, not as a refutation: the difference can come from the readin … |
dgn.core.4.412 |
core |
-\cos^32x |
the source answer is contradicted at x=1: the expression is exactly -4*sin(1)*6 - 1 - 4cos(1)*6 + 4cos(1)* … |
dgn.core.4.449 |
core |
(1 - a^2)/(2a) |
numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from … |
dgn.core.4.488 |
core |
A=B=|\cos\alpha\cos\beta\cos\gamma| |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
dgn.core.5.154 |
core |
x = \frac{2\pi}{3}(3k \pm 1) |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
dgn.core.5.155 |
core |
x_1=\frac{\pi}{2}(2k+1),\quad x_2=(-1)^k\frac{\pi}{6}+\pi k |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
dgn.core.5.156 |
core |
t_1=(-1)^k\frac{\pi}{6}+\pi k,\quad t_2=\frac{\pi}{2}(4k+1) |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
dgn.core.5.158 |
core |
x=\frac{\pi}{6}(6k\pm1) |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
dgn.core.5.186 |
core |
x_1=2\pi k,\ x_2=\frac{\pi}{2}(2k+1),\ x_3=\frac{\pi}{4}(4k- … |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
dgn.core.5.197 |
core |
x_1=\frac{\pi k}{3}; x_2=\frac{\pi}{12}(2k+1). |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
dgn.core.5.263 |
core |
t=\frac{\pi}{3}(3k\pm1) |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
dgn.core.5.293 |
core |
x=(-1)^{k+1}\frac{\pi}{12}+\pi k |
no solutions parsed - reported as suspected: the independent web calculator contradicts the printed answer, bu … |
dgn.core.5.301 |
core |
t_1 = 360^\circ \cdot k; \quad t_2 = 90^\circ (4k+1) |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
dgn.core.5.302 |
core |
t_1=90^\circ\cdot k;\quad t_2=\pm15^\circ+90^\circ\cdot k |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
dgn.core.5.311 |
core |
x=2\pi k. |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
dgn.core.5.337 |
core |
t=\pi k |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
dgn.core.5.414 |
core |
x_1 = \pi k, \ x_2 = \frac{\pi}{8}(2k+1) |
no solutions parsed - reported as suspected: the independent web calculator contradicts the printed answer, bu … |
dgn.core.5.418 |
core |
x=-1\pm\sqrt{1+\pi k},\quad k=0,1,2,\ldots |
error: TypeError - reported as suspected: the independent web calculator contradicts the printed answer, but t … |
dgn.core.5.462 |
core |
x=\frac{1\pm\sqrt{1+8\pi k}}{2}+2\pi k,\quad k=0,1,2,\ldots |
error: TypeError - reported as suspected: the independent web calculator contradicts the printed answer, but t … |
dgn.core.5.467 |
core |
x=\frac{\pi}{2}(4k+1) |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
dgn.core.5.473 |
core |
x_1=\frac\pi4(2k+1),\quad x_2=\frac\pi6(6k\pm1),\quad x_3= … |
no solutions parsed - reported as suspected: the independent web calculator contradicts the printed answer, bu … |
dgn.core.5.490 |
core |
x_1=\arccos\left(\sqrt5-2\right)+2\pi k,\quad x_2=\pi(2k+1). |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
dgn.core.6.088 |
core |
\approx 0.95 |
the source answer is contradicted at approx=1, x=-1/20: the expression is exactly 160000/168421, the source an … |
dgn.core.7.045 |
core |
890 |
the source answer is contradicted at : the expression is exactly sqrt(7) + 625 + 531441**(log(3)/log(6)), the … |
dgn.core.7.089 |
core |
x=5 |
no solutions parsed - reported as suspected: the independent web calculator contradicts the printed answer, bu … |
dgn.core.7.182 |
core |
z_1,z_2=\pm 2 |
error: TypeError - reported as suspected: the independent web calculator contradicts the printed answer, but t … |
dgn.core.7.188 |
core |
\frac{2a-1}{a+3} for a\ne -3,\ a\ne -2,\ a … |
error: TypeError - reported as suspected: the independent web calculator contradicts the printed answer, but t … |
dgn.core.7.218 |
core |
x=2 |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.7.317 |
core |
\log_n^2p, where \begin{cases}0<n<1,\\0<p<1\end{cases} or … |
the source answer is contradicted at m=2, n=2, p=2: the expression is exactly sqrt(3), the source answer gives … |
dgn.core.7.322 |
core |
x=\log_{0,4}\frac{\sqrt{5}-1}{2}, 0<x<1 |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.7.350 |
core |
x_1.2=1\pm\sqrt{2}, x_3=1. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.8.055 |
core |
[0, 4). |
the source answer excludes the upper endpoint 4, but it satisfies the inequality - reported as suspected, not … |
dgn.core.8.148 |
core |
[-3,-\sqrt{6})\cup(-\sqrt{6},-2]\cup[2,\sqrt{6})\cup(\sqrt … |
the source answer includes the lower endpoint -3, where equality holds - reported as suspected, not as a refut … |
dgn.core.8.185 |
core |
(a^4,a^{-1}) if 0<a<1; (a^{-1},a^4) if a>1 |
the source answer excludes the upper endpoint 2, but it satisfies the inequality (for a=1/2) - reported as sus … |
dgn.core.9.030 |
core |
x=8;\quad y=0,1,\ldots,7 |
error: TypeError - reported as suspected: the independent web calculator contradicts the printed answer, but t … |
dgn.core.9.032 |
core |
x=7 |
no solutions parsed - reported as suspected: the independent web calculator contradicts the printed answer, bu … |
dgn.core.10.006 |
core |
x_1=-\frac{2}{3},\ x_2=1. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.10.007 |
core |
x=1. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.10.080 |
core |
x_1^3+x_2^3=117 |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.10.081 |
core |
No |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.10.089 |
core |
x=1; the equation has no negative roots. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.10.093 |
core |
(\sqrt{x}-\sqrt{y})(\sqrt{x}-2\sqrt{y}) |
the source answer is contradicted at x=1, y=1: the expression is exactly 2, the source answer gives 0 - report … |
dgn.core.10.114 |
core |
(-\infty,2]\cup[6,\infty) |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.10.115 |
core |
(-\infty,2]\cup(4,6)\cup[8,\infty) |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.10.116 |
core |
(-\infty,-1)\cup[0,1) |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.10.138 |
core |
m=-1 and -\frac12<m<\infty |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.10.139 |
core |
0<p\le2/3 |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.10.140 |
core |
-\infty<c<-4 or -4<c\le-2. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.10.141 |
core |
-\infty<k<-4. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.10.144 |
core |
The equality is impossible. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.10.145 |
core |
x is an integer. |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
dgn.core.10.149 |
core |
(-\infty,-\sqrt{3}/3]\cup[\sqrt{3}/3,\infty) |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.10.150 |
core |
(\log_5 3+\log_5 z)\log_5 z=2+\frac{4}{3}\log_5 z |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.10.280 |
core |
No. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.10.318 |
core |
x=0 |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.10.347 |
core |
a=b/(b-1), where b>1 |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.core.11.078 |
core |
a>4 |
the source answer excludes the lower endpoint 4, but it satisfies the inequality - reported as suspected, not … |
dgn.core.11.252 |
core |
0.5\ln 2-1.5 |
numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from … |
dgn.algebra.2 |
algebra |
17\,964 |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.185 |
algebra |
x=2 |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.p30.192 |
algebra |
x = 1; y = -1. |
no candidate solution parsed - reported as suspected: the independent web calculator contradicts the printed a … |
dgn.algebra.195 |
algebra |
3; 4. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.2.1.4 Vieta's theorem |
algebra |
39. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.269 |
algebra |
x = \dfrac{-3 + \sqrt{17}}{2}. |
sides differ numerically |
dgn.algebra.381 |
algebra |
x = 2, y = 2 |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.403 |
algebra |
x = -2 + \sqrt{4 - \log_3 25}. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.513 |
algebra |
-\frac{17}{6} |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.540 |
algebra |
\{\tfrac{2}{5};\, 10\}. |
the two sides of the identity differ at every tested point |
dgn.algebra.542 |
algebra |
x_1 = 36; x_2 = \frac{3}{2} |
no relation - reported as suspected: the independent web calculator contradicts the printed answer, but the pa … |
dgn.algebra.556 |
algebra |
x_1 \cdot x_2 \cdot x_3 = 2^{\frac{3\sqrt{2}}{2}} |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.563 |
algebra |
x \geq 8; \ x = \frac{8}{3} |
no candidate solution parsed - reported as suspected: the independent web calculator contradicts the printed a … |
dgn.algebra.907 |
algebra |
x>\sqrt{3} |
the source answer excludes the lower endpoint 1.73205, but it satisfies the inequality |
dgn.algebra.1023 |
algebra |
4017 |
integer 4014 satisfies the inequality but is not listed |
dgn.algebra.1089 |
algebra |
two solutions: x = -\dfrac{1}{2}, y = 3 and x = 3, y = - … |
error: TypeError - reported as suspected: the independent web calculator contradicts the printed answer, but t … |
dgn.algebra.1135 |
algebra |
1 or \frac{10}{9} |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.1222 |
algebra |
8\sqrt{2} |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.p164.1284 |
algebra |
m = -12;\ n = 6. |
no candidate solution parsed - reported as suspected: the independent web calculator contradicts the printed a … |
dgn.algebra.1287 |
algebra |
x=2, y=1. |
no candidate solution parsed - reported as suspected: the independent web calculator contradicts the printed a … |
dgn.algebra.1306 |
algebra |
x = 0,\ y = 0 |
no candidate solution parsed - reported as suspected: the independent web calculator contradicts the printed a … |
dgn.algebra.1432 |
algebra |
x = 0; y = -1. |
no candidate solution parsed - reported as suspected: the independent web calculator contradicts the printed a … |
dgn.algebra.1437 |
algebra |
k = 6; l = 5. |
no candidate solution parsed - reported as suspected: the independent web calculator contradicts the printed a … |
dgn.algebra.1876 |
algebra |
0 < a < 2 - \sqrt{3}, a > 2 + \sqrt{3}. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.1894 |
algebra |
The admissible values are a \geq 0. If a = 0, the equation h … |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.1927 |
algebra |
a_1 = 1 + \log_5 \frac{8+\sqrt{55}}{3}, a_2 = 1 + \log_5 … |
no candidate solution parsed - reported as suspected: the independent web calculator contradicts the printed a … |
dgn.algebra.1950 |
algebra |
if -3<a\leq -2, the equation has 1 root; if -2<a … |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.1957 |
algebra |
a \leq -2. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.1959 |
algebra |
-\frac{5}{2} < a < 7; \frac{9 - \sqrt{211}}{2} \leq a … |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.p268.1964 |
algebra |
a<-\frac{1}{2e\ln 2}. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.1965 |
algebra |
a<0 or a\geq 3e\ln 3. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.1969 |
algebra |
a \in (0;5) \setminus \{1;4\} |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.1971 |
algebra |
`a = -\frac{1}{8};\ a = -2, | |
| when a = -\frac{1}{8}, x = 0;\ - …` | sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … | ||
dgn.algebra.1978 |
algebra |
if a = 1 - \sqrt{2}, a < -\frac{5}{4} or a … |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.1992 |
algebra |
a_1 = 0,\ a_2 = 3. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.2000 |
algebra |
a_1=-1,\ a_2=-3,\ a_3=-\frac{5}{7} |
no candidate solution parsed - reported as suspected: the independent web calculator contradicts the printed a … |
dgn.algebra.p274.2003 |
algebra |
b = \sqrt{3} |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.2004 |
algebra |
b = -\sqrt{2} |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.2010 |
algebra |
p=-\frac{1}{2} |
no candidate solution parsed - reported as suspected: the independent web calculator contradicts the printed a … |
dgn.algebra.p275.2012 |
algebra |
-4;\ -13;\ -\frac{17}{9}. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.2013 |
algebra |
a_1 = 3,\ a_2 = -\frac{1}{8},\ a_3 = \frac{15}{8}. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.2018 |
algebra |
z_{\max} = \sqrt{5} |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.2020 |
algebra |
a = \frac{2+2\sqrt{13}}{3} or -1 < a \le 0 |
no candidate solution parsed - reported as suspected: the independent web calculator contradicts the printed a … |
dgn.algebra.2031 |
algebra |
0 < a < \frac{1}{8} |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.2060 |
algebra |
a = \tfrac{3}{3}; a = \tfrac{3}{2}; a \in [2; 4) |
no candidate solution parsed - reported as suspected: the independent web calculator contradicts the printed a … |
dgn.algebra.2065 |
algebra |
a = -2. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.2066 |
algebra |
a = -3; s = 18. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.2067 |
algebra |
a = 7, x_1 = 2; x_2 = 4; x_3 = 8. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.2068 |
algebra |
a = -\frac{9}{82}. |
no candidate solution parsed - reported as suspected: the independent web calculator contradicts the printed a … |
dgn.algebra.2069 |
algebra |
p = 114,\ p = -6. |
no candidate solution parsed - reported as suspected: the independent web calculator contradicts the printed a … |
dgn.algebra.2070 |
algebra |
y_1 = 3, y_2 = -\frac{3}{7}. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.2071 |
algebra |
no such values exist |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.2072 |
algebra |
for any a |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.p284.2074 |
algebra |
a_1 = 1, \ a_2 = -1 |
error: TypeError - reported as suspected: the independent web calculator contradicts the printed answer, but t … |
dgn.algebra.2139 |
algebra |
\left[\tfrac{13}{7}; \tfrac{13}{5}\right] \cup \left[\tfrac … |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.2261 |
algebra |
2\sqrt{65}. |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.algebra.p446.1397 |
algebra |
**Answer:** \emptyset. |
the two sides of the identity differ at every tested point |
dgn.algebra.p519.11.9 |
algebra |
h < -2 - \sqrt{5},\ 2 - \sqrt{5} < h < 0,\ 0 < h < \sqrt{5} … |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.syllabus.p638.Example 7 |
syllabus |
a \leqslant -3, a = -2, a \geqslant -1. |
no candidate solution parsed - reported as suspected: the independent web calculator contradicts the printed a … |
dgn.syllabus.p644.15 |
syllabus |
a) |a|>1, x=1; b) |a|<1, x_1=1, x_2=\dfrac{a-5}{a+1}; c) … |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.syllabus.p700.Example 10 |
syllabus |
(5; 0),\ (-5; 0),\ (3; 2),\ (-3; 2),\ (3; -2),\ (-3; -2) |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.parameters.p14.3 |
parameters |
b = -\sqrt{2} |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p15.6 |
parameters |
a = 0; 1 |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p16.19 |
parameters |
For p = 0 there is one root x = 0, for p = 7 there is one ro … |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p16.20 |
parameters |
b \in (0; 1/50) \cup (25/2; +\infty) |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p18.30 |
parameters |
a \in (0; 1/54) |
the source answer excludes the upper endpoint 0.0185185, but it satisfies the inequality |
dgn.parameters.p22.31 |
parameters |
If a < 0, then x = (-1 + \sqrt{1 - 4a})/2; if a = 0, then x … |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.38 |
parameters |
If a = -2, then x = -1, 15/17, 17/15; if a = -1/8, then x = … |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p26.45 |
parameters |
For a \in (-\infty; -5/4) one solution; for a = -5/4 two sol … |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.59 |
parameters |
a \in \{3\} \cup [\sqrt{10}; \sqrt{11}] |
the answer was taken from the answer block of a neighbouring variant; the page was read again and the printed … |
dgn.parameters.71 |
parameters |
a \in (-\infty; 1] \cup \{5/4\} \cup [4/3, +\infty) |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.76 |
parameters |
a = 2 |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.85 |
parameters |
a = (1 - \sqrt{2})/2, x = 2 |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.88 |
parameters |
x = \sqrt{3} |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.91 |
parameters |
a = 0, a = 1 |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.97 |
parameters |
a \in (-2\sqrt{6}/3; 2\sqrt{6}/3) |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.101 |
parameters |
a = \pm\sqrt{2}, \pm(\sqrt{15} + 1)/4 |
the answer was taken from the answer block of a neighbouring variant; the page was read again and the printed … |
dgn.parameters.98 |
parameters |
x = -\sqrt{7}/5 |
the answer was taken from the answer block of a neighbouring variant; the page was read again and the printed … |
dgn.parameters.p57.110 |
parameters |
12\sqrt{2} |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p59.122 |
parameters |
a \in [-1 - 1/\sqrt{2}; -1], a = -2, 0, 1, 1/\sqrt{2} |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.125 |
parameters |
a = -5, a = -5/13 |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.128 |
parameters |
a = 13, the roots of the equation are 2, 6, 18 |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.132 |
parameters |
a = 0, a = 2\sin 1 |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.135 |
parameters |
b = \cot 1 |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.146 |
parameters |
a = 5\pi/6 + \pi l, \pi/18 + 2\pi m, 13\pi/18 + 2 … |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.159 |
parameters |
x = -1 |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p85.31 |
parameters |
a \in [1; 2) \cup (2; 3] |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.183 |
parameters |
[0; 1/2) \cup (1/2; 3/4] |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.187 |
parameters |
1. a \in (-3; 5). 2. a \in [2 - 2\sqrt{7}; -3] \cup \{5\} |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p111.214 |
parameters |
m = 0, x = 3; m = \pm 1, x = \pm(3 \pm 2\sqrt{2})/2; m = … |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p111.215 |
parameters |
m = 1, n = -1 and m = -1, n = 1 |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p111.217 |
parameters |
-13/3 < a \leqslant -19/5 |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p112.224 |
parameters |
m = 0, n = 0 |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p112.225 |
parameters |
x = 2, y = 1 |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p127.46 |
parameters |
p \in [-12; 0) \cup (0; 12] |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p130.257 |
parameters |
q \in [-15; 0) \cup (0; 15] |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p130.260 |
parameters |
a = -2 |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.278 |
parameters |
b = -1 |
the answer was taken from the answer block of a neighbouring variant; the page was read again and the printed … |
dgn.parameters.290 |
parameters |
6\sqrt{6} + 3 |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p163.7 |
parameters |
If a = 0, then the unique solution x = 0; if 0 < a < 1, then … |
the answer was taken from the answer block of a neighbouring variant; the page was read again and the printed … |
dgn.parameters.p164.7 |
parameters |
If a = 0, then the unique solution x = 0; if 0 < a < 1/2, th … |
the answer was taken from the answer block of a neighbouring variant; the page was read again and the printed … |
dgn.parameters.p172.6 |
parameters |
(1; 18 - 12\sqrt{2}] |
a=500.545 satisfies the inequality but is missing from the source answer - reported as suspected: the source h … |
dgn.parameters.p173.6 |
parameters |
(1; 8 - 4\sqrt{3}] |
a=500.565 satisfies the inequality but is missing from the source answer - reported as suspected: the source h … |
dgn.parameters.p175.3 |
parameters |
4. x_{1,2,3,4} = (A_{1,2} \pm \sqrt{A_{1,2}^{2} - 1})/2, … |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p182.5 |
parameters |
\left(\frac{1}{6}; \frac{2}{3}\right) \cup \{1\} \cup … |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.198_2 |
parameters |
207 |
the source answer is contradicted at x=1, y=1: the expression is exactly 2, the source answer gives 207 |
dgn.parameters.199.a |
parameters |
a) a \in (-5/2; 7), b) a \in [(9 - \sqrt{211})/2; -5/2] … |
the answer was taken from the answer block of a neighbouring variant; the page was read again and the printed … |
dgn.parameters.200.5.a |
parameters |
a) a \in (-3; 5), b) a \in [2 - 2\sqrt{7}; -3] \cup \{5\} |
the answer was taken from the answer block of a neighbouring variant; the page was read again and the printed … |
dgn.parameters.Task 5 (prev) |
parameters |
a) a \in (-7/2; 5), b) a \in [(3 - \sqrt{107})/2; -7/2] … |
the answer was taken from the answer block of a neighbouring variant; the page was read again and the printed … |
dgn.parameters.p211.1 |
parameters |
\frac{4}{3} |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.parameters.p217.5 |
parameters |
For a \in \left(-\infty; -\frac{5}{4} … |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p218.5 |
parameters |
For a \in \left(-\infty; -\frac{17}{4} … |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p238.1 |
parameters |
(-\infty; -5) \cup (-\frac{9}{7}; +\infty) |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p239.Task 1 |
parameters |
\left(-\frac{1}{2}; \frac{11}{6}\right) |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.Task 2 (Variant 2004, April, sociology fakultet) |
parameters |
1. 2. 2. (-\infty; 0) \cup \left(\frac{14}{3}; +\infty … |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p254.6 |
parameters |
a \in (0; 2c) |
the answer was taken from the answer block of a neighbouring variant; the page was read again and the printed … |
dgn.parameters.p256.7 |
parameters |
2\sqrt{65} |
sides differ but the question is not a proof request - reported as suspected: the independent web calculator c … |
dgn.parameters.p257.7 |
parameters |
2 |
the answer was taken from the answer block of a neighbouring variant; the page was read again and the printed … |
dgn.parameters.Variant 2005 - Task 3 |
parameters |
x \in [1;3) \cup (3;\sqrt{10}). |
the source answer excludes the upper endpoint 3.16228, but it satisfies the inequality |
dgn.parameters.p265.4 |
parameters |
x = 2, y = 1 |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p266.Task 5 |
parameters |
a \in [\pi/2;\, 3\pi/2 - 3] \cup (5\pi/2 - 6;\, 7\pi/2 - 9] |
the source answer excludes the lower endpoint 1.85398, but it satisfies the inequality |
dgn.parameters.p271.5 |
parameters |
x = 2, y = 0 |
no candidate solution parsed - reported as suspected: the independent web calculator contradicts the printed a … |
dgn.parameters.Variant 2005 (iyul), geology Faculty, 2, Task 4 |
parameters |
5\pi/2 |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p281.6 |
parameters |
a=2 |
no candidate solution parsed - reported as suspected: the independent web calculator contradicts the printed a … |
dgn.parameters.p282.6 |
parameters |
b = 1 |
the printed answer of this problem was missing; the answer block of the printed variant was read again and its … |
dgn.parameters.p285.7 |
parameters |
8\sqrt{2} |
the answer was taken from the answer block of a neighbouring variant; the page was read again and the printed … |
dgn.parameters.p289.8 |
parameters |
a \in (-\infty; 1] \cup \{5/4\} \cup [4/3; +\infty) |
the answer was taken from the answer block of a neighbouring variant; the page was read again and the printed … |
dgn.planimetry.6.37 |
planimetry |
AB \cdot CD + AD \cdot BC = AC \cdot BD |
the two sides of the identity differ at every tested point |
dgn.planimetry.13.31 |
planimetry |
\boldsymbol{e}_1 \sin\alpha + \boldsymbol{e}_2 \sin\beta + … |
the two sides of the identity differ at every tested point |
dgn.planimetry.31.45 |
planimetry |
a + c = 0 |
the two sides of the identity differ at every tested point |
Reproducing the audit
python verify_all3.py 12 dataset/core_math_dataset_v2.jsonl v33 # the main collection
python reverify_collections.py collv32 # the extended collections
python build_benchmark.py # assemble + split
python run_webcas_published.py # the independent calculator
python audit_pairing.py hf_dataset/benchmark_full.jsonl # question/answer integrity
The verification engine is verify3.py; verify.py holds the individual
checkers and verify2.py the newer ones (inequalities, limits, integrals,
factors).
Errors in the source that the engine cannot see
These are errors of the source material, found by reading the source itself. Two kinds occur: a wrong answer, which the symbolic engine does not reach because the exercise has several parts and a list of answers cannot be checked as a whole, and a misprinted statement, where the source states a problem that does not match its own answer.
| id | kind | part | what the source gives | what is correct | why |
|---|---|---|---|---|---|
dgn.trigonometry.p158.26.6 |
wrong answer | (d) | \left(\dfrac{\sqrt{3}}{3}; 1\right] |
\left(\dfrac{\sqrt{3}}{2}; 1\right] |
the exercise asks to solve arccos x < pi/6, whose solution is x in (cos(pi/6), 1] = (sqrt(3)/2, 1]; the source gives (sqrt(3)/3, 1] |
dgn.core.2.205 |
misprinted statement | second equation of the system | \begin{cases}x+y+\frac{x^2}{y^2}=7,\\ \frac{(x-y)x^2}{y^2}=12.\end{cases} |
\begin{cases}x+y+\frac{x^2}{y^2}=7,\\ \frac{(x+y)x^2}{y^2}=12.\end{cases} |
the source answer does not satisfy the second equation: (2; 1) gives (x-y)x^2/y^2 = 4, not 12, and (6; -3) gives 36, not 12. All four source answers satisfy x+y+x^2/y^2 = 7 and (x+y)x^2/y^2 = 12 exactly, so the source misprinted x+y as x-y. The system as written has exactly one real solution, (1.3863205; -0.5580341), whose coordinates are roots of a quintic and are therefore certainly not the intended answer. |
dgn.core.1.206 |
wrong answer | \frac{a}{a+1} |
\frac{a(a+1)}{2a+1} |
the expression (a+1)/(2*cuberoot(sqrt(3)-sqrt(2))6throot(5+2sqrt(6)) + 1/a) equals a(a+1)/(2a+1), because 2cuberoot(sqrt(3)-sqrt(2))*6throot(5+2sqrt(6)) = 2 exactly; the difference between the source answer and the true value is a^3/((a+1)(2a+1)), which is not an identity | |
dgn.core.1.202 |
wrong answer | \sqrt{mn} |
the expression has no closed form in m and n; its value at m = 8, n = 1 is exactly 40/3, so the source answer cannot be right. The cube root of mn is not the value either (it would be 2 at that point) |
the expression (cube root of m^2 + ncube root of m + n^2) * (cube root of m^4 - n^3 + n^2cube root of (m - mn)) / (mn^-1 + n - n^4m^-1 - n^2) does not have the square root of mn as its value. At m = 8 and n = 1 the expression is exactly 40/3, a rational number, while the source answer is sqrt(8) = 2sqrt(2), an irrational number, so the two can never be equal. The denominator is not zero there (it is 63/8), so the point lies inside the domain of the expression. The first factor and the denominator do factor neatly - the first one is (m - n^3)/(cube root of m - cube root of n) and the denominator is (m + n^2)(m - n^3)/(m*n) - which is why the statement was read carefully before the answer was called wrong | |
dgn.core.8.276 |
wrong answer | (\sqrt[3]{5},5) |
(\sqrt[5]{5},5) |
the source answer is (cuberoot(5), 5) = (5^(1/3), 5), but the answer of the inequality log_x^2 sqrt(5) - log_x 5 sqrt(5) + 1,25 < 0 is (5^(1/5), 5). With t = log_x 5 the inequality becomes t^2/4 - 3t/2 + 5/4 < 0, i.e. (t-1)(t-5) < 0, so 1 < t < 5 and x = 5^(1/t) runs from 5^(1/5) to 5; the cube root 5^(1/3) is only an interior point of that interval, not its lower endpoint. The root index in the source answer is a 3, not a 5: it was read from the page image at 800 dpi and the index glyph was compared pixel by pixel with the index of the cube root in 3sqrt35 printed on the same answer page; it matches the 3 far better than the 5 (IoU 0.61 against 0.51, and a 5 would have a horizontal top bar reaching the left edge and a narrow middle stem, neither of which the printed glyph has). | |
dgn.core.1.111 |
wrong answer | 100 |
the expression is negative at the stated values a = 25/7 and x = 7/25; under the transcribed structure its value is exactly -31250*sqrt(7)/343 - 84/25 |
the exercise asks for the value of the expression at a = 3 4/7 = 25/7 and x = 0,28 = 7/25. There ax = 1, so the dominant term -10a sqrt(x) sqrt((a x^-1)^2) is exactly -31250sqrt(7)/343 (about -241.1), and the remaining terms are bounded by 229 and 3, so the value is negative in every faithful reading of the printed expression (eight readings were evaluated exactly: the transcribed structure gives -31250sqrt(7)/343 - 84/25, the structure in which the first group is printed as ( ... )^3 gives about -1.39*10^7). A positive answer of 100 is therefore impossible. | |
dgn.algebra.777 |
wrong answer | $-\frac{19}{4} < x < -4$ |
$\left(-\frac{11}{4};\, -2\right) \cup \left(-2;\, +\infty\right)$ |
the source answer of log_{x+3}(1/2) > 1/2 lies entirely outside the domain of the logarithm, which needs x > -3 and x != -2. Solving the inequality gives x in (-11/4, -2) union (-2, +infinity): for 0 < x+3 < 1 the logarithm decreases, so 1/2 < (x+3)^(1/2) and x > -11/4, and for x+3 > 1 the inequality holds everywhere. The point x = -19/8 satisfies the inequality and is missing from the source answer. Page 99 was read at 400 dpi and shows exactly -19/4 < x < -4. | |
dgn.algebra.788 |
wrong answer | $-1 < x < -\frac{5}{6}$ |
$-1 < x < \frac{1}{\sqrt[3]{36}} - 1$ |
the source answer of log_6 log_{1/6th-root-of-6}(x+1) > 1 is too narrow. Writing u = log_{6^(-1/9)}(x+1), the condition is u > 6, and because the base is below 1 this gives 0 < x+1 < 6^(-2/3), so the solution is -1 < x < 6^(-2/3) - 1 = -0.6972.... The source upper end -5/6 = -0.8333 excludes the whole stretch from -0.8333 to -0.6972; the point x = -0.8 satisfies the inequality (log_6 log_{6^(-1/9)}(0.2) = 1.166 > 1) and is missing from the source answer. Page 100 was read at 400 dpi. | |
dgn.algebra.926 |
wrong answer | $[-18;-3] \cup (0;2)$ |
$[-18;-3) \cup (0;2)$ |
in sqrt((x^2+9x-162)/(x-2)) > x+9 equality holds at x = -3, because the radicand is 36 there and x+9 is 6 as well, so the strict inequality fails and -3 must not belong to the solution. The source closes the bracket at -3. The difference of the two sides is -x(x+3)(x+12)/(x-2), which changes sign exactly at -3. Page 116 was read at 400 dpi and shows the closed bracket. | |
dgn.core.2.343 |
wrong answer | first root | $x_1=3,\ x_2=\sqrt{3}/3,\ x_3=-2\sqrt{3}$ |
$x_1=\sqrt{3},\ x_2=\sqrt{3}/3,\ x_3=-2\sqrt{3}$ |
the source's own solution derives x_1 = sqrt(3) and x_1' = sqrt(3)/3, but its answer line prints x_1 = 3. The value 3 leaves the exact residual 18 + 24sqrt(3) in 3x^3 + 2sqrt(3)x^2 - 21x + 6sqrt(3), while sqrt(3), sqrt(3)/3 and -2sqrt(3) are exact roots |
dgn.core.2.356 |
wrong answer | $x_1=2,\ x_2=-2$ |
$x_1=\sqrt{2},\ x_2=-\sqrt{2}$ |
the printed pair is not a root pair of the equation: 2 leaves the exact residual 2 and -2 leaves the residual 34 in x^4 - 4x^3 + 3x^2 + 8x - 10. The equation factors as (x^2 - 2)(x^2 - 4x + 5), so its two real roots are sqrt(2) and -sqrt(2); the source appears to have lost the radical signs of its own answer | |
dgn.algebra.454 |
wrong answer | second root | $x_2 = \frac{4\log_3 2 - 1}{2\sqrt{3}(\log_3 5 - 1)}$ |
$x_2 = \frac{4\log_3 2 + 1}{2\sqrt{3}(\log_3 5 - 1)}$ |
the printed second root leaves the exact value -8/3 in the reduced quadratic r^2 - 5*sqrt(3)r + 12 with r = (5/3)^(sqrt(3) x), and the exact value -97.610355 in the equation itself, so it is not a solution. The sign of the 1 is wrong: the exact solution needs 4log_3(2) + 1 |
dgn.core.2.059 |
wrong answer | x=2 |
`` | Our text matches the print, but the printed answer x=2 gives LHS 5.162 != 6; the printed equation's root is about 3.4017. | |
dgn.core.2.295 |
wrong answer | x=65 |
`` | Our text matches the print, but the printed answer x=65 gives 0.33019 != 1/3; the printed equation's root is x=64 (1/12+1/4=1/3). | |
dgn.core.2.298 |
wrong answer | x=4 |
`` | Print reads 8,412th-root(x^-7) - 0,24th-root(x^-1)*cuberoot(x^2) = 12th-root(x^11); our text turned the root indices into powers 4^12 and 4^4. (The printed answer x=4 also fails the printed equation, residual -0.4.). After the corrected reading of the printed statement the printed answer still fails: the engine says mismatch. The statement was therefore misread by us and the answer is wrong in the source | |
dgn.core.7.001 |
wrong answer | x_1=-\frac{1}{2},\ x_2=6 |
`` | Our text matches the print, but neither printed root satisfies the printed equation (its true root is x=-11/4). | |
dgn.core.7.006 |
wrong answer | x=\frac{5}{3} |
`` | Our text matches the print, but the printed answer x=5/3 gives LHS 5.879 != 1; the true root is about 0.337. | |
dgn.core.7.013 |
wrong answer | x=10 |
x = \frac{70}{9} |
Print nests the powers as ((5th-root 27)^(x/4-sqrt(x)/3))^(x/4+sqrt(x)/3); our text invented an extra factor 4^(-x/4+sqrt(x)/3). (The printed answer x=10 also fails the printed equation; true root x=70/9.). After the corrected reading of the printed statement the printed answer still fails: the engine says mismatch. The statement was therefore misread by us and the answer is wrong in the source | |
dgn.core.7.026 |
wrong answer | x_1=3,\quad x_2=\frac{3}{\log_2 6}=3\log_6 2 |
`` | Our text matches the print, but neither printed root satisfies the printed equation (the source's own reduction to 2^(6/x)-8*2^(3/x)+12=0 is not equivalent to it). | |
dgn.core.7.029 |
wrong answer | x_1=-\frac13,\quad x_2=0 |
`` | Our text matches the print, but neither printed root satisfies it (x=0 gives 0.75 != 1); the printed equation has no real solution. | |
dgn.core.7.035 |
wrong answer | x_1=1\sqrt{10},\quad x_2=100 |
the printed equation has no real solution |
Printed problem matches our text, but the printed answer x_1=1/sqrt(10) was recorded by us as 1*sqrt(10). (The printed equation itself has no real solution, so the source is also defective.). After the corrected reading of the printed statement the printed answer still fails: the engine says mismatch. The statement was therefore misread by us and the answer is wrong in the source | |
dgn.core.7.110 |
wrong answer | x = 5 |
`` | Our text matches the print, but the printed answer x=5 gives LHS -1.875 vs RHS 2; the source's solution works with a different (division) form, and the printed root is about 3.003. | |
dgn.core.7.133 |
wrong answer | x=\sqrt[3]{2} |
`` | Our text matches the print, but the printed answer x=cuberoot(2) gives 4.08-184 != 8; the true root is about 6.7. | |
dgn.core.7.144 |
wrong answer | x=-2 |
`` | Our text matches the print, but the printed answer x=-2 gives LHS 4.585 vs RHS -0.585; the true root is about -2.71. | |
dgn.core.7.251 |
wrong answer | x = 1/3. |
`` | Our text matches the print, but the printed answer x=1/3 gives LHS -5.099 != 0; the printed equation's root is x=3^(4/9)=1.6295. The source's own solution silently uses log_{sqrt3}9=2 instead of the printed log_{sqrt3}3=1/2. | |
dgn.core.7.252 |
wrong answer | x=4 |
`` | Our text matches the print, but the printed answer x=4 gives 1/4-1=-3/4 != 0 and the left side is identically -3/4, so the printed equation has no solution at all. | |
dgn.core.7.258 |
wrong answer | x = 1. |
`` | Our text matches the print, but the printed answer x=1 gives LHS 0 vs RHS -3; the printed equation's root is about 2.4255. | |
dgn.core.7.265 |
wrong answer | x=256. |
`` | Our text matches the print, but the printed answer x=256 gives LHS -1.5 vs RHS -1.08907; the printed equation's solution is about 2.384e14. | |
dgn.core.7.348 |
wrong answer | x=100 |
`` | Our text is numerically equivalent to the print (5*(3*...) vs 5(3)*...), but the printed answer x=100 gives LHS 16 vs RHS 45; the printed equation's root is about 4.762e4. | |
dgn.syllabus.p148.30 |
wrong answer | x = -\frac{\pi}{4} + \pi n, \ x = -\operatorname{arctg}(3 + \sqrt{10}) + \pi n, \ x = \operatorname{arctg}(\sqrt{10} - 3) + \pi n, \ n \in \mathbf{Z} |
x = -\frac{\pi}{4} + \pi n, \ x = -\arctan(2 + \sqrt{3}) + \pi n, \ x = -\arctan(2 - \sqrt{3}) + \pi n, \ n \in \mathbf{Z} |
the printed answer does not solve the printed equation 3tan(x + pi/4) + cot 2x = 0: four of the roots of the equation are missing from the answer and the answer adds four roots that the equation does not have. The printed family is the correct answer for 4tan(x + pi/4) + cot 2x = 0, so the source misprinted either the coefficient 3 or the answer; the published row keeps the printed exercise and the recomputed answer, which agree with each other exactly | |
dgn.syllabus.p149.52 |
wrong answer | x = \frac{\pi n}{2}, \ x = \frac{\pi}{8} + \frac{\pi n}{4}, \ n \in \mathbf{Z} |
x = \frac{\pi n}{2}, \ n \in \mathbf{Z} |
the printed equation 8(sin^5 x cos x + cos^5 x sin x) + 3 sin 2x = 0 reduces to (2cos^2 2x + 5) sin 2x = 0, and 2cos^2 2x + 5 is never zero, so its solution set is exactly x = pi n / 2. The printed answer adds the family pi/8 + pi n/4, whose eight values in one period do not satisfy the equation | |
dgn.syllabus.p149.61 |
wrong answer | x = \pm \dfrac{\pi}{6} + \pi n, \ x = (-1)^{n+1} \dfrac{\pi n}{12} + \dfrac{\pi n}{2}, \ n \in \mathbf{Z} |
x = \pi n, \ x = \frac{\pi}{6} + \pi n, \ x = \frac{5\pi}{6} + \pi n, \ x = \frac{7\pi}{12} + \pi n, \ x = \frac{11\pi}{12} + \pi n, \ n \in \mathbf{Z} |
the printed answer describes a superset of the solution set: it contains all ten roots of the printed equation but also fourteen values that do not solve it, and it misses the family x = pi n. The printed equation cos x - 2 cos 3x + cos 5x = tan x cos 3x reduces to cos 3x (4 sin x cos x + 1) = 0, so its roots are the roots of cos 3x = 0 (the two of them where the equation is not defined left out), the roots of sin x = 0 and the roots of 2 sin 2x = -1. The published answer is that set, and it reproduces the root set of the printed equation exactly | |
dgn.syllabus.p149.65 |
wrong answer | x = \pm \dfrac{\pi}{4} + \pi n, \ x = \arctan \dfrac{1}{2} + (-1)^n \arcsin \dfrac{1}{\sqrt{10}} + \pi n, \ n \in \mathbf{Z} |
x = \pm\frac{\pi}{4} + 2\pi n, \ x = \arctan\frac{1}{2} + (-1)^n \arcsin\frac{1}{\sqrt{10}} + \pi n, \ n \in \mathbf{Z} |
the printed first family reads +/- pi/4 + pi n, which adds 3pi/4 and 5pi/4; neither satisfies the printed equation 2 sin 2x - cos 2x = 2 sqrt(2) sin x. With 2 pi n in place of pi n the printed answer is exactly right, and that is what is published | |
dgn.syllabus.p149.76 |
wrong answer | x = \dfrac{\pi}{2} \pi n, \ n \in \mathbf{Z} |
x = \frac{\pi}{2} + 2\pi n, \ n \in \mathbf{Z} |
the printed answer is a garbled product, (pi/2) pi n. Read literally it describes 61 values and not one of them satisfies the printed equation, which has the single root x = pi/2 in one period. The published answer x = pi/2 + 2 pi n reproduces it exactly | |
dgn.syllabus.p149.77 |
wrong answer | x = \dfrac{\pi}{8} + \dfrac{\pi n}{4}, \ x = \pm \dfrac{2\pi}{3} + 2\pi n, \ n \in \mathbf{Z} |
x = \frac{\pi}{8} + \frac{\pi n}{4}, \ x = \pm\frac{\pi}{3} + 2\pi n, \ n \in \mathbf{Z} |
the printed second family reads +/- 2pi/3 instead of +/- pi/3. The values 2pi/3 and 4pi/3 do not satisfy the printed equation, while pi/3 and 5pi/3 do; the published answer reproduces the root set of the printed equation exactly | |
dgn.syllabus.p151.143 |
wrong answer | x = \pi n, \ x = \frac{\pi}{4} + \frac{\pi n}{2}, \ x = (-1)^n \frac{\pi}{6} + \pi n, \ n \in \mathbf{Z} |
x = \pi n, \ x = \pm 0.205560277 + 2\pi n, \ x = \pm 2.019445231 + 2\pi n, \ x = \pm 2.765378489 + 2\pi n, \ n \in \mathbf{Z} |
the printed answer keeps only 0 and pi of the eight roots of the printed equation and adds six values that do not satisfy it. The solution set of sin^3 x cos 3x / cos x + cos^2 x sin 3x = sin 8x - tan 2x has no elementary closed form: cleared of its denominators it becomes a palindromic polynomial of degree 18 in exp(i x), hence a degree-9 polynomial in 2 cos x, whose six roots in (0, pi) are 0.205560277, 2.019445231 and 2.765378489 together with their negatives. The published answer therefore lists the roots numerically | |
dgn.syllabus.p149.89 |
wrong answer | x = \frac{\pi n}{7}, \ x = \frac{\pi}{9} + \frac{2\pi k}{9}, \ n \in \mathbf{Z}, \ k \in \mathbf{Z}, \ n \neq 7p, \ (p \in \mathbf{Z}), \ 2k + 1 \neq 9m, \ (m \in \mathbf{Z}) |
x = \frac{2\pi n}{7}, \ x = \frac{\pi}{9} + \frac{2\pi k}{9}, \ n \in \mathbf{Z}, \ k \in \mathbf{Z}, \ n \neq 7p, \ (p \in \mathbf{Z}), \ 2k + 1 \neq 9m, \ (m \in \mathbf{Z}) |
the printed first family reads x = pi n / 7, but the printed equation 8 cos 4x cos 2x cos x = 1 has the identity 8 cos 4x cos 2x cos x = sin 8x / sin x, so it is equivalent to sin 8x = sin x and its solution set is x = 2 pi n / 7 together with x = pi/9 + 2 pi k / 9. The printed family pi n / 7 solves the equation only for the even values of n and fails for the odd ones, where the left-hand side is -1 instead of 1, and the printed exclusion n != 7p is the one that belongs to the family 2 pi n / 7, so the source dropped a 2 in the first family. With 2 pi n / 7 in place of pi n / 7 the printed answer is exactly right, and that is what is published | |
dgn.parameters.p176.Task 1 |
mispaired answer | $-4$ |
-1 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Faculty of Fundamental Medicine, 2.; answer block on page 177) | |
dgn.parameters.p176.Task 4 |
mispaired answer | $30^\circ$ |
\sqrt{3} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Faculty of Fundamental Medicine, 2.; answer block on page 177) | |
dgn.parameters.p177.1 |
mispaired answer | $x = \pi/4 + \pi n,\ n \in \mathbb{Z}$ |
\frac{\pi}{15} + \frac{\pi}{5}n, n \in \mathbb{Z} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geography Faculty, Faculty of Bioengineering and Bioinformatics, 1.; answer block on page 178) | |
dgn.parameters.p177.2 |
mispaired answer | $\sqrt{3}$ |
3(\sqrt{2} - 1) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geography Faculty, Faculty of Bioengineering and Bioinformatics, 1.; answer block on page 178) | |
dgn.parameters.p177.3 |
mispaired answer | $\pi/4 + \pi n,\ n \in \mathbb{Z}$ |
[-3; -\frac{3}{\sqrt{2}}) \cup (-\frac{3}{\sqrt{2}}; 0) \cup (0; 3] |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geography Faculty, Faculty of Bioengineering and Bioinformatics, 1.; answer block on page 178) | |
dgn.parameters.p177.4 |
answer recovered | `` | 3100 \text{ m}^{2}/\text{day} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geography Faculty, Faculty of Bioengineering and Bioinformatics, 1.; answer block on page 178) | |
dgn.parameters.p9.2 |
mispaired answer | $x \in [a; a+1)$ for any $a$ |
x \in [a; a + 1) for any a |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 9) | |
dgn.parameters.p9.3 |
mispaired answer | Part of the answer: if $a < 1$, then $x \in (-\infty;a) \cup (1;+\infty)$ |
If a < 1, then x \in (-\infty; a) \cup (1; +\infty); if a = 1, then x \in (-\infty; 1) \cup (1; +\infty); if a > 1, then x \in (-\infty; 1) \cup (a; +\infty). |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 10) | |
dgn.parameters.p10.4 |
mispaired answer | If $a < 0$, then $x \in (-a; +\infty)$; if $a = 0$, then $x \in \varnothing$; if $a > 0$, then $x \in (-\infty; -a)$. Let's consider another simple example. |
If a < 0, then x \in (-a; +\infty); if a = 0, then x \in \varnothing; if a > 0, then x \in (-\infty; -a). |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 10) | |
dgn.parameters.p10.5 |
answer recovered | `` | If a < 0, then x \in (0; -a); if a = 0, then x \in \varnothing; if a > 0, then x \in (-a; 0). |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 11) | |
dgn.parameters.p11.5 |
answer recovered | `` | If a < 1, then x \in (a; (a + 1)/2) \cup (1; +\infty); if a = 1, then x \in (1; +\infty); if a < 1, then x \in (1; (a + 1)/2) \cup (a; +\infty). |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 12) | |
dgn.parameters.p12.12 |
mispaired answer | If $a<1$, then $x\in(a;(a+1)/2)\cup(1;+\infty)$; if $a=1$, then $x\in(1;+\infty)$; if $a>1$, then $x\in(1;(a+1)/2)\cup(a;+\infty)$. |
If a < 1, then x \in (a; (a + 1)/2) \cup (1; +\infty); if a = 1, then x \in (1; +\infty); if a < 1, then x \in (1; (a + 1)/2) \cup (a; +\infty). |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 12) | |
dgn.parameters.p13.7 |
answer recovered | `` | If a < -4, then x \in (5 - \left(\frac{a+3}{a+4}\right)^2; 5]; if a \in [-4; -3), then x \leqslant 5; if a \geqslant -3, then x < 5 - \left(\frac{a+3}{a+4}\right)^2. |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 14) | |
dgn.parameters.p14.1 |
answer recovered | `` | a \in (0; 1) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.p14.2 |
answer recovered | `` | b = 125 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.p14.3 |
answer recovered | `` | b = -\sqrt{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.p15.10 |
answer recovered | `` | For c \in (-\infty; 1) there are no solutions; if c \in [1; 2), then x \in [-2\sqrt{c} - 1; 2\sqrt{c} - 1]; if c \in [2; +\infty), then x \in [-c; c] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.11 |
answer recovered | `` | If a < 1, then x \in [-1; ((a-2)/(a-1))^2 - 1); if a \in [1; 2), then x \geqslant -1; if a \geqslant 2, then x > ((a-2)/(a-1))^2 - 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.p15.12 |
answer recovered | `` | p = 7 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.p15.4 |
answer recovered | `` | a \in (1; 2) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.p15.5 |
answer recovered | `` | (1; -1), (-1/5; 7/5) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.p15.6 |
answer recovered | `` | a = 0; 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.p15.7 |
answer recovered | `` | c \in [1; 5/2) \cup [4; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.p15.8 |
answer recovered | `` | If b \leqslant -1, then x \in (-\infty; -1] \cup [1; +\infty); if -1 < b \leqslant 0, then x \in (-1/\sqrt{1 - b^2}; -1] \cup [1; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.p15.9 |
answer recovered | `` | 1. If a \in (0; 1/2), then x \in (-3^a; -1) \cup (1; 3^a); if a > 1/2, then x \in (-\infty; -3^a) \cup (3^a; +\infty). 2. For a = 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.13 |
answer recovered | `` | If b \in (0; 1), then x \in (0; 1) \cup (1; b^{-3}); if b \in (1; +\infty), then x \in (b^{-3}; 1) \cup (1; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.p16.14 |
answer recovered | `` | If a \in (0; 1], then x = \pm((1 - a^2)/2a)^2; for other a there are no solutions |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.p16.15 |
answer recovered | `` | 1. If a \in (-3; -2), then x \in \mathbb{R}; if a \in (-\infty; -2] \cup [-3; +\infty), then x \in (-\infty; 3 - \sqrt{a^2 + 5a + 6}) \cup (3 + \sqrt{a^2 + 5a + 6}; +\infty). 2. a \in ((-5 - \sqrt{13})/2; -3] \cup [-2; (-5 + \sqrt{13})/2) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.p16.16 |
answer recovered | `` | x = 0, a for any a, x = (a \pm \sqrt{a^2 - 72})/6 for |a| \geqslant 6\sqrt{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.17 |
answer recovered | `` | a \in (7; 7,5) \cup (7,5; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.18 |
answer recovered | `` | a \in (1/15; 1/8) \cup (1/8; 4/15] \cup \{1/2\} \cup [1; 4) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.p16.19 |
answer recovered | `` | For p = 0 there is one root x = 0, for p = 7 there is one root x = 7^4/2, for p > 7 there are two positive roots |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.p16.20 |
answer recovered | `` | b \in (0; 1/50) \cup (25/2; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.21 |
answer recovered | `` | (-\infty; 0,7] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.22 |
answer recovered | `` | a = 0, a \in [-9; -1/4] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.23 |
answer recovered | `` | p \in (-\infty; 0] \cup [3; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.24 |
answer recovered | `` | If a \in (-\infty; -5), then there are no solutions; if a = -5, then x = 0; if a \in (-5; 1), then x \in [0; (a + 5)^2]; if a \in [1; +\infty), then x \in [(a - 1)^2; (a + 5)^2] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.25 |
answer recovered | `` | a \in \{-1\} \cup [-1/2; 0) \cup (0; 1/2] \cup \{1\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.26 |
answer recovered | `` | a \in [-9/10; 0) \cup (2; 9/4) \cup (9/4; 19/8) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.27 |
answer recovered | `` | a = 2, b = 3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.p18.28 |
mispaired answer | $a = -1{,}2$; $a = -0{,}67$ |
a = -1,2; a = -0,67 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 18) | |
dgn.parameters.p19.8 |
mispaired answer | If $a < 0$, then $x \in (-\infty; +\infty)$; if $a \geqslant 0$, then $x \in (-\infty; -2a) \cup (0; +\infty)$. |
If a < 0, then x \in (-\infty; +\infty); if a \geqslant 0, then x \in (-\infty; -2a) \cup (0; +\infty). |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 19) | |
dgn.parameters.p19.9 |
answer recovered | `` | If a < 0, then x \in (-a/2; -a) \cup [-a; +\infty) = (-a/2; +\infty); if a \geqslant 0, then x \in \varnothing. |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 20) | |
dgn.parameters.Example 10 |
answer recovered | `` | 1. The equation does not have an infinite set of solutions for any value of a. 2. For a \in (-\infty; -8) \cup (0; +\infty) the equation has no solutions. |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 22) | |
dgn.parameters.p21.3 |
answer recovered | `` | 1. The equation does not have an infinite set of solutions for any value of a. 2. For a \in (-\infty; -8) \cup (0; +\infty), the equation has no solutions. |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 22) | |
dgn.parameters.p21.4 |
answer recovered | `` | 1. The equation does not have an infinite set of solutions for any value of a. 2. For a \in (-\infty; -8) \cup (0; +\infty), the equation has no solutions. |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 22) | |
dgn.parameters.p22.31 |
answer recovered | `` | If a < 0, then x = (-1 + \sqrt{1 - 4a})/2; if a = 0, then x = 0, -1; if a \in (0; 1/4), then x = \frac{-1 - \sqrt{1 + 4a}}{2}, x = \frac{-1 \pm \sqrt{1 - 4a}}{2}; if a = 1/4, then x = (-1 - \sqrt{2})/2, x = -1/2; if a > 1/4, then x = (-1 - \sqrt{1 + 4a})/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 23) | |
dgn.parameters.p22.32 |
answer recovered | `` | If a < -1, then x \in (0; -a - \sqrt{a^2 - 1}] \cup [-a + \sqrt{a^2 - 1}; -a + \sqrt{a^2 + 1}]; if a \geqslant -1, then x \in (0; a + \sqrt{a^2 + 1}] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 23) | |
dgn.parameters.p22.33 |
answer recovered | `` | a \in [4/3; 2] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 23) | |
dgn.parameters.34 |
answer recovered | `` | If a < -1, then x = 4; if a = -1, then x \geqslant 4; if a \in (-1; 1), then x = 4, x = 4(a - 2)/(a + 1); if a = 1, then x \in [-2; 4]; if a > 1, then x = 4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 23) | |
dgn.parameters.p22.35 |
answer recovered | `` | a = 0, a = 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 23) | |
dgn.parameters.p22.36 |
answer recovered | `` | a \geqslant 2/3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 23) | |
dgn.parameters.37 |
answer recovered | `` | 1. The equation has no solutions for k \in (-23; 0). 2. The equation has a finite nonempty set of solutions for k \in (-\infty; -23) \cup (0; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 23) | |
dgn.parameters.38 |
answer recovered | `` | If a = -2, then x = -1, 15/17, 17/15; if a = -1/8, then x = -1/136, 0, 1/120 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 23) | |
dgn.parameters.p23.39 |
answer recovered | `` | 1. a \in (-5/2; 7). 2. a \in [(9 - \sqrt{211})/2; -5/2] \cup \{7\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 23) | |
dgn.parameters.40 |
answer recovered | `` | (\pi/2 + 2\pi n; 0), n \in \mathbb{Z}, (-3\pi/2; t), t \in \mathbb{R} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 23) | |
dgn.parameters.p23.41 |
answer recovered | `` | a = 2 \pm \sqrt{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 23) | |
dgn.parameters.Example 11 |
answer recovered | `` | [3 - \sqrt{6}; 2] \cup [5; 3 + \sqrt{6}] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.3. Solving inverse problems; answer block on page 26) | |
dgn.parameters.43 |
answer recovered | `` | a \in (-\infty; -1/2) \cup [2/3; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.4. Trigonometric equations and inequalities with a parameter; answer block on page 29) | |
dgn.parameters.p26.44 |
answer recovered | `` | a \in [2/5; 11/2] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.4. Trigonometric equations and inequalities with a parameter; answer block on page 29) | |
dgn.parameters.p26.45 |
answer recovered | `` | For a \in (-\infty; -5/4) one solution; for a = -5/4 two solutions; for a \in (-5/4; -1) three solutions; for a \in [-1; 1 - \sqrt{2}) two solutions; for a = 1 - \sqrt{2} one solution; for a \in (1 - \sqrt{2}; 5) two solutions; for a \in [5; +\infty) one solution |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.4. Trigonometric equations and inequalities with a parameter; answer block on page 29) | |
dgn.parameters.p28.46 |
answer recovered | `` | q \in (-\infty; 0] \cup [12; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.4. Trigonometric equations and inequalities with a parameter; answer block on page 29) | |
dgn.parameters.p28.47 |
answer recovered | `` | a \in [-9; 2] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.4. Trigonometric equations and inequalities with a parameter; answer block on page 29) | |
dgn.parameters.p30.30 |
answer recovered | `` | If a = \arccos(-2/\sqrt{7}) + 2\pi l, l \in \mathbb{Z}, then x = -\pi/6 + 2\pi k, k \in \mathbb{Z}; if a = -\arccos(-2/\sqrt{7}) + 2\pi m, m \in \mathbb{Z}, then x = -5\pi/6 + 2\pi n, n \in \mathbb{Z}; for other values of the parameter a there are no solutions. |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.4. Trigonometric equations and inequalities with a parameter; answer block on page 32) | |
dgn.parameters.p31.31 |
answer recovered | `` | If a = \arccos(-2/\sqrt{7}) + 2\pi l, l \in \mathbb{Z}, then x = -\pi/6 + 2\pi k, k \in \mathbb{Z}; if a = -\arccos(-2/\sqrt{7}) + 2\pi m, m \in \mathbb{Z}, then x = -5\pi/6 + 2\pi n, n \in \mathbb{Z}; for other values of the parameter a there are no solutions. |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.4. Trigonometric equations and inequalities with a parameter; answer block on page 32) | |
dgn.parameters.p32.14 |
answer recovered | `` | -1/3, -5/3, \pm 1, \pm \sqrt{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.4. Trigonometric equations and inequalities with a parameter; answer block on page 35) | |
dgn.parameters.p35.50 |
answer recovered | `` | k \in [1; 2) \cup (2; 3] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p35.51 |
answer recovered | `` | a \in (-5; -\sqrt{24}) \cup (-\sqrt{24}; -3) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.52 |
mispaired answer | $a \in (15/2; 8) \cup (8; 12)$. |
a \in (15/2; 8) \cup (8; 12) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p36.53 |
answer recovered | `` | a \in [-1; 2) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.54 |
mispaired answer | $a = \pm 1/6, \pm \sqrt{2}/6$. |
a = \pm 1/6, \pm \sqrt{2}/6 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p36.55 |
answer recovered | `` | a \in (-\infty; -3) \cup (1; 6) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p36.56 |
answer recovered | `` | a \in [-12/5; 0] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p36.57 |
answer recovered | `` | If b = 5\pi/6 + 2\pi l, l \in \mathbb{Z}, then x = \pi/6 + 2\pi k, k \in \mathbb{Z}; if b = -5\pi/6 + 2\pi m, m \in \mathbb{Z}, then x = 5\pi/6 + 2\pi n, n \in \mathbb{Z}; for other values of the parameter b there are no solutions |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p36.58 |
answer recovered | `` | a = [1/3; 33/32] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.59 |
mispaired answer | $a \in \{3\} \cup [\sqrt{10}; \sqrt{11}]$. |
a \in \{3\} \cup [\sqrt{10}; \sqrt{11}] |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p37.60 |
answer recovered | `` | x = \pi n, y = \pi n - 1, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p37.61 |
answer recovered | `` | a = 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.62 |
mispaired answer | $-1/6$, $-1/2$, $\pm 3/4$, $\pm \dfrac{1}{4\sqrt{39}/7}$. |
-1/6, -1/2, \pm 3/4, \pm \frac{1}{4\sqrt{39}/7} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p37.63 |
answer recovered | `` | b = 2/5 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.64 |
mispaired answer | $[0; 1)$. |
[0; 1) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p38.65 |
answer recovered | `` | a \in [0; 1) \cup (1; 2] \cup \{3\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p38.66 |
answer recovered | `` | [11\pi/24; \pi/2) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p38.67 |
answer recovered | `` | If p = 9, then x = 3\pi/2; for p < 9 there are no solutions |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p38.68 |
answer recovered | `` | \alpha = 2\pi k, k \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p39.50 |
mispaired answer | $k \in [1;2) \cup (2;3]$. |
k \in [1; 2) \cup (2; 3] |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p39.51 |
mispaired answer | $a \in (-5; -\sqrt{24}) \cup (-\sqrt{24}; -3)$. |
a \in (-5; -\sqrt{24}) \cup (-\sqrt{24}; -3) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p39.53 |
mispaired answer | $a \in [-1; 2)$. |
a \in [-1; 2) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p39.55 |
mispaired answer | $a \in (-\infty; -3) \cup (1; 6)$. |
a \in (-\infty; -3) \cup (1; 6) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p39.56 |
mispaired answer | $a \in [-12/5; 0]$. |
a \in [-12/5; 0] |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p39.57 |
mispaired answer | If $b = 5\pi/6 + 2\pi l$, $l \in \mathbb{Z}$, then $x = \pi/6 + 2\pi k$, $k \in \mathbb{Z}$; if $b = -5\pi/6 + 2\pi m$, $m \in \mathbb{Z}$, then $x = 5\pi/6 + 2\pi n$, $n \in \mathbb{Z}$; for other values of the parameter $b$ there are no solutions. |
If b = 5\pi/6 + 2\pi l, l \in \mathbb{Z}, then x = \pi/6 + 2\pi k, k \in \mathbb{Z}; if b = -5\pi/6 + 2\pi m, m \in \mathbb{Z}, then x = 5\pi/6 + 2\pi n, n \in \mathbb{Z}; for other values of the parameter b there are no solutions |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p39.58 |
mispaired answer | $a = [1/3; 33/32]$. |
a = [1/3; 33/32] |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p39.60 |
mispaired answer | $x = \pi n$, $y = \pi n - 1$, $n \in \mathbb{Z}$. |
x = \pi n, y = \pi n - 1, n \in \mathbb{Z} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p39.61 |
mispaired answer | $a = 1$. |
a = 1 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p39.63 |
mispaired answer | $b = 2/5$. |
b = 2/5 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p39.65 |
mispaired answer | $a \in [0; 1) \cup (1; 2] \cup \{3\}$. |
a \in [0; 1) \cup (1; 2] \cup \{3\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p39.66 |
mispaired answer | $[11\pi/24; \pi/2)$. |
[11\pi/24; \pi/2) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p39.67 |
mispaired answer | If $p = 9$, then $x = 3\pi/2$; for $p < 9$ there are no solutions. |
If p = 9, then x = 3\pi/2; for p < 9 there are no solutions |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p39.68 |
mispaired answer | $\alpha = 2\pi k$, $k \in \mathbb{Z}$. |
\alpha = 2\pi k, k \in \mathbb{Z} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 39) | |
dgn.parameters.p41.16 |
answer recovered | `` | For a \leqslant -1/\sqrt[4]{12} there are no solutions; if -1/\sqrt[4]{12} < a < 0, then x \in \left(\frac{-1+\sqrt{1-12a^4}}{2a}; \frac{-1-\sqrt{1-12a^4}}{2a}\right); if a = 0, then x \in (0; +\infty); if 0 < a \leqslant 1/\sqrt[4]{12}, then x \in \left(-\infty; \frac{-1-\sqrt{1-12a^4}}{2a}\right) \cup \left(\frac{-1+\sqrt{1-12a^4}}{2a}; +\infty\right); if a > 1/\sqrt[4]{12}, then x \in (-\infty; +\infty). |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 43) | |
dgn.parameters.p42.42 |
answer recovered | `` | For a \leqslant -1/\sqrt[4]{12} there are no solutions; if -1/\sqrt[4]{12} < a < 0, then x \in \left(\frac{-1+\sqrt{1-12a^4}}{2a}; \frac{-1-\sqrt{1-12a^4}}{2a}\right); if a = 0, then x \in (0; +\infty); if 0 < a \leqslant 1/\sqrt[4]{12}, then x \in \left(-\infty; \frac{-1-\sqrt{1-12a^4}}{2a}\right) \cup \left(\frac{-1+\sqrt{1-12a^4}}{2a}; +\infty\right); if a > 1/\sqrt[4]{12}, then x \in (-\infty; +\infty). |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 43) | |
dgn.parameters.p44.44 |
answer recovered | `` | b \in [3/4; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.5. Equations that reduce to the study of a quadratic; answer block on page 45) | |
dgn.parameters.69 |
answer recovered | `` | a = 12 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 48) | |
dgn.parameters.70 |
answer recovered | `` | a = 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 48) | |
dgn.parameters.71 |
answer recovered | `` | a \in (-\infty; 1] \cup \{5/4\} \cup [4/3, +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 48) | |
dgn.parameters.72 |
answer recovered | `` | |a| \leqslant 3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 48) | |
dgn.parameters.73 |
answer recovered | `` | a \in (-\infty; 20] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 48) | |
dgn.parameters.74 |
answer recovered | `` | a \in (0; 1/8) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 48) | |
dgn.parameters.75 |
answer recovered | `` | a > -2\sqrt{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 48) | |
dgn.parameters.76 |
answer recovered | `` | a = 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 48) | |
dgn.parameters.77 |
answer recovered | `` | 1. (1; (2 + \sqrt{13})/4]; 2. b = 7/3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 48) | |
dgn.parameters.78 |
answer recovered | `` | a \in (2; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 48) | |
dgn.parameters.79 |
answer recovered | `` | (-\infty; -3/2] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 48) | |
dgn.parameters.80 |
answer recovered | `` | For a < 0 there are no solutions; if a = 0, then x \in (0; +\infty); if a > 0, then x \in [-a/3; 0), (8a; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 48) | |
dgn.parameters.81 |
answer recovered | `` | b \in (-\infty; -1/(2\sqrt{2})] \cup [-1/4; 1/4] \cup [1/(2\sqrt{2}); +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 48) | |
dgn.parameters.82 |
answer recovered | `` | a \in (-1/3; 3/10] \cup \{1\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 48) | |
dgn.parameters.83 |
answer recovered | `` | a) 2 + \sqrt{2}, b) (-\infty; 2 - \sqrt{2}) \cup \{1\} \cup (2 + \sqrt{2}; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 48) | |
dgn.parameters.Example 18 |
answer recovered | `` | If a = 2/3, then x = 2; for other a there are no solutions. |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 49) | |
dgn.parameters.p49.19 |
mispaired answer | a = 4, a = -8 |
For a = 4, a = -8. |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 49) | |
dgn.parameters.84 |
answer recovered | `` | a \in (-\infty; -2) \cup (2; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 52) | |
dgn.parameters.85 |
answer recovered | `` | a = (1 - \sqrt{2})/2, x = 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 52) | |
dgn.parameters.86 |
answer recovered | `` | x = -1, y = -2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 52) | |
dgn.parameters.87 |
answer recovered | `` | x = 0, y = \pm 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 52) | |
dgn.parameters.88 |
answer recovered | `` | x = \sqrt{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 52) | |
dgn.parameters.89 |
answer recovered | `` | (1; 5; 0), (1; -5; 0), (-1; 5; 0), (-1; -5; 0) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 52) | |
dgn.parameters.90 |
answer recovered | `` | a = -3, a = 9 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 52) | |
dgn.parameters.91 |
answer recovered | `` | a = 0, a = 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 52) | |
dgn.parameters.92 |
answer recovered | `` | x = -1, -1 + \pi/2 + 2\pi n/3, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 52) | |
dgn.parameters.93 |
answer recovered | `` | For b = -1/2 the solution is (1/\sqrt[3]{8}; -\pi/4 + \pi k), k \in \mathbb{Z}; for b = 1/2 the solution is (1/\sqrt[3]{8}; \pi/4 + \pi k), k \in \mathbb{Z}; for b \neq \pm 1/2 there are no solutions |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 52) | |
dgn.parameters.p51.94 |
answer recovered | `` | For a = -1/2 the solution is (\pi/2 + \pi k; 1/\sqrt[3]{2}), k \in \mathbb{Z}; for a = 1/2 the solution is (\pi k; 1/\sqrt[3]{2}), k \in \mathbb{Z}; for a \neq \pm 1/2 there are no solutions |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 52) | |
dgn.parameters.95 |
answer recovered | `` | (3; 3), (-3; -3), (2\sqrt{3}; \sqrt{3}), (-2\sqrt{3}; -\sqrt{3}) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 52) | |
dgn.parameters.96 |
answer recovered | `` | (x; y) = (-2/3; 1), (-1 - 1/(l - 1); l^2 + l - 1), (-1 + 1/(l + 2); l^2 + l - 1), l \in \mathbb{Z}, l \neq -5, -2, 1, 4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 52) | |
dgn.parameters.97 |
answer recovered | `` | a \in (-2\sqrt{6}/3; 2\sqrt{6}/3) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 52) | |
dgn.parameters.100 |
mispaired answer | $\alpha \in (3/2; +\infty)$. **Hint.** Introduce the substitution $u = 2^x$, $v = 2^y$. Eliminate $v$ from the first inequality and prove that under the resulting restriction on $\alpha$ a solution always exists. |
\alpha \in (3/2; +\infty) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 52) | |
dgn.parameters.101 |
mispaired answer | $a = \pm\sqrt{2}$, $\pm(\sqrt{15} + 1)/4$. |
a = \pm\sqrt{2}, \pm(\sqrt{15} + 1)/4 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 52) | |
dgn.parameters.p52.102 |
mispaired answer | If $a = 0$, then $x = \pi m$, $2\pi n/3$, $m, n \in \mathbb{Z}$; if $a \neq 0$ there are no solutions. |
If a = 0, then x = \pi m, 2\pi n/3, m, n \in \mathbb{Z}; for a \neq 0 there are no solutions |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 52) | |
dgn.parameters.98 |
mispaired answer | $x = -\sqrt{7/5}$. **Hint.** Complete the square with respect to $z$ first. |
x = -\sqrt{7}/5 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 52) | |
dgn.parameters.99 |
mispaired answer | $t = 1$, $x = -\pi/30 + 2\pi n/5$, $n \in \mathbb{Z}$. |
t = 1, x = -\pi/30 + 2\pi n/5, n \in \mathbb{Z} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.6. Completing the square and non-negative expressions; answer block on page 52) | |
dgn.parameters.p54.20 |
mispaired answer | If $a \leqslant -1/\sqrt[4]{12}$, no solutions; if $-1/\sqrt[4]{12} < a < 0$, $x \in \left(\dfrac{-1+\sqrt{1-12a^4}}{2a};\,\dfrac{-1-\sqrt{1-12a^4}}{2a}\right)$; if $a = 0$, $x \in (0;+\infty)$; if $0 < a \leqslant \dfrac{1}{\sqrt[4]{12}}$, $x \in \left(-\infty;\,\dfrac{-1-\sqrt{1-12a^4}}{2a}\right) \cup \left(\dfrac{-1+\sqrt{1-12a^4}}{2a};+\infty\right)$; if $a > 1/\sqrt[4]{12}$, $x \in (-\infty;+\infty)$. |
For a \leqslant -1/\sqrt[4]{12} there are no solutions; if -1/\sqrt[4]{12} < a < 0, then x \in \left(\frac{-1+\sqrt{1-12a^4}}{2a}; \frac{-1-\sqrt{1-12a^4}}{2a}\right); if a = 0, then x \in (0; +\infty); if 0 < a \leqslant 1/\sqrt[4]{12}, then x \in \left(-\infty; \frac{-1-\sqrt{1-12a^4}}{2a}\right) \cup \left(\frac{-1+\sqrt{1-12a^4}}{2a}; +\infty\right); if a > 1/\sqrt[4]{12}, then x \in (-\infty; +\infty). |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.7. Factorisation; answer block on page 54) | |
dgn.parameters.Example 21 |
answer recovered | `` | If a \leqslant -1/4, then x \in [-3 - 1/\sqrt{3}; -3) \cup (-3; -3 + 1/\sqrt{3}]; if a \in (-1/4; 1/10), then x \in (-4; -3 - f_a] \cup [-3 - 1/\sqrt{3}; -3) \cup (-3; -3 + 1/\sqrt{3}] \cup [-3 + f_a; -2); if a = 1/10, then x \in (-4; -3) \cup (-3; -2); if a \in (1/10; 1/3), then x \in (-4; -3 - 1/\sqrt{3}] \cup [-3 - f_a; -3) \cup (-3; -3 + f_a] \cup [-3 + 1/\sqrt{3}; -2); if a \geqslant 1/3, then x \in (-4; -3 - 1/\sqrt{3}] \cup [-3 + 1/\sqrt{3}; -2), where f_a = ((1 - 3a)/(a + 2))^{\log_3 \sqrt{(a+2)/(1-3a)}}. |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.7. Factorisation; answer block on page 56) | |
dgn.parameters.107 |
answer recovered | `` | a \in [1; 5/2] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.7. Factorisation; answer block on page 59) | |
dgn.parameters.108 |
answer recovered | `` | If a \leqslant -3, then x \in (a + 1; 0) \cup (-(a + 3); +\infty); if a \in (-3; -2), then x \in (a + 1; -(a + 3)) \cup (0; +\infty); if a = -2, then x \in (0; +\infty); if a \in (-2; -1), then x \in (-(a + 3); a + 1) \cup (0; +\infty); if a \geqslant -1, then x \in (-(a + 3); 0) \cup (a + 1; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.7. Factorisation; answer block on page 59) | |
dgn.parameters.p57.109 |
answer recovered | `` | If a < -2, then x \in (-\infty; a) \cup (-2; +\infty); if a = -2, then x \in \mathbb{R}\setminus\{-2\}; if a \in (-2; 1/2), then x \in (-\infty; -2) \cup (a; +\infty); if a = 1/2, then x \in (-\infty; -2) \cup (1/2; 1) \cup (1; +\infty); if a > 1/2, then x \in (-\infty; -2) \cup (a; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.p57.110 |
answer recovered | `` | 12\sqrt{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.p57.111 |
answer recovered | `` | a = 1, a = \sqrt{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.p57.112 |
answer recovered | `` | If a \leqslant -3/2, then x \in (3; 4 - 1/\sqrt{2}] \cup [1/\sqrt{2}; 5); if a \in (-3/2; -1), then x \in (3; 4 - 1/\sqrt{2}] \cup [4 - f_a; 4) \cup (4; 4 + f_a] \cup [4 + 1/\sqrt{2}; 5); if a = -1, then x \in (3; 4) \cup (4; 5); if a \in (-1; -2/3), then x \in (3; 4 - f_a] \cup [4 - 1/\sqrt{2}; 4) \cup (4; 4 + 1/\sqrt{2}] \cup [4 + f_a; 5); if a \geqslant -2/3, then x \in [4 - 1/\sqrt{2}; 4) \cup (4; 4 + 1/\sqrt{2}], where f_a = ((2a + 3)/(1 - a))^{\log_2 \sqrt{(1-a)/(2a+3)}} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.p57.113 |
answer recovered | `` | For a < 0 there are no solutions, for a = 0 the solution is the unique one x = -1, for a > 0 the solutions are x = -1 \pm \sqrt{a} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.p58.114 |
answer recovered | `` | For a \leqslant -\sqrt{2} there are no solutions; if -\sqrt{2} < a < 0, then x \in ((-2 + \sqrt{4 - a^4})/a; (-2 - \sqrt{4 - a^4})/a); if a = 0, then x \in (0; +\infty); if 0 < a \leqslant \sqrt{2}, then x \in (-\infty; (-2 - \sqrt{4 - a^4})/a) \cup ((-2 + \sqrt{4 - a^4})/a; +\infty); if a > \sqrt{2}, then x \in (-\infty; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.p58.115 |
answer recovered | `` | \beta = 9/16 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.p58.116 |
answer recovered | `` | (1; -2); (-1; -2); (t; 2), t \in \mathbb{R} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.p58.117 |
answer recovered | `` | [3; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.p58.118 |
answer recovered | `` | y \in [5/6; 1) \cup (1; 3/2] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.p58.119 |
answer recovered | `` | a = -\sqrt{3}/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.p59.120 |
answer recovered | `` | a \in (44 - 24\sqrt{2}; 44 + 24\sqrt{2}) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.p59.121 |
answer recovered | `` | a = \pm 1/6, a = \pm\sqrt{2}/6 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.p59.122 |
answer recovered | `` | a \in [-1 - 1/\sqrt{2}; -1], a = -2, 0, 1, 1/\sqrt{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.p59.123 |
answer recovered | `` | a = 1/3, 4/3, -6/11, 6/11, 18/11 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.p60.109 |
answer recovered | `` | If a < -2, then x \in (-\infty; a) \cup (-2; +\infty); if a = -2, then x \in \mathbb{R}\setminus\{-2\}; if a \in (-2; 1/2), then x \in (-\infty; -2) \cup (a; +\infty); if a = 1/2, then x \in (-\infty; -2) \cup (1/2; 1) \cup (1; +\infty); if a > 1/2, then x \in (-\infty; -2) \cup (a; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.p60.112 |
answer recovered | `` | If a \leqslant -3/2, then x \in (3; 4 - 1/\sqrt{2}] \cup [1/\sqrt{2}; 5); if a \in (-3/2; -1), then x \in (3; 4 - 1/\sqrt{2}] \cup [4 - f_a; 4) \cup (4; 4 + f_a] \cup [4 + 1/\sqrt{2}; 5); if a = -1, then x \in (3; 4) \cup (4; 5); if a \in (-1; -2/3), then x \in (3; 4 - f_a] \cup [4 - 1/\sqrt{2}; 4) \cup (4; 4 + 1/\sqrt{2}] \cup [4 + f_a; 5); if a \geqslant -2/3, then x \in [4 - 1/\sqrt{2}; 4) \cup (4; 4 + 1/\sqrt{2}], where f_a = ((2a + 3)/(1 - a))^{\log_2 \sqrt{(1-a)/(2a+3)}} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.p60.113 |
answer recovered | `` | For a < 0 there are no solutions, for a = 0 there is one solution x = -1, for a > 0 the solutions are x = -1 \pm \sqrt{a} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.p60.114 |
answer recovered | `` | For a \leqslant -\sqrt{2} there are no solutions; if -\sqrt{2} < a < 0, then x \in ((-2 + \sqrt{4 - a^4})/a; (-2 - \sqrt{4 - a^4})/a); if a = 0, then x \in (0; +\infty); if 0 < a \leqslant \sqrt{2}, then x \in (-\infty; (-2 - \sqrt{4 - a^4})/a) \cup ((-2 + \sqrt{4 - a^4})/a; +\infty); if a > \sqrt{2}, then x \in (-\infty; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.p60.121 |
mispaired answer | $a = \pm 1/6, \, a = \pm \sqrt{2}/6$ |
a = \pm 1/6, a = \pm\sqrt{2}/6 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 60) | |
dgn.parameters.Example 23 |
answer recovered | `` | a = 2, b = 3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.8. Vieta's theorem for an equation of high degree; answer block on page 65) | |
dgn.parameters.124 |
answer recovered | `` | (-3; 9), (2; 4) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.9. Problems on uniqueness and on the number of solutions; answer block on page 66) | |
dgn.parameters.125 |
answer recovered | `` | a = -5, a = -5/13 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.9. Problems on uniqueness and on the number of solutions; answer block on page 66) | |
dgn.parameters.126 |
answer recovered | `` | x = \pm\sqrt{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.9. Problems on uniqueness and on the number of solutions; answer block on page 66) | |
dgn.parameters.127 |
answer recovered | `` | u = 6, v = 4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.9. Problems on uniqueness and on the number of solutions; answer block on page 66) | |
dgn.parameters.128 |
answer recovered | `` | a = 13, the roots of the equation are 2, 6, 18 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.9. Problems on uniqueness and on the number of solutions; answer block on page 66) | |
dgn.parameters.129 |
answer recovered | `` | a = 2, b = 4, c = 64 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.9. Problems on uniqueness and on the number of solutions; answer block on page 66) | |
dgn.parameters.Example 25 |
answer recovered | `` | a = 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.9. Problems on uniqueness and on the number of solutions; answer block on page 69) | |
dgn.parameters.130 |
answer recovered | `` | a = -2, a = 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 72) | |
dgn.parameters.131 |
answer recovered | `` | b = \pi n, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 72) | |
dgn.parameters.132 |
answer recovered | `` | a = 0, a = 2\sin 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 72) | |
dgn.parameters.133 |
answer recovered | `` | a = 3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 72) | |
dgn.parameters.134 |
answer recovered | `` | b = 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 72) | |
dgn.parameters.135 |
answer recovered | `` | b = \operatorname{ctg} 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 72) | |
dgn.parameters.136 |
answer recovered | `` | a = 2, b = \pi/2 + 2\pi n, n \in \mathbb{Z}; a = -2, b \in \mathbb{R} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 72) | |
dgn.parameters.137 |
answer recovered | `` | b = \sqrt{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 72) | |
dgn.parameters.138 |
answer recovered | `` | b \in (-2; 0) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 72) | |
dgn.parameters.139 |
answer recovered | `` | a = \pm 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 72) | |
dgn.parameters.140 |
answer recovered | `` | a = 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 72) | |
dgn.parameters.141 |
answer recovered | `` | a = 4/3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 72) | |
dgn.parameters.142 |
answer recovered | `` | a = -3, a = -2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 72) | |
dgn.parameters.143 |
answer recovered | `` | a = -4, 4, 6 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 72) | |
dgn.parameters.144 |
answer recovered | `` | a = \sqrt{3}/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 72) | |
dgn.parameters.Example 26 |
mispaired answer | a = 1/8; x = y = 1/2 |
For a = 1/8 the system of inequalities has the unique solution x = y = 1/2. |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 73) | |
dgn.parameters.Example 27 |
answer recovered | `` | a = 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 75) | |
dgn.parameters.145 |
answer recovered | `` | a = \pm\sqrt{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 77) | |
dgn.parameters.146 |
answer recovered | `` | a = 5\pi/6 + \pi l, \pi/18 + 2\pi m, 13\pi/18 + 2\pi n, l, m, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 77) | |
dgn.parameters.147 |
answer recovered | `` | a = 0, a = -2/5 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 77) | |
dgn.parameters.148 |
answer recovered | `` | a = 0, a = (3 \pm \sqrt{5})/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 77) | |
dgn.parameters.149 |
answer recovered | `` | b = 1/3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 77) | |
dgn.parameters.150 |
answer recovered | `` | If b = \frac{-1}{2\sqrt{2}}, then there is one solution (1/\sqrt[3]{5}; 0; 0). If b = -1/2 + \sqrt{3}/8, then there are two solutions (1/\sqrt[3]{5}; 1; \pi/4) and (1/\sqrt[3]{5}; -1; -\pi/4) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 77) | |
dgn.parameters.151 |
answer recovered | `` | a = -1/32, a = -1/4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 77) | |
dgn.parameters.152 |
answer recovered | `` | b = 3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 77) | |
dgn.parameters.153 |
answer recovered | `` | a = 2/3, a = 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 77) | |
dgn.parameters.154 |
answer recovered | `` | b = -1/4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 77) | |
dgn.parameters.155 |
answer recovered | `` | a = -2, a = -1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 77) | |
dgn.parameters.156 |
answer recovered | `` | a = 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 77) | |
dgn.parameters.p77.157 |
answer recovered | `` | x \in [3\sqrt{2}; 6] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 77) | |
dgn.parameters.p79.28 |
answer recovered | `` | x = 26, y = 6 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 80) | |
dgn.parameters.p80.29 |
mispaired answer | x = 26, y = 6 |
a = 2, a = 16/3 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 82) | |
dgn.parameters.p82.157 |
mispaired answer | a = 2, a = 16/3 |
x \in [3\sqrt{2}; 6] |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 77) | |
dgn.parameters.158 |
answer recovered | `` | [17; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 84) | |
dgn.parameters.159 |
answer recovered | `` | x = -1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 84) | |
dgn.parameters.160 |
answer recovered | `` | For a \in (-1; 0) there are no solutions. For a = -1, a = 0, the solution is x = y = 1. For a \in (-\infty; -1) \cup (0; +\infty), the solutions are x_1 = y_1 = 2^2\sqrt{(a^2+a)/2}, x_2 = y_2 = 2^{-2}\sqrt{(a^2+a)/2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 84) | |
dgn.parameters.161 |
answer recovered | `` | a = 1/16 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 84) | |
dgn.parameters.162 |
answer recovered | `` | x = \pi/4 + \pi n, y = \pi/4 + \pi k, n, k \in \mathbb{Z}; x = -\pi/4 + \pi n, y = -\pi/4 + \pi k, n, k \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 84) | |
dgn.parameters.164 |
answer recovered | `` | a = -2, a = -16/3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 84) | |
dgn.parameters.165 |
answer recovered | `` | x = \pi/4 + \pi n/2, y = \pi/2 + 2\pi k, n, k \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 84) | |
dgn.parameters.166 |
answer recovered | `` | \pm\pi/6 + \pi n, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 84) | |
dgn.parameters.167 |
answer recovered | `` | x = 1/2 + (1/2)\log_4 3, y = 1/2 - (1/2)\log_4 3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 84) | |
dgn.parameters.168 |
answer recovered | `` | (2\sqrt{2}/(2\sqrt{2} - 1)), (2\sqrt{2}/(2\sqrt{2} + 1)) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 84) | |
dgn.parameters.169 |
answer recovered | `` | b = -\sqrt{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 84) | |
dgn.parameters.170 |
answer recovered | `` | x = 0; x = \sqrt{5a}/2; x = -\sqrt{5a}/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 84) | |
dgn.parameters.p85.31 |
answer recovered | `` | a \in [1; 2) \cup (2; 3] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 87) | |
dgn.parameters.173 |
answer recovered | `` | x = 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 89) | |
dgn.parameters.174 |
answer recovered | `` | p \in (-\infty; -11) \cup (5; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 89) | |
dgn.parameters.175 |
answer recovered | `` | x = 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 89) | |
dgn.parameters.176 |
answer recovered | `` | If a = 0, then x = -1, 0, if a \neq 0, then x = 0 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 89) | |
dgn.parameters.177 |
answer recovered | `` | If q = -4, then the solution is (1; 0), if q = 4, then the solution is (-3; 0) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 89) | |
dgn.parameters.178 |
answer recovered | `` | p = -2, p = 1/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 89) | |
dgn.parameters.179 |
answer recovered | `` | For a \in (0; 4 - 2\sqrt{2}] the minimum equals -a^2, and for a \in (4 - 2\sqrt{2}; 2) the minimum equals 8(1 - a) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 89) | |
dgn.parameters.180 |
answer recovered | `` | x = -1, y = \pm 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 89) | |
dgn.parameters.181 |
answer recovered | `` | 3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 89) | |
dgn.parameters.182 |
answer recovered | `` | a \in [3 \cdot 10^6; 3.125 \cdot 10^6] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 89) | |
dgn.parameters.183 |
answer recovered | `` | [0; 1/2) \cup (1/2; 3/4] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 89) | |
dgn.parameters.184 |
answer recovered | `` | a \in [-1/\sqrt{10}; 1/\sqrt{10}] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 89) | |
dgn.parameters.p92.32 |
answer recovered | `` | a \in [-8; 6] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 93) | |
dgn.parameters.p93.33 |
answer recovered | `` | x \in (-\infty; (4\pi + 18)/5] \cup [8\pi - 18; 18 - 3\pi] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 95) | |
dgn.parameters.p94.94 |
answer recovered | `` | x \in (-\infty; (4\pi + 18)/5] \cup [8\pi - 18; 18 - 3\pi] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 95) | |
dgn.parameters.186 |
answer recovered | `` | The smallest value is 1/5, attained at a = \pm 2/5, b = 4/5 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 97) | |
dgn.parameters.187 |
answer recovered | `` | 1. a \in (-3; 5). 2. a \in [2 - 2\sqrt{7}; -3] \cup \{5\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 97) | |
dgn.parameters.188 |
answer recovered | `` | a \in [-6; 4] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 97) | |
dgn.parameters.189 |
answer recovered | `` | a \in (-7; 5) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 97) | |
dgn.parameters.195 |
mispaired answer | $a \in [1;3]$. |
a \in [1; 3] |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 97) | |
dgn.parameters.196 |
mispaired answer | $n=k$, $a=10{,}15-k$, where $k \in \mathbb{Z}$, $1 \leqslant k \leqslant 10$. |
n = k, a = 10, 15 - k, where k \in \mathbb{Z}, 1 \leqslant k \leqslant 10 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 97) | |
dgn.parameters.p97.197 |
mispaired answer | $a \in [2;3) \cup (3;4]$. |
a \in [2; 3) \cup (3; 4] |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 97) | |
dgn.parameters.p97.198 |
mispaired answer | $a \in [\pi-3-7/5;\pi-3+7/5) \cup \{\pi-3+\sqrt{74}/5\}$. |
a \in [\pi - 3 - 7/5; \pi - 3 + 7/5) \cup \{\pi - 3 + \sqrt{74}/5\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 97) | |
dgn.parameters.p97.199 |
mispaired answer | $a \in [0;1]$. |
a \in [0; 1] |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 97) | |
dgn.parameters.p99.35 |
answer recovered | `` | For a < 0 there are no solutions; if a \in [0; 1/8], then x \in (0; 1-2\sqrt{2a}); if a \in (1/8; 1/2], then x \in (1-2\sqrt{2a}; 0); if a > 1/2, then x \in [-2a; 0). |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 101) | |
dgn.parameters.p101.36 |
answer recovered | `` | S_{ABCD} = 12 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 104) | |
dgn.parameters.p102.102 |
answer recovered | `` | S_{ABCD} = 12 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 104) | |
dgn.parameters.200 |
answer recovered | `` | a \in (-1/2; 2) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 106) | |
dgn.parameters.201 |
answer recovered | `` | a_1 \in (2; 2,5) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 106) | |
dgn.parameters.202 |
answer recovered | `` | 9(\pi + 1)/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 106) | |
dgn.parameters.203 |
answer recovered | `` | S = \pi/3 + 2\sqrt{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 106) | |
dgn.parameters.p104.204 |
answer recovered | `` | P = 3\pi/\sqrt{2} + 2\sqrt{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 106) | |
dgn.parameters.205 |
answer recovered | `` | S = 2\pi + 7 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 106) | |
dgn.parameters.206 |
answer recovered | `` | 10\pi + 16 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 106) | |
dgn.parameters.207 |
answer recovered | `` | [-1; 5] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 106) | |
dgn.parameters.208 |
answer recovered | `` | S = 15/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 106) | |
dgn.parameters.209 |
answer recovered | `` | p = 6 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 106) | |
dgn.parameters.210 |
answer recovered | `` | a \in [-6; 1 - \sqrt{13}] \cup [\sqrt{13} - 1; 6] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 106) | |
dgn.parameters.211 |
answer recovered | `` | (x - 2)^2 + (y - 5)^2 = 20 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 106) | |
dgn.parameters.212 |
answer recovered | `` | If |b| \in [0; 1/4], then x \in (0; 1 - 4|b|); if |b| \in (1/4; 1/2], then x \in (1 - 4|b|; 0); if |b| > 1/2, then x \in [-4b^2; 0) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 106) | |
dgn.parameters.213 |
answer recovered | `` | If a = -1, then x \in (2; +\infty); if a \in (-1; -1/2), then x \in (1; a + 2] \cup (1 - a; +\infty); if a = -1/2, then x \in (1; 3/2) \cup (3/2; +\infty); if a \in (-1/2; 0), then x \in (1; 1 - a) \cup [a + 2; +\infty); if a = 0, then x \in [2; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 106) | |
dgn.parameters.p110.39 |
mispaired answer | $(x; y)=(2; 0)$ |
x = 2, y = 0 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 111) | |
dgn.parameters.p111.214 |
answer recovered | `` | m = 0, x = 3; m = \pm 1, x = \pm(3 \pm 2\sqrt{2})/2; m = \pm 2, x = (3 \pm \sqrt{5})/2; m = \pm 3, x = \pm 3/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p111.215 |
answer recovered | `` | m = 1, n = -1 and m = -1, n = 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p111.216 |
transcription error | Item 216 of the «Ответы» block on p. 114: «216. 72. Указание. Упростите выражение и воспользуйтесь оценкой |x| < √(x(x+2)) < |x+1|.» |
72 |
the printed answer was mis-transcribed during extraction (the page prints the integer 72, not a square root) - The page shows «216. 72.» — the answer is the integer x = 72 — but we publish «\sqrt{72}» (which the engine then read as 6√2 and compared with the expression's value). The printed answer is correct: the expression simplifies to (x+2)/2·(1+√(x/(x+2))), and for x = 1…99 the closest value to 73 is attained at x = 72, where the value is 37+6√37 ≈ 73.4966 (x = 71 gives 72.4965, x = 73 gives 74.4966). | |
dgn.parameters.p111.217 |
answer recovered | `` | -13/3 < a \leqslant -19/5 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p111.218 |
answer recovered | `` | \{(9; 9)\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p112.219 |
answer recovered | `` | (4; 3), (6; 13), (14; 5) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p112.220 |
answer recovered | `` | (0, 8; 0, 98) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p112.221 |
answer recovered | `` | (7k; 3k; 2k), k \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p112.222 |
answer recovered | `` | m = 4, n = 9; m = 5, n = 8 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p112.223 |
answer recovered | `` | [-3; -2) \cup \{1\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p112.224 |
answer recovered | `` | m = 0, n = 0 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p112.225 |
answer recovered | `` | x = 2, y = 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p112.226 |
answer recovered | `` | (1; 6), (1; 7), (2; 7) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p113.227 |
answer recovered | `` | |a| \in (1; \sqrt{2}] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p113.228 |
answer recovered | `` | (0; 0), (2; 2), (0; 3), (3; 0) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p113.229 |
answer recovered | `` | u = -187; v = -819 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p113.230 |
answer recovered | `` | 1984 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p113.231 |
answer recovered | `` | (x; y; z) = (5; 4; 4) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p113.232 |
answer recovered | `` | (0; 2), (-2; 0), (0; 3), (2; 1) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p113.233 |
answer recovered | `` | (2; 3), (2; -3), (-2; 3), (-2; -3) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p113.234 |
answer recovered | `` | a = -2, b = 4; 5; \ldots; a = -1, b = 3; 4; 5; \ldots |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p114.215 |
mispaired answer | $m=1$, $n=-1$ and $m=-1$, $n=1$ |
m = 1, n = -1 and m = -1, n = 1 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p114.216 |
mispaired answer | $72$ |
\sqrt{72} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p114.218 |
mispaired answer | $(9; 9)$ |
\{(9; 9)\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p114.220 |
mispaired answer | $(0{,}8; 0{,}98)$ |
(0, 8; 0, 98) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 114) | |
dgn.parameters.p117.42 |
answer recovered | `` | x = 587 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 118) | |
dgn.parameters.236 |
answer recovered | `` | \operatorname{ctg} 1 > \{\pi/2\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 119) | |
dgn.parameters.237 |
answer recovered | `` | x = k, k \in \mathbb{Z}, x \in [m; \sqrt{m^2 + 1}), m = 0, 1, 2, 3, \ldots |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 119) | |
dgn.parameters.238 |
answer recovered | `` | [2; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 119) | |
dgn.parameters.239 |
answer recovered | `` | 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 119) | |
dgn.parameters.p120.44 |
answer recovered | `` | x = \sin(\pi/12) = (\sqrt{6} - \sqrt{2})/4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 121) | |
dgn.parameters.p121.45 |
answer recovered | `` | (x; y; z) = (\tg \alpha; \tg 2\alpha; \tg 4\alpha), where \alpha = \pm\pi/7, \pm2\pi/7, \pm3\pi/7. |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 122) | |
dgn.parameters.242 |
answer recovered | `` | [0; (3 - \sqrt{5})/6) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 124) | |
dgn.parameters.243 |
wrong answer | page 124, item 243 of the 'Ответы' block: x₁ = −1/√2, x₂ = √2 − √3/2 (printed problem, page 122: 'Задача 243. Найдите все решения уравнения √((1 + 2x√(1−x²))/2) + 2x² = 1.') |
x_1 = -1/\sqrt{2},\quad x_2 = (\sqrt{6}-\sqrt{2})/4 |
With x = sin t, |x| ⩽ 1, the equation becomes |sin t + cos t|/√2 = cos 2t; it has exactly two real roots. x₁ = −1/√2 checks out (radicand 0, 2x² = 1). The printed x₂ = √2 − √3/2 = 0.5481882 does NOT: 2x² + √((1+2x√(1−x²))/2) − 1 = +0.5800404 ≠ 0. The true second root is (√6 − √2)/4 = sin(π/12) = 0.2588190451, for which 2x√(1−x²) = 1/2, the radicand is 3/4 and the left side is √3/2 + (2−√3)/2 = 1. A sign-change sweep over [−1, 1] returns only −0.7071067812 and 0.2588190451, matching the corrected pair. The engine's complaint is valid. | |
dgn.parameters.244 |
answer recovered | `` | \pm\sqrt{2}/6, (\sqrt{2} + \sqrt{6})/12 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 124) | |
dgn.parameters.245 |
answer recovered | `` | x_1 = \cos(\pi/9), x_2 = \cos(2\pi/7), x_3 = \cos(\pi/3) = 1/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 124) | |
dgn.parameters.246 |
answer recovered | `` | -10 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 124) | |
dgn.parameters.247 |
answer recovered | `` | a \in [-5; 5] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 124) | |
dgn.parameters.248 |
answer recovered | `` | \pi/16 + \pi n/4, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 124) | |
dgn.parameters.249 |
answer recovered | `` | (\frac{\operatorname{tg}\alpha}{3}; \frac{\operatorname{tg}2\alpha}{3}; \operatorname{tg}4\alpha), \alpha = \pm\frac{\pi}{7}, \pm\frac{2\pi}{7}, \pm\frac{3\pi}{7} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 124) | |
dgn.parameters.250 |
answer recovered | `` | a \in (-\infty; -\sqrt{3}/2) \cup (\sqrt{3}/2; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 124) | |
dgn.parameters.251 |
answer recovered | `` | 2\sqrt{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 124) | |
dgn.parameters.252 |
answer recovered | `` | (\operatorname{tg}\alpha; \frac{\operatorname{tg}2\alpha}{2}; \frac{\operatorname{tg}4\alpha}{2}), \alpha = \pm\frac{\pi}{7}, \pm\frac{2\pi}{7}, \pm\frac{3\pi}{7} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 124) | |
dgn.parameters.253 |
answer recovered | `` | 4\sqrt{6}/3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 124) | |
dgn.parameters.254 |
answer recovered | `` | a \in (-\infty; -\sqrt{13}] \cup [11/3; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 124) | |
dgn.parameters.255 |
answer recovered | `` | x = 1, -1/3, -1/2, -3/4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 124) | |
dgn.parameters.256 |
answer recovered | `` | (3 + \sqrt{8} - \sqrt{11})^2/4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 124) | |
dgn.parameters.p127.46 |
answer recovered | `` | p \in [-12; 0) \cup (0; 12] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 128) | |
dgn.parameters.p129.47 |
answer recovered | `` | p = 4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 129) | |
dgn.parameters.p130.257 |
answer recovered | `` | q \in [-15; 0) \cup (0; 15] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 132) | |
dgn.parameters.p130.258 |
answer recovered | `` | x \in (0; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 132) | |
dgn.parameters.p130.259 |
answer recovered | `` | a \in [5; 12] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 132) | |
dgn.parameters.p130.260 |
answer recovered | `` | a = -2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 132) | |
dgn.parameters.p130.261 |
answer recovered | `` | a \in (-\infty; 1) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 132) | |
dgn.parameters.p131.262 |
answer recovered | `` | p = -3/2, p = -1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 132) | |
dgn.parameters.p131.263 |
answer recovered | `` | x \in (-2/3; 0) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 132) | |
dgn.parameters.p131.264 |
answer recovered | `` | -1/5 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 132) | |
dgn.parameters.p131.265 |
answer recovered | `` | For a \in (-\infty; 0) \cup [1; +\infty) there are no solutions; for a = 0 there is one solution x = 1/3; for a \in (0; (3 - \sqrt{3})/2) there are two solutions: x = (1/3) \cdot (1 \pm \sqrt{2a/(1 - a)}); for a = (3 - \sqrt{3})/2 there are three solutions: x = -\frac{1}{\sqrt{3}}, x = \frac{1 \pm \sqrt[4]{12}}{3}; for a \in ((3 - \sqrt{3})/2; 2/3) there are four solutions: x = \frac{-a \pm \sqrt{-2a^2 + 6a - 3}}{3(1 - a)}, x = (1/3) \cdot (1 \pm \sqrt{2a/(1 - a)}); for a = 2/3 there are three solutions: x = -1/3, x = \pm 1; for a \in (2/3; 1) there are two solutions: x = (1/3) \cdot (1 + \sqrt{2a/(1 - a)}), x = (-a + \sqrt{-2a^2 + 6a - 3})/(3(1 - a)) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 132) | |
dgn.parameters.p131.266 |
answer recovered | `` | 1/4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 132) | |
dgn.parameters.p131.267 |
answer recovered | `` | x \in (-13 - \sqrt{57}; 8) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 132) | |
dgn.parameters.p131.268 |
answer recovered | `` | a \in [-1; 1/5) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 132) | |
dgn.parameters.p132.265 |
answer recovered | `` | For a \in (-\infty; 0) \cup [1; +\infty) there are no solutions; for a = 0 there is one solution x = 1/3; for a \in (0; (3 - \sqrt{3})/2) there are two solutions: x = (1/3) \cdot (1 \pm \sqrt{2a/(1 - a)}); for a = (3 - \sqrt{3})/2 there are three solutions: x = -\frac{1}{\sqrt{3}}, x = \frac{1 \pm \sqrt[4]{12}}{3}; for a \in ((3 - \sqrt{3})/2; 2/3) there are four solutions: x = \frac{-a \pm \sqrt{-2a^2 + 6a - 3}}{3(1 - a)}, x = (1/3) \cdot (1 \pm \sqrt{2a/(1 - a)}); for a = 2/3 there are three solutions: x = -1/3, x = \pm 1; for a \in (2/3; 1) there are two solutions: x = (1/3) \cdot (1 + \sqrt{2a/(1 - a)}), x = (-a + \sqrt{-2a^2 + 6a - 3})/(3(1 - a)) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.1. Simplest equations and inequalities with a parameter; answer block on page 132) | |
dgn.parameters.p135.50 |
mispaired answer | $a = \pm 1$, $b = 0$; $a = 0$, $b = 0$. |
a = \pm 1, b = 0; a = 0, b = 0 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 135) | |
dgn.parameters.p135.51 |
answer recovered | `` | a = 5/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 136) | |
dgn.parameters.273 |
answer recovered | `` | a \in (-\infty; -2 - \sqrt{6}) \cup (\sqrt{2}; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 139) | |
dgn.parameters.274 |
answer recovered | `` | a = 1, 2, 3, 4, 5, 6, 7 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 139) | |
dgn.parameters.275 |
answer recovered | `` | a > 3/82 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 139) | |
dgn.parameters.276 |
answer recovered | `` | |a| \leqslant 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 139) | |
dgn.parameters.277 |
answer recovered | `` | 1) a \in [-\sqrt{26} - 1; \sqrt{26} + 1], 2) a \in [-\sqrt{26} + 1; \sqrt{26} - 1] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 139) | |
dgn.parameters.278 |
mispaired answer | Continuation of a previous variant's solutions (Problems 6–7) and answers to problems 1–7, followed by a new header. Problem 6. The bisector $CD$ of angle $ACB$ at the base of an isosceles triangle $ABC$ ($AB = AC$) divides the side $AB$ in such a way that $AD = BC$. Find the length of the bisector $CD$ and the area of triangle $ABC$, if its base $BC$ is equal to $2$. Problem 7. For which integers $a$ does the inequality $$2\log_{1/2} a - 3 + 2x\log_{1/2} a - x^2 < 0$$ hold for every value of $x$? Answers. 1. $0, \pm 1, \pm 2, \pm 3$. 2. $240$. 3. $(-3/4; 0) \cup (3; +\infty)$. 4. $2\pi$. 5. $\pi/2 + 2\pi n, \ n \in \mathbb{Z}$. 6. $l = 2, \ S = \operatorname{tg}(2\pi/5) = \sqrt{5 + 2\sqrt{5}}$. 7. $1, 2, 3, 4, 5, 6, 7$. |
b = -1 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 139) | |
dgn.parameters.279 |
answer recovered | `` | a \geqslant 0 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 139) | |
dgn.parameters.280 |
answer recovered | `` | a \in (15/2; 8) \cup (12; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 139) | |
dgn.parameters.281 |
answer recovered | `` | a = 4/3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 139) | |
dgn.parameters.282 |
answer recovered | `` | (a; b) = (0; 0), (a; b) = (1; 0) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 139) | |
dgn.parameters.283 |
answer recovered | `` | a = 50 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 139) | |
dgn.parameters.p138.284 |
answer recovered | `` | b = 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 139) | |
dgn.parameters.285 |
answer recovered | `` | a = -1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 139) | |
dgn.parameters.286 |
answer recovered | `` | [-7/2; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 139) | |
dgn.parameters.288 |
answer recovered | `` | 15/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 141) | |
dgn.parameters.289 |
answer recovered | `` | \sqrt{38} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 141) | |
dgn.parameters.290 |
answer recovered | `` | 6\sqrt{6} + 3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 141) | |
dgn.parameters.291 |
answer recovered | `` | 2\pi\sqrt{10}\ \text{cm} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 141) | |
dgn.parameters.292 |
answer recovered | `` | 8 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 141) | |
dgn.parameters.293 |
answer recovered | `` | \arccos(6 - \sqrt{33}) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 141) | |
dgn.parameters.Primer 54 |
answer recovered | `` | 13 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 143) | |
dgn.parameters.p143.55 |
answer recovered | `` | x = 5/2, y = 1/4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 144) | |
dgn.parameters.p144.56 |
answer recovered | `` | a = 4, \sqrt{45} - 3 < |b| < \sqrt{45} + 3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 146) | |
dgn.parameters.p146.57 |
mispaired answer | $a = 4$, $\sqrt{45} - 3 < |b| < \sqrt{45} + 3$. |
4 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 148) | |
dgn.parameters.294 |
answer recovered | `` | 15 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 149) | |
dgn.parameters.295 |
answer recovered | `` | x = -1/2, y = 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 149) | |
dgn.parameters.296 |
answer recovered | `` | x = (217 - 5\sqrt{415})/29, y = (180 + 2\sqrt{415})/29 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 149) | |
dgn.parameters.297 |
mispaired answer | $x = 3/2$, $y = 1/4$. |
x = 3/2, y = 1/4 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 149) | |
dgn.parameters.298 |
mispaired answer | $a \in [-2; 1) \cup (1; 4]$. |
a \in [-2; 1) \cup (1; 4] |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 149) | |
dgn.parameters.299 |
mispaired answer | $b = -1$, $\sqrt{90} - 4 < |a| < \sqrt{90} + 4$. |
b = -1, \sqrt{90} - 4 < |a| < \sqrt{90} + 4 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 149) | |
dgn.parameters.300 |
mispaired answer | $5$. |
5 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: §1.2. Simplest problems with a modulus; answer block on page 149) | |
dgn.parameters.p150.1 |
answer recovered | `` | 54 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Mechanics and Mathematics Faculty, 1.; answer block on page 151) | |
dgn.parameters.(Task 5 prodolzhenif) |
answer recovered | `` | 6 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Mechanics and Mathematics Faculty, 2.; answer block on page 152) | |
dgn.parameters.Variant 2003 (mart), Mechanical-Math, 2 - Task 1 |
answer recovered | `` | 44 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Mechanics and Mathematics Faculty, 2.; answer block on page 152) | |
dgn.parameters.Variant 2003 (mart), Mechanical-Math, 2 - Task 2 |
answer recovered | `` | x = 0, x \in [\sqrt[3]{4}; \sqrt[4]{32/5}] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Mechanics and Mathematics Faculty, 2.; answer block on page 152) | |
dgn.parameters.Variant 2003 (mart), Mechanical-Math, 2 - Task 3 |
answer recovered | `` | 3\sqrt{2}, 9(1 + \sqrt{2}) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Mechanics and Mathematics Faculty, 2.; answer block on page 152) | |
dgn.parameters.Variant 2003 (mart), Mechanical-Math, 2 - Task 4 |
answer recovered | `` | S = 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Mechanics and Mathematics Faculty, 2.; answer block on page 152) | |
dgn.parameters.Variant 2003 (mart), Mechanical-Math, 2 - Task 5 |
answer recovered | `` | 6 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Mechanics and Mathematics Faculty, 2.; answer block on page 152) | |
dgn.parameters.p151.Task 6 |
mispaired answer | $\alpha = \pi n, \; n \in \mathbb{Z}$. |
\alpha = \pi n, n \in \mathbb{Z} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Mechanics and Mathematics Faculty, 1.; answer block on page 151) | |
dgn.parameters.p152.1 |
answer recovered | `` | 0 < x \leqslant 1, x \neq 3^{-4} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Mechanics and Mathematics Faculty, 1.; answer block on page 153) | |
dgn.parameters.p152.2 |
answer recovered | `` | \alpha = \pi/2 + \pi n, \alpha = -\arccos(2/5) + 2\pi k, n, k \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Mechanics and Mathematics Faculty, 1.; answer block on page 153) | |
dgn.parameters.p152.3 |
answer recovered | `` | 90^\circ, 20^\circ |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Mechanics and Mathematics Faculty, 1.; answer block on page 153) | |
dgn.parameters.p152.4 |
answer recovered | `` | 4800 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Mechanics and Mathematics Faculty, 1.; answer block on page 153) | |
dgn.parameters.p152.5 |
answer recovered | `` | 96\sqrt{3}/125 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Mechanics and Mathematics Faculty, 1.; answer block on page 153) | |
dgn.parameters.p152.6 |
answer recovered | `` | \alpha = \pi/2 + \pi n, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Mechanics and Mathematics Faculty, 2.; answer block on page 152) | |
dgn.parameters.Variant 2003 (may), Mechmath, 2 - Task 3 |
mispaired answer | $90^{\circ},\ 80^{\circ}$ |
90^\circ, 80^\circ |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Mechanics and Mathematics Faculty, 2.; answer block on page 153) | |
dgn.parameters.p155.1 |
answer recovered | `` | 3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Mechanics and Mathematics Faculty, 2.; answer block on page 155) | |
dgn.parameters.p155.2 |
answer recovered | `` | 1/8 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Mechanics and Mathematics Faculty, 2.; answer block on page 155) | |
dgn.parameters.p155.3 |
answer recovered | `` | 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Mechanics and Mathematics Faculty, 2.; answer block on page 155) | |
dgn.parameters.p155.4 |
answer recovered | `` | 2 : 1, identical |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Mechanics and Mathematics Faculty, 2.; answer block on page 155) | |
dgn.parameters.p155.5 |
answer recovered | `` | 2\pi - 4 - 5/6 < \alpha \leqslant 2\pi - 4 + 5/6, \alpha = 2\pi - 4 + \sqrt{61}/6 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Mechanics and Mathematics Faculty, 2.; answer block on page 155) | |
dgn.parameters.p155.6 |
answer recovered | `` | 6(2 - \sqrt{2}) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Mechanics and Mathematics Faculty, 2.; answer block on page 155) | |
dgn.parameters.2003 April |
answer recovered | `` | 1. q = -1; \pm \sqrt[10]{(\sqrt{5} - 1)/2}; 2. \angle A = \angle C = \alpha, \angle B = \pi - 2\alpha, where \alpha = 2\arcsin \frac{\sqrt{8k^2 + 1} - 1}{4k}; k > 0; 3. \{-5\pi/2\} \cup \{3\pi/2\} \cup (-\pi/4; 0] \cup [-9\pi/4 - \arcsin \frac{\sqrt{2} - 1}{\sqrt{2}}; -2\pi] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Faculty of Computational Mathematics and Cybernetics, 1.; answer block on page 157) | |
dgn.parameters.2003 Mechanics |
mispaired answer | 1. 24. 2. x = 3, y = z = 1; no. 3. n = 0, \,-5. 4. The minor arc AB (without the endpoints) of the circle tangent to the sides of angle \angle AOB at the points A and B; 8\pi/(3\sqrt{3}). |
1. 24; 2. x = 3, y = z = 1; no; 3. n = 0,-5; 4. The minor arc AB (without endpoints) of the circle tangent to the sides of angle AOB at points A and B; 8\pi/(3\sqrt{3}) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003, Mechanics and Mathematics Faculty, oral exam problems.; answer block on page 156) | |
dgn.parameters.p157.1 |
mispaired answer | $a \in (44 - 24\sqrt{2}; 44 + 24\sqrt{2})$ |
\{-1; \pm \sqrt[6]{(\sqrt{17} - 1)/2}\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Faculty of Computational Mathematics and Cybernetics, 2.; answer block on page 158) | |
dgn.parameters.p157.2 |
mispaired answer | $\angle A = \angle C = \alpha, \angle B = \pi - 2\alpha$, where $\alpha = 2\arcsin \dfrac{\sqrt{8k^2+1}-1}{4k}; \ k > 0$ |
\angle A = \alpha, \angle B = \frac{\pi}{2} - \alpha, where \alpha = 2\operatorname{arctg}\frac{1}{n\sqrt{2 + 1}}; n > 0 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Faculty of Computational Mathematics and Cybernetics, 2.; answer block on page 158) | |
dgn.parameters.p157.3 |
wrong answer | page 158, item 3 of the 'Ответы' block for ВАРИАНТ 2003 (апрель), факультет вычислительной математики и кибернетики, 2.: {π} ∪ {3π} ∪ [π/2; 3π/4) ∪ [5π/2; 11π/4 − arcsin((1−√2)/√2)] (printed problem, page 157: 'Задача 3. Найдите все решения системы неравенств √(−cos x) ⩽ √(sin³x + cos²x), π/4 < |x − π| ⩽ 2π.') |
{-\pi} \cup \{3\pi\} \cup [\pi/2; 3\pi/4) \cup [5\pi/2; 11\pi/4 - \arcsin((1-\sqrt{2})/\sqrt{2})] |
The printed system is satisfied at x = −π (constraint |x−π| = 2π is allowed, and the first inequality is an equality: sin³(−π)+cos²(−π) = 1 = −cos(−π)), but NOT at x = π, where the constraint π/4 < |x−π| fails (|π−π| = 0). The source printed {π} where the answer key must print {−π}; every other element of the printed answer is correct. With F(x) = sin³x + cos²x + cos x: F(8.936631158693137) = 3.3e−16 (the printed endpoint 11π/4 − arcsin((1−√2)/√2)), F(π/2) = 1, F(5π/2) = 1, F(3π) = 0, F(−π) = F(π) = 0; 3π/4 is open because |3π/4−π| = π/4 is excluded by the strict constraint, not by the inequality. The engine's complaint is false: the radicand printed on the page is sin³x + cos²x (only this reading reproduces the printed endpoints 5π/2 and 8.9366; the engine's sin(x³) reading gives F = −1.756 at the printed root), and for the printed reading F(1.9635) = +0.5523 > 0, i.e. x = 1.9635 satisfies the inequality and in fact lies inside the printed interval [π/2; 3π/4). | |
dgn.parameters.p157.5 |
answer recovered | `` | a \in (44 - 24\sqrt{2}; 44 + 24\sqrt{2}) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Faculty of Computational Mathematics and Cybernetics, 1.; answer block on page 157) | |
dgn.parameters.p158.2003.1 |
answer recovered | `` | (1; -1) \cup [2; 3) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 1.; answer block on page 159) | |
dgn.parameters.p158.2003.2 |
answer recovered | `` | \{\pm\frac{\pi}{4} + 2\pi k, k \in \mathbb{Z}\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 1.; answer block on page 159) | |
dgn.parameters.2003.3 |
answer recovered | `` | [-2; -\frac{3}{2}) \cup [-1; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 1.; answer block on page 159) | |
dgn.parameters.p159.1 |
answer recovered | `` | (1/4; 1/3) \cup [2; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 2.; answer block on page 160) | |
dgn.parameters.p159.2 |
answer recovered | `` | \{\pi/8 + 2\pi k; 5\pi/8 + 2\pi n; k, n \in \mathbb{Z}\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 2.; answer block on page 160) | |
dgn.parameters.p159.3 |
answer recovered | `` | (-\infty; -2] \cup (1/2; 1] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 2.; answer block on page 160) | |
dgn.parameters.p159.5 |
mispaired answer | $\alpha\in\left(\dfrac{3}{2};+\infty\right)$ |
\alpha \in (\frac{3}{2}; +\infty) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 1.; answer block on page 159) | |
dgn.parameters.p160.1 |
wrong answer | page 161, item 1 of the 'Ответы' block for ВАРИАНТ 2003 (июль), отделение бакалавров, 1.: [−8/7; 0] (printed problem, page 160: 'Задача 1. Решите неравенство 3|x + 2| − 4|x + 1| ⩽ 2.') |
(-\infty; -8/7] \cup [0; +\infty) |
sympy: solve(3|x+2|-4|x+1| <= 2, x) returns (0 <= x) | (x <= -8/7), i.e. (-inf; -8/7] u [0; +inf). The printed answer [-8/7; 0] is exactly the solution set of the REVERSED inequality (sympy: solve(3|x+2|-4|x+1| >= 2, x) = (-8/7 <= x) & (x <= 0)), so the source's answer key is inconsistent with the printed sign. The engine's probe is legitimate: x=-12/11 lies inside [-8/7;0] and 3|x+2|-4|x+1|-2 = 4/11 = 0.3636 > 0, so it violates '⩽ 2'. | |
dgn.parameters.p160.2 |
wrong answer | page 161, item 2 of the 'Ответы' block for ВАРИАНТ 2003 (июль), факультет вычислительной математики и кибернетики (отделение бакалавров), 1.: (−4/3; −1) ∪ [0; 1/2) (printed problem, page 160: 'Задача 2. Решите неравенство log₍(1−2x)/3₎ (12/(3x+4)) ⩽ −1.') |
(-1; 0] |
Domain: base (1−2x)/3 > 0 and ≠ 1, argument 12/(3x+4) > 0, i.e. x ∈ (−4/3, 1/2), x ≠ −1. For x < −1 the base is > 1 (log increasing): log ⩽ −1 ⟺ 12/(3x+4) ⩽ 3/(1−2x) ⟺ 12−24x ⩽ 9x+12 ⟺ x ⩾ 0, impossible in (−4/3, −1) — the whole branch is empty. For −1 < x < 1/2 the base is in (0,1) (log decreasing): log ⩽ −1 ⟺ 12/(3x+4) ⩾ 3/(1−2x) ⟺ x ⩽ 0, giving (−1; 0]. The printed answer (−4/3; −1) ∪ [0; 1/2) is exactly the solution set of the REVERSED inequality '⩾ −1', so the key contradicts the printed problem. The engine's probe is legitimate: x = −1.31667 lies inside the printed (−4/3; −1) and gives log₍(1−2x)/3₎(12/(3x+4)) = 28.6146 > −1, violating '⩽ −1'. | |
dgn.parameters.p160.3 |
answer recovered | `` | \{5\pi/4 + 2\pi k; 7\pi/4 + 2\pi n; k, n \in \mathbb{Z}\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 1.; answer block on page 161) | |
dgn.parameters.p160.4 |
answer recovered | `` | 9\sqrt{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 2.; answer block on page 160) | |
dgn.parameters.p160.5 |
answer recovered | `` | \beta \in (2/3; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 2.; answer block on page 160) | |
dgn.parameters.p160.6 |
answer recovered | `` | 5\pi/12 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 2.; answer block on page 160) | |
dgn.parameters.320 |
answer recovered | `` | 6\sqrt{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 1.; answer block on page 161) | |
dgn.parameters.321 |
mispaired answer | $\frac{5\pi}{12}$ |
5\pi/12 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 1.; answer block on page 161) | |
dgn.parameters.p161.322 |
wrong answer | Item 1 of the «Ответы» block on p. 162: «1. [1; 17/7].» |
(-\infty;\, -\frac{53}{3}] \cup [\frac{73}{7};\, +\infty) |
The printed problem (p. 161, confirmed at 600 dpi: «5|x − 2| − 2|x − 31| ⩾ 1») has solution set x ≤ −53/3 or x ≥ 73/7, so the printed key «[1; 17/7]» does not answer it (at x=1: 5·|1−2| − 2·|1−31| = 5−60 = −55 < 1, i.e. F−1 = −56, exactly the engine's number). The printed key is the solution of the older-looking statement 5|x−2| − 2|x−3| ⩽ 1, so the source is internally inconsistent; the answer that matches the printed problem is (−∞;−53/3] ∪ [73/7;+∞). | |
dgn.parameters.p161.323 |
wrong answer | page 162, item 2 of the 'Ответы' block for ВАРИАНТ 2003 (июль), отделение бакалавров, 2.: (5/2; 8/3) ∪ [6; +∞) (printed problem, page 161: 'Задача 2. Решите неравенство log₍₄/(3x−4)₎ (2/(2x−5)) ⩽ 1.') |
(8/3; 6] |
Domain: x > 5/2 (argument > 0), x ≠ 8/3 (base = 1). For x in (5/2, 8/3) the base is > 1, so log ≤ 1 requires 2/(2x-5) ≤ 4/(3x-4), i.e. x ≥ 6 — empty; for x > 8/3 the base is < 1, so log ≤ 1 requires 2/(2x-5) ≥ 4/(3x-4), i.e. x ≤ 6. Hence the solution is (8/3; 6] (at x = 6 the argument equals the base, log = 1, included). The printed answer (5/2; 8/3) ∪ [6; +∞) is exactly the solution set of the reversed inequality '⩾ 1', so the key does not match the printed problem. | |
dgn.parameters.p161.324 |
answer recovered | `` | \{\frac{\pi}{8} + 2\pi k; \frac{13\pi}{8} + 2\pi n; k, n \in \mathbb{Z}\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 2.; answer block on page 162) | |
dgn.parameters.p161.325 |
answer recovered | `` | If d \leqslant 0, then x = \log_{2/7}(2 + \sqrt{4 - d}); if 0 < d < 4, then x = \log_{2/7}(2 \pm \sqrt{4 - d}); if d = 4, then x = \log_{2/7} 2; if d > 4, then there are no solutions |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 2.; answer block on page 162) | |
dgn.parameters.p161.326 |
answer recovered | `` | (-\infty; -1) \cup (1; 3] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 2.; answer block on page 162) | |
dgn.parameters.p161.327 |
answer recovered | `` | 6\sqrt{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 1.; answer block on page 161) | |
dgn.parameters.Variant 2003 (mart), physics Faculty, 1 |
answer recovered | `` | 1. -\pi/2 + 2\pi n, (-1)^{n+1}\pi/12 + \pi n/2, n \in \mathbb{Z}; 2. x < -2, -2 < x < -1/2; 3. 0 < x \leqslant 1/2; x \geqslant \log_2 3; 4. (1 + \sqrt{33})/2; 5. (-2; 3), (2; 3) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Physics Faculty, 1.; answer block on page 163) | |
dgn.parameters.p163.1 |
mispaired answer | $-\pi/2 + 2\pi n,\ (-1)^{n+1}\pi/12 + \pi n/2,\ n \in \mathbb{Z}$ |
(-1)^{n+1}\pi/12 + \pi n/2, 2\pi n, n \in \mathbb{Z} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Physics Faculty, 2.; answer block on page 164) | |
dgn.parameters.p163.2 |
mispaired answer | $x < -2,\ -2 < x < -1/2$ |
-1 < x < 1, x > 1 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Physics Faculty, 2.; answer block on page 164) | |
dgn.parameters.p163.3 |
mispaired answer | $0 < x \leqslant 1/2;\ x \geqslant \log_2 3$ |
0 < x \leqslant \log_3 7; x \geqslant 3 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Physics Faculty, 2.; answer block on page 164) | |
dgn.parameters.p163.4 |
mispaired answer | $(1+\sqrt{33})/2$ |
3 + \sqrt{57}/4 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Physics Faculty, 2.; answer block on page 164) | |
dgn.parameters.p163.7 |
mispaired answer | If $a=0$, the unique solution is $x=0$; if $0 < a < 1$, there are no solutions; if $1 \leqslant a < 2^{2/3}$, there are two solutions $x = \left(a \pm \sqrt{2\sqrt{a}-a^2}\right)^2/4$; if $a=2^{2/3}$, the unique solution is $x=2^{-2/3}$; if $a > 2^{2/3}$, there are no solutions. |
If a = 0, then the unique solution x = 0; if 0 < a < 1, then there are no solutions; if 1 \leqslant a < 2^{2/3}, then there are two solutions x = (a \pm \sqrt{2\sqrt{a} - a^2})^2/4; if a = 2^{2/3}, then the unique solution x = 2^{-2/3}; if a > 2^{2/3}, then there are no solutions |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Physics Faculty, 1.; answer block on page 163) | |
dgn.parameters.1 (Variant 2003, May, Physics Faculty, 1) |
answer recovered | `` | \pm\arccos(\pi/18) + 2\pi n, \pm\arccos(5\pi/18) + 2\pi n, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Physics Faculty, 1.; answer block on page 165) | |
dgn.parameters.2 (Variant 2003, May, Physics Faculty, 1) |
answer recovered | `` | \pm\arccos(\pi/18) + 2\pi n, \pm\arccos(5\pi/18) + 2\pi n, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Physics Faculty, 1.; answer block on page 165) | |
dgn.parameters.3 (Variant 2003, May, Physics Faculty, 1) |
answer recovered | `` | \pm\arccos(\pi/18) + 2\pi n, \pm\arccos(5\pi/18) + 2\pi n, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Physics Faculty, 1.; answer block on page 165) | |
dgn.parameters.p164.5 |
answer recovered | `` | (3; -2), (3; 2) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Physics Faculty, 2.; answer block on page 164) | |
dgn.parameters.p164.7 |
mispaired answer | If $a = 0$, the unique solution $x = 0$; if $0 < a < 1/2$, there are no solutions; if $1/2 \leqslant a < 2^{-1/3}$, then the two solutions $x = \left(2a \pm \sqrt{2\sqrt{2a} - 4a^2}\right)^2/4$; if $a = 2^{-1/3}$, then the unique solution $x = 2^{-2/3}$; if $a > 2^{-1/3}$, there are no solutions. |
If a = 0, then the unique solution x = 0; if 0 < a < 1/2, then there are no solutions; if 1/2 \leqslant a < 2^{-1/3}, then there are two solutions x = (2a \pm \sqrt{2\sqrt{2a} - 4a^2})^2/4; if a = 2^{-1/3}, then the unique solution x = 2^{-2/3}; if a > 2^{-1/3}, then there are no solutions |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Physics Faculty, 2.; answer block on page 164) | |
dgn.parameters.Variant 2003 (may), physics Faculty, 2. Task 1 |
answer recovered | `` | (-1)^n \arcsin\left(\frac{\pi}{6}\right) + \pi n, (-1)^{n+1} \arcsin\left(\frac{\pi}{6}\right) + \pi n, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Physics Faculty, 2.; answer block on page 166) | |
dgn.parameters.Variant 2003 (may), physics Faculty, 2. Task 2 |
answer recovered | `` | \left(0; \frac{1}{2}\right) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Physics Faculty, 2.; answer block on page 166) | |
dgn.parameters.Variant 2003 (may), physics Faculty, 2. Task 3 |
answer recovered | `` | x < -11; \frac{-7-\sqrt{5}}{2} < x < -4; -3 < x < \frac{-7+\sqrt{5}}{2}; x > 4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Physics Faculty, 2.; answer block on page 166) | |
dgn.parameters.Variant 2003 (may), physics Faculty, 2. Task 4 |
answer recovered | `` | \frac{c+a}{c+b} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Physics Faculty, 2.; answer block on page 166) | |
dgn.parameters.Variant 2003 (may), physics Faculty, 2. Task 5 |
answer recovered | `` | \frac{1}{2} \leqslant x \leqslant 3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Physics Faculty, 2.; answer block on page 166) | |
dgn.parameters.p165.Task 7 |
mispaired answer | If $0 < a < 1$, then $a^3 < x < 1$, $x > 1$; if $a > 1$, then $0 < x < 1$, $1 < x < a^3$. |
If 0 < a < 1, then a^3 < x < 1, x > 1; if a > 1, then 0 < x < 1, 1 < x < a^3 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Physics Faculty, 1.; answer block on page 165) | |
dgn.parameters.p166.1 |
mispaired answer | $(-1)^n \arcsin\!\left(\dfrac{\pi}{6}\right) + \pi n,\ \, (-1)^{n+1} \arcsin\!\left(\dfrac{\pi}{6}\right) + \pi n,\ \, n \in \mathbb{Z}$ |
\pm\frac{\pi}{3} + \pi n, n \in \mathbb{Z} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Physics Faculty, 1.; answer block on page 167) | |
dgn.parameters.p166.2 |
mispaired answer | $\left(0;\ \dfrac{1}{2}\right)$ |
x \leqslant -\frac{2}{3}, x \geqslant \frac{1}{2} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Physics Faculty, 1.; answer block on page 167) | |
dgn.parameters.p166.3 |
mispaired answer | $x < -11;\ \, \dfrac{-7 - \sqrt{5}}{2} < x < -4;\ \, -3 < x < \dfrac{-7 + \sqrt{5}}{2};\ \, x > 4$ |
\log_{5}626 - 4 < x < \log_{5}26 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Physics Faculty, 1.; answer block on page 167) | |
dgn.parameters.p166.4 |
mispaired answer | $\dfrac{c + a}{c + b}$ |
2 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Physics Faculty, 1.; answer block on page 167) | |
dgn.parameters.p166.5 |
mispaired answer | $\dfrac{1}{2} \leqslant x \leqslant 3$ |
x = 5, y = 1; x = \frac{1}{3}, y = -\frac{11}{3} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Physics Faculty, 1.; answer block on page 167) | |
dgn.parameters.p166.7 |
mispaired answer | If $0 < b < 1$, then $0 < x < 1$ and $1 < x < b^{-3}$; if $b > 1$, then $b^{-3} < x < 1$ and $x > 1$ |
If 0 < b < 1, then 0 < x < 1 and 1 < x < b^{-3}; if b > 1, then b^{-3} < x < 1 and x > 1 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Physics Faculty, 2.; answer block on page 166) | |
dgn.parameters.2.8 |
answer recovered | `` | \frac{1125}{224} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Physics Faculty, 1.; answer block on page 167) | |
dgn.parameters.2003.faculty-of-physics.2.1 |
answer recovered | `` | \pm\frac{\pi}{3} + \pi n, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Physics Faculty, 2.; answer block on page 168) | |
dgn.parameters.2003.faculty-of-physics.2.2 |
answer recovered | `` | -\frac{1}{4} \leqslant x \leqslant \frac{9}{32} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Physics Faculty, 2.; answer block on page 168) | |
dgn.parameters.2003.faculty-of-physics.2.3 |
answer recovered | `` | \log_{2}65 - 4 < x < \log_{2}12 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Physics Faculty, 2.; answer block on page 168) | |
dgn.parameters.2003.faculty-of-physics.2.4 |
answer recovered | `` | 3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Physics Faculty, 2.; answer block on page 168) | |
dgn.parameters.2003.faculty-of-physics.2.5 |
answer recovered | `` | x = 13, y = 3; x = \frac{25}{3}, y = \frac{23}{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Physics Faculty, 2.; answer block on page 168) | |
dgn.parameters.2003.faculty-of-physics.2.6 |
answer recovered | `` | \frac{a}{2\sin\left(\frac{\beta}{2}\right)} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Physics Faculty, 2.; answer block on page 168) | |
dgn.parameters.2003.faculty-of-physics.2.7 |
answer recovered | `` | If a = \frac{1}{4}, then x < -4, \frac{1}{4} < x < 2, x > 2; if a < -4, then x < a and x > -4; if a = -4, then x < -4 and x > -4; if -4 < a < \frac{1}{4}, a > \frac{1}{4}, then x < -4 and x > a |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Physics Faculty, 2.; answer block on page 168) | |
dgn.parameters.2003.faculty-of-physics.2.8 |
answer recovered | `` | \frac{63}{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Physics Faculty, 2.; answer block on page 168) | |
dgn.parameters.2003_1 |
mispaired answer | Answers. 1. $a=0$, $a=1$. 2. $(0;\log_2 3]\cup[2;+\infty)$. 3. $x=1$, $x=(1\pm\sqrt{5})/2$, $x=(5\pm\sqrt{5})/10$. 4. $3/2$. 5. $(3;0)$, $(3;2)$, $(3;-2)$. 6. $1/2$. |
1. a = 0, a = 1; 2. (0; \log_{2}3] \cup [2; +\infty); 3. x = 1, x = (1 \pm \sqrt{5})/2, x = (5 \pm \sqrt{5})/10; 4. 3/2; 5. (3; 0), (3; 2), (3; -2); 6. 1/2 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Chemistry Faculty, 1.; answer block on page 168) | |
dgn.parameters.continuation |
mispaired answer | $a = 2$, $b = 3$. |
\frac{63}{2} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Physics Faculty, 2.; answer block on page 168) | |
dgn.parameters.p169.1 |
mispaired answer | $b = -1$, $b = 0$ |
a \in (0; 1) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Chemistry Faculty, Faculty of Materials Science, 1.; answer block on page 170) | |
dgn.parameters.p169.2 |
mispaired answer | $(0; \log_3 2] \cup [3; +\infty)$ |
(0; \log_{3}2] \cup [3; +\infty) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Chemistry Faculty, 2.; answer block on page 169) | |
dgn.parameters.p170.2003.1 |
answer recovered | `` | b \in (-2; 0) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Chemistry Faculty, Faculty of Materials Science, 2.; answer block on page 171) | |
dgn.parameters.p170.2003.2 |
answer recovered | `` | \pi n, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Chemistry Faculty, Faculty of Materials Science, 2.; answer block on page 171) | |
dgn.parameters.2003-1-1 |
answer recovered | `` | 1. 2, \log_{2}5; 2. 2; 3. [3/7; 1) \cup (3; +\infty); 4. 3\sqrt{2}a, 7a/2, 9a/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Faculty of Materials Science, 1.; answer block on page 172) | |
dgn.parameters.p171.3 |
answer recovered | `` | 3. x \in (-1; 1); 4. x = 0; 5. 2; 6. a \in [-6; 6] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Chemistry Faculty, Faculty of Materials Science, 2.; answer block on page 171) | |
dgn.parameters.p171.6 |
mispaired answer | $b \in (-2;\,0)$ |
a \in [-6; 6] |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Chemistry Faculty, Faculty of Materials Science, 2.; answer block on page 171) | |
dgn.parameters.2003-2-1 |
mispaired answer | $2, \log_2 5$ |
2, 3\log_{3}4 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Faculty of Materials Science, 2.; answer block on page 173) | |
dgn.parameters.2003-2-2 |
mispaired answer | $2$ |
3 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Faculty of Materials Science, 2.; answer block on page 173) | |
dgn.parameters.2003-2-3 |
mispaired answer | $[3/7; 1) \cup (3; +\infty)$ |
[4/5; 2) \cup (4; +\infty) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Faculty of Materials Science, 2.; answer block on page 173) | |
dgn.parameters.2003-2-4 |
mispaired answer | $3\sqrt{2}a,\ 7a/2,\ 9a/2$ |
4\sqrt{19}a/3, 4\sqrt{2}a, \sqrt{217}a/3 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Faculty of Materials Science, 2.; answer block on page 173) | |
dgn.parameters.p173.1 |
answer recovered | `` | (1 + \sqrt{3})/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Biology Faculty, 1.; answer block on page 174) | |
dgn.parameters.p173.2 |
answer recovered | `` | [3/4; 1] \cup (7/4; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Biology Faculty, 1.; answer block on page 174) | |
dgn.parameters.p173.3 |
answer recovered | `` | 51/100 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Biology Faculty, 1.; answer block on page 174) | |
dgn.parameters.2003_april_biology_2.1 |
answer recovered | `` | (1 + \sqrt{10})/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Biology Faculty, 2.; answer block on page 175) | |
dgn.parameters.2003_april_biology_2.2 |
answer recovered | `` | [11/4; 3] \cup (13/4; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Biology Faculty, 2.; answer block on page 175) | |
dgn.parameters.2003_april_biology_2.3 |
answer recovered | `` | 51/100 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Biology Faculty, 1.; answer block on page 174) | |
dgn.parameters.p174.4 |
answer recovered | `` | x_{1,2} = (A \pm \sqrt{A^{2} - 1})/2, \text{where } A = \log_{3}\left((3 - \sqrt{6})/2\right) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Biology Faculty, 1.; answer block on page 174) | |
dgn.parameters.p174.5 |
answer recovered | `` | (144q^{2})/(25 - 24\sqrt{1 - q^{2}}) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Biology Faculty, 1.; answer block on page 174) | |
dgn.parameters.p174.6 |
answer recovered | `` | a \in [2101608; 4000000] \cup [600 \cdot 10^{4}; 625 \cdot 10^{4}] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Biology Faculty, 1.; answer block on page 174) | |
dgn.parameters.7/100 |
answer recovered | `` | 51/100 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Biology Faculty, 1.; answer block on page 174) | |
dgn.parameters.p175.3 |
answer recovered | `` | 4. x_{1,2,3,4} = (A_{1,2} \pm \sqrt{A_{1,2}^{2} - 1})/2, \text{where } A_{1} = \log_{2}((9 - \sqrt{31})/10), A_{2} = \log_{2}(3/10); 5. (2a)/\sqrt{100 + a^{2} - 8\sqrt{6}a}; 6. a \in [286624; 1000000] \cup [3000 \cdot 10^{3}; 3125 \cdot 10^{3}] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Biology Faculty, 2.; answer block on page 175) | |
dgn.parameters.Variant 2003 (May), Faculty of Fundamental Medicine, 1 |
answer recovered | `` | 1. -4; 2. \pi/4 + \pi n, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Faculty of Fundamental Medicine, 1.; answer block on page 176) | |
dgn.parameters.2003-1 |
answer recovered | `` | \frac{\pi}{18} + \frac{\pi}{3}n, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geography Faculty, Faculty of Bioengineering and Bioinformatics, 2.; answer block on page 179) | |
dgn.parameters.2003-2 |
answer recovered | `` | 2 - \sqrt{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geography Faculty, Faculty of Bioengineering and Bioinformatics, 2.; answer block on page 179) | |
dgn.parameters.2003-3 |
answer recovered | `` | [-5; 0) \cup \left(0; \frac{5\sqrt{2}}{2}\right) \cup \left(\frac{5\sqrt{2}}{2}; 5\right] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geography Faculty, Faculty of Bioengineering and Bioinformatics, 2.; answer block on page 179) | |
dgn.parameters.2003-4 |
answer recovered | `` | 3100 \text{ m}^{2}/\text{day} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geography Faculty, Faculty of Bioengineering and Bioinformatics, 1.; answer block on page 178) | |
dgn.parameters.p178.5 |
mispaired answer | $24:1$. |
24 : 1 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geography Faculty, Faculty of Bioengineering and Bioinformatics, 1.; answer block on page 178) | |
dgn.parameters.p178.6 |
mispaired answer | $[-4; 1)$. |
[-4; 1) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geography Faculty, Faculty of Bioengineering and Bioinformatics, 1.; answer block on page 178) | |
dgn.parameters.p179.1 |
answer recovered | `` | \frac{\pi}{4} + \pi n, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Biology Faculty, Faculty of Fundamental Medicine, Faculty of Bioengineering and Bioinformatics, 1.; answer block on page 180) | |
dgn.parameters.p179.2 |
answer recovered | `` | (-\infty; -\sqrt{2}] \cup [\sqrt{2}; 2) \cup [(8 + \sqrt{10})/3; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Biology Faculty, Faculty of Fundamental Medicine, Faculty of Bioengineering and Bioinformatics, 1.; answer block on page 180) | |
dgn.parameters.p179.3 |
answer recovered | `` | 2^{\frac{-1-\sqrt{33}}{2}} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Biology Faculty, Faculty of Fundamental Medicine, Faculty of Bioengineering and Bioinformatics, 1.; answer block on page 180) | |
dgn.parameters.p179.4 |
mispaired answer | 1700 \text{ m}^3/\text{day} |
1700 \text{ m}^{3}/\text{day} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geography Faculty, Faculty of Bioengineering and Bioinformatics, 2.; answer block on page 179) | |
dgn.parameters.p180.1 |
mispaired answer | $\frac{\pi}{4} + \pi n,\; n \in \mathbb{Z}$ |
\pi/8 + \pi n, n \in \mathbb{Z} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Biology Faculty, Faculty of Fundamental Medicine, Faculty of Bioengineering and Bioinformatics, 2.; answer block on page 181) | |
dgn.parameters.p180.2 |
mispaired answer | $(-\infty;-\sqrt{2}] \cup [\sqrt{2};2) \cup \left[\frac{8+\sqrt{10}}{3};+\infty\right)$ |
[-4; (3 - \sqrt{23})/2] \cup (3; 4] |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Biology Faculty, Faculty of Fundamental Medicine, Faculty of Bioengineering and Bioinformatics, 2.; answer block on page 181) | |
dgn.parameters.p180.3 |
mispaired answer | $2^{\frac{-1-\sqrt{33}}{2}}$ |
5^{5+2\sqrt{6}} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Biology Faculty, Faculty of Fundamental Medicine, Faculty of Bioengineering and Bioinformatics, 2.; answer block on page 181) | |
dgn.parameters.p181.4 |
answer recovered | `` | 40 \text{ minutes} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Biology Faculty, Faculty of Fundamental Medicine, Faculty of Bioengineering and Bioinformatics, 2.; answer block on page 181) | |
dgn.parameters.p181.5 |
answer recovered | `` | \arcsin((\sqrt{17} - 3)/4) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Biology Faculty, Faculty of Fundamental Medicine, Faculty of Bioengineering and Bioinformatics, 2.; answer block on page 181) | |
dgn.parameters.p181.6 |
answer recovered | `` | (-2; 3] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Biology Faculty, Faculty of Fundamental Medicine, Faculty of Bioengineering and Bioinformatics, 2.; answer block on page 181) | |
dgn.parameters.Variant 2003 (iyul), geography Faculty, 1 |
answer recovered | `` | -5^{-\frac{5}{3}} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geography Faculty, 1.; answer block on page 182) | |
dgn.parameters.p181.Task 2 |
answer recovered | `` | (-\infty; -6] \cup [-1; 0) \cup (0; (\sqrt{73} - 7)/2] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geography Faculty, 1.; answer block on page 182) | |
dgn.parameters.p181.Task 3 |
answer recovered | `` | 14 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geography Faculty, 1.; answer block on page 182) | |
dgn.parameters.p181.Task 4 |
answer recovered | `` | 2\sqrt{2}(\sqrt{3} + 1), \text{less} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geography Faculty, 1.; answer block on page 182) | |
dgn.parameters.p182.2003 |
answer recovered | `` | 1. -3^{-\frac{7}{5}}; 2. [(-\sqrt{113} + 9)/2; 0) \cup (0; 1] \cup [8; +\infty); 3. 15; 4. 8\sqrt{2} + \sqrt{3}, \text{less}; 5. \left(\frac{1}{6}; \frac{2}{3}\right) \cup \{1\} \cup \left(\frac{4}{3}; 6\right] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geography Faculty, 2.; answer block on page 182) | |
dgn.parameters.p182.5 |
answer recovered | `` | \left(\frac{1}{6}; \frac{2}{3}\right) \cup \{1\} \cup \left(\frac{4}{3}; 6\right] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geography Faculty, 2.; answer block on page 182) | |
dgn.parameters.183 a |
mispaired answer | 1. $x = 1$. 2. $\pi/12 + \pi n/2$, $n \in \mathbb{Z}$. 3. $74$. 4. $(7k; 3k; 2k)$, $k \in \mathbb{Z}$. 5. $(1; 5/4) \cup (5/4; 3/2)$. 6. $P = a(\sqrt{2} + \sqrt{5})$, $S = \sqrt{6}a^2/4$. |
1. x = 1; 2. \pi/12 + \pi n/2, n \in \mathbb{Z}; 3. 74; 4. (7k; 3k; 2k), k \in \mathbb{Z}; 5. (1; 5/4) \cup (5/4; 3/2); 6. P = a(\sqrt{2} + \sqrt{5}), S = \sqrt{6}a^{2}/4 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Faculty of Soil Science, 1.; answer block on page 183) | |
dgn.parameters.183 b |
answer recovered | `` | 1. x = 1; 2. \pi/6 + \pi n/2, n \in \mathbb{Z}; 3. 41; 4. (2k; 7k; 3k), k \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Faculty of Soil Science, 2.; answer block on page 184) | |
dgn.parameters.p184.1 |
answer recovered | `` | \pi/4 + \pi n/2, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Soil Science, 1.; answer block on page 184) | |
dgn.parameters.p184.2 |
answer recovered | `` | (0; -1); (1; 0) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Soil Science, 1.; answer block on page 184) | |
dgn.parameters.p184.3 |
wrong answer | page 184, item 3 of the SECOND 'Ответы' block (the one following ВАРИАНТ 2003 (июль), факультет почвоведения, 1.): 3. (−∞; 2/3] ∪ {0} ∪ [2/3; +∞) (printed problem, page 184: 'Задача 3. Решите неравенство 3x²/2 − |x| ⩾ 0.') |
(-\infty; -2/3] \cup \{0\} \cup [2/3; +\infty) |
Putting u = |x| ≥ 0 the inequality 3u²/2 − u ≥ 0 becomes u(3u/2 − 1) ≥ 0, i.e. u = 0 or u ≥ 2/3 (sympy: roots of 6u²+u−5 are −1 and 5/6), so x = 0 or |x| ≥ 2/3, i.e. (−∞; −2/3] ∪ {0} ∪ [2/3; +∞). The source prints '−∞; 2/3]' — the minus sign of the first endpoint is missing on the page — so the printed answer wrongly includes e.g. x = −1/3, where 3x²/2 − |x| = −1/6 < 0 (exactly the engine's counterexample). | |
dgn.parameters.p184.4 |
answer recovered | `` | 4\sqrt{2} - 3\sqrt{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Soil Science, 1.; answer block on page 184) | |
dgn.parameters.p184.5 |
answer recovered | `` | 2\operatorname{arctg}(1/4) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Soil Science, 1.; answer block on page 184) | |
dgn.parameters.p184.6 |
answer recovered | `` | (-\infty; -6) \cup (-1/3; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Soil Science, 1.; answer block on page 184) | |
dgn.parameters.Variant 2003 (iyul), Faculty soil science, 2 |
answer recovered | `` | 1. \pi/10 + 2\pi n/5, n \in \mathbb{Z}; 2. (0; 3); (3; 0); 3. (-\infty; -3/4] \cup \{0\} \cup [3/4; +\infty); 4. \sqrt{39} - 4\sqrt{2}; 5. 2\operatorname{arctg}(1/4); 6. (-\infty; -5/3) \cup (-2/11; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Soil Science, 2.; answer block on page 185) | |
dgn.parameters.Variant 2003 (may), geology Faculty, 1 |
answer recovered | `` | 1. \frac{\pi}{3} + 2\pi k, -\frac{\pi}{9} + \frac{2\pi n}{3}; k, n \in \mathbb{Z}; 2. (-\infty; -6] \cup (-3; -2); 3. 39 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geology Faculty, 1.; answer block on page 186) | |
dgn.parameters.p186.1 |
answer recovered | `` | -\frac{5\pi}{6} + 2\pi k, \frac{5\pi}{18} + \frac{2\pi n}{3}; k, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geology Faculty, 2.; answer block on page 187) | |
dgn.parameters.p186.2 |
answer recovered | `` | (-\infty; -8] \cup (-4; -3) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geology Faculty, 2.; answer block on page 187) | |
dgn.parameters.p186.4 |
mispaired answer | $(-\infty;-2]\cup\{-1\}\cup(7;8]$. |
(-\infty; -2] \cup \{-1\} \cup (7; 8] |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geology Faculty, 1.; answer block on page 186) | |
dgn.parameters.p186.5 |
mispaired answer | $\angle BAC=\operatorname{arctg}\frac{\sqrt{3}}{2}-\frac{\pi}{12}$ (alternative form of the answer: $\angle BAC=\frac{\pi}{4}-\operatorname{arctg}\frac{\sqrt{3}}{5}$). |
\angle BAC = \operatorname{arctg}\frac{\sqrt{3}}{2} - \frac{\pi}{12} \ (\text{another form of the answer: } \angle BAC = \frac{\pi}{4} - \operatorname{arctg}\frac{\sqrt{3}}{5}) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geology Faculty, 1.; answer block on page 186) | |
dgn.parameters.p186.6 |
mispaired answer | $\left\{0;-\frac{2}{3}\right\}$. |
\{0; -\frac{2}{3}\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geology Faculty, 1.; answer block on page 186) | |
dgn.parameters.p186.7 |
mispaired answer | $\left(\frac{3}{2};\frac{1}{4}\right)$. |
\left(\frac{3}{2}; \frac{1}{4}\right) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geology Faculty, 1.; answer block on page 186) | |
dgn.parameters.p186.8 |
mispaired answer | $\frac{3}{\sqrt{31}}$. |
\frac{3}{\sqrt{31}} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geology Faculty, 1.; answer block on page 186) | |
dgn.parameters.p187.1 |
answer recovered | `` | [-7; -2) \cup (-2; -1] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geology Faculty, 1.; answer block on page 188) | |
dgn.parameters.p187.3 |
answer recovered | `` | 21 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geology Faculty, 2.; answer block on page 187) | |
dgn.parameters.p187.4 |
answer recovered | `` | (-\infty; -3] \cup \{-2\} \cup (6; 7] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geology Faculty, 2.; answer block on page 187) | |
dgn.parameters.p187.5 |
answer recovered | `` | \angle FEG = \frac{5\pi}{12} - \operatorname{arctg}\frac{1}{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geology Faculty, 2.; answer block on page 187) | |
dgn.parameters.p187.6 |
answer recovered | `` | \{0; -\frac{2}{5}\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geology Faculty, 2.; answer block on page 187) | |
dgn.parameters.p187.7 |
answer recovered | `` | \left(\frac{5}{2}; \frac{1}{4}\right) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geology Faculty, 2.; answer block on page 187) | |
dgn.parameters.p187.8 |
answer recovered | `` | \frac{3}{2\sqrt{13}} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (May), Geology Faculty, 2.; answer block on page 187) | |
dgn.parameters.p188.1 |
wrong answer | Item 1 of the «Ответы» block on p. 189: «1. [1; 3) ∪ (3; 5].» |
(-\infty;\, 1] \cup [5;\, +\infty) |
The printed problem (p. 188, confirmed at 600 dpi: «(x+3)/|x−3| − (3x−7)/(x−3) ⩽ 0») gives F = 4(x−1)/(3−x) for x<3 and F = 2(5−x)/(x−3) for x>3, so ⩽0 holds for x ≤ 1 and x ≥ 5; the printed key [1;3) ∪ (3;5] is instead the answer of the ⩾0 version of the very same expression (and at x=1.1, F = 0.2105 > 0, the engine's number). So the printed answer does not solve the printed problem; the answer matching the printed problem is (−∞;1] ∪ [5;+∞). | |
dgn.parameters.p188.2 |
mispaired answer | $x = \{1/2\}$ |
\{1/2\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geology Faculty, 1.; answer block on page 188) | |
dgn.parameters.p188.3 |
mispaired answer | The first pedestrian covered the distance in 1 hour, the second in 5 hours. |
The first pedestrian covered the distance in 1 hour, the second — in 5 hours |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geology Faculty, 1.; answer block on page 188) | |
dgn.parameters.p188.4 |
mispaired answer | $(x; y) = (0; \sqrt{3}); (0; -\sqrt{3}); \left(\sqrt{\frac{3}{119}}; 11\sqrt{\frac{3}{119}}\right); \left(-\sqrt{\frac{3}{119}}; -11\sqrt{\frac{3}{119}}\right)$. |
(x; y) = (0; \sqrt{3}); (0; -\sqrt{3}); \left(\sqrt{\frac{3}{119}}; 11\sqrt{\frac{3}{119}}\right); \left(-\sqrt{\frac{3}{119}}; -11\sqrt{\frac{3}{119}}\right) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geology Faculty, 1.; answer block on page 188) | |
dgn.parameters.p188.5 |
mispaired answer | $S = \frac{9}{19}\sqrt{2}$. |
S = \frac{9}{19}\sqrt{2} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geology Faculty, 1.; answer block on page 188) | |
dgn.parameters.p188.6 |
mispaired answer | $\{\pm\pi/6 + \pi n; \pi/4 - \sqrt{3}/2 + 2\pi k; k, n \in \mathbb{Z}\}$. |
\{\pm\pi/6 + \pi n; \pi/4 - \sqrt{3}/2 + 2\pi k; k, n \in \mathbb{Z}\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geology Faculty, 1.; answer block on page 188) | |
dgn.parameters.p188.7 |
mispaired answer | $a \in \{\log_3(18 \pm 9\sqrt{3}); \log_3(10 \pm \sqrt{19})\}$. |
a \in \{\log_{3}(18 \pm 9\sqrt{3}); \log_{3}(10 \pm \sqrt{19})\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geology Faculty, 1.; answer block on page 188) | |
dgn.parameters.p188.8 |
mispaired answer | $V_{SABM} : V_{SAMCD} = 1 : 41$. |
V_{SABM} : V_{SAMCD} = 1 : 41 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geology Faculty, 1.; answer block on page 188) | |
dgn.parameters.p189.1 |
answer recovered | `` | (-2; 0) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Philology Faculty.; answer block on page 190) | |
dgn.parameters.p189.3 |
mispaired answer | The first pedestrian took 2 hours for the trip, the second took 1 hour. |
The first pedestrian spent 2 hours on the journey, the second — 1 hour |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geology Faculty, 2.; answer block on page 189) | |
dgn.parameters.p189.4 |
mispaired answer | $(x;y) = (0;\sqrt{2});\ (0;-\sqrt{2});\ \left(\sqrt{\frac{2}{47}};\,7\sqrt{\frac{2}{47}}\right);\ \left(-\sqrt{\frac{2}{48}};\,-7\sqrt{\frac{2}{47}}\right).$ |
(x; y) = (0; \sqrt{2}); (0; -\sqrt{2}); \left(\sqrt{\frac{2}{47}}; 7\sqrt{\frac{2}{47}}\right); \left(-\sqrt{\frac{2}{47}}; -7\sqrt{\frac{2}{47}}\right) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geology Faculty, 2.; answer block on page 189) | |
dgn.parameters.p189.5 |
mispaired answer | $S = \frac{36}{19}\sqrt{2}.$ |
S = \frac{36}{19}\sqrt{2} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geology Faculty, 2.; answer block on page 189) | |
dgn.parameters.p189.6 |
mispaired answer | $\{\pi/4 + \pi n;\ \pi/2 + \pi m;\ \pi/4 + \sqrt{2}/2 + \pi k;\ k,m,n \in \mathbb{Z}\}.$ |
\{\pi/4 + \pi n; \pi/2 + \pi m; \pi/4 + \sqrt{2}/2 + \pi k; k, m, n \in \mathbb{Z}\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geology Faculty, 2.; answer block on page 189) | |
dgn.parameters.p189.7 |
mispaired answer | $a \in \{\log_7(20 \pm 3\sqrt{39});\ \log_7(8 \pm \sqrt{15})\}.$ |
a \in \{\log_{7}(20 \pm 3\sqrt{39}); \log_{7}(8 \pm \sqrt{15})\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geology Faculty, 2.; answer block on page 189) | |
dgn.parameters.p189.8 |
mispaired answer | $V_{SAMCD} : V_{SABM} = 57 : 1.$ |
V_{SAMCD} : V_{SABM} = 57 : 1 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Geology Faculty, 2.; answer block on page 189) | |
dgn.parameters.p190.4 |
mispaired answer | $\frac{1}{2} a^2 \cdot \dfrac{\sin\delta \cdot \sin(\gamma+\delta)}{\sin\gamma}$ |
\frac{1}{2}(a^{2})\frac{\sin\delta\cdot\sin(\gamma+\delta)}{\sin\gamma} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Philology Faculty.; answer block on page 190) | |
dgn.parameters.p190.5 |
mispaired answer | a) no; b) 21 |
a) no; b) 21 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (March), Philology Faculty.; answer block on page 190) | |
dgn.parameters.Variant 2003-1 |
answer recovered | `` | 1. \frac{\pi n}{3}, n \in \mathbb{Z}; 2. (1, 2\pi k); \left(\frac{1}{4}, -\frac{\pi}{2} + 2\pi m\right); \left(4, \frac{\pi}{2} + 2\pi n\right), \text{where } k, m, n \in \mathbb{Z}; 3. \angle A = \angle C = \operatorname{arctg} 3; \angle B = \pi - 2\operatorname{arctg} 3, \frac{1}{4}; 4. 28 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Philology Faculty, 1.; answer block on page 191) | |
dgn.parameters.Variant 2003 (July), Philology Faculty, 2 |
answer recovered | `` | 1. \frac{\pi}{4} + \frac{\pi n}{2}, (-1)^{k+1}\frac{\pi}{12} + \frac{\pi k}{2}, n, k \in \mathbb{Z}; 2. (1, \frac{\pi}{2} + 2\pi k); (9, 2\pi n); (1/9, \pi + 2\pi m), \text{where } k, n, m \in \mathbb{Z}; 3. \angle A = \angle C = \operatorname{arctg} 3; \angle B = \pi - 2\operatorname{arctg} 3, 4; 4. 34; 5. \frac{n}{7} - \frac{1}{14}, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Philology Faculty, 2.; answer block on page 192) | |
dgn.parameters.p192.1 |
answer recovered | `` | 4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Economics Faculty (Management Division), 1.; answer block on page 192) | |
dgn.parameters.p192.2 |
answer recovered | `` | 90\% |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Economics Faculty (Management Division), 1.; answer block on page 192) | |
dgn.parameters.p192.3 |
answer recovered | `` | \frac{\sqrt{3}}{16} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Economics Faculty (Management Division), 1.; answer block on page 192) | |
dgn.parameters.p192.4 |
answer recovered | `` | \left\{\pm\frac{\sqrt{2}}{6}; \frac{\sqrt{2}+\sqrt{6}}{12}\right\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Economics Faculty (Management Division), 1.; answer block on page 192) | |
dgn.parameters.p192.5 |
answer recovered | `` | b = 3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Economics Faculty (Management Division), 1.; answer block on page 192) | |
dgn.parameters.p193.1 |
mispaired answer | 1. $0$. 2. $30\%$. 3. $\frac{15\sqrt{3}}{32}$. 4. $\left\{\pm \frac{\sqrt{2}}{4}; -\frac{\sqrt{2}+\sqrt{6}}{8}\right\}$. 5. $a = 1$. |
1. 0; 2. 30\%; 3. \frac{15\sqrt{3}}{32}; 4. \left\{\pm\frac{\sqrt{2}}{4}; -\frac{\sqrt{2}+\sqrt{6}}{8}\right\}; 5. a = 1 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Economics Faculty (Management Division), 2.; answer block on page 193) | |
dgn.parameters.p193.2 |
answer recovered | `` | 1. [-2; 1]; 2. -207 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Economics Faculty (Economics Division), 1. √; answer block on page 194) | |
dgn.parameters.194.3 |
answer recovered | `` | \left\{(\log_{7}3; 2); (\log_{7}(5/3); -1)\right\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Economics Faculty (Economics Division), 1. √; answer block on page 194) | |
dgn.parameters.194.4 |
answer recovered | `` | 24 \text{ and } 80 \text{ tons}; m = 4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Economics Faculty (Economics Division), 1. √; answer block on page 194) | |
dgn.parameters.194.5 |
answer recovered | `` | 4\sqrt{13} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Economics Faculty (Economics Division), 1. √; answer block on page 194) | |
dgn.parameters.194.6 |
answer recovered | `` | \left(-\infty; -\frac{1}{2\sqrt{2}}\right] \cup \left[-\frac{1}{4}; \frac{1}{4}\right] \cup \left[\frac{1}{2\sqrt{2}}; +\infty\right) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Economics Faculty (Economics Division), 1. √; answer block on page 194) | |
dgn.parameters.194.7 |
answer recovered | `` | 36 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Economics Faculty (Economics Division), 1. √; answer block on page 194) | |
dgn.parameters.2003.econ.1 |
answer recovered | `` | [1; 4] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Economics Faculty (Economics Division), 2. √; answer block on page 195) | |
dgn.parameters.2003.econ.2 |
answer recovered | `` | 7 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Economics Faculty (Economics Division), 2. √; answer block on page 195) | |
dgn.parameters.2003.econ.3 |
answer recovered | `` | \left\{\left(-\frac{1}{2}; \log_{9}\frac{9}{2}\right); \left(\log_{4}\frac{4}{3}; -\frac{1}{2}\right)\right\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Economics Faculty (Economics Division), 2. √; answer block on page 195) | |
dgn.parameters.p195.1 |
mispaired answer | $[1;4]$ |
\{-2\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Economics Faculty (Management Division), 1. √; answer block on page 196) | |
dgn.parameters.p195.3 |
mispaired answer | $\left\{\left(-\frac{1}{2}; \log_9 \frac{9}{2}\right); \left(\log_4 \frac{4}{3}; -\frac{1}{2}\right)\right\}$ |
\left\{\left(\frac{1}{2}; \log_{3}3\sqrt{2}\right)\right\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Economics Faculty (Management Division), 1. √; answer block on page 196) | |
dgn.parameters.p195.4 |
mispaired answer | 180 rub. and 240 rub.; $n=3$ |
180 \text{ rubles} \text{ and } 240 \text{ rubles}; n = 3 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Economics Faculty (Economics Division), 2. √; answer block on page 195) | |
dgn.parameters.p195.6 |
mispaired answer | $\left(-1; -\sqrt{\frac{2}{3}}\right)\cup\left(\sqrt{\frac{2}{3}}; 1\right)$ |
\left(-1; -\sqrt{\frac{2}{3}}\right] \cup \left(\sqrt{\frac{2}{3}}; 1\right) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Economics Faculty (Economics Division), 2. √; answer block on page 195) | |
dgn.parameters.p196.2003 |
mispaired answer | 1. $\{1\}$. 2. $-60$. 3. $\left\{\left(\frac{3}{2}\log_3 2; -1/2\right)\right\}$. 4. $120$ rub.; $n = 3$. 5. $2\arcsin(1/3)\;(= \arccos(7/9))$. 6. $\left(-\frac{2}{\sqrt{5}}; -\sqrt{\frac{3}{5}}\right) \cup \left(\sqrt{\frac{3}{5}}; \frac{2}{\sqrt{5}}\right)$. |
1. \{1\}; 2. -60; 3. \left\{\left(\frac{3}{2}\log_{3}2; -\frac{1}{2}\right)\right\}; 4. 120 \text{ rubles}; n = 3; 5. 2\arcsin(1/3)\ (= \arccos(7/9)); 6. \left(-\frac{2}{\sqrt{5}}; -\sqrt{\frac{3}{5}}\right] \cup \left(\sqrt{\frac{3}{5}}; \frac{2}{\sqrt{5}}\right) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Economics Faculty (Management Division), 2. √; answer block on page 196) | |
dgn.parameters.p197.197 |
answer recovered | `` | 1. [1; 4]; 2. -210; 3. \left\{\left(\frac{3}{2}\log_{3}2; -\frac{1}{2}\right)\right\}; 4. 3 \text{ ha}; n = 4; 5. 3\sqrt{3}; 6. \left(-1; -\sqrt{\frac{2}{3}}\right] \cup \left(\sqrt{\frac{2}{3}}; 1\right); 7. 5 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Economics Faculty (Evening Division), 1. √; answer block on page 197) | |
dgn.parameters.p197.198 |
answer recovered | `` | [-2; 1] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Economics Faculty (Evening Division), 2. √; answer block on page 198) | |
dgn.parameters.198_3 |
mispaired answer | \left(\dfrac{1}{2};\, \log_3 3\sqrt{2}\right) |
\left(\frac{1}{2}; \log_{3}3\sqrt{2}\right) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Economics Faculty (Evening Division), 2. √; answer block on page 198) | |
dgn.parameters.198_4 |
mispaired answer | 8 \text{ million rubles};\; m = 5 |
8 \text{ million rubles}; m = 5 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Economics Faculty (Evening Division), 2. √; answer block on page 198) | |
dgn.parameters.p199.199 |
mispaired answer | $x = (\sqrt{5} - 3)/3$ |
d = 5, S_{ABCD} = 975\sqrt{3}/196 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Psychology Faculty, 1.; answer block on page 199) | |
dgn.parameters.199.a |
mispaired answer | $a \in (-5/2; 7)$ |
a) a \in (-5/2; 7), b) a \in [(9 - \sqrt{211})/2; -5/2] \cup \{7\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Psychology Faculty, 1.; answer block on page 199) | |
dgn.parameters.199.b |
mispaired answer | $a \in [(9 - \sqrt{211})/2; -5/2] \cup \{7\}$ |
a) a \in (-5/2; 7), b) a \in [(9 - \sqrt{211})/2; -5/2] \cup \{7\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Psychology Faculty, 1.; answer block on page 199) | |
dgn.parameters.200.5.a |
mispaired answer | $a \in (-3; 5)$ |
a) a \in (-3; 5), b) a \in [2 - 2\sqrt{7}; -3] \cup \{5\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Psychology Faculty, 2.; answer block on page 199) | |
dgn.parameters.200.5.b |
mispaired answer | $a \in [2 - 2\sqrt{7}; -3] \cup \{5\}$ |
a) a \in (-3; 5), b) a \in [2 - 2\sqrt{7}; -3] \cup \{5\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Psychology Faculty, 2.; answer block on page 199) | |
dgn.parameters.201.1 |
answer recovered | `` | x = 2 + 1/\sqrt{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Psychology Faculty, 3.; answer block on page 200) | |
dgn.parameters.201.2 |
answer recovered | `` | (\pm \arccos((1 - \sqrt{10})/4) + 2\pi n)/4, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Psychology Faculty, 3.; answer block on page 200) | |
dgn.parameters.201.3 |
answer recovered | `` | x = \{-3/4\} \cup [-1/2; 13/4] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Psychology Faculty, 3.; answer block on page 200) | |
dgn.parameters.p200.Variant 2003 |
mispaired answer | 1. $x = (5 - \sqrt{5})/3$; 2. $(\pm \arccos((\sqrt{5} - 1)/4) + 2\pi n)/4$, $n \in \mathbb{Z}$; 3. $x = \{-6/5\} \cup [-1; 19/5]$; 4. $d = 8$, $S_{ABCD} = 768/25$; 5. a) $a \in (-7/2; 5)$, b) $a \in [(3 - \sqrt{107})/2; -7/2] \cup \{5\}$ |
1. x = (5 - \sqrt{5})/3; 2. (\pm \arccos((\sqrt{5} - 1)/4) + 2\pi n)/4, n \in \mathbb{Z}; 3. x = \{-6/5\} \cup [-1; 19/5]; 4. d = 8, S_{ABCD} = 768/25; 5. a) a \in (-7/2; 5), b) a \in [(3 - \sqrt{107})/2; -7/2] \cup \{5\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Psychology Faculty, 4.; answer block on page 200) | |
dgn.parameters.Task 4 (prev) |
answer recovered | `` | d = 7, S_{ABCD} = 2156\sqrt{3}/169 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Psychology Faculty, 3.; answer block on page 200) | |
dgn.parameters.Task 5 (prev) |
mispaired answer | a) $a \in (-2; 6)$, b) $a \in [4 - \sqrt{38}; -2] \cup \{6\}$ |
a) a \in (-7/2; 5), b) a \in [(3 - \sqrt{107})/2; -7/2] \cup \{5\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Psychology Faculty, 4.; answer block on page 200) | |
dgn.parameters.Variant 2003 (iyul), ISAA, 1, Task 2 |
mispaired answer | $\left\{11;\, \dfrac{-11-\sqrt{161}}{2}\right\}$ |
\{11; (-11 - \sqrt{161})/2\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), ISAA, 1.; answer block on page 201) | |
dgn.parameters.Variant 2003 (iyul), ISAA, 1, Task 3 |
mispaired answer | $\dfrac{\sqrt{3}}{6}\, a^2$ |
\frac{\sqrt{3}}{6}a^{2} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), ISAA, 1.; answer block on page 201) | |
dgn.parameters.Variant 2003 (iyul), ISAA, 1, Task 4 |
mispaired answer | $\left\{\dfrac{\pi}{2};\, \pi + \operatorname{arctg}\dfrac{\sqrt{2}}{2};\, \dfrac{3\pi}{2}\right\}$ |
\{\pi/2; \pi + \operatorname{arctg}\frac{\sqrt{2}}{2}; 3\pi/2\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), ISAA, 1.; answer block on page 201) | |
dgn.parameters.Variant 2003 (iyul), ISAA, 1, Task 5 |
mispaired answer | $(-\infty;-5] \cup \left[-3;-\dfrac{1}{\sqrt[3]{4}}\right] \cup \left\{-\dfrac{1}{32}\right\} \cup \left\{\dfrac{1}{8}\right\} \cup \left[\dfrac{1}{\sqrt[3]{4}};+\infty\right)$ |
(-\infty; -5] \cup \left[-3; -\frac{1}{3\sqrt{4}}\right] \cup \{-\frac{1}{32}\} \cup \{\frac{1}{8}\} \cup \left[\frac{1}{3\sqrt{4}}; +\infty\right) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), ISAA, 1.; answer block on page 201) | |
dgn.parameters.Variant 2003 (iyul), ISAA, 1, Task 6 |
mispaired answer | Segment $[0;2]$ at $\begin{cases} c=0 \\ d=1 \end{cases}$ or $\begin{cases} c=-1 \\ d=-2 \end{cases}$ |
Segment [0; 2] for c = 0, d = 1 or c = -1, d = -2 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), ISAA, 1.; answer block on page 201) | |
dgn.parameters.Variant 2003 (iyul), ISAA, 2, Task 1 |
answer recovered | `` | A \in [1/32; 22] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), ISAA, 2.; answer block on page 202) | |
dgn.parameters.p202.2 |
answer recovered | `` | \{5; (-3 - \sqrt{33})/2\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), ISAA, 2.; answer block on page 202) | |
dgn.parameters.p202.3 |
answer recovered | `` | 2r^{2}\sqrt{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), ISAA, 2.; answer block on page 202) | |
dgn.parameters.p202.4 |
answer recovered | `` | \{-\pi/2; -\pi + \operatorname{arctg}\frac{\sqrt{3}}{2}\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), ISAA, 2.; answer block on page 202) | |
dgn.parameters.p202.5 |
answer recovered | `` | (-\infty; -\frac{1}{3\sqrt{9}}] \cup \{-\frac{1}{9}\} \cup \{\frac{1}{27}\} \cup \left[\frac{1}{3\sqrt{9}}; 2\right)[5; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), ISAA, 2.; answer block on page 202) | |
dgn.parameters.p202.6 |
answer recovered | `` | The segment [0; 2] for a = -1, b = 0 or a = 1/3, b = 2/3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), ISAA, 2.; answer block on page 202) | |
dgn.parameters.Variant 2003 (July), Sociology Faculty, 1 |
answer recovered | `` | 3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Sociology Faculty, 1. √ √ √; answer block on page 203) | |
dgn.parameters.Variant 2003 (July), Sociology Faculty, 2 |
answer recovered | `` | \{5; (-3 - \sqrt{33})/2\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), ISAA, 2.; answer block on page 202) | |
dgn.parameters.Variant 2003 (iyul), sociology Faculty, 2 - Task 1 |
answer recovered | `` | 4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Sociology Faculty, 2. √ √ √; answer block on page 204) | |
dgn.parameters.Variant 2003 (iyul), sociology Faculty, 2 - Task 2 |
answer recovered | `` | [3; 12) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Sociology Faculty, 2. √ √ √; answer block on page 204) | |
dgn.parameters.Variant 2003 (iyul), sociology Faculty, 2 - Task 3 |
answer recovered | `` | 2\sqrt{7} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Sociology Faculty, 2. √ √ √; answer block on page 204) | |
dgn.parameters.Variant 2003 (iyul), sociology Faculty, 2 - Task 4 |
answer recovered | `` | 33\frac{1}{3}\%,\ 33\frac{1}{3}\%,\ 66\frac{2}{3}\% |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Sociology Faculty, 2. √ √ √; answer block on page 204) | |
dgn.parameters.p203.Task 3 |
answer recovered | `` | \sqrt{7} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Sociology Faculty, 1. √ √ √; answer block on page 203) | |
dgn.parameters.p203.Task 4 |
answer recovered | `` | 33\frac{1}{3}\%,\ 33\frac{1}{3}\%,\ 66\frac{2}{3}\% |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Sociology Faculty, 2. √ √ √; answer block on page 204) | |
dgn.parameters.p203.Task 5 |
answer recovered | `` | The first — 10, the second — 12 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Sociology Faculty, 1. √ √ √; answer block on page 203) | |
dgn.parameters.p203.Task 6 |
answer recovered | `` | a = 7; x_1 = 2, x_2 = 4, x_3 = 8 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Sociology Faculty, 1. √ √ √; answer block on page 203) | |
dgn.parameters.p204.204 |
answer recovered | `` | 1. 4; 2. [3; 12); 3. 2\sqrt{7}; 4. 33\frac{1}{3}\%,\ 33\frac{1}{3}\%,\ 66\frac{2}{3}\%; 5. The first — 15, the second — 18; 6. a = 13; x_1 = 2, x_2 = 6, x_3 = 18 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Sociology Faculty, 2. √ √ √; answer block on page 204) | |
dgn.parameters.Variant 2003 (iyul), Faculty gosudarstvennogo management, 2 |
answer recovered | `` | [-4; 14/3] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Public Administration, 2.; answer block on page 206) | |
dgn.parameters.p205.Task 1 |
answer recovered | `` | In part B |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Public Administration, 2.; answer block on page 206) | |
dgn.parameters.p205.Task 2 |
answer recovered | `` | [-4; 14/3] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Public Administration, 2.; answer block on page 206) | |
dgn.parameters.p205.Task 3 |
answer recovered | `` | 100 thousand u. e. for enterprise D and 10 thousand u. e. — for E |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Public Administration, 2.; answer block on page 206) | |
dgn.parameters.p205.Task 4 |
answer recovered | `` | 9\sqrt{5}/10 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Public Administration, 2.; answer block on page 206) | |
dgn.parameters.p205.Task 5 |
answer recovered | `` | There are no solutions for a \leqslant 0, b < 0; (-\infty; +\infty) for a \geqslant 0, b \geqslant 0; (-\infty; -\sqrt{-b/a}] \cup [\sqrt{-b/a}; +\infty) for a > 0, b < 0; [-\sqrt{-b/a}; \sqrt{-b/a}] for a < 0, b \geqslant 0 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Public Administration, 2.; answer block on page 206) | |
dgn.parameters.p205.Task 7 |
answer recovered | `` | They will make it in time |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Public Administration, 1.; answer block on page 205) | |
dgn.parameters.Otvet |
answer recovered | `` | \{(2; 2; -2)\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Public Administration, 1.; answer block on page 205) | |
dgn.parameters.1 (Variant 2003, April, Higher School of Business, 1) |
answer recovered | `` | \{0; 2\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Higher School of Business, 1. √; answer block on page 207) | |
dgn.parameters.2 (Variant 2003, April, Higher School of Business, 1) |
answer recovered | `` | \{0; 2\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Higher School of Business, 1. √; answer block on page 207) | |
dgn.parameters.3 (Variant 2003, April, Higher School of Business, 1) |
answer recovered | `` | [-4; 5) \cup (21; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Higher School of Business, 1. √; answer block on page 207) | |
dgn.parameters.4 (Variant 2003, April, Higher School of Business, 1) |
answer recovered | `` | The center is at the point (-2; -5), R = 5 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Higher School of Business, 1. √; answer block on page 207) | |
dgn.parameters.5 (Variant 2003, April, Higher School of Business, 1) |
answer recovered | `` | (4; 8) \cup \{2\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Higher School of Business, 1. √; answer block on page 207) | |
dgn.parameters.6 (Variant 2003, April, Higher School of Business, 1) |
answer recovered | `` | \{0; 2\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Higher School of Business, 1. √; answer block on page 207) | |
dgn.parameters.p206.7 |
mispaired answer | Yes, they will make it. |
Will they make it |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (July), Faculty of Public Administration, 2.; answer block on page 206) | |
dgn.parameters.p211.3 |
mispaired answer | $\frac{19}{6};\ \frac{13}{2}$ |
\frac{19}{6}, \frac{13}{2} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (March), Mechanics and Mathematics Faculty, 2.; answer block on page 211) | |
dgn.parameters.p212.2 |
mispaired answer | $(-\infty;\,-2-\sqrt{3}] \cup (-0{,}3;\,-2+\sqrt{3}] \cup \{1\} \cup (2;\infty)$ |
(-\infty; -2 - \sqrt{3}] \cup (-0, 3; -2 + \sqrt{3}] \cup \{1\} \cup (2; \infty) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Mechanics and Mathematics Faculty, 1.; answer block on page 212) | |
dgn.parameters.p212.5 |
mispaired answer | $16$ km |
16 \text{ km} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Mechanics and Mathematics Faculty, 1.; answer block on page 212) | |
dgn.parameters.p213.Variant 2004 |
mispaired answer | Answers (collected): **1.** $(-1; 0) \cup (0; 1)$. **2.** $(-\infty; -5 - \sqrt{21}] \cup (-0.5; -5 + \sqrt{21}] \cup \{2\} \cup (3; \infty)$. **3.** 13. **4.** $-\tfrac{\pi}{4} + (-1)^n \tfrac{\pi}{4} + \pi n,\, -\tfrac{\pi}{4} + (-1)^k \tfrac{\pi}{3} + \pi k,\, n, k \in \mathbb{Z}$. **5.** 18 km. **6.** $\dfrac{4(5\sqrt{3}+1)}{3\sqrt{11}}$. |
1. (-1; 0) \cup (0; 1); 2. (-\infty; -5 - \sqrt{21}] \cup (-0, 5; -5 + \sqrt{21}] \cup \{2\} \cup (3; \infty); 3. 13; 4. -\frac{\pi}{4} + (-1)^{n}\frac{\pi}{4} + \pi n, -\frac{\pi}{4} + (-1)^{k}\frac{\pi}{3} + \pi k, n, k \in \mathbb{Z}; 5. 18 \text{ km}; 6. \frac{4(5\sqrt{3} + 1)}{3\sqrt{11}} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Mechanics and Mathematics Faculty, 2.; answer block on page 213) | |
dgn.parameters.p215.Task 1 |
mispaired answer | $\{2, 4, 8, 16\}$ |
\{3, -6, 12, -24\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (April), Faculty of Computational Mathematics and Cybernetics, 2.; answer block on page 216) | |
dgn.parameters.p215.Task 2 |
mispaired answer | $\{2; 3\}$ |
\mathbb{R} \setminus \{4 - \sqrt{3}; 4 + \sqrt{3}\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (April), Faculty of Computational Mathematics and Cybernetics, 2.; answer block on page 216) | |
dgn.parameters.p215.Task 3 |
mispaired answer | $\{-15\}$ |
\{13\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (April), Faculty of Computational Mathematics and Cybernetics, 2.; answer block on page 216) | |
dgn.parameters.p215.Task 5 |
mispaired answer | $\frac{1}{4}$ or $\frac{5}{4}$ |
\frac{1}{4} \text{ or } \frac{5}{4} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (April), Faculty of Computational Mathematics and Cybernetics, 1.; answer block on page 215) | |
dgn.parameters.p215.Task 6 |
mispaired answer | For $a \in (-\infty; 0) \cup [1/2; +\infty)$ there are no solutions; for $a \in (0; (3-\sqrt{3})/4)$ two solutions: $x = 1 \pm 2\sqrt{a/(1-2a)}$; for $a = (3-\sqrt{3})/4$ three solutions: $x = 2+\sqrt{3}$ and $x = 1 \pm \sqrt[4]{12}$; for $a \in ((3-\sqrt{3})/4; 1/3)$ four solutions: $x = (2(a-1) \pm \sqrt{-8a^2+12a-3})/(2a-1)$, $x = 1 \pm 2\sqrt{a/(1-2a)}$; for $a = 1/3$ three solutions: $x = -1$, $x = 3$, $x = 5$; for $a \in (1/3; 1/2)$ two solutions: $x = 1 - 2\sqrt{a/(1-2a)}$, $x = (2(a-1) - \sqrt{-8a^2+12a-3})/(2a-1)$. |
For a \in (-\infty; 0) \cup [1/2; +\infty) there are no solutions; for a \in (0; (3 - \sqrt{3})/4) there are two solutions: x = 1 \pm 2\sqrt{a/(1 - 2a)}; for a = (3 - \sqrt{3})/4 there are three solutions: x = 2 + \sqrt{3} and x = 1 \pm \sqrt[4]{12}; for a \in ((3 - \sqrt{3})/4; 1/3) there are four solutions: x = (2(a - 1) \pm \sqrt{-8a^{2} + 12a - 3})/(2a - 1), x = 1 \pm 2\sqrt{a/(1 - 2a)}; for a = 1/3 there are three solutions: x = -1, x = 3, x = 5; for a \in (1/3; 1/2) there are two solutions: x = 1 - 2\sqrt{a/(1 - 2a)}, x = (2(a - 1) - \sqrt{-8a^{2} + 12a - 3})/(2a - 1) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (April), Faculty of Computational Mathematics and Cybernetics, 1.; answer block on page 215) | |
dgn.parameters.p216.1 |
answer recovered | `` | \mathbb{R} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 1.; answer block on page 217) | |
dgn.parameters.p216.2 |
answer recovered | `` | (-3; -2] \cup \left[\frac{3 - \sqrt{31}}{2}; \frac{\sqrt{23} - 1}{2}\right] \cup [2; 5) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 1.; answer block on page 217) | |
dgn.parameters.p216.4 |
answer recovered | `` | \{27; 1/27\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (April), Faculty of Computational Mathematics and Cybernetics, 2.; answer block on page 216) | |
dgn.parameters.p216.5 |
answer recovered | `` | \{1/4; 7/4\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (April), Faculty of Computational Mathematics and Cybernetics, 2.; answer block on page 216) | |
dgn.parameters.p216.6 |
answer recovered | `` | For a \in (-\infty; 0) \cup [1; +\infty) there are no solutions; for a = 0 there is one solution x = 1/3; for a \in \left(0; \frac{3 - \sqrt{3}}{2}\right) there are two solutions: x = 1/3 \cdot \left(1 \pm \sqrt{\frac{2a}{1 - a}}\right); for a = \frac{3 - \sqrt{3}}{2} there are three solutions: x = -\frac{1}{\sqrt{3}}, x = \frac{1 \pm \sqrt[4]{12}}{3}; for a \in \left(\frac{3 - \sqrt{3}}{2}; 2/3\right) there are four solutions: x = \frac{-a \pm \sqrt{-2a^{2} + 6a - 3}}{3(1 - a)}, x = 1/3 \cdot \left(1 \pm \sqrt{\frac{2a}{1 - a}}\right); for a = 2/3 there are three solutions: x = -\frac{1}{3}, x = \pm 1; for a \in \left(\frac{2}{3}; 1\right) there are two solutions: x = \frac{1}{3} \cdot \left(1 + \sqrt{\frac{2a}{1 - a}}\right), x = \frac{-a + \sqrt{-2a^{2} + 6a - 3}}{3(1 - a)} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (April), Faculty of Computational Mathematics and Cybernetics, 2.; answer block on page 216) | |
dgn.parameters.p217.1 |
answer recovered | `` | \mathbb{R} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 2.; answer block on page 218) | |
dgn.parameters.p217.2 |
answer recovered | `` | \left[-3; \frac{2 - \sqrt{34}}{2}\right] \cup \left[\frac{\sqrt{46} - 4}{2}; 3\right] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 2.; answer block on page 218) | |
dgn.parameters.p217.3 |
answer recovered | `` | \left\{\frac{15\pi}{16} + 2\pi k; k \in \mathbb{Z}\right\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 1.; answer block on page 217) | |
dgn.parameters.p217.4 |
answer recovered | `` | 65 : 144 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 1.; answer block on page 217) | |
dgn.parameters.p217.5 |
answer recovered | `` | For a \in \left(-\infty; -\frac{5}{4}\right) there is one solution; for a = -\frac{5}{4} there are two solutions; for a \in \left(-\frac{5}{4}; -1\right) there are three solutions; for a \in [-1; 1 - \sqrt{2}) there are two solutions; for a = 1 - \sqrt{2} there is one solution; for a \in (1 - \sqrt{2}; 5) there are two solutions; for a \in [5; +\infty) there is one solution |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 1.; answer block on page 217) | |
dgn.parameters.p218.1 |
answer recovered | `` | \left(\frac{3}{4}; \frac{7}{8}\right] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 1.; answer block on page 219) | |
dgn.parameters.p218.2 |
answer recovered | `` | \mathbb{R} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 1.; answer block on page 219) | |
dgn.parameters.p218.3 |
answer recovered | `` | \left\{\frac{17\pi}{14} + 2\pi n; n \in \mathbb{Z}\right\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 2.; answer block on page 218) | |
dgn.parameters.p218.4 |
answer recovered | `` | 3 : 5 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 2.; answer block on page 218) | |
dgn.parameters.p218.5 |
answer recovered | `` | For a \in \left(-\infty; -\frac{17}{4}\right) there are no solutions; for a = -\frac{17}{4} there is one solution; for a \in \left(-\frac{17}{4}; \frac{3}{4}\right) there are two solutions; for a = \frac{3}{4} there are three solutions; for a \in \left(\frac{3}{4}; 2\right) there are four solutions; for a \in [2; 3) there are three solutions; for a \in [3; +\infty) there are two solutions |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 2.; answer block on page 218) | |
dgn.parameters.p218.6 |
answer recovered | `` | NA : KN = 1 : 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Specialist Division), 2.; answer block on page 218) | |
dgn.parameters.p219.1 |
answer recovered | `` | \left(0; \frac{1}{6}\right] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 2.; answer block on page 220) | |
dgn.parameters.p219.2 |
answer recovered | `` | \mathbb{R} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 2.; answer block on page 220) | |
dgn.parameters.p219.3 |
answer recovered | `` | \left[\frac{3 - \sqrt{137}}{4}; -2\right] \cup \left[\frac{-5 + \sqrt{153}}{4}; 2\right] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 1.; answer block on page 219) | |
dgn.parameters.p219.4 |
answer recovered | `` | If d \in (-\infty; -4), then there are no solutions; if d = -4, then x = 1; if d \in (-4; 0), then x \in [\log_{2}(2 - \sqrt{d + 4}); \log_{2}(2 + \sqrt{d + 4})]; x \in (-\infty; \log_{2}(2 + \sqrt{d + 4})], if d \in [0; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 1.; answer block on page 219) | |
dgn.parameters.p219.5 |
answer recovered | `` | \left\{\frac{23\pi}{14} + 2\pi m; m \in \mathbb{Z}\right\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 1.; answer block on page 219) | |
dgn.parameters.p219.6 |
answer recovered | `` | 4 : 5 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 1.; answer block on page 219) | |
dgn.parameters.p219.7 |
answer recovered | `` | NA : KN = 1 : 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 1.; answer block on page 219) | |
dgn.parameters.p220.1 |
answer recovered | `` | \frac{\pi}{4} + (-1)^{n} \arcsin \frac{1}{3} + \pi n, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (March), Physics Faculty, 1.; answer block on page 221) | |
dgn.parameters.p220.3 |
mispaired answer | $(-4;-3) \cup \left[\frac{9-\sqrt{233}}{4};\frac{\sqrt{161}-3}{4}\right] \cup [3;7)$. |
(-4; -3) \cup \left[\frac{9 - \sqrt{233}}{4}; \frac{\sqrt{161} - 3}{4}\right] \cup [3; 7) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 2.; answer block on page 220) | |
dgn.parameters.p220.4 |
mispaired answer | If $c \in (-\infty;-9]$, then $x$ — any number; $x \in (-\infty;\log_3(3-\sqrt{c+9})) \cup [\log_3(3+\sqrt{c+9});+\infty)$, if $c \in (-9;0)$; $x \in [\log_3(3+\sqrt{c+9});+\infty)$, if $c \in [0;+\infty)$. |
If c \in (-\infty; -9], then x is any number; x \in (-\infty; \log_{3}(3 - \sqrt{c + 9})] \cup [\log_{3}(3 + \sqrt{c + 9}); +\infty), if c \in (-9; 0); x \in [\log_{3}(3 + \sqrt{c + 9}); +\infty), if c \in [0; +\infty) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 2.; answer block on page 220) | |
dgn.parameters.p220.5 |
mispaired answer | $\left\{\frac{7\pi}{10}+2\pi k;\ k \in \mathbb{Z}\right\}$. |
\left\{\frac{7\pi}{10} + 2\pi k; k \in \mathbb{Z}\right\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 2.; answer block on page 220) | |
dgn.parameters.p220.6 |
mispaired answer | $25:156$. |
25 : 156 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 2.; answer block on page 220) | |
dgn.parameters.p220.7 |
mispaired answer | $DK:AD = 1:2$. |
DK : AD = 1 : 2 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 2.; answer block on page 220) | |
dgn.parameters.p221.2 |
answer recovered | `` | \frac{1 - \sqrt{5}}{2} < x < 0, 1 < x < \frac{1 + \sqrt{5}}{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (March), Physics Faculty, 1.; answer block on page 221) | |
dgn.parameters.p221.3 |
answer recovered | `` | -\sqrt{5} < x < -2, -1 < x < 0, 0 < x < 1, 2 < x < \sqrt{5} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (March), Physics Faculty, 1.; answer block on page 221) | |
dgn.parameters.p221.4 |
answer recovered | `` | \frac{6}{11} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (March), Physics Faculty, 1.; answer block on page 221) | |
dgn.parameters.p221.5 |
answer recovered | `` | x_{1} = \frac{5}{2}, y_{1} = \frac{3}{2}; x_{2} = \frac{7}{2} - \log_{2} 3, y_{2} = \frac{1}{2} + \log_{2} 3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (March), Physics Faculty, 1.; answer block on page 221) | |
dgn.parameters.p221.6 |
answer recovered | `` | \frac{35}{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (March), Physics Faculty, 1.; answer block on page 221) | |
dgn.parameters.p221.7 |
answer recovered | `` | If a = 0, then x > 0; if a \neq 0, then x = \frac{3a + \sqrt{9a^{2} + 4}}{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (March), Physics Faculty, 1.; answer block on page 221) | |
dgn.parameters.p221.8 |
answer recovered | `` | \operatorname{arctg} \frac{2}{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (March), Physics Faculty, 1.; answer block on page 221) | |
dgn.parameters.2004 |
mispaired answer | 1. $\frac{\pi}{10}\pm \arccos\frac{2}{5}+2\pi n,\ n\in \mathbb{Z}$. 2. $1-\sqrt{2}<x<0,\ 2<x<1+\sqrt{2}$. 3. $-\sqrt{13}<x<-3,\ -2<x<0,\ 0<x<2,\ 3<x<\sqrt{13}$. 4. $\frac{60}{143}$. 5. $x_1=\frac{1}{2},\ y_1=\frac{3}{2};\ x_2=\frac{1}{2}+\log_5 3,\ y_2=\frac{3}{2}-\log_5 3$. 6. $\frac{35}{11}$. 7. If $a=0$, then $x>0$; if $a\ne 0$, then $x=2a+\sqrt{4a^2+1}$. 8. $\arctg 2$. |
1. \frac{\pi}{10} \pm \arccos \frac{2}{5} + 2\pi n, n \in \mathbb{Z}; 2. 1 - \sqrt{2} < x < 0, 2 < x < 1 + \sqrt{2}; 3. -\sqrt{13} < x < -3, -2 < x < 0, 0 < x < 2, 3 < x < \sqrt{13}; 4. \frac{60}{143}; 5. x_{1} = \frac{1}{2}, y_{1} = \frac{3}{2}; x_{2} = \frac{1}{2} + \log_{5} 3, y_{2} = \frac{3}{2} - \log_{5} 3; 6. \frac{35}{11}; 7. If a = 0, then x > 0; if a \neq 0, then x = 2a + \sqrt{4a^{2} + 1}; 8. \operatorname{arctg} 2 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (March), Physics Faculty, 2.; answer block on page 222) | |
dgn.parameters.Variant 2004 (iyul), physics Faculty, 1 |
answer recovered | `` | 1. \frac{\pi}{12} + \frac{\pi n}{6}, \frac{\pi}{2} + \pi n, n \in \mathbb{Z}; 2. -5 < x < -1, 3 < x < 7; 3. -2 < x < -1 + \sqrt{2}; 4. -3 + \sqrt{24}; 5. x = \frac{1 + \sqrt{13}}{2}, y = \frac{-5 + \sqrt{13}}{4}; 6. 96; 7. -\frac{13}{30} \leqslant a < -\frac{9}{30}, \frac{11}{30} < a \leqslant \frac{15}{30}; 8. \frac{\sqrt{14}}{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Physics Faculty, 1.; answer block on page 223) | |
dgn.parameters.2004 (iyul), physics Faculty, 2 |
mispaired answer | 1. $\frac{\pi n}{6}$, $\frac{\pi}{2} + \pi n$, $n \in \mathbb{Z}$. 2. $-10 < x < -1$, $5 < x < 14$. 3. $-3 < x < -2 + \sqrt{5}$. 4. $-2 + \sqrt{20}$. 5. $x = \frac{1-\sqrt{13}}{6}$, $y = \frac{-5 - \sqrt{13}}{4}$. 6. $80$. 7. $-\frac{11}{30} \leqslant a < -\frac{7}{30}$, $\frac{5}{30} < a \leqslant \frac{9}{30}$. 8. $\frac{1}{2}$. |
1. \frac{\pi n}{6}, \frac{\pi}{2} + \pi n, n \in \mathbb{Z}; 2. -10 < x < -1, 5 < x < 14; 3. -3 < x < -2 + \sqrt{5}; 4. -2 + \sqrt{20}; 5. x = \frac{1 - \sqrt{13}}{6}, y = \frac{-5 - \sqrt{13}}{4}; 6. 80; 7. -\frac{11}{30} \leqslant a < -\frac{7}{30}, \frac{5}{30} < a \leqslant \frac{9}{30}; 8. \frac{1}{2} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Physics Faculty, 2.; answer block on page 224) | |
dgn.parameters.Variant 2003 (aprel), Faculty Sciences o Materials, 1, Task 1 |
mispaired answer | 2.5 L |
2,5 \text{ l |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Faculty of Materials Science, 1.; answer block on page 225) | |
dgn.parameters.Variant 2003 (aprel), Faculty Sciences o Materials, 1, Task 4 |
mispaired answer | $\dfrac{16}{3}$ and $\dfrac{21}{25}$ or $3$ and $\dfrac{112}{75}$ |
\frac{16}{3} \text{ and } \frac{21}{25} \text{ or } 3 \text{ and } \frac{112}{75} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Faculty of Materials Science, 1.; answer block on page 225) | |
dgn.parameters.Variant 2003 (aprel), Faculty Sciences o Materials, 1, Task 5 |
mispaired answer | $\pi k;\ (-1)^m \arcsin\left(\dfrac{1-\sqrt{5}}{4}\right)+\pi m;\ (-1)^{n+1}\dfrac{\pi}{6}+\pi n,\ k,m,n\in\mathbb{Z}$ |
\pi k; (-1)^{m} \arcsin\left(\frac{1 - \sqrt{5}}{4}\right) + \pi m; (-1)^{n+1}\frac{\pi}{6} + \pi n, k, m, n \in \mathbb{Z} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Faculty of Materials Science, 1.; answer block on page 225) | |
dgn.parameters.Variant 2003 (aprel), Faculty Sciences o Materials, 2, Task 1 |
answer recovered | `` | 4 \text{ l |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Faculty of Materials Science, 2.; answer block on page 226) | |
dgn.parameters.p226.322 |
mispaired answer | $\left(-2;\,-\frac{\sqrt{17}+1}{4}\right] \cup \left[\frac{\sqrt{17}-3}{4};\,1\right) \cup \left\{\frac{1}{2}\right\}.$ |
\left(-2; -\frac{\sqrt{17} + 1}{4}\right] \cup \left[\frac{\sqrt{17} - 3}{4}; 1\right) \cup \{\frac{1}{2}\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Faculty of Materials Science, 2.; answer block on page 226) | |
dgn.parameters.p226.323 |
mispaired answer | $\pm 1,\, \pm 3.$ |
\pm 1, \pm 3 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Faculty of Materials Science, 2.; answer block on page 226) | |
dgn.parameters.p226.324 |
mispaired answer | $12$ and $\frac{181}{100}$, or $12$ and $\frac{675}{28}.$ |
12 \text{ and } \frac{181}{100} \text{ or } 12 \text{ and } \frac{675}{28} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Faculty of Materials Science, 2.; answer block on page 226) | |
dgn.parameters.p226.325 |
mispaired answer | $\frac{\pi}{2}+\pi k;\ \pm \arccos\left(\frac{1-\sqrt{5}}{4}\right)+2\pi m;\ \pm \frac{2\pi}{3}+2\pi n,\ k,m,n\in\mathbb{Z}.$ |
\frac{\pi}{2} + \pi k; \pm \arccos\left(\frac{1 - \sqrt{5}}{4}\right) + 2\pi m; \pm \frac{2\pi}{3} + 2\pi n, k, m, n \in \mathbb{Z} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Faculty of Materials Science, 2.; answer block on page 226) | |
dgn.parameters.p226.326 |
mispaired answer | $-\sqrt{3}.$ |
-\sqrt{3} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Faculty of Materials Science, 2.; answer block on page 226) | |
dgn.parameters.p226.327 |
answer recovered | `` | x \in \left(\frac{3 - \sqrt{5}}{2}; 2\right) \cup \left(\frac{3 + \sqrt{5}}{2}; +\infty\right) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (May), Chemistry Faculty, test exam1 .; answer block on page 227) | |
dgn.parameters.end of problem 1 (continuation) |
mispaired answer | 4 L. |
4 \text{ l} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2003 (April), Faculty of Materials Science, 2.; answer block on page 226) | |
dgn.parameters.p227.327 |
answer recovered | `` | x \in (0; 1) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (May), Chemistry Faculty, test exam1 .; answer block on page 227) | |
dgn.parameters.328 |
answer recovered | `` | \pi n, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (May), Chemistry Faculty, test exam1 .; answer block on page 227) | |
dgn.parameters.329 |
answer recovered | `` | S = \frac{48}{5} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (May), Chemistry Faculty, test exam1 .; answer block on page 227) | |
dgn.parameters.330 |
answer recovered | `` | x = \cos\left(\frac{5\pi}{12}\right) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (May), Chemistry Faculty, test exam1 .; answer block on page 227) | |
dgn.parameters.331 |
answer recovered | `` | \frac{40}{\sqrt{41}} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (May), Chemistry Faculty, test exam1 .; answer block on page 227) | |
dgn.parameters.332 |
answer recovered | `` | x = \pi/6 + \pi n/3, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (2 July), exam for the winners of the olympiads at the faculties of chemistry, biology, materials science and fundamental medicine, 1.; answer block on page 228) | |
dgn.parameters.333 |
answer recovered | `` | x \in (1; 3) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (2 July), exam for the winners of the olympiads at the faculties of chemistry, biology, materials science and fundamental medicine, 1.; answer block on page 228) | |
dgn.parameters.334 |
answer recovered | `` | (x; y) = (2a; a), (-2a; a), a \in \mathbb{R} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (2 July), exam for the winners of the olympiads at the faculties of chemistry, biology, materials science and fundamental medicine, 1.; answer block on page 228) | |
dgn.parameters.335 |
answer recovered | `` | DC = 2, DC = 4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (2 July), exam for the winners of the olympiads at the faculties of chemistry, biology, materials science and fundamental medicine, 1.; answer block on page 228) | |
dgn.parameters.p228.1 |
answer recovered | `` | x = \pi n/3, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (2 July), exam for the winners of the olympiads at the faculties of chemistry, biology, materials science and fundamental medicine, 2.; answer block on page 229) | |
dgn.parameters.p228.2 |
wrong answer | page 229, item 2 of the 'Ответы' block for ВАРИАНТ 2004 (2 июля), экзамен для победителей олимпиад на факультетах: химическом, биологическом, наук о материалах и фундаментальной медицины, 2.: x ∈ (1; 7) (printed problem, page 228: 'Задача 2. Решите неравенство log₂(7−x) − log₂(x+1) > log₍√2₎√3.') |
x \in (-1; 1) |
log₍√2₎√3 = ln√3/ln√2 = log₂3 = 1.5849625, so the printed inequality is log₂((7−x)/(x+1)) > log₂3 on the domain −1 < x < 7, i.e. (7−x)/(x+1) > 3 ⟺ 7−x > 3x+3 ⟺ x < 1. The solution is therefore (−1; 1). The printed answer (1; 7) is exactly the solution set of the REVERSED inequality '<', so the key contradicts the printed problem. The engine's probe is legitimate and reproduces exactly: F(1.3) = log₂(5.7/2.3) − log₍√2₎√3 = 1.308939 − 1.584963 = −0.275634 (the engine's value), i.e. x = 1.3 violates '>'. | |
dgn.parameters.p228.3 |
answer recovered | `` | (x; y) = (3b; b), (-3b; b), b \in \mathbb{R} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (2 July), exam for the winners of the olympiads at the faculties of chemistry, biology, materials science and fundamental medicine, 2.; answer block on page 229) | |
dgn.parameters.p228.4 |
answer recovered | `` | KN = 1, KN = 5 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (2 July), exam for the winners of the olympiads at the faculties of chemistry, biology, materials science and fundamental medicine, 2.; answer block on page 229) | |
dgn.parameters.p228.5 |
mispaired answer | $\mathbf{5}.$ $(54;60)$, $(54;63)$, $(55;60)$, $(56;63)$. |
(48; 54), (48; 56), (49; 56), (50; 55) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (2 July), exam for the winners of the olympiads at the faculties of chemistry, biology, materials science and fundamental medicine, 2.; answer block on page 229) | |
dgn.parameters.p228.6 |
mispaired answer | $\mathbf{6}.$ $SD_1:SD = 12:13$, $V_{SA_1B_1C_1D_1}/V_{SABCD} = 220/1911$. |
SD_{1} : SD = 12 : 13, V_{SA_{1}B_{1}C_{1}D_{1}}/V_{SABCD} = 220/1911 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (2 July), exam for the winners of the olympiads at the faculties of chemistry, biology, materials science and fundamental medicine, 1.; answer block on page 228) | |
dgn.parameters.p229.5 |
mispaired answer | $SN_1 : SN = 1 : 2,\ V_{SK_1 L_1 M_1 N_1}/V_{SKLMN} = 20/297$. |
2\sqrt{22}/11 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Chemistry Faculty and Faculty of Materials Science, 1.; answer block on page 229) | |
dgn.parameters.p229.6 |
mispaired answer | Problem 1: $(-\infty; -1] \cup (1; 2) \cup [4; +\infty)$. Problem 2: $\pi n,\ n \in \mathbb{Z}$. Problem 3: $x > 5^{2\sqrt{21}/3}$. Problem 4: $BE = 14/3$. Problem 5: $2\sqrt{22}/11$. Problem 6: $(a; b) = (1; 2),\ (-1; -2),\ (7/\sqrt{31}; -4/\sqrt{31}),\ (-7/\sqrt{31}; 4/\sqrt{31})$. |
1. (-\infty; -1] \cup (1; 2) \cup [4; +\infty); 2. \pi n, n \in \mathbb{Z}; 3. x > 5^{2\sqrt{21}/3}; 4. BE = 14/3; 5. 2\sqrt{22}/11; 6. (a; b) = (1; 2), (-1; -2), (7/\sqrt{31}; -4/\sqrt{31}), (-7/\sqrt{31}; 4/\sqrt{31}) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Chemistry Faculty and Faculty of Materials Science, 1.; answer block on page 229) | |
dgn.parameters.p230.7 |
answer recovered | `` | (3 \pm \sqrt{13})/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Biology Faculty, Faculty of Fundamental Medicine, Faculty of Bioengineering and Bioinformatics, 1. √; answer block on page 231) | |
dgn.parameters.p230.8 |
answer recovered | `` | \pi/10 + \pi m/5, \pi/2 + \pi n, m, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Biology Faculty, Faculty of Fundamental Medicine, Faculty of Bioengineering and Bioinformatics, 1. √; answer block on page 231) | |
dgn.parameters.p230.9 |
answer recovered | `` | 15\sqrt{2}/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Biology Faculty, Faculty of Fundamental Medicine, Faculty of Bioengineering and Bioinformatics, 1. √; answer block on page 231) | |
dgn.parameters.p231.3 |
mispaired answer | $78\sqrt{5}$ |
15\sqrt{2}/2 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Biology Faculty, Faculty of Fundamental Medicine, Faculty of Bioengineering and Bioinformatics, 1. √; answer block on page 231) | |
dgn.parameters.p231.4 |
mispaired answer | $(-\infty; \log_3(3-\sqrt{2})] \cup (\log_3 6; +\infty)$ |
(-\infty; \log_{3}(3 - \sqrt{2})] \cup (\log_{3} 6; +\infty) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Biology Faculty, Faculty of Fundamental Medicine, Faculty of Bioengineering and Bioinformatics, 2. √; answer block on page 231) | |
dgn.parameters.p231.5 |
mispaired answer | $6/(1+\sqrt{5})$ |
13/2 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Biology Faculty, Faculty of Fundamental Medicine, Faculty of Bioengineering and Bioinformatics, 2. √; answer block on page 231) | |
dgn.parameters.1 (Variant 2004 (July), Geography Faculty, 2) |
answer recovered | `` | \{-1\} \cup \{1\} \cup (2; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geography Faculty, 2.; answer block on page 233) | |
dgn.parameters.p232.3 |
mispaired answer | a) $60$ km; b) $\frac{195}{2}$ km |
a) 60 \text{ km}, b) \frac{195}{2} \text{ km} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geography Faculty, 1.; answer block on page 232) | |
dgn.parameters.p232.4 |
mispaired answer | a) $12$; b) $\frac{5}{2}$ |
a) 12, b) \frac{5}{2} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geography Faculty, 1.; answer block on page 232) | |
dgn.parameters.Task 1 (Variant 2004, May, Faculty pochvovedeniya) |
answer recovered | `` | x = -(\sqrt{6} + \sqrt{2})/4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (May), Faculty of Soil Science, 1.; answer block on page 234) | |
dgn.parameters.p233.Task 2 |
answer recovered | `` | \frac{\pi}{4} + \frac{\pi n}{2}, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geography Faculty, 2.; answer block on page 233) | |
dgn.parameters.p233.Task 3 |
answer recovered | `` | a) 30 \text{ km}, b) \frac{195}{4} \text{ km} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geography Faculty, 2.; answer block on page 233) | |
dgn.parameters.p233.Task 4 |
answer recovered | `` | a) 9, b) 7 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geography Faculty, 2.; answer block on page 233) | |
dgn.parameters.p233.Task 5 |
answer recovered | `` | -2, -\frac{16}{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geography Faculty, 2.; answer block on page 233) | |
dgn.parameters.p233.Task 6 |
answer recovered | `` | 6990 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geography Faculty, 2.; answer block on page 233) | |
dgn.parameters.Variant 2004 (iyul), Faculty soil science, 1 - Task 1 |
answer recovered | `` | \max y = 9, \min y = -9/8 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Soil Science, 1.; answer block on page 235) | |
dgn.parameters.Variant 2004 (may), Faculty soil science, 2 - Task 2 |
mispaired answer | $42$ |
26 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (May), Faculty of Soil Science, 2.; answer block on page 234) | |
dgn.parameters.Variant 2004 (may), Faculty soil science, 2 - Task 3 |
mispaired answer | $x \in (-\infty; \log_2 3)$ (listed answer: $x \in (\log_6 2; +\infty)$) |
x \in (\log_{6} 2; +\infty) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (May), Faculty of Soil Science, 2.; answer block on page 234) | |
dgn.parameters.Variant 2004 (may), Faculty soil science, 2 - Task 5 |
mispaired answer | $\frac{am}{3a - 2m}$ |
am/(3a - 2m) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (May), Faculty of Soil Science, 2.; answer block on page 234) | |
dgn.parameters.p234.Task 2 |
answer recovered | `` | 42 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (May), Faculty of Soil Science, 1.; answer block on page 234) | |
dgn.parameters.p234.Task 3 |
mispaired answer | $x \in (-\infty; \log_2 3)$ |
x \in (-\infty; \log_{2} 3) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (May), Faculty of Soil Science, 1.; answer block on page 234) | |
dgn.parameters.p234.Task 5 |
mispaired answer | $\frac{3ab}{a + 2b}$ |
3ab/(a + 2b) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (May), Faculty of Soil Science, 1.; answer block on page 234) | |
dgn.parameters.Variant 2004 (iyul), Faculty soil science, 2 |
answer recovered | `` | x \in (-3; 0) \cup (0; 3) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Soil Science, 2.; answer block on page 236) | |
dgn.parameters.p235.Task 1 |
answer recovered | `` | \max y = 12, \min y = -25/8 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Soil Science, 2.; answer block on page 236) | |
dgn.parameters.p235.Task 2 |
answer recovered | `` | x = -1/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Soil Science, 1.; answer block on page 235) | |
dgn.parameters.p235.Task 3 |
answer recovered | `` | 0; 8 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Soil Science, 2.; answer block on page 236) | |
dgn.parameters.p235.Task 4 |
answer recovered | `` | x \in (\log_{2}(3/2); 1] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Soil Science, 1.; answer block on page 235) | |
dgn.parameters.p235.Task 5 |
answer recovered | `` | 0, -\pi/6, -\pi/3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Soil Science, 2.; answer block on page 236) | |
dgn.parameters.p235.Task 6 |
answer recovered | `` | \sqrt{14}, \sqrt{7} \pm \sqrt{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Soil Science, 2.; answer block on page 236) | |
dgn.parameters.p235.Task 7 |
answer recovered | `` | (-2; -8) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Soil Science, 1.; answer block on page 235) | |
dgn.parameters.p236.1 |
answer recovered | `` | [-\sqrt{5}; 2) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geology Faculty, 1.; answer block on page 237) | |
dgn.parameters.p236.2 |
answer recovered | `` | \{-1; \frac{1}{2}\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geology Faculty, 1.; answer block on page 237) | |
dgn.parameters.p236.3 |
answer recovered | `` | \{-21\} \cup [0; 21] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geology Faculty, 1.; answer block on page 237) | |
dgn.parameters.p236.4 |
answer recovered | `` | \operatorname{arctg} 3; 10 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geology Faculty, 1.; answer block on page 237) | |
dgn.parameters.p236.5 |
answer recovered | `` | [1; 2) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geology Faculty, 1.; answer block on page 237) | |
dgn.parameters.p236.6 |
answer recovered | `` | \frac{1}{9}; m = 3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geology Faculty, 1.; answer block on page 237) | |
dgn.parameters.p237.1 |
answer recovered | `` | [-\sqrt{6}; \sqrt{6}] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geology Faculty, 2.; answer block on page 238) | |
dgn.parameters.p237.2 |
answer recovered | `` | \{0; -\sqrt{3}/2\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geology Faculty, 2.; answer block on page 238) | |
dgn.parameters.p237.3 |
answer recovered | `` | \{-17\} \cup [0; 17] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geology Faculty, 2.; answer block on page 238) | |
dgn.parameters.p237.4 |
answer recovered | `` | \operatorname{arctg} 4; 17 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geology Faculty, 2.; answer block on page 238) | |
dgn.parameters.p237.5 |
answer recovered | `` | [3; 4) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geology Faculty, 2.; answer block on page 238) | |
dgn.parameters.p237.7 |
wrong answer | page 237, item 7 of the 'Ответы' block for ВАРИАНТ 2004 (июль), геологический факультет, 1.: { (π/4 + 2πm; 3π/4 + 2πk); (−π/4 + 2πm; π/4 + 2πk); k, m ∈ ℤ } (printed problem, page 237: 'Задача 7. Решите систему уравнений sin x/cos(x+y) = −1/√2, cos x/cos(x+y) = 1/√2.') |
\{ (-\pi/4 + 2\pi m;\ \pi/4 + 2\pi k);\ (3\pi/4 + 2\pi m;\ \pi/4 + 2\pi k);\ k, m \in \mathbb{Z} \} |
Adding and subtracting the two equations gives sin x + cos x = 0 (so tan x = −1) and cos x − sin x = √2·cos(x+y), hence cos(x+y) = ±1 and x+y ∈ πℤ; the parity of the branch fixes y ≡ π/4 (mod 2π), while x ≡ −π/4 or 3π/4 (mod 2π). The source's FIRST family (π/4 + 2πm; 3π/4 + 2πk) has the two coordinates swapped: substituting x = π/4, y = 3π/4 gives cos(x+y) = −1, so eq1 = 0 (satisfied) but eq2 = −√2 (violated); the corrected first family is (3π/4 + 2πm; π/4 + 2πk), which satisfies both equations with residual (0, 0). The source's SECOND family (−π/4 + 2πm; π/4 + 2πk) is valid, so the engine's claim that 'no assignment of the source tuple satisfies the system' is false — at x = −π/4, y = π/4 the residuals are (0, 1.1e−16). Verdict is still source_error because the printed key contains one invalid family. | |
dgn.parameters.p238.1 |
answer recovered | `` | (-\infty; -5) \cup (-\frac{9}{7}; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Philology Faculty, 1.; answer block on page 239) | |
dgn.parameters.p238.2 |
answer recovered | `` | (-1)^{n}\frac{\pi}{18} + \frac{\pi n}{3}, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Philology Faculty, 1.; answer block on page 239) | |
dgn.parameters.p238.3 |
answer recovered | `` | -4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Philology Faculty, 1.; answer block on page 239) | |
dgn.parameters.p238.6 |
answer recovered | `` | \frac{2}{9}; m = \frac{4}{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geology Faculty, 2.; answer block on page 238) | |
dgn.parameters.p238.7 |
wrong answer | page 238, item 7 of the 'Ответы' block: 7. { (5π/6 + 2πm; 2π/3 + 2πk); (−5π/6 + 2πm; π/3 + 2πk); k, m ∈ Z } (printed problem, page 238: 'Задача 7. Решите систему уравнений sin x / cos(x−y) = 1/√3, cos x / cos(x−y) = −1/√3.') |
\varnothing \text{ (the system has no solutions)} |
With c = cos(x−y) the two printed equations give sin x = c/√3 and cos x = −c/√3, so sin²x + cos²x = 2c²/3 = 1 forces c² = 3/2 > 1 — impossible for a cosine, hence the printed system has no solutions at all. The printed family indeed fails the second equation: for (5π/6, 2π/3) it gives cos x/cos(x−y) = −1 and for (−5π/6, π/3) it gives +1, never −1/√3 (that family would be correct if the second numerator were 'cos y'). The engine's complaint is legitimate. | |
dgn.parameters.p238.8 |
answer recovered | `` | 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Geology Faculty, 2.; answer block on page 238) | |
dgn.parameters.p239.Task 1 |
answer recovered | `` | \left(-\frac{1}{2}; \frac{11}{6}\right) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Philology Faculty, 2.; answer block on page 240) | |
dgn.parameters.p239.Task 2 |
answer recovered | `` | \pm \frac{\pi}{12} + \frac{\pi n}{2}, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Philology Faculty, 2.; answer block on page 240) | |
dgn.parameters.p239.Task 4 |
answer recovered | `` | 1. 6. 2. 3\sqrt{5} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Philology Faculty, 1.; answer block on page 239) | |
dgn.parameters.p239.Task 5 |
answer recovered | `` | 58.5 \text{ min} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Philology Faculty, 1.; answer block on page 239) | |
dgn.parameters.p239.Task 6 |
answer recovered | `` | 1. (\pm 1; 3). 2. 0 < y < \frac{x^{2}}{12} \text{ for } x > 0 \text{ and } -\frac{x^{2}}{12} < y < 0 \text{ for } x < 0 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Philology Faculty, 1.; answer block on page 239) | |
dgn.parameters.p240.1 |
answer recovered | `` | \frac{1}{\sqrt{3}} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Economics Faculty (Economics Division), 1.; answer block on page 241) | |
dgn.parameters.p240.4 |
mispaired answer | 1. $2\sqrt{3}$. 2. $6$. |
1. 2\sqrt{3}. 2. 6 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Philology Faculty, 2.; answer block on page 240) | |
dgn.parameters.p240.5 |
mispaired answer | $1$ hour $57$ minutes. |
1 \text{ h } 57 \text{ min} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Philology Faculty, 2.; answer block on page 240) | |
dgn.parameters.p240.6 |
mispaired answer | 1. $(\pm 1; \pm 2)$. 2. $0 < y < \frac{x^2}{8}$ for $x > 0$ and $-\frac{x^2}{8} < y < 0$ for $x < 0$. |
1. (\pm 1; \pm 2). 2. 0 < y < \frac{x^{2}}{8} \text{ for } x > 0 \text{ and } -\frac{x^{2}}{8} < y < 0 \text{ for } x < 0 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Philology Faculty, 2.; answer block on page 240) | |
dgn.parameters.p241.2 |
answer recovered | `` | \left\{\frac{5}{4}; \frac{3}{2}\right\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Economics Faculty (Economics Division), 1.; answer block on page 241) | |
dgn.parameters.p241.3 |
answer recovered | `` | \frac{9}{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Economics Faculty (Economics Division), 1.; answer block on page 241) | |
dgn.parameters.p241.4 |
answer recovered | `` | \left(-\frac{5}{2}; -2\right) \cup (2; 3] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Economics Faculty (Economics Division), 1.; answer block on page 241) | |
dgn.parameters.p241.5 |
answer recovered | `` | 17100 \text{ rubles} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Economics Faculty (Economics Division), 1.; answer block on page 241) | |
dgn.parameters.p241.6 |
answer recovered | `` | -14 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Economics Faculty (Economics Division), 1.; answer block on page 241) | |
dgn.parameters.p241.7 |
answer recovered | `` | \frac{1}{6} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Economics Faculty (Economics Division), 1.; answer block on page 241) | |
dgn.parameters.p242.Variant 2004 |
answer recovered | `` | 1. 1; 2. \{1; 3\}; 3. \frac{12}{5}; 4. (-\infty; -3) \cup \left(0; \frac{1}{2}\right); 5. 132 \text{ u. e.}; 6. 7; 7. \frac{2}{15} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Economics Faculty (Economics Division), 2.; answer block on page 243) | |
dgn.parameters.Var2004 (July), Economics Faculty (Management Division), No. 1 |
mispaired answer | 1. $\sqrt{3}$. 2. $\left\{\frac{5}{4};\frac{3}{2}\right\}$. 3. $1$. 4. $\left(0;\frac{1}{2}\right)$. 5. $90$ thousand rubles. 6. $2$. |
\sqrt{3} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Economics Faculty (Management Division), 1.; answer block on page 243) | |
dgn.parameters.Variant 2004 (iyul), economics Faculty (Evening otdelenif), 1 |
answer recovered | `` | \sqrt{\frac{7}{3}} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Economics Faculty (Evening Division), 1.; answer block on page 245) | |
dgn.parameters.Variant 2004 (iyul), economics Faculty (Department menedzhmenta), 2 |
mispaired answer | 1. $\frac{4\sqrt{6}}{3}$; 2. $\left\{\frac{3}{2};\frac{5}{2}\right\}$; 3. $\frac{3}{2}$; 4. $\left(-\frac{7}{2}; -3\right)$; 5. $66$ thousand c.u.; 6. $-3$ |
1. \frac{4\sqrt{6}}{3}; 2. \left\{\frac{3}{2}; \frac{5}{2}\right\}; 3. \frac{3}{2}; 4. \left(-\frac{7}{2}; -3\right); 5. 66 \text{ thousand currency units}; 6. -3 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Economics Faculty (Management Division), 2.; answer block on page 244) | |
dgn.parameters.p246.1 |
answer recovered | `` | -\sqrt{\frac{2}{3}} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Economics Faculty (Evening Division), 2.; answer block on page 247) | |
dgn.parameters.p246.2 |
answer recovered | `` | \left\{\frac{7}{3}; \frac{8}{3}\right\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Economics Faculty (Evening Division), 2.; answer block on page 247) | |
dgn.parameters.p246.3 |
answer recovered | `` | \frac{10}{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Economics Faculty (Evening Division), 2.; answer block on page 247) | |
dgn.parameters.p246.4 |
answer recovered | `` | \left(-\frac{5}{2}; -2\right) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Economics Faculty (Evening Division), 2.; answer block on page 247) | |
dgn.parameters.p246.5 |
answer recovered | `` | 85 \text{ thousand rubles} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Economics Faculty (Evening Division), 2.; answer block on page 247) | |
dgn.parameters.p246.6 |
answer recovered | `` | 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Economics Faculty (Evening Division), 2.; answer block on page 247) | |
dgn.parameters.p246.7 |
answer recovered | `` | \frac{1}{6} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Economics Faculty (Evening Division), 1.; answer block on page 245) | |
dgn.parameters.Problem 7 |
mispaired answer | $-\sqrt{\frac{2}{3}}$ |
\frac{\sqrt{2}}{12} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Economics Faculty (Evening Division), 2.; answer block on page 247) | |
dgn.parameters.Variant 2004 (iyul), psychology Faculty, 1 |
answer recovered | `` | 1. x = \log_{2}(6 + \sqrt{37}); 2. x \in [-1; 2) \cup (5; 6] \cup (8; +\infty); 3. \frac{\pi}{2} + 2\pi m, m \in \mathbb{Z}; x = -\frac{\pi}{3} + (-1)^{n+1}\frac{\pi}{6} + \pi n, n \in \mathbb{Z}; \text{the sum of roots belonging to the segment } [-\pi; \pi] \text{ equals } \frac{5\pi}{6}; 4. \angle ABD = \arcsin\left(\frac{5}{6}\right), AC = \frac{225}{16\sqrt{11}}; 5. a = 0; a = 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Psychology Faculty, 1.; answer block on page 247) | |
dgn.parameters.Variant 2004 (iyul), psychology Faculty, 2. Task 1 |
answer recovered | `` | x = \log_{5}(5 + \sqrt{28}) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Psychology Faculty, 2.; answer block on page 248) | |
dgn.parameters.Variant 2004 (iyul), psychology Faculty, 2. Task 2 |
answer recovered | `` | x \in [-5; 1) \cup (4; 5] \cup (7; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Psychology Faculty, 2.; answer block on page 248) | |
dgn.parameters.Variant 2004 (iyul), psychology Faculty, 2. Task 3 |
answer recovered | `` | \pi/2 + \pi m, m \in \mathbb{Z}; x = \pi/6 + 2\pi n, n \in \mathbb{Z}; \text{the sum of roots belonging to the segment } [-\pi; \pi] \text{ equals } \pi/6 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Psychology Faculty, 2.; answer block on page 248) | |
dgn.parameters.Variant 2004 (iyul), psychology Faculty, 2. Task 4 |
answer recovered | `` | \angle KMN = \arcsin(7/8), KL = 784/(33\sqrt{15}) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Psychology Faculty, 2.; answer block on page 248) | |
dgn.parameters.Variant 2004 (iyul), psychology Faculty, 2. Task 5 |
mispaired answer | 1. $x = \log_{5}(5 + \sqrt{28})$. 2. $x \in [-5; 1) \cup (4; 5] \cup (7; +\infty)$. 3. $\pi/2 + \pi m$, $m \in \mathbb{Z}$; $x = \pi/6 + 2\pi n$, $n \in \mathbb{Z}$; the sum of the roots belonging to $[-\pi; \pi]$ equals $\pi/6$. 4. $\angle KMN = \arcsin(7/8)$, $KL = 784/(33\sqrt{15})$. 5. $a = 0$; $a = -4$. |
a = 0; a = -4 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Psychology Faculty, 2.; answer block on page 248) | |
dgn.parameters.Variant 2004 (iyul), psychology Faculty, 3. Task 1 |
wrong answer | page 249, item 1 of the 'Ответы' block for ВАРИАНТ 2004 (июль), психологический факультет, 3.: x = log₂(6 + √37) (printed problem, page 248: 'Задача 1. Решите уравнение 16^x − 8 · 4^x − 2 = 0.') |
x = \log_{4}(4 + 3\sqrt{2}) |
With t = 4^x > 0 the equation becomes t^2 - 8t - 2 = 0, whose only positive root is t = 4 + 3√2, hence x = log_4(4 + 3√2) = 1.52155 (residual 16^x - 8·4^x - 2 = 0 exactly). The printed value x = log_2(6 + √37) = 3.59488 leaves a residual of about +20144 in the printed equation, so it is not a root; it would be the root of 4^x - 12·2^x - 1 = 0. The engine's complaint is legitimate. | |
dgn.parameters.Variant 2004 (iyul), psychology Faculty, 3. Task 2 |
wrong answer | Item 2 of the «Ответы» block on p. 249: «2. x ∈ [6; 8).» |
[-5;\, -1) \cup (3;\, 4] \cup (7;\, +\infty) |
For log_{(x+1)/(2x−6)}((x+6)/4) ⩽ 1 the domain splits into (−6;−1) (base<1), (3;7) (base>1) and (7;+∞) (base<1), and the comparison reduces to (x+5)(x−4) ≤ 0 or ≥ 0 respectively, giving [−5;−1) ∪ (3;4] ∪ (7;+∞) — not [6;8). The engine's check is right (at x=6: log_{7/6}(3) − 1 = 6.1269 > 0); the printed key is stale — the very same key block is printed verbatim for the psychology-faculty-4 variant at the bottom of p. 249, while the faculty-2 key on p. 248 does match its own problem. | |
dgn.parameters.Variant 2004 (iyul), psychology Faculty, 3. Task 3 |
answer recovered | `` | \pi/2 + 2\pi m, m \in \mathbb{Z}; x = -\pi/3 + (-1)^{n+1}\pi/6 + \pi n, n \in \mathbb{Z}; \text{the sum of roots belonging to the segment } [-\pi; \pi] \text{ equals } 5\pi/6 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Psychology Faculty, 3.; answer block on page 249) | |
dgn.parameters.2004 (July) |
mispaired answer | 4. $\angle MKN = \arcsin(6/7)$, $ML = 1764/(95\sqrt{13})$ |
1. x = \log_{2}(6 + \sqrt{37}); 2. x \in [6; 8); 3. \pi/2 + 2\pi m, m \in \mathbb{Z}; x = -\pi/3 + (-1)^{n+1}\pi/6 + \pi n, n \in \mathbb{Z}; \text{the sum of the roots belonging to the segment } [-\pi; \pi] \text{ equals } 5\pi/6; 4. \angle MKN = \arcsin(6/7), ML = 1764/(95\sqrt{13}); 5. a = 0; a = 1 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Psychology Faculty, 4.; answer block on page 249) | |
dgn.parameters.p249.4 |
mispaired answer | 4. $\angle CBD = \arcsin(5/6)$, $CA = 225/(16\sqrt{11})$ |
4. \angle CBD = \arcsin(5/6), CA = 225/(16\sqrt{11}); 5. a = 0; a = 1 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Psychology Faculty, 3.; answer block on page 249) | |
dgn.parameters.Variant 2004 (iyul), ISAA, 1. Task 2 |
mispaired answer | $\left\{\pi n; -\dfrac{\pi}{2} + \pi k;\ n, k \in \mathbb{Z}\right\}$ |
\left\{\pi + 2\pi n; \frac{\pi}{2} + \pi k; n, k \in \mathbb{Z}\right\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), ISAA, 2. √; answer block on page 251) | |
dgn.parameters.Variant 2004 (iyul), ISAA, 1. Task 3 |
mispaired answer | $\{(0;0;0); (2;2;2); (2;-2;-2); (-2;2;-2); (-2;-2;2)\}$ |
\{(0; 0; 0); (1; 1; 1); (1; -1; -1); (-1; 1; -1); (-1; -1; 1)\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), ISAA, 2. √; answer block on page 251) | |
dgn.parameters.Variant 2004 (iyul), ISAA, 1. Task 4 |
mispaired answer | $\log_2\left(8 \pm \sqrt{63}\right)$ |
\log_{2}(8 \pm \sqrt{63}) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), ISAA, 1. √; answer block on page 250) | |
dgn.parameters.Variant 2004 (iyul), ISAA, 2. Task 1 |
answer recovered | `` | (-\infty; -7) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), ISAA, 2. √; answer block on page 251) | |
dgn.parameters.Variant 2004 (iyul), ISAA, 2. Task 2 |
answer recovered | `` | \left\{\pi + 2\pi n; \frac{\pi}{2} + \pi k; n, k \in \mathbb{Z}\right\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), ISAA, 2. √; answer block on page 251) | |
dgn.parameters.Variant 2004 (iyul), ISAA, 2. Task 3 |
answer recovered | `` | \{(0; 0; 0); (1; 1; 1); (1; -1; -1); (-1; 1; -1); (-1; -1; 1)\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), ISAA, 2. √; answer block on page 251) | |
dgn.parameters.Task 1 (Variant 2004, April, sociology fakultet) |
answer recovered | `` | 3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (April), Sociology Faculty, 1.; answer block on page 252) | |
dgn.parameters.Task 2 (Variant 2004, April, sociology fakultet) |
answer recovered | `` | 1. 2. 2. (-\infty; 0) \cup \left(\frac{14}{3}; +\infty\right) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (April), Sociology Faculty, 1.; answer block on page 252) | |
dgn.parameters.Task 3 (Variant 2004, April, sociology fakultet) |
answer recovered | `` | 14\% |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (April), Sociology Faculty, 1.; answer block on page 252) | |
dgn.parameters.p251.Task 4 |
mispaired answer | $$\log_3\frac{27\pm5\sqrt{29}}{2}.$$ |
\left\{\log_{3}\frac{27 \pm 5\sqrt{29}}{2}\right\} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), ISAA, 2. √; answer block on page 251) | |
dgn.parameters.p251.Task 5 |
mispaired answer | $$\frac{9\pi}{14\pi-6\sqrt{3}}.$$ |
\frac{9\pi}{14\pi - 6\sqrt{3}} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), ISAA, 2. √; answer block on page 251) | |
dgn.parameters.p251.Task 6 |
mispaired answer | $$\frac{3641}{3136}\pm\frac{\sqrt{505}}{28}.$$ |
\frac{3641}{3136} \pm \frac{\sqrt{505}}{28} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), ISAA, 2. √; answer block on page 251) | |
dgn.parameters.p252.Task 4 |
answer recovered | `` | \arccos \frac{23}{4\sqrt{37}} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (April), Sociology Faculty, 1.; answer block on page 252) | |
dgn.parameters.p252.Task 5 |
answer recovered | `` | -\frac{\pi}{3}, \frac{\pi^{2} - \pi}{3}, \frac{4\pi^{2} - \pi}{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (April), Sociology Faculty, 1.; answer block on page 252) | |
dgn.parameters.1 (Variant 2004, July, Sociology Faculty) |
answer recovered | `` | [-5; -3] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Sociology Faculty, 1.; answer block on page 254) | |
dgn.parameters.2 (Variant 2004, July, Sociology Faculty) |
answer recovered | `` | \pm \frac{\pi}{9} + \frac{\pi k}{3}, k \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Sociology Faculty, 1.; answer block on page 254) | |
dgn.parameters.2.2 |
answer recovered | `` | \pm \frac{\pi}{9} + \frac{\pi k}{3}, k \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Sociology Faculty, 1.; answer block on page 254) | |
dgn.parameters.3 (Variant 2004, July, Sociology Faculty) |
answer recovered | `` | \text{increased by } \frac{500}{9}\% |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Sociology Faculty, 1.; answer block on page 254) | |
dgn.parameters.4 (Variant 2004, July, Sociology Faculty) |
answer recovered | `` | \frac{6\sqrt{3}}{\sqrt{19}} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Sociology Faculty, 1.; answer block on page 254) | |
dgn.parameters.p253.5 |
mispaired answer | $\dfrac{\pi}{4}$, $\dfrac{\pi^2+\pi}{4}$, $\dfrac{4\pi^2+\pi}{4}$. |
\frac{\pi}{4}, \frac{\pi^{2} + \pi}{4}, \frac{4\pi^{2} + \pi}{4} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (April), Sociology Faculty, 2.; answer block on page 253) | |
dgn.parameters.5 (Variant 2004, July, Sociology Faculty) |
answer recovered | `` | \arccos \frac{7}{54} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (April), Sociology Faculty, 2.; answer block on page 253) | |
dgn.parameters.p254.6 |
mispaired answer | $a \in (0; 2c)$. |
a \in (0; 2c) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Sociology Faculty, 1.; answer block on page 254) | |
dgn.parameters.Variant 2004 (iyul), sociology Faculty, 1 |
mispaired answer | $[-5; -3]$. |
a \in (0; 2c) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Sociology Faculty, 1.; answer block on page 254) | |
dgn.parameters.Variant 2004 (iyul), sociology Faculty, 2 |
mispaired answer | $\pm \frac{\pi}{9} + \frac{\pi k}{3}, k \in \mathbb{Z}$. |
\pm \frac{\pi}{18} + \frac{\pi k}{3}, k \in \mathbb{Z} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Sociology Faculty, 2.; answer block on page 255) | |
dgn.parameters.Variant 2004 (iyul), sociology Faculty, 3 |
mispaired answer | grew by $\frac{500}{9}\%$. |
\text{Increased by } \frac{700}{11}\% |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Sociology Faculty, 2.; answer block on page 255) | |
dgn.parameters.Variant 2004 (iyul), sociology Faculty, 4 |
mispaired answer | $\frac{6\sqrt{3}}{\sqrt{19}}$. |
\frac{3}{\sqrt{17}} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Sociology Faculty, 2.; answer block on page 255) | |
dgn.parameters.Variant 2004 (iyul), sociology Faculty, 5 |
mispaired answer | $2\sqrt{3}, 2\sqrt{3}, 2\sqrt{3}$. |
2\sqrt{3}, 2\sqrt{3}, 2\sqrt{3} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Sociology Faculty, 1.; answer block on page 254) | |
dgn.parameters.V 2004 (iyul), Dept. State. Management., 2 |
answer recovered | `` | 1. 14; 2. \left\{\frac{\pi k}{3}; k \in \mathbb{Z}\right\}; 3. \sqrt{7}; 4. 4920 \text{ dollars}; 5. \left(-3; -\frac{5}{2}\right) \cup \left[\frac{\sqrt{73} - 1}{4}; +\infty\right) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Public Administration, 2.; answer block on page 257) | |
dgn.parameters.p257.5 |
mispaired answer | 8 hours 15 min |
8 \text{ h } 15 \text{ min} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Higher School of Business, 1.; answer block on page 257) | |
dgn.parameters.p257.7 |
mispaired answer | 10 |
2 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Faculty of Public Administration, 2.; answer block on page 257) | |
dgn.parameters.p258.258 |
mispaired answer | 1. $(-\infty;-2) \cup (-2;0]$; 2. $\{\pi/3+2\pi n; n\in\mathbb{Z}\}$; 3. $\{(28;6); (-7;-29)\}$; 4. $32$; 5. $7$ ч $30$ мин; 6. $\{(10;9)\}$; 7. $-10$; 8. $a\in[-6;-5)\cup(-2;6]$ |
1. (-\infty; -2) \cup (-2; 0]; 2. \left\{\pi/3 + 2\pi n; n \in \mathbb{Z}\right\}; 3. \{(28; 6); (-7; -29)\}; 4. 32; 5. 7 \text{ h } 30 \text{ min}; 6. \{(10; 9)\}; 7. -10; 8. a \in [-6; -5) \cup (-2; 6] |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2004 (July), Higher School of Business, 2.; answer block on page 258) | |
dgn.parameters.p259.3 |
answer recovered | `` | S_{ABCD} = 25 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (May), "Lomonosov-2005"1 , 1.; answer block on page 260) | |
dgn.parameters.p259.8 |
answer recovered | `` | a \in [-8; 6] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (May), "Lomonosov-2005"1 , 1.; answer block on page 260) | |
dgn.parameters.p259.9 |
answer recovered | `` | From A to B; 8 \text{ km/h} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (May), "Lomonosov-2005"1 , 1.; answer block on page 260) | |
dgn.parameters.p260.1 |
answer recovered | `` | -27 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (May), "Lomonosov-2005", 2.; answer block on page 261) | |
dgn.parameters.p260.10 |
answer recovered | `` | 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (May), "Lomonosov-2005"1 , 1.; answer block on page 260) | |
dgn.parameters.p260.2 |
answer recovered | `` | x \in (0; \log_{5} 3] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (May), "Lomonosov-2005", 2.; answer block on page 261) | |
dgn.parameters.p260.3 |
answer recovered | `` | S_{ABCD} = 18 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (May), "Lomonosov-2005", 2.; answer block on page 261) | |
dgn.parameters.p260.4 |
answer recovered | `` | -\frac{\pi}{6} + \pi n, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (May), "Lomonosov-2005", 2.; answer block on page 261) | |
dgn.parameters.p260.5 |
answer recovered | `` | BE = 10 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (May), "Lomonosov-2005", 2.; answer block on page 261) | |
dgn.parameters.p260.6 |
answer recovered | `` | x \in \left[-3; \frac{\sqrt{192} - 19}{13}\right] \cup \{0\} \cup \left[\frac{11}{13}; 1\right] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (May), "Lomonosov-2005", 2.; answer block on page 261) | |
dgn.parameters.p261.1 |
answer recovered | `` | -125 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (May), "Lomonosov-2005", 3.; answer block on page 262) | |
dgn.parameters.p261.10 |
mispaired answer | $4$. |
4 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (May), "Lomonosov-2005", 2.; answer block on page 261) | |
dgn.parameters.p261.2 |
answer recovered | `` | x \in (0; \log_{7} 3] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (May), "Lomonosov-2005", 3.; answer block on page 262) | |
dgn.parameters.p261.7 |
mispaired answer | $V = 108\sqrt{3}$. |
V = 108\sqrt{3} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (May), "Lomonosov-2005", 2.; answer block on page 261) | |
dgn.parameters.p261.8 |
mispaired answer | $a \in (-7;5)$. |
a \in (-7; 5) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (May), "Lomonosov-2005", 2.; answer block on page 261) | |
dgn.parameters.p261.9 |
mispaired answer | From $A$ to $B$; $5$ km/h. |
From A to B; 5 \text{ km/h} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (May), "Lomonosov-2005", 2.; answer block on page 261) | |
dgn.parameters.p262.9 |
mispaired answer | From $A$ to $B$; $4.5$ km/h |
From A to B; 4.5 \text{ km/h} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (May), "Lomonosov-2005", 3.; answer block on page 262) | |
dgn.parameters.Variant 2005 |
mispaired answer | 1. 32. 2. 2. 3. $x \in [1;3) \cup (3;\sqrt{10})$. 4. $18$, $(1/3;1) \cup (1;5/3)$. 5. $0$, $-4 - 2/\sqrt{3}$. 6. $2\sqrt{6}$. |
1. 32; 2. 2; 3. x \in [1; 3) \cup (3; \sqrt{10}); 4. 18, (1/3; 1) \cup (1; 5/3); 5. 0, -4 - 2/\sqrt{3}; 6. 2\sqrt{6} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Mechanics and Mathematics Faculty, 2.; answer block on page 264) | |
dgn.parameters.p265.1 |
answer recovered | `` | Yes |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005, Mechanics and Mathematics Faculty, oral exam problems.; answer block on page 265) | |
dgn.parameters.p265.2 |
answer recovered | `` | cos 3 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005, Mechanics and Mathematics Faculty, oral exam problems.; answer block on page 265) | |
dgn.parameters.p265.3 |
answer recovered | `` | 4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005, Mechanics and Mathematics Faculty, oral exam problems.; answer block on page 265) | |
dgn.parameters.p265.4 |
answer recovered | `` | x = 2, y = 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005, Mechanics and Mathematics Faculty, oral exam problems.; answer block on page 265) | |
dgn.parameters.p265.5 |
mispaired answer | 1. Yes. 2. \cos 3. 3. 4. 4. x = 2, y = 1. |
x = 2, y = 1 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005, Mechanics and Mathematics Faculty, oral exam problems.; answer block on page 265) | |
dgn.parameters.p265.6 |
answer recovered | `` | (0; (5 - \sqrt{17})/2) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Computational Mathematics and Cybernetics, 1.; answer block on page 266) | |
dgn.parameters.p265.7 |
answer recovered | `` | \pi + \operatorname{arcctg} 2 + 2\pi n, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Computational Mathematics and Cybernetics, 1.; answer block on page 266) | |
dgn.parameters.p265.8 |
answer recovered | `` | S_{100} = 15050, b_{40} = 81 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Computational Mathematics and Cybernetics, 1.; answer block on page 266) | |
dgn.parameters.p266.Task 1 |
mispaired answer | $(0;\, (5 - \sqrt{17})/2)$ |
(0; 5 - 2\sqrt{5}) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Computational Mathematics and Cybernetics, 2.; answer block on page 267) | |
dgn.parameters.p266.Task 2 |
mispaired answer | $\pi + \operatorname{arcctg} 2 + 2\pi n,\ n \in \mathbb{Z}$ |
\operatorname{arctg} 2 + 2\pi n, n \in \mathbb{Z} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Computational Mathematics and Cybernetics, 2.; answer block on page 267) | |
dgn.parameters.p266.Task 3 |
mispaired answer | $S_{100} = 15050,\ b_{40} = 81$ |
S_{90} = 16470, a_{35} = 71 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Computational Mathematics and Cybernetics, 2.; answer block on page 267) | |
dgn.parameters.p266.Task 4 |
mispaired answer | $4\sqrt{10} - 2\sqrt{15}$ |
R = 2\sqrt{15} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Computational Mathematics and Cybernetics, 2.; answer block on page 267) | |
dgn.parameters.p267.1 |
answer recovered | `` | -150 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 1.; answer block on page 268) | |
dgn.parameters.p267.2 |
wrong answer | page 268, item 2 of the 'Ответы' block for ВАРИАНТ 2005 (июль), факультет вычислительной математики и кибернетики (отделение бакалавров), 1.: (0; 1) (printed problem, page 267: 'Задача 2. Решите неравенство log₃((x² + |2x−5| + 5)/(x+2)) − |log₃x − 1| > log₃x + 1.') |
\varnothing |
Domain x > 0. Put A(x) = (x² + |2x−5| + 5)/(x+2); for 0 < x ⩽ 2.5 this is (x²−2x+10)/(x+2) ∈ (3, 5), and for x > 2.5 it is exactly x. If |log₃x − 1| = 1 − log₃x (x < 3), the inequality collapses to log₃ A > 2, i.e. A > 9 — impossible since A < 5. If x > 3, |log₃x − 1| = log₃x − 1 and the inequality becomes A > x², which for x > 2.5 means x > x², impossible. Hence the solution set is EMPTY and the printed '(0; 1)' is wrong (it looks like the result of mishandling |log₃x − 1| on (0, 1)). A dense sweep over (0, 50) with step 1e−3 finds no point with F > 0. The engine's probe is legitimate: x = 0.05 gives F = −0.566421 (exactly the engine's value) < 0. | |
dgn.parameters.p267.3 |
answer recovered | `` | \operatorname{arcctg} 3 + 2\pi n, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 1.; answer block on page 268) | |
dgn.parameters.p267.4 |
answer recovered | `` | (x; y) = ((-13 - \sqrt{481})/10; (-13 + \sqrt{481})/10), ((-13 + \sqrt{481})/10; (-13 - \sqrt{481})/10) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 1.; answer block on page 268) | |
dgn.parameters.p267.5 |
answer recovered | `` | 4\sqrt{14} - 2\sqrt{7} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 1.; answer block on page 268) | |
dgn.parameters.p267.6 |
answer recovered | `` | 72 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Computational Mathematics and Cybernetics, 2.; answer block on page 267) | |
dgn.parameters.p268.Task 1 |
answer recovered | `` | 10 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 2.; answer block on page 269) | |
dgn.parameters.p268.Task 2 |
answer recovered | `` | (0; 5 - \sqrt{23}) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 2.; answer block on page 269) | |
dgn.parameters.p268.Task 3 |
answer recovered | `` | \pi + \operatorname{arctg}(1/3) + 2\pi n, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 2.; answer block on page 269) | |
dgn.parameters.p268.Task 4 |
answer recovered | `` | (x; y) = ((7 - \sqrt{301})/6; (7 + \sqrt{301})/6), ((7 + \sqrt{301})/6; (7 - \sqrt{301})/6) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 2.; answer block on page 269) | |
dgn.parameters.p268.Task 5 |
answer recovered | `` | R = 2\sqrt{7} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Computational Mathematics and Cybernetics (Bachelor's Division), 2.; answer block on page 269) | |
dgn.parameters.p268.Task 6 |
wrong answer | Item 6 of the «Ответы» block on p. 269: «6. a ∈ [0; 2π − 6] ∪ [2π − 4; π].» |
a \in [0;\, 2\pi - 6] \cup (2\pi - 4;\, \pi] |
On [1;π] the right-hand side is 1 for x∈[1;2)∪[3;π] and 0 for x∈[2;3); counting the roots of cos(2x+a)=1 (x = πn − a/2) and of cos(2x+a)=0 (x = π/4+πn/2 − a/2) per sub-interval gives exactly two solutions iff a∈[0;2π−6] ∪ (2π−4;π]. At a = 2π−4 the candidate x = 2 of the «cos = 1» branch falls into [x]=2, where the right-hand side is 0 (cos(4+a) = 1 ≠ 0), so the equation has only ONE solution (x = 2+π/4) and the printed closed endpoint 2π−4 is wrong. The engine's point x = −500.035 is outside [1;π] and the problem is an equation, so its complaint is void. | |
dgn.parameters.Variant 2005 (iyul), physics Faculty, 1 |
mispaired answer | **1.** $\pi/4 + \pi k/2, \; 2\pi m/5, \; 2\pi n/9, \; k, m, n \in \mathbb{Z}$. **2.** $[-1; 0)$. **3.** $(1/\sqrt{3})\log_5\sqrt[3]{3}, \; (1/\sqrt{3})\log_5\sqrt[3]{4\sqrt{3}}$. **4.** $(qs - t^2)/t$. **5.** $x = 5/3, \; y = -7/3$. **6.** $8$. **7.** If $a = 2$, there are no solutions; if $a \in (\sqrt{2}; \sqrt{3})$, then $x > 1/a$; if $a \in (\sqrt{3}; 2) \cup (2; +\infty)$, then $0 < x < 1/a$. **8.** $5\sqrt{6}/3$. |
1. \pi/4 + \pi k/2, 2\pi m/5, 2\pi n/9, k, m, n \in \mathbb{Z}; 2. [-1; 0); 3. (1/\sqrt{3}) \log_{5/3} \sqrt{3}, (1/\sqrt{3}) \log_{5/3} 4\sqrt{3}; 4. (qs - t^{2})/t; 5. x = 5/3, y = -7/3; 6. 8; 7. If a = 2, then there are no solutions; if a \in (\sqrt{2}; \sqrt{3}), then x > 1/a; if a \in (\sqrt{3}; 2) \cup (2; +\infty), then 0 < x < 1/a; 8. 5\sqrt{6}/3 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Physics Faculty, 1.; answer block on page 269) | |
dgn.parameters.Variant 2005 (iyul), physics Faculty, 2 |
answer recovered | `` | 1. \pi k/3, 2\pi m, \pi/7 + 2\pi n/7, k, m, n \in \mathbb{Z}; 2. [-2; 0); 3. (1/\sqrt{7}) \log_{3/7} \sqrt{7}, (1/\sqrt{7}) \log_{3/7} 2\sqrt{7}; 4. (d^{2} - ac)/d; 5. x = 32/5, y = 13/5; 6. 5; 7. If a = \sqrt{6}, then there are no solutions; if a \in (\sqrt{5}; \sqrt{6}), then x > a; if a \in (\sqrt{6}; 3) \cup (3; +\infty), then 0 < x < a; 8. 7\sqrt{102}/10 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Physics Faculty, 2.; answer block on page 270) | |
dgn.parameters.Variant 2005 (iyul), chemistry Faculty, 1 |
answer recovered | `` | 1. \pm 1; 2. [1/4; +\infty); 3. \pi/10 + \pi n/5, n \in \mathbb{Z}; 4. 10 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Chemistry Faculty, 1.; answer block on page 271) | |
dgn.parameters.p271.4 |
mispaired answer | 13 |
10 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Chemistry Faculty, 1.; answer block on page 271) | |
dgn.parameters.p272.1 |
answer recovered | `` | \pm 2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Biology Faculty, Faculty of Bioengineering and Bioinformatics, Faculty of Fundamental Medicine, Faculty of Materials Science, 2.; answer block on page 273) | |
dgn.parameters.p272.3 |
mispaired answer | $16\sqrt{5}$ |
6\sqrt{10}/7 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Biology Faculty, Faculty of Bioengineering and Bioinformatics, Faculty of Fundamental Medicine, Faculty of Materials Science, 2.; answer block on page 273) | |
dgn.parameters.p272.5 |
mispaired answer | 4 times |
4 times |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Biology Faculty, Faculty of Bioengineering and Bioinformatics, Faculty of Fundamental Medicine, Faculty of Materials Science, 1.; answer block on page 272) | |
dgn.parameters.p273.2 |
mispaired answer | $\sin(\sqrt{3}\pi/6)$ |
1/(2\sqrt{2}) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Biology Faculty, Faculty of Bioengineering and Bioinformatics, Faculty of Fundamental Medicine, Faculty of Materials Science, 2.; answer block on page 273) | |
dgn.parameters.p273.4 |
mispaired answer | 10 hours |
10 hours |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Soil Science, 1.; answer block on page 273) | |
dgn.parameters.p273.6 |
mispaired answer | $x = 3/2,\; y = 2$ |
1 : 3 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Soil Science, 1.; answer block on page 273) | |
dgn.parameters.continued |
answer recovered | `` | 160 m/min, 120 m/min |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Biology Faculty, Faculty of Bioengineering and Bioinformatics, Faculty of Fundamental Medicine, Faculty of Materials Science, 2.; answer block on page 273) | |
dgn.parameters.p274.1 |
mispaired answer | 2 |
\{-1\} \cup [1/2; 1] |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Geology Faculty, 1.; answer block on page 275) | |
dgn.parameters.p274.2 |
mispaired answer | $\pm \sin(\sqrt{2}\,\pi/4)$ |
[-3; (\sqrt{41} - 5)/4] |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Geology Faculty, 1.; answer block on page 275) | |
dgn.parameters.p274.3 |
mispaired answer | $y \geqslant -1$ |
S_{\triangle ABC} = 1, R = \frac{\sqrt{85}}{2} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Geology Faculty, 1.; answer block on page 275) | |
dgn.parameters.p274.4 |
mispaired answer | 18 hours |
18 hours |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Soil Science, 2.; answer block on page 274) | |
dgn.parameters.p274.5 |
mispaired answer | 9 |
S_6 = 0, 3 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Geology Faculty, 1.; answer block on page 275) | |
dgn.parameters.p274.7 |
answer recovered | `` | [-1; 5] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Geology Faculty, 1.; answer block on page 275) | |
dgn.parameters.Variant 2005 (iyul), geology Faculty, 2, Task 1 |
answer recovered | `` | \{-2\} \cup [1/2; 2] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Geology Faculty, 2.; answer block on page 275) | |
dgn.parameters.Variant 2005 (iyul), geology Faculty, 2, Task 2 |
answer recovered | `` | [-2; (\sqrt{41} - 1)/4] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Geology Faculty, 2.; answer block on page 275) | |
dgn.parameters.Variant 2005 (iyul), geology Faculty, 2, Task 3 |
answer recovered | `` | S_{\triangle ABC} = 1, R = \frac{\sqrt{85}}{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Geology Faculty, 2.; answer block on page 275) | |
dgn.parameters.Variant 2005 (iyul), geology Faculty, 2, Task 4 |
answer recovered | `` | 5\pi/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Geology Faculty, 2.; answer block on page 275) | |
dgn.parameters.Variant 2005 (iyul), geology Faculty, 2, Task 5 |
answer recovered | `` | S_8 = 0, 4 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Geology Faculty, 2.; answer block on page 275) | |
dgn.parameters.Variant 2005 (iyul), geology Faculty, 2, Task 6 |
answer recovered | `` | [3/4; 1) \cup [4; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Geology Faculty, 2.; answer block on page 275) | |
dgn.parameters.Variant 2005 (iyul), geology Faculty, 2, Task 7 |
answer recovered | `` | [-5; 3] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Geology Faculty, 2.; answer block on page 275) | |
dgn.parameters.Variant 2005 (iyul), geology Faculty, 2, Task 8 |
answer recovered | `` | 2\sqrt{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Geology Faculty, 2.; answer block on page 275) | |
dgn.parameters.p275.Task 8 |
answer recovered | `` | 12/5 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Geology Faculty, 1.; answer block on page 275) | |
dgn.parameters.Variant 2005 (iyul), geography Faculty, 1 |
mispaired answer | 1. $(x;y) = (-\pi/2 \pm \pi/3 + 2\pi k;\ 2\pi m),\ k,m \in \mathbb{Z}$. 2. 150. 3. $[4/5;\,1)$. 4. $(3\pi+4)\sqrt{2}/2$. 5. 26. 6. $\sqrt{7/2}\,\pi$. |
1. (x; y) = (-\pi/2 \pm \pi/3 + 2\pi k; 2\pi m), k, m \in \mathbb{Z}; 2. 150; 3. [4/5; 1); 4. (3\pi + 4)\sqrt{2}/2; 5. 26; 6. \sqrt{7}/2 \pi |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Geography Faculty, 1.; answer block on page 276) | |
dgn.parameters.Variant 2005 (iyul), geography Faculty, 2 |
answer recovered | `` | 1. (x; y) = (\pm \pi/3 + 2\pi k; -\pi/2 + 2\pi m), k, m \in \mathbb{Z}; 2. 34 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Geography Faculty, 2.; answer block on page 277) | |
dgn.parameters.p277.1 |
mispaired answer | $(x; y) = (\pm \pi/3 + 2\pi k; -\pi/2 + 2\pi m),\ k, m \in \mathbb{Z}$ |
0, \pm 1, \pm 2, \pm 3 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Philology Faculty, 1.; answer block on page 278) | |
dgn.parameters.p277.2 |
mispaired answer | $34$ |
240 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Philology Faculty, 1.; answer block on page 278) | |
dgn.parameters.2005 |
answer recovered | `` | 1. 0, \pm 1, \pm 4, \pm 5; 2. 250; 3. (-2; -17/9) \cup (-1; 7); 4. 3\pi; 5. \pi/2 + 2\pi n, n \in \mathbb{Z}; 6. l = 4, P = 4 + 4/\cos(2\pi/5) = 8 + 4\sqrt{5}; 7. \{1, 2, 3, 4, 5, 6, 7, 8\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Philology Faculty, 2.; answer block on page 278) | |
dgn.parameters.2.3 |
answer recovered | `` | 1. \frac{9\pi(4k+1)}{8}, k \in \mathbb{Z}; 2. 18; 3. 962500 rubles; 4. [1 - \sqrt{23}; -\frac{7}{2}) \cup (-7/2; -\sqrt{34/3}] \cup [\sqrt{34/3}; 4); 5. 2\sqrt{3} + 4, 2\sqrt{3} + 4, 4\sqrt{3} + 6; 6. (x; y) = (-\frac{2}{3}; 1), (-1 - 1/(l - 1); l^{2} + l - 1), (-1 + 1/(l + 2); l^{2} + l - 1), l \in \mathbb{Z}, l \neq -5, -2, 1, 4; 7. (0; 16\pi) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Economics Faculty (Economics Division), 1.; answer block on page 279) | |
dgn.parameters.p280.6 |
mispaired answer | $(x;y) = (-1;\,-2),\ (2l^2 + 2l - 1;\, -1 - 1/(l+1)),\ (2l^2 + 2l - 1;\, -1 - 1/l),\ l \in \mathbb{Z},\ l \neq -6,\,-1,\,0,\,5$ |
(x; y) = (-1; -2), (2l^{2} + 2l - 1; -1 - 1/(l + 1)), (2l^{2} + 2l - 1; -1 - 1/l), l \in \mathbb{Z}, l \neq -6, -1, 0, 5 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Economics Faculty (Economics Division), 2.; answer block on page 280) | |
dgn.parameters.p281.1 |
mispaired answer | $x=-2$ |
-2 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Economics Faculty (Management Division), 1.; answer block on page 281) | |
dgn.parameters.p281.3 |
mispaired answer | 10 million 2 thousand conventional units |
10 million 2 thousand USD |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Economics Faculty (Management Division), 1.; answer block on page 281) | |
dgn.parameters.p282.3 |
answer recovered | `` | 2 810 000 rubles |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Economics Faculty (Management Division), 2.; answer block on page 282) | |
dgn.parameters.p282.4 |
answer recovered | `` | [-\sqrt{17/3}; -1) \cup (2; \sqrt{17/3}] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Economics Faculty (Management Division), 2.; answer block on page 282) | |
dgn.parameters.p282.5 |
answer recovered | `` | 1/(1 + \sqrt{3}) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Economics Faculty (Management Division), 2.; answer block on page 282) | |
dgn.parameters.p282.6 |
answer recovered | `` | b = 1 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Economics Faculty (Management Division), 2.; answer block on page 282) | |
dgn.parameters.Variant 2005 (iyul), Faculty psychology, 1 |
answer recovered | `` | 1. \pm 3; 2. x \in (-\infty; 1] \cup (\log_{2} 11; +\infty); 3. -\pi/12 + 2\pi m, -11\pi/12 + 2\pi n, m, n \in \mathbb{Z}; 4. \pi - \arcsin(5/8), 4\sqrt{22}/13; 5. 5 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Psychology Faculty, 1.; answer block on page 283) | |
dgn.parameters.top-problem |
answer recovered | `` | 5 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Psychology Faculty, 1.; answer block on page 283) | |
dgn.parameters.Variant 2005 (iyul), Faculty psychology, 2 |
answer recovered | `` | 1. 4, -2; 2. x \in (-\infty; 1] \cup (\log_{5} 6; +\infty); 3. -\pi/8 + 2\pi m, -7\pi/8 + 2\pi n, m, n \in \mathbb{Z}; 4. \pi - \arcsin(2/3), 3\sqrt{41}/20; 5. 15/17 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Psychology Faculty, 2.; answer block on page 284) | |
dgn.parameters.p284.284 |
answer recovered | `` | 15/17 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Psychology Faculty, 2.; answer block on page 284) | |
dgn.parameters.p284.6 |
mispaired answer | $|a| \leqslant 5/2$, $a \neq 0$. |
|a| \leqslant 5/2, a \neq 0 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Psychology Faculty, 2.; answer block on page 284) | |
dgn.parameters.Variant 2005 (iyul), ISAA, 1 |
answer recovered | `` | 1. (-\infty; -1) \cup (-1; 5]; 2. \pi/4 + \pi m/2, \pm \arccos((1 - \sqrt{3})/2) + 2\pi n, n, m \in \mathbb{Z}; 3. (-3 + \sqrt{5})/2; 4. 5\%; 5. (-1/3; -1/4) \cup (-1/6; -1/10); 6. \pi/6, \pi/3, 7\pi/12, 11\pi/12 \text{ or } \pi/2, 2\pi/3, \alpha, 5\pi/6 - \alpha, \text{ where } \alpha = \arccos\left(\frac{\sqrt{3}}{2\sqrt{4 - \sqrt{3}}}\right); 7. 12\sqrt{2} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), ISAA, 1. 2; answer block on page 285) | |
dgn.parameters.p285.1 |
mispaired answer | $(-\infty; -4) \cup (-4; 3]$. |
(-\infty; -4) \cup (-4; 3] |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), ISAA, 2. 2; answer block on page 285) | |
dgn.parameters.p285.2 |
mispaired answer | $\pi m/2, \; (-1)^n \arcsin\bigl((-1+\sqrt{3})/2\bigr) + \pi n, \; n, m \in \mathbb{Z}$. |
\pi m/2, (-1)^{n} \arcsin((-1 + \sqrt{3})/2) + \pi n, n, m \in \mathbb{Z} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), ISAA, 2. 2; answer block on page 285) | |
dgn.parameters.p285.3 |
mispaired answer | $(-1+\sqrt{5})/2$. |
(-1 + \sqrt{5})/2 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), ISAA, 2. 2; answer block on page 285) | |
dgn.parameters.p285.4 |
mispaired answer | $2\%$. |
2\% |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), ISAA, 2. 2; answer block on page 285) | |
dgn.parameters.p285.5 |
mispaired answer | $(-1/2; -1/3) \cup (-1/5; 0) \cup (0; +\infty)$. |
(-1/2; -1/3) \cup (-1/5; 0) \cup (0; +\infty) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), ISAA, 2. 2; answer block on page 285) | |
dgn.parameters.p285.6 |
mispaired answer | $\pi/4, \pi/3, \pi/2, 11\pi/12$ or $\pi/2, 7\pi/12, \alpha, 11\pi/12 - \alpha$, where $\alpha = \arccos\left(\dfrac{\sqrt{3}-1}{2\sqrt{8+2\sqrt{3}}}\right)$. |
\pi/4, \pi/3, \pi/2, 11\pi/12 \text{ or } \pi/2, 7\pi/12, \alpha, 11\pi/12 - \alpha, \text{ where } \alpha = \arccos\left(\frac{\sqrt{3} - 1}{2\sqrt{8 + 2\sqrt{3}}}\right) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), ISAA, 2. 2; answer block on page 285) | |
dgn.parameters.p285.7 |
mispaired answer | $8\sqrt{2}$. |
8\sqrt{2} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), ISAA, 2. 2; answer block on page 285) | |
dgn.parameters.previous |
mispaired answer | 1. $(-\infty; -1) \cup (-1; 5]$. 2. $\pi/4 + \pi m/2, \pm \arccos\bigl((1-\sqrt{3})/2\bigr) + 2\pi n, \; n, m \in \mathbb{Z}$. 3. $(-3+\sqrt{5})/2$. 4. $5\%$. 5. $(-1/3; -1/4) \cup (-1/6; -1/10)$. 6. $\pi/6, \pi/3, 7\pi/12, 11\pi/12$ or $\pi/2, 2\pi/3, \alpha, 5\pi/6 - \alpha$, where $\alpha = \arccos\bigl(\sqrt{3}/(2\sqrt{4-\sqrt{3}})\bigr)$. 7. $12\sqrt{2}$. |
12\sqrt{2} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), ISAA, 1. 2; answer block on page 285) | |
dgn.parameters.variant_header |
answer recovered | `` | (-1 + \sqrt{5})/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), ISAA, 2. 2; answer block on page 285) | |
dgn.parameters.p286.1 |
mispaired answer | $0, 4$ |
0, 16 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Sociology Faculty, 2. √; answer block on page 287) | |
dgn.parameters.p286.2 |
mispaired answer | $\pm 1/2$ |
\pm 1/3 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Sociology Faculty, 2. √; answer block on page 287) | |
dgn.parameters.p286.4 |
mispaired answer | $11/3$ |
11/12 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Sociology Faculty, 2. √; answer block on page 287) | |
dgn.parameters.p286.6 |
mispaired answer | $20, 18000$ rubles |
20, 18000 rubles |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Sociology Faculty, 1. √; answer block on page 286) | |
dgn.parameters.VARIANT 2005 |
answer recovered | `` | 1. It is possible. From 8 rubles 50 kopeks to 16 rubles 00 kopeks inclusive.; 2. (5; +\infty); 3. E(0; 0); 4. 1 + 2\pi n, n \in \mathbb{Z}; 5. 69 rubles 43 kopeks.; 6. a \geqslant 0; 7. It will be found. |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Public Administration, 1.; answer block on page 288) | |
dgn.parameters.p288.1 |
answer recovered | `` | Yes. From 2,5 g to 14 g inclusive. |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Public Administration, 2.; answer block on page 289) | |
dgn.parameters.p288.2 |
answer recovered | `` | (11/2; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Public Administration, 2.; answer block on page 289) | |
dgn.parameters.p288.3 |
answer recovered | `` | T(0; 0) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Public Administration, 2.; answer block on page 289) | |
dgn.parameters.p288.4 |
answer recovered | `` | 1 + \pi + 2\pi n, n \in \mathbb{Z} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Public Administration, 2.; answer block on page 289) | |
dgn.parameters.p288.5 |
answer recovered | `` | 28 rubles 64 kopeks. |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Public Administration, 2.; answer block on page 289) | |
dgn.parameters.p288.6 |
answer recovered | `` | a \geqslant 0 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Public Administration, 2.; answer block on page 289) | |
dgn.parameters.p288.7 |
answer recovered | `` | There will be one. |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Public Administration, 1.; answer block on page 288) | |
dgn.parameters.p289.7 |
mispaired answer | Such a point exists. |
There will be one. |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Public Administration, 2.; answer block on page 289) | |
dgn.parameters.p289.8 |
mispaired answer | $a\in(-\infty;1]\cup\{5/4\}\cup[4/3;+\infty)$ |
a \in (-\infty; 1] \cup \{5/4\} \cup [4/3; +\infty) |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Moscow School of Economics, 1.; answer block on page 289) | |
dgn.parameters.p290.1 |
mispaired answer | [-3/2; +\infty) |
126 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Global Processes, 1.; answer block on page 291) | |
dgn.parameters.p290.2 |
wrong answer | (-3.2; -\sqrt{3.2}) \cup [-1; 0) \cup (0; \sqrt{3/2}) |
(-3/2; -\sqrt{3/2}) \cup [-1; 0) \cup (0; \sqrt{3/2}) |
the printed answer writes the decimal 3.2 where the same answer block writes the fraction 3/2; the page was read at 600 dpi and the digits stand on the baseline, so the first two components are a misprint - the solution set of the printed equation is (-3/2; -sqrt(3/2)) u [-1; 0) u (0; sqrt(3/2)) | |
dgn.parameters.Task 1 (Variant 2005 July Faculty Global protsessoc) |
answer recovered | `` | 62 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Global Processes, 2.; answer block on page 292) | |
dgn.parameters.Task 2 (Variant 2005 July Faculty Global protsessoc) |
answer recovered | `` | (-1/\sqrt{2}; 0) \cup (0; 1/2) \cup (1/\sqrt{2}; 1] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Global Processes, 2.; answer block on page 292) | |
dgn.parameters.p291.Task 3 |
answer recovered | `` | 1/2 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Global Processes, 1.; answer block on page 291) | |
dgn.parameters.p291.Task 4 |
answer recovered | `` | 4\sqrt{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Global Processes, 1.; answer block on page 291) | |
dgn.parameters.p291.Task 5 |
answer recovered | `` | The statement is true. |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Global Processes, 1.; answer block on page 291) | |
dgn.parameters.p291.Task 6 |
answer recovered | `` | 20\% |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Global Processes, 1.; answer block on page 291) | |
dgn.parameters.p291.Task 7 |
answer recovered | `` | \pi/2, \arccos(5/6) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Global Processes, 1.; answer block on page 291) | |
dgn.parameters.p291.Task 8 |
answer recovered | `` | a \in (-\infty; -\sqrt{3}/2) \cup (\sqrt{3}/2; +\infty) |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Global Processes, 1.; answer block on page 291) | |
dgn.parameters.p292.4 |
mispaired answer | $120^\circ$ |
120^{\circ} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Global Processes, 2.; answer block on page 292) | |
dgn.parameters.p292.5 |
mispaired answer | The statement is true. |
The statement is true. |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2005 (July), Faculty of Global Processes, 2.; answer block on page 292) | |
dgn.parameters.p293.Task 2 |
mispaired answer | $[-3; (5+\sqrt{21})/2)$ |
x \in \{-2\} \cup [0; 2] |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2006 (April), "Conquer the Sparrow Hills-2006"1 , 1. √; answer block on page 293) | |
dgn.parameters.p293.Task 3 |
mispaired answer | $x \in \{-2\} \cup [0; 2]$ |
\arccos(1/4) + 2\pi l, l \in \mathbb{Z} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2006 (April), "Conquer the Sparrow Hills-2006", 2. √; answer block on page 293) | |
dgn.parameters.p293.Task 5 |
mispaired answer | $S_1 = S_2 = S_3 = 4$, $S_4 = \sqrt{55}/2$ |
S_1 = S_2 = S_3 = 2\sqrt{3}, S_4 = \sqrt{39}/2 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2006 (April), "Conquer the Sparrow Hills-2006", 2. √; answer block on page 293) | |
dgn.parameters.p293.Task 6 |
mispaired answer | $x = 0$ for any $a$, $x = (a \pm \sqrt{a^2-72})/6$ for $|a| \geqslant 6\sqrt{2}$ |
x = 0 and for any a, x = (a \pm \sqrt{a^{2} - 72})/6 for |a| \geqslant 6\sqrt{2} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2006 (April), "Conquer the Sparrow Hills-2006"1 , 1. √; answer block on page 293) | |
dgn.parameters.Task 6 (Variant 2006, aprel) |
mispaired answer | $x = 0$ for any $a$, $x = (a \pm \sqrt{a^2-16})/4$ for $|a| \geqslant 4$ |
x = 0, a for any a, x = (a \pm \sqrt{a^{2} - 16})/4 for |a| \geqslant 4 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2006 (April), "Conquer the Sparrow Hills-2006", 2. √; answer block on page 293) | |
dgn.parameters.p294.1 |
answer recovered | `` | -19 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2006 (May), "Lomonosov-2006"1 , 1.; answer block on page 295) | |
dgn.parameters.p294.2 |
answer recovered | `` | The smaller root of the equation is greater. |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2006 (May), "Lomonosov-2006"1 , 1.; answer block on page 295) | |
dgn.parameters.p294.3 |
answer recovered | `` | \pm\sqrt{2\pi k}, -1 \pm \sqrt{1 + 2\pi n}, k, n = 0, 1, 2, \ldots |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2006 (May), "Lomonosov-2006"1 , 1.; answer block on page 295) | |
dgn.parameters.p294.4 |
answer recovered | `` | 36, 8 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2006 (May), "Lomonosov-2006"1 , 1.; answer block on page 295) | |
dgn.parameters.p294.5 |
answer recovered | `` | 3/5 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2006 (May), "Lomonosov-2006"1 , 1.; answer block on page 295) | |
dgn.parameters.p294.6 |
answer recovered | `` | [0; 4] |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2006 (May), "Lomonosov-2006"1 , 1.; answer block on page 295) | |
dgn.parameters.p294.7 |
answer recovered | `` | (-\infty; -2] \cup \{-1/2\} \cup [0; 1/2] \cup \{2\} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2006 (May), "Lomonosov-2006"1 , 1.; answer block on page 295) | |
dgn.parameters.p294.8 |
answer recovered | `` | 2, 2\sqrt{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2006 (May), "Lomonosov-2006"1 , 1.; answer block on page 295) | |
dgn.parameters.p295.1 |
mispaired answer | $2, 2\sqrt{3}$ |
-38 |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2006 (May), "Lomonosov-2006", 2.; answer block on page 296) | |
dgn.parameters.p295.2 |
mispaired answer | $3/5$ |
The smaller root of the trinomial is greater. |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2006 (May), "Lomonosov-2006", 2.; answer block on page 296) | |
dgn.parameters.p295.3 |
mispaired answer | $[0; 4]$ |
\pm\sqrt{2\pi k}, -1 \pm \sqrt{1 + \pi + 2\pi n}, k, n = 0, 1, 2, \ldots |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2006 (May), "Lomonosov-2006", 2.; answer block on page 296) | |
dgn.parameters.p295.4 |
answer recovered | `` | 20/3, 5 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2006 (May), "Lomonosov-2006", 2.; answer block on page 296) | |
dgn.parameters.p295.5 |
answer recovered | `` | 3/7 |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2006 (May), "Lomonosov-2006", 2.; answer block on page 296) | |
dgn.parameters.p295.8 |
answer recovered | `` | 2, 2\sqrt{3} |
the page-image extraction did not attach the printed answer to this problem; the answer block of the printed variant was read again and the printed answer of this problem is published instead (variant: Variant 2006 (May), "Lomonosov-2006"1 , 1.; answer block on page 295) | |
dgn.parameters.p296.10 |
mispaired answer | $x\in[\pi/4+\pi k;\,3\pi/4+\pi k],\ k\in\mathbb{Z}$ |
x \in [\pi/4 + \pi k; 3\pi/4 + \pi k], k \in \mathbb{Z} |
the answer was taken from the answer block of the neighbouring exam variant printed on the same page; the page was read again and the printed answer of this problem is published instead (variant: Variant 2006 (May), "Lomonosov-2006", 2.; answer block on page 296) | |
dgn.syllabus.p668.8 |
answer recovered | `` | a \in (-\infty; -7/3] \cup [15 + 4\sqrt{17}; +\infty) |
the exercise carried no answer in the extraction; the answer was recomputed with the symbolic engine and confirmed by an independent computation of the range of the parameter | |
dgn.parameters.p280.4 |
transcription error | Item 4 of the «Ответы» block on p. 280: «4. (−∞; 1 − √14] ∪ [−√(17/3); −2) ∪ (1; √(17/3)].» (the radical bar covers 17/3) |
(-\infty;\, 1-\sqrt{14}] \cup [-\sqrt{\frac{17}{3}};\, -2) \cup (1;\, \sqrt{\frac{17}{3}}] |
the printed answer was mis-transcribed during extraction (the radical covers 17/3, not 17/3 with the root outside the fraction) - Our published answer writes \sqrt{17}/3, but the 600 dpi crop of p. 280 shows the radical bar over «17/3», i.e. √(17/3) — our transcription moved the «/3» out of the radical. The printed answer is mathematically right: putting c = √2−1, 3+√8 = c⁻², 3−√8 = c², 17−6√8 = c⁴, the inequality is equivalent to x²+x−2 ≤ (3−x)|2x+5| on x<3, x≠−5/2, x<−2 or x>1, i.e. x ≤ 1−√14 (for x<−5/2) or 3x² ≤ 17 (for x>−5/2), giving exactly (−∞;1−√14] ∪ [−√(17/3);−2) ∪ (1;√(17/3)]. The engine's complaint (x = −2.19 missing) is therefore an artefact of our mis-transcription: −2.19 does lie in [−√(17/3);−2). | |
dgn.parameters.Task 1 (Variant 2006, aprel) |
wrong answer | page 293, item 1 of the 'Ответы' block for ВАРИАНТ 2006 (апрель): [−2; (7+√21)/21) (printed problem, page 293: 'Задача 1. Решите неравенство √(x+2) > x−3.') |
[-2; (7+\sqrt{21})/2) |
√(x+2) > x−3 has domain x ≥ −2. For x ≤ 3 the right side is ≤ 0 while the left side is ≥ 0, so every x ∈ [−2, 3] satisfies it; for x > 3 both sides are positive and squaring gives x+2 > x²−6x+9 ⟺ x²−7x+7 < 0 ⟺ x ∈ ((7−√21)/2, (7+√21)/2), so the whole solution is [−2, (7+√21)/2). The source printed the upper endpoint as (7+√21)/21 = 0.5515512 instead of (7+√21)/2 = 5.7912878 (a typesetting slip in the denominator — the companion variant on the same page prints (5+√21)/2 for its own variant). The engine's complaint is valid: 0.5515512 satisfies the inequality (√(2.5515512) − 0.5515512 = 4.0458 > 0) yet is excluded from the printed set. | |
dgn.parameters.p289.2 |
wrong answer | page 289, item 2 of the 'Ответы' block for ВАРИАНТ 2005 (июль), московская школа экономики, 1.: (−∞; 2] (printed problem, page 289: 'Задача 2. Решите неравенство √(x²+2x) − x > 1.') |
(-\infty; -2] |
Domain x²+2x ≥ 0, i.e. x ⩽ −2 or x ≥ 0. On x ⩽ −2 the left side is √(x²+2x) − x ⩾ −x ⩾ 2 > 1, so the whole branch x ⩽ −2 satisfies the inequality (at x = −2 the value is 1 + 2 = 3). On x ≥ 0 both sides are non-negative and squaring gives x²+2x > x²+2x+1, i.e. 0 > 1 — impossible, so that branch is empty. The solution is (−∞; −2]; the printed answer (−∞; 2] has the sign of the endpoint wrong. The engine's probe is legitimate and reproduces exactly: F(2) = √8 − 2 − 1 = −0.17157288 (the engine's value) < 0, so the printed endpoint 2 does not satisfy the printed inequality. | |
dgn.parameters.p263.3 |
wrong answer | page 263, item 3 of the 'Ответы' block for ВАРИАНТ 2005 (июль), механико-математический факультет, 1.: x ∈ [1; 2) ∪ (2; √5) (printed problem, page 263: 'Задача 3. Решите неравенство (3 − x − √(5−x²))/(cos((2x−7)/4) − cos((x−5)/4)) ⩾ 0.') |
x \in [1; 2) \cup (2; \sqrt{5}] |
The domain is x ∈ [−√5, √5] with the denominator non-zero. The numerator N = 3 − x − √(5−x²) has N > 0 for x < 1 and for x > 2, N < 0 for 1 < x < 2 (its zeros are x = 1 and x = 2). The denominator D = cos((2x−7)/4) − cos((x−5)/4) = −2·sin(3(x−4)/8)·sin((x−2)/8) is negative for x < 2, zero at x = 2 and positive for 2 < x ⩽ √5 (D(√5) = 0.0362478 ≠ 0). The quotient is therefore ⩾ 0 exactly on [1; 2) ∪ (2; √5], and √5 must be INCLUDED (N(√5) = 3 − √5 = 0.7639 > 0, D(√5) = 0.0362 > 0, quotient = 21.0753). The printed answer leaves the upper endpoint √5 open, so it omits a genuine solution — the engine's complaint is valid. | |
dgn.parameters.p271.2 |
wrong answer | page 271, item 2 of the SECOND 'Ответы' block (the one following ВАРИАНТ 2005 (июль), химический факультет, 2.): 2. (−∞; 1/4] (printed problem, page 271: 'Задача 2. Решите неравенство √5 · 9^x ⩽ √3 · 25^x.') |
[1/4; +\infty) |
√5·9^x ⩽ √3·25^x is equivalent to 3^(2x−1/2) ≤ 5^(2x−1/2); since 3 < 5 this holds iff 2x − 1/2 ≥ 0, i.e. x ≥ 1/4 (equality at x = 1/4, both sides = √15). The printed answer (−∞; 1/4] is the solution set of the reversed inequality '⩾'. The engine's probe x = −0.75 lies in the printed interval and gives √5·9^(−0.75) − √3·25^(−0.75) = +0.275412 (matching its F) > 0, so the complaint is valid. | |
dgn.parameters.p274.6 |
wrong answer | 3 |
1 : 3 |
the question asks in what ratio the point K divides the diagonal AC, and the printed key gives the single number 3; the diagonals AC and BD meet at K = A + (1/4)(C - A), so AK : KC = 1 : 3, and the same problem of the neighbouring variant is keyed 1 : 3 on the printed page 273 |