url stringclasses 51
values | page_number int64 1 451 | extracted_answer_choices sequencelengths 1 20 | extracted_solution stringclasses 1
value | improved_question_solution stringlengths 26 9.11k | r1_distill_70b_response stringlengths 279 77.9k | __original_row_idx int64 0 5k |
|---|---|---|---|---|---|---|
https://ia801305.us.archive.org/8/items/ProblemsInCalculusOfOneVariableI.A.Maron/Problems%20in%20Calculus%20of%20One%20Variable%20-%20I.%20A.%20Maron.pdf | 205 | [
"\"I = (x³ + 1) sin x - 3I₁\"",
"\"I₁ = - x² cos x + 2I₂\"",
"\"I₂ = x sin x + cos x + C.\"",
"\"I = (x³ - 6x + 1) sin x + (3x² - 6) cos x + C.\""
] | NO ANSWER DETECTED | Let us put
u = x³ + 1; dv = cos x dx,
whence
du = 3x² dx; v = sin x.
I = (x³ + 1) sin x - 3 ∫ x² sin x dx = (x³ + 1) sin x - 3I₁,
where I₁ = ∫ x² sin x dx.
Integrating by parts again, we get
I₁ = - x² cos x + 2I₂,
where I₂ = ∫ x cos x dx.
Integrating by parts again, we obtain
I₂ = x sin x + cos x + ... | <think>
Okay, so I need to solve the integral I = ∫ (x³ + 1) cos x dx using integration by parts. Hmm, I remember that integration by parts is based on the formula ∫ u dv = uv - ∫ v du. So I need to choose which part of the integrand to set as u and which as dv.
Looking at the integrand (x³ + 1) cos x, I can either ch... | 1 |
https://ia801305.us.archive.org/8/items/ProblemsInCalculusOfOneVariableI.A.Maron/Problems%20in%20Calculus%20of%20One%20Variable%20-%20I.%20A.%20Maron.pdf | 262 | [
"\"free response\""
] | NO ANSWER DETECTED | If \( m(b - a) \leq \int f(x) dx \leq M(b - a) \), where \( m \) is the least value and \( M \) the greatest value of the function \( f(x) \) on the interval \([a, b]\), then this represents an estimation of the integral of the function \( f(x) \) over the interval from \( a \) to \( b \).
What is the estimation of t... | <think>
Okay, so I have this problem about estimating the integral of a function \( f(x) \) over an interval \([a, b]\). The statement given is that \( m(b - a) \leq \int f(x) dx \leq M(b - a) \), where \( m \) is the minimum value and \( M \) is the maximum value of \( f(x) \) on that interval. The question is asking ... | 8 |
https://jeeadv.ac.in/past_qps/2013_1.pdf | 3 | [
"(A) 0.3 × 10^-17 kg ms-1",
"(B) 1.0 × 10^-17 kg ms-1",
"(C) 3.0 × 10^-17 kg ms-1",
"(D) 9.0 × 10^-17 kg ms-1"
] | NO ANSWER DETECTED | A pulse of light of duration 100 ns is absorbed completely by a small object initially at rest. Power of the pulse is 30 mW and the speed of light is 3 × 10^8 ms-1. The final momentum of the object is:
(A) 0.3 × 10^-17 kg ms-1
(B) 1.0 × 10^-17 kg ms-1
(C) 3.0 × 10^-17 kg ms-1
(D) 9.0 × 10^-17 kg ms-1
What is ... | <think>
Okay, so I've got this physics problem here, and I need to figure out the final momentum of a small object after it completely absorbs a pulse of light. The pulse has a duration of 100 nanoseconds (ns), a power of 30 milliwatts (mW), and the speed of light is given as 3 × 10^8 meters per second (ms⁻¹). The opti... | 6 |
https://www.asi.edu.au/wp-content/uploads/2022/12/2017-asoe-chemistry-exam.pdf | 6 | [
"\"A: 1 mol\"",
"\"B: 2 mol\"",
"\"C: 4 mol\"",
"\"D: 8 mol\"",
"\"E: 16 mol\""
] | NO ANSWER DETECTED | Aqueous Fe2+ ions are readily oxidised to Fe3+ ions by oxygen gas in acidic solutions. Consider the following relevant reduction half equations:
Fe3+(aq) + e¯ → Fe2+(aq)
4H+(aq) + O2(g) + 4e¯ → 2H2O(l)
What chemical amount (in mol) of H2O would be produced from the reaction of 2 mol of Fe2+ with excess oxygen un... | <think>
Okay, so I have this chemistry problem here. It's about the oxidation of Fe²+ ions to Fe³+ ions using oxygen gas in an acidic solution. I need to find out how many moles of H2O are produced when 2 moles of Fe²+ react with excess oxygen. The options are from (a) 1 mol up to (e) 16 mol. Hmm, let's break this down... | 12 |
https://jeeadv.ac.in/past_qps/2017_1.pdf | 31 | [
"[A] (I) (iii) (P)",
"[B] (II) (iv) (Q)",
"[C] (III) (i) (R)",
"[D] (II) (iii) (P)"
] | NO ANSWER DETECTED | Here's the text extraction from the PDF, including the text within images:
**Page 1:**
* <bold>Answer Q.52, Q.53 and Q.54 by appropriately matching the information given in the three</bold>
<bold>columns of the following table.</bold>
* Let f (x) = x + loge x - x loge x, x ∈ (0,∞).
* • Column 1 contains inf... | <think>
Alright, let's tackle Q.54 step by step. So, the question is asking which combination is the only incorrect one. The options are A, B, C, and D, each pairing different statements from Columns 1, 2, and 3.
First, let me recall what each column represents:
- **Column 1** has statements about the zeros of f(x), ... | 15 |
https://ia801305.us.archive.org/8/items/ProblemsInCalculusOfOneVariableI.A.Maron/Problems%20in%20Calculus%20of%20One%20Variable%20-%20I.%20A.%20Maron.pdf | 238 | [
"NO CHOICES AVAILABLE"
] | NO ANSWER DETECTED | To evaluate the integral of the form I, we have:
I = ∫ dx / √(5 + 2x + x²).
We can rewrite the expression under the square root as follows: 5 + 2x + x² = 4 + (x + 1)².
Thus, we can substitute x + 1 = t, transforming the integral:
I = ∫ dt / √(4 + t²).
This leads to the substitution of t = 2 tan z.
What is the i... | <think>
Okay, so I have this integral I need to solve: I = ∫ dx / √(5 + 2x + x²). Hmm, it looks like a standard integral, but I'm not exactly sure how to approach it. Let me think.
First, I remember that when integrating expressions with square roots of quadratics, completing the square might be helpful. So maybe I s... | 0 |
https://ia801305.us.archive.org/8/items/ProblemsInCalculusOfOneVariableI.A.Maron/Problems%20in%20Calculus%20of%20One%20Variable%20-%20I.%20A.%20Maron.pdf | 342 | [
"NO CHOICES AVAILABLE"
] | NO ANSWER DETECTED | It is obvious that the abscissa of the point M equals 2a (the value of x at φ= 0), the abscissa of the point K being the minimum of the function x = a (1 + cos φ) cos φ. Let us find this minimum: dx = – a sin φ (1 + 2 cos φ) = 0, dφ φ₁ = 0; φ₂ = 2/3 π. At φ₁ = 0 we obtain xM = 2a, at φ₂ = 2/3 π, xK = –a/4. Find the min... | <think>
Okay, so I need to find the minimum value of the function \( x = a(1 + \cos \phi) \cos \phi \). Alright, let me think about how to approach this. I remember that to find the minimum or maximum of a function, we usually take its derivative and set it equal to zero. Then, we solve for the variable, which in this ... | 7 |
https://www.asi.edu.au/wp-content/uploads/2022/12/2014-asoe-chemistry-exam.pdf | 5 | [
"\"A: CO2\"",
"\"B: H2O2\"",
"\"C: H2S\"",
"\"D: NaCl\"",
"\"E: SO2\""
] | NO ANSWER DETECTED | A small spacecraft of capacity 10 m³ is connected to another of capacity 30 m³. Prior to connection, the pressure inside the smaller craft is 50 kPa and that inside the larger is 100 kPa. If all measurements are made at the same temperature, what is the pressure in the combined arrangement after connection?
Which one... | <think>
Okay, so I have two problems here. The first one is about connecting two spacecraft, and the second one is about identifying a substance with linear molecules under certain conditions. I'll tackle them one by one.
Starting with the spacecraft problem: There's a small spacecraft with a capacity of 10 m³ connect... | 2 |
https://ia801305.us.archive.org/8/items/ProblemsInCalculusOfOneVariableI.A.Maron/Problems%20in%20Calculus%20of%20One%20Variable%20-%20I.%20A.%20Maron.pdf | 159 | [
"\"free response\""
] | NO ANSWER DETECTED | As a result of n measurements of an unknown quantity x the numbers x₁, x₂, . . . , xₙ are obtained. It is required to find at what value of x the sum of the squares of the errors
f (x) = (x-x₁)² + (x - x₂)² +...+(x-xₙ)²
will be the least.
At what value of x will the sum of the squares of the errors be the least? | <think>
Alright, so I need to figure out at what value of x the sum of the squares of the errors is the least. The function given is f(x) = (x - x₁)² + (x - x₂)² + ... + (x - xₙ)². Hmm, okay, so this looks like a problem where I have to find the minimum of this function.
First, since this is a sum of squares, I remem... | 22 |
https://jeeadv.ac.in/past_qps/2019_2_English.pdf | 7 | [
"NO CHOICES AVAILABLE"
] | NO ANSWER DETECTED | A ball is thrown from ground at an angle Θ with horizontal and with an initial speed uo. For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is V₁. After hitting the ground, the ball rebounds at the same angle Θ but with a reduced... | <think>
Alright, let me see. So there's this problem about projectile motion. A ball is thrown upwards at an angle Θ with an initial speed u₀. It hits the ground, then rebounds with the same angle but a reduced speed u₀/α. I need to find α such that the average velocity over the entire duration is 0.8 V₁, where V₁ is t... | 20 |
https://ia801305.us.archive.org/8/items/ProblemsInCalculusOfOneVariableI.A.Maron/Problems%20in%20Calculus%20of%20One%20Variable%20-%20I.%20A.%20Maron.pdf | 166 | [
"NO CHOICES AVAILABLE"
] | NO ANSWER DETECTED | Find the intervals in which the graphs of the following functions are concave or convex and locate the points of inflection: (h) y = 2 - |x⁵ - 1|.
This function is part of a section discussing the convexity and concavity of curves, where it's stated that if f''(x) < 0 (or > 0) on an interval (a, b), then the curve y ... | <think>
Alright, so I need to figure out where the graph of y = 2 - |x⁵ - 1| is concave or convex, and also find the points of inflection. Hmm, okay. Let's start by understanding the function.
The function is y = 2 - |x⁵ - 1|. That means it's a transformation of the absolute value function applied to x⁵ - 1, and then ... | 13 |
https://ia801305.us.archive.org/8/items/ProblemsInCalculusOfOneVariableI.A.Maron/Problems%20in%20Calculus%20of%20One%20Variable%20-%20I.%20A.%20Maron.pdf | 342 | [
"NO CHOICES AVAILABLE"
] | NO ANSWER DETECTED | The relevant text states:
```text
7.6.18. Compute the volume of the solid generated by revolving the figure bounded by the following lines:
(a) xy = 4, x=1, x = 4, y=0 about the x-axis;
(b) y=2x-x², y=0 about the x-axis;
```
Compute the volume of the solid generated by revolving the figure bounded by the following lin... | <think>
Alright, so I have two calculus problems here where I need to compute the volume of solids of revolution. Let's take them one at a time, starting with part (a).
Problem (a) is bounded by the curves xy = 4, x=1, x=4, and y=0, revolved around the x-axis. Hmm, okay. So, let's visualize this region. The curve xy =... | 21 |
https://ia801305.us.archive.org/8/items/ProblemsInCalculusOfOneVariableI.A.Maron/Problems%20in%20Calculus%20of%20One%20Variable%20-%20I.%20A.%20Maron.pdf | 49 | [
"NO CHOICES AVAILABLE"
] | NO ANSWER DETECTED | Find lim x, if x = n(-1)^n / (1 - 2n) n² + 1. | <think>
Okay, so I need to find the limit of x as n approaches something. Wait, actually, looking at the problem again, it says "Find lim x, if x = n(-1)^n / (1 - 2n) n² + 1." Hmm, this might be a bit confusing. Is it x as n approaches infinity? I think so, because usually, limits like this are about n approaching infi... | 31 |
https://www.asi.edu.au/wp-content/uploads/2022/12/2012-asoe-chemistry-exam.pdf | 5 | [
"\"4.53\"",
"\"5.53\"",
"\"7.00\"",
"\"8.47\"",
"\"9.47\""
] | NO ANSWER DETECTED | A certain buffer solution contains equal concentration of weak acid (HX) and its conjugate base (X¯). Ka for HX is 2.9 × 10¯5. What is the pH of the buffer?
a. 4.53
b. 5.53
c. 7.00
d. 8.47
e. 9.47 | <think>
Okay, so I have this buffer solution problem here, and I need to figure out the pH. Let me see. The problem says it's a buffer with equal concentrations of a weak acid, HX, and its conjugate base, X⁻. The Ka for HX is 2.9 × 10⁻⁵. The options are a through e, with pH values ranging from around 4.5 to 9.5. Hmm, b... | 3 |
https://www.asi.edu.au/wp-content/uploads/2022/12/2014-asoe-chemistry-exam.pdf | 5 | [
"\"A: 0.8%\"",
"\"B: 8.0%\"",
"\"C: 20%\"",
"\"D: 80%\"",
"\"E: 92%\""
] | NO ANSWER DETECTED | The relative atomic mass of boron, which consists of the isotopes 10B and 11B is 10.8. What is the percentage of 11B atoms in the isotopic mixture?
(a) 0.8% (b) 8.0% (c) 20% (d) 80% (e) 92% | <think>
Okay, so I have this problem here about the relative atomic mass of boron, which is made up of two isotopes: 10B and 11B. The relative atomic mass is given as 10.8, and I need to find the percentage of 11B atoms in the mixture. The options are (a) 0.8%, (b) 8.0%, (c) 20%, (d) 80%, and (e) 92%. Hmm, alright.
Fi... | 24 |
https://ia801305.us.archive.org/8/items/ProblemsInCalculusOfOneVariableI.A.Maron/Problems%20in%20Calculus%20of%20One%20Variable%20-%20I.%20A.%20Maron.pdf | 49 | [
"NO CHOICES AVAILABLE"
] | NO ANSWER DETECTED | Find lim x, if x = Vn / n. | <think>
</think>
To find the limit of \( x \) as \( n \) approaches infinity, where \( x = \frac{V_n}{n} \), I need more information about \( V_n \). Could you clarify or provide the context or expression for \( V_n \)? This will help me determine how to proceed with the calculation. | 19 |
https://www.asi.edu.au/wp-content/uploads/2022/12/2012-asoe-chemistry-exam.pdf | 3 | [
"\"a. A minimum of 1.21 volts is applied to the cell and electrons move from the silver electrode to the iron electrode.\"",
"\"b. A minimum of 1.21 volts is applied to the cell and electrons move from the iron electrode to the silver electrode.\"",
"\"c. A minimum of 0.39 volts is applied to the cell and elect... | NO ANSWER DETECTED | A working galvanic cell is constructed using an iron electrode in an iron(II) nitrate solution and a silver electrode in a silver(I) nitrate solution. When the cell runs down, it can be charged up again. The standard electrode potentials are given as E°(Ag+(aq)/Ag(s) = +0.80 V and E°(Fe2+(aq)/Fe(s) = -0.41 V). Which of... | <think>
Okay, so I need to figure out what happens during the charging process of this galvanic cell. The cell uses an iron electrode in iron(II) nitrate and a silver electrode in silver(I) nitrate. I remember that when a cell is discharged, it's just operating normally as a galvanic cell, but when it's charged, it's b... | 23 |
https://jeeadv.ac.in/past_qps/2015_2.pdf | 14 | [
"\"3\""
] | NO ANSWER DETECTED | The molar conductivity of a solution of a weak acid HX (0.01 M) is 10 times smaller than the molar conductivity of a solution of a weak acid HY (0.10 M). If the degree of ionization of both acids is considered to be much less than 1, what is the difference in their pKa values, pKa (HX) - pKa (HY)? | <think>
Okay, so I have this problem about molar conductivity of weak acids, HX and HY. Let me try to figure it out step by step. Hmm, the problem says that HX is 0.01 M and its molar conductivity is 10 times smaller than HY, which is 0.10 M. Both are weak acids, so their degree of ionization is much less than 1. I nee... | 14 |
https://ia801305.us.archive.org/8/items/ProblemsInCalculusOfOneVariableI.A.Maron/Problems%20in%20Calculus%20of%20One%20Variable%20-%20I.%20A.%20Maron.pdf | 175 | [
"\"x₁= -1",
"x2=0",
"x₃ = 1\""
] | NO ANSWER DETECTED | Here's the text extraction from the provided PDF images, with formatting indicated:
**Page 1**
<bold>§ 3.11. Investigating Functions and Sketching Graphs</bold>
<highlighted>175</highlighted>
<highlighted>3. Test the function for continuity, find out the discontinuities</highlighted>
<highlighted>and their character.... | <think>
To determine the critical points of the function \( y = x^6 - 3x^4 + 3x^2 - 5 \), I'll follow these steps:
First, I need to find the derivative of the function with respect to \( x \). The derivative will help identify where the function has horizontal tangents or doesn't change direction, which are potential ... | 81 |
https://www.asi.edu.au/wp-content/uploads/2022/12/2017-asoe-chemistry-exam.pdf | 7 | [
"\"A: Covalent bonding\"",
"\"B: Hydrogen bonding\"",
"\"C: Ionic bonding\"",
"\"D: Intermolecular forces\"",
"\"E: Metallic bonding\""
] | NO ANSWER DETECTED | Solid carbon dioxide, also known as dry ice, sublimes to form gaseous carbon dioxide. What type of bonding is significantly weakened because of this phase change?
(a) Covalent bonding
(b) Hydrogen bonding
(c) Ionic bonding
(d) Intermolecular forces
(e) Metallic bonding | <think>
Okay, so I've got this question about dry ice and what happens when it sublimes. Let me try to work through this step by step.
First, I know that dry ice is solid carbon dioxide. I've seen it before; it's that stuff that looks like ice but smokes a lot because it turns straight into gas. So, when it sublimes, ... | 65 |
https://ia801305.us.archive.org/8/items/ProblemsInCalculusOfOneVariableI.A.Maron/Problems%20in%20Calculus%20of%20One%20Variable%20-%20I.%20A.%20Maron.pdf | 221 | [
"\"√ax^2 + bx + c = t + x√a\"",
"\"√ax^2 + bx + c = tx - √c\"",
"\"√ax^2 + bx + c = (x - α) t\""
] | NO ANSWER DETECTED | Integrals of the form ∫ R(x, √ax² + bx + c) dx are calculated with the aid of one of the three Euler substitutions:
(1) √ax² + bx + c = t + x√a if a > 0;
(2) √ax² + bx + c = tx - √c if c > 0;
(3) √ax² + bx + c = (x - α)t if ax² + bx + c = a(x - α)(x - β), i.e. if α is a real root of the trinomial ax² + bx + c.
... | <think>
Alright, so I need to figure out what the third Euler substitution is used for when dealing with integrals of the form ∫ R(x, √(ax² + bx + c)) dx, specifically for quadratic expressions that have a real root. Hmm, I remember Euler substitutions are techniques to simplify integrals involving square roots of quad... | 16 |
https://www.asi.edu.au/wp-content/uploads/2022/12/2009-asoe-chemistry-exam.pdf | 16 | [
"NO CHOICES AVAILABLE"
] | NO ANSWER DETECTED | To obtain a single product, sometimes “protections" are employed to “mask” certain functional groups to prevent them from taking part in a reaction. Some of these protections include converting reactive amines, alcohols, aldehydes and carboxylic acids into relatively unreactive esters, ethers and amides. These protecti... | <think>
Okay, so I'm trying to figure out the answer to the question about protections used to mask functional groups in organic chemistry. I remember that in synthesis, sometimes certain groups are too reactive, so they need to be protected to prevent them from interfering in the reaction. The user mentioned that Tabl... | 74 |
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