conversations listlengths 2 2 | question stringlengths 11 25k β |
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"value": "11.1 In the following figure, $A B C D$ is a square of length $10 \\mathrm{~cm} . A E B, F E D$ and $F B C$ are straight lines. The area of $\\triangle A E D$ is larger than that of $\\triangle F E B$ by $10 \\mathrm{~cm}^{2}$. If the area of $\\triangle D F B$ is $P \\mathrm{~cm... | 11.1 In the following figure, $A B C D$ is a square of length $10 \mathrm{~cm} . A E B, F E D$ and $F B C$ are straight lines. The area of $\triangle A E D$ is larger than that of $\triangle F E B$ by $10 \mathrm{~cm}^{2}$. If the area of $\triangle D F B$ is $P \mathrm{~cm}^{2}$, find the value of $P$.
I1.2 Workman A... |
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"value": "If the calculation from a 2Γ2 contingency table yields $\\chi^2=4.073$, then we can be confident to what degree that there is a relationship between the two variables? \\[Given P(\\chi^2\\geq3.841)\\approx0.05, P(\\chi^2\\geq5.024)\\approx0.025\\]."
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"from": "assistan... | If the calculation from a 2Γ2 contingency table yields $\chi^2=4.073$, then we can be confident to what degree that there is a relationship between the two variables? \[Given P(\chi^2\geq3.841)\approx0.05, P(\chi^2\geq5.024)\approx0.025\]. |
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"value": "Given that $M(x_{0},y_{0})$ is a point on the hyperbola $C$: $\\frac{x^{2}}{2}-y^{2}=1$, and $F_{1}$, $F_{2}$ are the left and right foci of $C$, if $\\overrightarrow{MF_{1}} \\cdot \\overrightarrow{MF_{2}} < 0$, then the range of $y_{0}$ is $(\\quad\\quad)$.\n\nA: $(-\\frac{\\sq... | Given that $M(x_{0},y_{0})$ is a point on the hyperbola $C$: $\frac{x^{2}}{2}-y^{2}=1$, and $F_{1}$, $F_{2}$ are the left and right foci of $C$, if $\overrightarrow{MF_{1}} \cdot \overrightarrow{MF_{2}} < 0$, then the range of $y_{0}$ is $(\quad\quad)$.
A: $(-\frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$
B: $(-\frac{\sqrt... |
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"value": "Given $x \\in (-1, 3)$, then the range of the function $y = (x - 2)^2$ is ( )\nA: $(1, 4)$\nB: $[0, 9)$\nC: $[0, 9]$\nD: $[1, 4)$"
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"value": "<think>\nOkay, let's see. The problem says that x is in the interval (-1, 3), and we need to find t... | Given $x \in (-1, 3)$, then the range of the function $y = (x - 2)^2$ is ( )
A: $(1, 4)$
B: $[0, 9)$
C: $[0, 9]$
D: $[1, 4)$ |
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"value": "The tangent line to the function $f(x)=\\ln x-ax$ at $x=1$ is perpendicular to the $y$-axis.\n\n$(1)$ Find $a$; \n$(2)$ Determine the intervals of monotonicity for the function $f(x)$."
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"value": "<think>\nOkay, so I need to solve this proble... | The tangent line to the function $f(x)=\ln x-ax$ at $x=1$ is perpendicular to the $y$-axis.
$(1)$ Find $a$;
$(2)$ Determine the intervals of monotonicity for the function $f(x)$. |
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"value": "Given the sets $A=\\{x|x\\leq -2 \\text{ or } x\\geq 2\\}$, $B=\\{x|1<x<5\\}$, and $C=\\{x|m-1\\leq x\\leq 3m\\}$.\n**(I)** Find $A\\cap B$ and $(\\complement_{\\mathbb{R}}A) \\cup B$.\n**(II)** If $B \\cap C = C$, find the range of the real number $m$."
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"from": "ass... | Given the sets $A=\{x|x\leq -2 \text{ or } x\geq 2\}$, $B=\{x|1<x<5\}$, and $C=\{x|m-1\leq x\leq 3m\}$.
**(I)** Find $A\cap B$ and $(\complement_{\mathbb{R}}A) \cup B$.
**(II)** If $B \cap C = C$, find the range of the real number $m$. |
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"value": "1. At each vertex of a cube with an edge length of 1, there is the center of a sphere. All spheres are the same, and each touches three adjacent spheres. Find the length of that part of the spatial diagonal of the cube that lies outside the spheres."
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"from": "assista... | 1. At each vertex of a cube with an edge length of 1, there is the center of a sphere. All spheres are the same, and each touches three adjacent spheres. Find the length of that part of the spatial diagonal of the cube that lies outside the spheres. |
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"value": "The equation of the asymptotes of the hyperbola $\\dfrac{x^2}{16} - \\dfrac{y^2}{9} = -1$ is (γγ).\nA: $y = \\pm \\dfrac{3}{4}x$\nB: $y = \\pm \\dfrac{4}{3}x$\nC: $y = \\pm \\dfrac{16}{9}x$\nD: $y = \\pm \\dfrac{9}{16}x$"
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"value": "<think>\nO... | The equation of the asymptotes of the hyperbola $\dfrac{x^2}{16} - \dfrac{y^2}{9} = -1$ is (γγ).
A: $y = \pm \dfrac{3}{4}x$
B: $y = \pm \dfrac{4}{3}x$
C: $y = \pm \dfrac{16}{9}x$
D: $y = \pm \dfrac{9}{16}x$ |
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"value": "If point P $(\\cos\\alpha, \\sin\\alpha)$ lies on the line $y = -2x$, then $\\sin\\alpha \\cdot \\cos\\alpha =$ οΌγγοΌ\nA: $-\\frac{9}{10}$\nB: $-\\frac{2}{5}$\nC: $\\frac{1}{2}$\nD: $\\frac{9}{10}$"
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"value": "<think>\nOkay, so I need to solve ... | If point P $(\cos\alpha, \sin\alpha)$ lies on the line $y = -2x$, then $\sin\alpha \cdot \cos\alpha =$ οΌγγοΌ
A: $-\frac{9}{10}$
B: $-\frac{2}{5}$
C: $\frac{1}{2}$
D: $\frac{9}{10}$ |
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"value": "$A$ walks $30 \\mathrm{~km}$ per day from $P$ to $Q$. After walking $72 \\mathrm{~km}$, $B$ starts from $Q$ towards $P$ on the same road, and walks $\\frac{1}{10}$ of the entire distance each day. After $B$ has walked for as many days as the $\\frac{1}{8}$ of the distance he walk... | $A$ walks $30 \mathrm{~km}$ per day from $P$ to $Q$. After walking $72 \mathrm{~km}$, $B$ starts from $Q$ towards $P$ on the same road, and walks $\frac{1}{10}$ of the entire distance each day. After $B$ has walked for as many days as the $\frac{1}{8}$ of the distance he walks per day, he meets $A$. What is the distanc... |
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"value": "Given $p: (x-1)(x-2) < 0$, $q: x-a < 0$, if $p$ is a sufficient condition for $q$, find the range of values for $a$."
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"value": "<think>\nOkay, let's see. The problem says that p is a sufficient condition for q. So that means if p is true, the... | Given $p: (x-1)(x-2) < 0$, $q: x-a < 0$, if $p$ is a sufficient condition for $q$, find the range of values for $a$. |
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"value": "The equation of one of the axes of symmetry in the graph of the function $$y=\\sin(2x+ \\frac{\\pi}{3})$$ is (γγ)\n\nA: $$x= \\frac{\\pi}{12}$$\n\nB: $$x= \\frac{\\pi}{6}$$\n\nC: $$x= \\frac{\\pi}{3}$$\n\nD: $$x=- \\frac{\\pi}{12}$$"
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"value":... | The equation of one of the axes of symmetry in the graph of the function $$y=\sin(2x+ \frac{\pi}{3})$$ is (γγ)
A: $$x= \frac{\pi}{12}$$
B: $$x= \frac{\pi}{6}$$
C: $$x= \frac{\pi}{3}$$
D: $$x=- \frac{\pi}{12}$$ |
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"value": "5. (20 points) A voltmeter connected to the terminals of a current source with an EMF of $12 \\mathrm{~V}$ shows a voltage $U=9 \\mathrm{~V}$. Another identical voltmeter is connected to the terminals of the source. Determine the readings of the voltmeters. (The internal resistan... | 5. (20 points) A voltmeter connected to the terminals of a current source with an EMF of $12 \mathrm{~V}$ shows a voltage $U=9 \mathrm{~V}$. Another identical voltmeter is connected to the terminals of the source. Determine the readings of the voltmeters. (The internal resistance of the source is non-zero, the resistan... |
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