problem stringlengths 14 4.09k | solution stringlengths 1 802k | task_type stringclasses 3
values | problem_tokens int64 5 895 |
|---|---|---|---|
Given that the domain and range of the function $f(x)$ are both $(0, +\infty)$, and the derivative of $f(x)$ is $f'(x)$, where $2f(x) \lt f'(x) \lt 3f(x)$, then the range of $\frac{{f(2023)}}{{f(2024)}}$ is ______. | (e^{-3}, e^{-2}) | math | 85 |
Given the equation of a circle $x^2+y^2-2x-4y+m=0$.
1. If the circle intersects with the line $x+2y-4=0$ at points M and N, and $OM \perp ON$ (O is the origin), find the value of $m$;
2. Under the condition of (1), find the equation of the circle with MN as its diameter. | (x- \frac{4}{5})^2+(y- \frac{8}{5})^2= \frac{16}{5} | math | 92 |
In right triangle $DEF$ with $\angle D = 90^\circ$, we have $DE = 8$ and $EF = 17$. Find $\sin F$. | \frac{8}{17} | math | 38 |
The left and right foci of the ellipse $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$ are $F_{1}$ and $F_{2}$, respectively, and its eccentricity is $\dfrac {1}{2}$. Point $P$ is a moving point on the ellipse, and the maximum area of $\triangle F_{1}PF_{2}$ is $\sqrt {3}$.
$(1)$ Find the equation of the ... | 0 | math | 241 |
Given the function $y=2x+\cos x$, find the derivative of $y$ with respect to $x$. | 2-\sin x | math | 25 |
In \\(\triangle ABC\\), let the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) be \\(a\\), \\(b\\), and \\(c\\) respectively. Let vector \\( \overrightarrow{m}=(\cos A+ \sqrt {2},\sin A)\\) and vector \\( \overrightarrow{n}=(-\sin A,\cos A)\\). If \\(| \overrightarrow{m}+ \overrightarrow{n}|=2\\),
\\((1)\\) fin... | 16 | math | 169 |
Given the area of a circle is doubled when its radius $r$ is increased by $n$, determine the relationship between $r$ and $n$. | n(\sqrt{2} + 1) | math | 31 |
Find all real $x$ such that \[\left\lfloor 2x \lfloor x \rfloor \right\rfloor = 58.\] | [5.8, 5.9) | math | 35 |
One less than the reciprocal of a certain number is $\frac{5}{4}$. What is the original number expressed as a common fraction? | -4 | math | 29 |
Given the function \( f(x)=\frac{a x}{2 x+3} \), if \( f(f(x))=x \) is always true, then the value of the real number \( a \) is ____ . | -3 | math | 48 |
Given functions $f(x)=-2x$ for $x<0$ and $g(x)=\frac{x}{\ln x}+x-2$. If $f(x_{1})=g(x_{2})$, find the minimum value of $x_{2}-2x_{1}$. | 4\sqrt{e}-2 | math | 64 |
Let the function be $$f(x)= \begin{cases} x^{2}-4x+6, & x\geq0 \\ x+6, & x<0\end{cases}.$$ Then, the solution set of the inequality $f(x) > f(1)$ is \_\_\_\_\_\_. | (-3, 1) \cup (3, +\infty) | math | 68 |
In the figure, $ABCD$ is a rectangle, $AZ=WC=6$ units, $AB=12$ units and the area of trapezoid $ZWCD$ is 120 square units. What is the area of triangle $BQW$? [asy]
draw((0,0)--(12,0)--(12,20)--(0,20)--(0,0)--(12,20));
draw((0,14)--(12,6));
label("$A$",(0,20),W);
label("$Z$",(0,14),W);
label("$D$",(0,0),W);
label("$Q$"... | 42 | math | 201 |
Distribute four students, named A, B, C, and D, into three different classes, with each class having at least one student. Students A and B cannot be in the same class. The number of different ways to distribute the students is. | 30 | math | 51 |
The measure of angle $ACB$ is 70 degrees. If ray $CA$ is rotated 600 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new obtuse angle $ACB$, in degrees? | 170 | math | 55 |
Thirty teams are participating in a knockout tournament where each match results in a win or lose outcome only, and the losing team is eliminated. Each team has a $50 \%$ chance of winning any game it plays. The probability that the final sequence of winners is in the exact increasing order of their initial arbitrary l... | 29 | math | 103 |
Li Fang has 4 shirts of different colors, 3 skirts of different patterns, and 2 dresses of different styles. Calculate the total number of different choices she has for the May Day celebration. | 14 | math | 40 |
The smallest integral value of $k$ such that $2x(kx-4)-x^2+6=0$ has no real roots. | 2 | math | 31 |
Given the number $-\frac{1}{2}$, find its opposite. | \frac{1}{2} | math | 16 |
The teacher asked Adam and Eve to calculate the perimeter of a trapezoid, where the longer base measures $30 \mathrm{~cm}$, the height is $24 \mathrm{~cm}$, and the legs are $25 \mathrm{~cm}$ and $30 \mathrm{~cm}$. Adam and Eve obtained different perimeters, yet the teacher praised both for correct solutions.
Determin... | 90 \, \text{cm} \text{ and } 104 \, \text{cm} | math | 95 |
Find the sum of all integral values of $c$ with $c \leq 30$ for which the equation $y = x^2 - 9x - c$ has two rational roots. | -28 | math | 43 |
Given a rectangular parallelepiped. The perimeters of each of its three mutually perpendicular faces are equal to the sides of a new rectangular parallelepiped. What can be the minimum ratio of the volume of the new parallelepiped to the volume of the original one? | 64 | math | 54 |
Given the ellipse $\frac{x^{2}}{a\_1^{2}} + \frac{y^{2}}{b\_1^{2}} = 1$ and the hyperbola $\frac{x^{2}}{a\_2^{2}} - \frac{y^{2}}{b\_2^{2}} = 1$, for which the ellipse has $a\_1 > b\_1 > 0$ and the hyperbola has $a\_2 > 0, b\_2 > 0$, and share the same foci $F\_1$, $F\_2$, point $P$ is a common point of the two curves, ... | 8 | math | 170 |
In a geometric sequence, the first term is $12$ and the second term is $-24$. What is the $102^{nd}$ term of this sequence? | -2^{101} \times 12 | math | 38 |
Use the Horner's method to calculate the value of the polynomial $f(x) = 2x^4 - x^3 + 3x^2 + 7$ when $x = 3$, the value of $v_3$ is \_\_\_\_\_\_. | 54 | math | 60 |
Let $k$ be a positive integer, and the coefficient of the fourth term in the expansion of $(1+ \frac{x}{k})^{k}$ is $\frac{1}{16}$. Consider the functions $y= \sqrt{8x-x^{2}}$ and $y= \frac{1}{4}kx$, and let $S$ be the shaded region enclosed by their graphs. Calculate the probability that the point $(x,y)$ lies within ... | \frac{\pi}{4} - \frac{1}{2} | math | 126 |
Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, when $a=2b$, point $P$ is on the ellipse and $PF_{1} \perp PF_{2}$, $|PF_{1}| \cdot |PF_{2}|=2$. Find the equation of the ellipse. | \frac{x^{2}}{4}+y^{2}=1 | math | 110 |
Three people, A, B, and C, independently attempt to crack a code. Their respective probabilities of success are $\frac{1}{2}$, $\frac{3}{5}$, and $\frac{3}{4}$. Find:
(1) The probability that exactly one person succeeds when all three attempt to crack the code simultaneously;
(2) The probability that the code is su... | 3 | math | 118 |
Given vectors $\overrightarrow{a}=(2\sin \frac {4}{3}\pi,\cos \frac {5}{6}\pi)$ and $\overrightarrow{b}=(k,1)$. If $\overrightarrow{a} \parallel \overrightarrow{b}$, then $k=$ ______. | 2 | math | 66 |
The Evil League of Evil plans to set out from their headquarters at (5,1) to poison two pipes: one along the line \( y = x \) and the other along the line \( x = 7 \). They wish to determine the shortest distance they can travel to visit both pipes and then return to their headquarters. | 4\sqrt{5} | math | 67 |
A convex hexagon has interior angles with measures $x+2$, $2x-1$, $3x+1$, $4x-2$, $5x+3$, and $6x-4$ degrees. What is the measure of the largest angle? | 202^\circ | math | 56 |
Given $f\left(x\right)=\frac{\sqrt{{x}^{2}+1}+x-1}{\sqrt{{x}^{2}+1}+x+1}+ax^{3}+bx-8$ and $f\left(-2\right)=10$, find $f\left(2\right)=\_\_\_\_\_\_$. | -26 | math | 84 |
Given the sequence $\{a\_n\}$, where $a\_1=1$ and $a\_n=a_{n-1}+3$, $(n\geqslant 2,n\in\mathbb{N}^{*})$, find the expression for $a\_n$. | 3n-2 | math | 64 |
A food factory regularly purchases flour. It is known that the factory needs 6 tons of flour per day, the price of each ton of flour is 1800 yuan, and the storage and other costs for flour are an average of 3 yuan per ton per day. Each time flour is purchased, a shipping fee of 900 yuan is required. How often should th... | 10 | math | 90 |
If positive numbers $x$ and $y$ satisfy the equation $3x+y=5xy$, find the minimum value of $4x+3y$. | 5 | math | 33 |
The product of four different positive integers is equal to $5^4$. What is the sum of these four integers? | 156 | math | 24 |
Given the parabola C: y²=4x, its focus is F, and the line passing through point P(2,0) intersects the parabola at points A and B.
(1) If $\overrightarrow {FA}\cdot \overrightarrow {FB}=-11$, find the equation of line AB.
(2) Find the minimum area of △ABF. | 2 \sqrt {2} | math | 82 |
Given a geometric sequence with positive terms $\{a_n\}$, the product of its first $n$ terms is denoted as $\pi_n$. It is known that $a_{m-1} \cdot a_{m+1} = 2a_m$ and $\pi_{2m-1} = 2048$. Find the value of $m$. | 6 | math | 79 |
A rectangular field was allocated for corn. After some time, the length of this field was increased by $35\%$, and the width was decreased by $14\%. By what percentage did the area of the field change? | 16.1\% | math | 48 |
Find the area of triangle $JKL$, where $\angle J = 90^\circ$, $\angle K = 45^\circ$, and the length of side $KL$ is 24 units. | 144 | math | 44 |
Given that two of the roots of the cubic equation \[ax^3 + bx^2 + cx + d = 0\] are \(1 + i\) and \(1 - i\), and \(a \neq 0\), compute \(\frac{b + c}{a}\). | 2 | math | 63 |
Cora and Dana play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $g(n)$ points, where
\[ g(n) = \left\{
\begin{array}{cl}
8 & \text{ if } n \text{ is a multiple of 3 and 2}, \\
4 & \text{ if } n \text{ is a multiple of 3 but not of 2}, \\
3 & \text{ if } n \text{ is only a... | 480 | math | 218 |
Given two concentric circles, where one circle has a radius $r$ and the other has a radius $2r$, determine the number of common tangents that these two circles can have. | 0 | math | 39 |
The minimum value of the function $f(x) = x - 2\cos{x}$ in the interval $\left[-\frac{\pi}{2}, 0\right]$ can be found by determining the critical points and evaluating the function at these points and the endpoints of the interval. | -\frac{\pi}{6} - \sqrt{3} | math | 59 |
$(1)$ Given proposition $p:∃x>-1,\frac{x^{2}+3x+6}{x+1}<a$, if proposition $p$ is false, find the range of real number $a$. <br/> $(2)$ If positive numbers $a$ and $b$ satisfy $a+2b=1$, find the minimum value of $2a+\frac{1}{a}+4b+\frac{8}{b}$. | 27 | math | 98 |
If $x$ cows produce $x+2$ cans of milk in $x+4$ days, determine the number of days it will take $x+4$ cows to produce $x+6$ cans of milk. | \frac{x(x+4)(x+6)}{(x+2)(x+4)} | math | 47 |
Find the degree measure of the smallest positive angle $\theta$ for which
\[\sin 10^\circ = \cos 40^\circ - \cos \theta.\] | 30^\circ | math | 38 |
A certain train accelerates at an average speed of $v$ km/h. In the same amount of time, the train travels $s$ km before accelerating, and after accelerating, it travels 50 km more than before. The average speed of the train before accelerating can be calculated as ____ km/h. | \dfrac{sv}{50} | math | 63 |
(a) Find the smallest number of lines drawn on the plane so that they produce exactly 2022 points of intersection. (Note: For 1 point of intersection, the minimum is 2; for 2 points, minimum is 3; for 3 points, minimum is 3; for 4 points, minimum is 4; for 5 points, the minimum is 4, etc.)
(b) What happens if the lines... | k = 65 | math | 101 |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = 2\sqrt{2}, |\overrightarrow{b}| = \sqrt{2}$, and $\overrightarrow{a}\cdot\overrightarrow{b} = 1$, find $| \overrightarrow{a} - 2\overrightarrow{b} |$. | 2\sqrt{3} | math | 83 |
Given $x \gt 0$, $y \gt 0$, and $\frac{xy}{2y+3x}=1$, the inequality $\frac{x}{2}+\frac{y}{3}\geqslant m$ always holds. The range of real number $m$ is ____. | (-\infty, 4] | math | 64 |
If a line is tangent to the circle $x^2+y^2-2x-4y+a=0$ and the graph of the function $y= \frac {x^{2}}{4}$ at the same point, the value of $a$ is \_\_\_\_\_\_. | 3 | math | 63 |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and they satisfy $b=7a\sin B$. Find $\sin A=\_\_\_\_\_\_$. If $B=60^{\circ}$, find $\sin C=\_\_\_\_\_\_$. | \sin C = \frac{13}{14} | math | 78 |
Let \( A, B, C \) be the angles of a triangle where \( A = 45^\circ \) and \( A + B + C = 180^\circ \). Compute:
\[
\begin{vmatrix}
\tan A & 1 & 1 \\
1 & \tan B & 1 \\
1 & 1 & \tan C
\end{vmatrix}.
\] | 2 | math | 91 |
(6 points) (2015•Lishui Mock Test) Suppose the sequence $\{a_n\}$ is an arithmetic sequence with a common difference of $d$, and $a_1+a_3+a_5=105$, $a_2+a_4+a_6=99$. Find the value of $d$, the general term $a_n$, and the value of $n$ for which the sum of the first $n$ terms $S_n$ is maximized. | 20 | math | 107 |
Let \( h(x) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 \). What is the remainder when the polynomial \( h(x^{10}) \) is divided by the polynomial \( h(x) \)? | 7 | math | 59 |
Given that $\{a_n\}$ is an increasing arithmetic sequence, and $a_2$, $a_4$ are the roots of the equation $x^2-5x+6=0$.
$(1)$ Find the general formula for $\{a_n\}$;
$(2)$ Find the sum of the first $n$ terms of the sequence $\left\{\dfrac{a_n}{2^n}\right\}$. | 2- \dfrac{n+4}{2^{n+1}} | math | 92 |
Suppose that \(a\) and \(b\) are positive integers such that \(a-b=8\) and \(\text{gcd}\left(\frac{a^3+b^3}{a+b}, ab\right) = 16\). Find the smallest possible value of \(b\). | 4 | math | 62 |
Real numbers \( x \) and \( y \) satisfy the equation \( 4x^{2} - 5xy + 4y^{2} = 5 \). Let \( S = x^{2} + y^{2} \). Find the value of \( \frac{1}{S_{\max}} + \frac{1}{S_{\min}} \). | \frac{8}{5} | math | 80 |
In triangle $ABC$, where $A$ is at (0, 4), $B$ is at (0, 0), and $C$ is at (3, 0), find $\tan A$. | \frac{3}{4} | math | 45 |
Suppose $\mathbf{a}$ and $\mathbf{b}$ are unit vectors with an angle of $\frac{\pi}{4}$ between them. Calculate the volume of the parallelepiped formed by the vectors $\mathbf{a}, \mathbf{b},$ and $\mathbf{b} + \mathbf{b} \times \mathbf{a}$. | \frac{1}{2} | math | 80 |
Given the regression equation $\hat y=2x+1$, and a set of data obtained from the experiment is $(2,4.9)$, $(3,7.1)$, $(4,9.1)$, then the sum of squared residuals is ______. | 0.03 | math | 56 |
Given the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$, calculate how many different subsets of two elements can be eliminated such that the average of the remaining elements is 7. | 3 | math | 68 |
Given the polynomial $f(x) = x^5 + 4x^4 + x^2 + 20x + 16$, evaluate $f(-2)$ using the Qin Jiushao algorithm to find the value of $v_2$. | -4 | math | 54 |
Let $z = a + bi,$ where $a$ and $b$ are positive real numbers. If
\[z^3 + |z|^2 + z = 0,\]then enter the ordered pair $(a,b).$ | \left( \frac{1}{2}, \frac{\sqrt{7}}{2} \right) | math | 50 |
Let $p,$ $q,$ and $r$ be real numbers, and let $A,$ $B,$ $C$ be points such that the midpoint of $\overline{BC}$ is $(p, p, 0),$ the midpoint of $\overline{AC}$ is $(0, q, q),$ and the midpoint of $\overline{AB}$ is $(r, 0, r).$ Find
\[\frac{AB^2 + AC^2 + BC^2}{p^2 + q^2 + r^2}.\] | 8 | math | 118 |
Given a parabola $C$ that passes through the point $(4,4)$ and its focus lies on the $x$-axis.
$(1)$ Find the standard equation of parabola $C$.
$(2)$ Let $P$ be any point on parabola $C$. Find the minimum distance between point $P$ and the line $x - y + 4 = 0$. | \frac{3\sqrt{2}}{2} | math | 85 |
The constant term in the expansion of $(x^{2}+1)\left( \dfrac {1}{ \sqrt {x}}-2\right)^{5}$ is to be found. | -42 | math | 41 |
Given circles $C\_1$: $x^{2}+y^{2}=4$ and $C\_2$: $x^{2}+y^{2}-4x+2y+4=0$, determine the number of their common tangent lines. | 2 | math | 54 |
What is the least integer value of $b$ such that $-10$ is not in the range of $y = x^2 + bx + 20$? | -10 | math | 37 |
Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, the point $(n, \frac{S_{n}}{n})$ lies on the line $y=\frac{1}{2}x+\frac{11}{2}$. Another sequence $\{b_n\}$ satisfies the relation $b_{n+2} - 2b_{n+1} + b_n = 0$ ($n\in \mathbb{N}^*$), and it is given that $b_3=11$ and the sum of the first 9 terms is... | m = 11 | math | 372 |
Given the quadratic function $f(x)=x^{2}+ax+b$ satisfies $f(0)=6$, $f(1)=5$
$(1)$ Find the expression for the function $f(x)$
$(2)$ Find the maximum and minimum values of the function $f(x)$ when $x \in [-2,2]$. | 14 | math | 72 |
John's quiz scores so far are: 85, 88, 90, 92, and 83. What score does he need to get on his sixth quiz to make the arithmetic mean of the six scores equal to 90? | 102 | math | 55 |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=1$, $(\overrightarrow{a}+\overrightarrow{b}) \perp \overrightarrow{a}$, and $(2\overrightarrow{a}+\overrightarrow{b}) \perp \overrightarrow{b}$, calculate the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{3\pi}{4} | math | 93 |
In triangle $ABC$, $\angle C = 90^\circ$, $AC = 5$ and $BC = 12$. Point $D$ is on $\overline{AB}$, point $E$ is on $\overline{BC}$, and point $F$ is on $\overline{AC}$ such that $\angle FED = 90^\circ$. If $DE = 5$ and $DF = 3$, then what is the length of $BD$? | BD = 10 | math | 105 |
Points $A$, $B$, and $C$ are on the same line. Given that $AB=5$ cm and $BC=4$ cm, find the distance between points $A$ and $C$. | 1\, \text{cm} \text{ or } 9\, \text{cm} | math | 45 |
Two thieves stole an open chain with $2k$ white beads and $2m$ black beads. They want to share the loot equally, by cutting the chain to pieces in such a way that each one gets $k$ white beads and $m$ black beads. What is the minimal number of cuts that is always sufficient? | 2 | math | 75 |
If the total sum of squares for a sample is $256$ and the residual sum of squares is $32$, calculate the regression sum of squares. | 224 | math | 33 |
Calculate:<br/>$(1){({-\frac{1}{2}})^{-2}}-\sqrt{3}\cos60°-|{2-\sqrt{3}}|-{({3-π})^0}-\sqrt{12}$;<br/>$(2)$ Simplify first, then evaluate: $({\frac{{x+2}}{{{x^2}-2x}}-\frac{{x-1}}{{{x^2}-4x+4}}})÷\frac{{x-4}}{x}$, where $x=\tan 30^{\circ}+2$. | 3 | math | 128 |
Find all real \(x\) such that \(\left\lfloor x \lfloor x \rfloor\right \rfloor = 20.\) | [5, 5.25) | math | 33 |
If 700 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 30 | math | 35 |
Given a set of positive numbers $(x\_1)$, $(x\_2)$, $(x\_3)$ with a variance of $s^{2}= \frac {1}{3}(x\_1^{2}+x\_2^{2}+x\_3^{2}-12)$, find the average of the data $(x\_1+1)$, $(x\_2+1)$, $(x\_3+1)$. | 3 | math | 93 |
Given that the ellipse $C$ passes through point $A(2,3)$, and point $F(2,0)$ is its right focus, find the standard equation of the ellipse $C$. | \frac{x^{2}}{16}+\frac{y^{2}}{12}=1 | math | 42 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\cos B = \frac{4}{5}$.
(1) If $c = 2a$, find the value of $\frac{\sin B}{\sin C}$;
(2) If $C - B = \frac{\pi}{4}$, find the value of $\sin A$. | \sin A = \frac{31\sqrt{2}}{50} | math | 97 |
Let $p(x)$ be the product of all digits of the decimal integer $x$. Find all positive integers $x$ such that $p(x) = x^2 - 10x - 22$. | 12 | math | 45 |
The radius of the base of a cone is $1$, and the slant height is $3$. Calculate the central angle of its lateral surface development diagram. | 120 | math | 32 |
Given the set $A=\{x\in R|ax^{2}-3x+2=0\}$.
(1) If $A=\varnothing$, find the range of values for the real number $a$.
(2) If $A$ is a singleton set, find the value of $a$ and the set $A$. | k=\frac{9}{8}, A=\{\frac{4}{3}\} | math | 73 |
$n$ coins lies in the circle. If two neighbour coins lies both head up or both tail up, then we can flip both. How many variants of coins are available that can not be obtained from each other by applying such operations? | 2 | math | 49 |
Let $a_0 = -3, b_0 = 2$, and for $n \geq 0$, let
\begin{align*}
a_{n+1} &= 2a_n + 2b_n + \sqrt{a_n^2 + b_n^2}, \\
b_{n+1} &= 2a_n + 2b_n - \sqrt{a_n^2 + b_n^2}.
\end{align*}
Find $\frac{1}{a_{2012}} + \frac{1}{b_{2012}}.$ | \frac{1}{6} | math | 128 |
Given a right-angle coordinate system with its origin $O$ as the pole, the positive semi-axis of the $x$-axis as the polar axis, and both coordinate systems having equal length units $.$, the parametric equation of line $l$ is $ \begin{cases} x=t\sin \phi \\ y=1+t\cos \phi \end{cases} (t$ is the parameter, $0 < φ < π$)... | 4 | math | 182 |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{a}-\overrightarrow{b}|=\sqrt{3}$, $\overrightarrow{a}•\overrightarrow{b}=1$, calculate the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{3} | math | 79 |
In the book "Nine Chapters on the Mathematical Art," a tetrahedron with all four faces being right-angled triangles is called a "biēnào." Given that tetrahedron $ABCD$ is a "biēnào," $AB\bot $ plane $BCD$, $BC\bot CD$, and $AB=\frac{1}{2}BC=\frac{1}{3}CD$. If the volume of this tetrahedron is $1$, then the surface area... | 14\pi | math | 114 |
Let $f_{n}(x)=\cos x\cos \frac{x}{2}\cos \frac{x}{4}\cdot \ldots \cdot \cos \frac{x}{{{2^{n-1}}}$, evaluate $f_5\left(\frac{{8\pi}}{3}\right)$. | -\frac{\sqrt{3}}{32} | math | 67 |
A line is parameterized by a parameter $t,$ so that the vector on the line at $t = 1$ is $\begin{pmatrix} 2 \\ 4 \\ 9 \end{pmatrix},$ and the vector on the line at $t = -1$ is $\begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}.$ Find the vector on the line at $t = 0.$ | \begin{pmatrix} 1/2 \\ 5/2 \\ 11/2 \end{pmatrix} | math | 96 |
Two numbers, \( x \) and \( y \), are randomly selected from the interval \((0,1)\). What is the probability that \(\left\lfloor \log _{2} x \right\rfloor = \left\lfloor \log _{2} y \right\rfloor \)? If necessary, round the answer to two decimal places. Note that \([a]\) denotes the greatest integer less than or equal ... | 0.33 | math | 99 |
Given the function $f(x)=\sin (\omega x-\varphi)$ $(\omega > 0,|\varphi| < \frac {\pi}{2})$ whose graph intersects with the x-axis at points that are a distance of $\frac {\pi}{2}$ apart, and it passes through the point $(0,- \frac {1}{2})$
$(1)$ Find the analytical expression of the function $f(x)$;
$(2)$ Let the ... | \frac {1}{2} | math | 144 |
Expanding $(1+0.2)^{1000}$ by the binomial theorem and doing no further manipulation gives
\[{1000 \choose 0}(0.2)^0+{1000 \choose 1}(0.2)^1+{1000 \choose 2}(0.2)^2+\cdots+{1000 \choose 1000}(0.2)^{1000}= A_0 + A_1 + A_2 + \cdots + A_{1000},\]where $A_k = {1000 \choose k}(0.2)^k$ for $k = 0,1,2,\ldots,1000.$ For which ... | 166 | math | 179 |
A student made a mistake when simplifying $\frac{1}{2}x-2(x-\frac{1}{3}y^{2})+(-\frac{3}{2}x+\frac{1}{3}y^{2})$. The solution process is as follows:
Original expression $=\frac{1}{2}x-(2x-\frac{2}{3}y^{2})+(-\frac{3}{2}x+\frac{1}{3}y^{2})$ (Step 1)
$=\frac{1}{2}x-2x-\frac{2}{3}y^{2}-\frac{3}{2}x+\frac{1}{3}y^{2}$ (... | -3x + y^{2} | math | 214 |
The minimum value of the distance $|AB|$ is to be found, where points $A$ and $B$ are the intersections of the line $y=m$ with the curves $y = 2(x+1)$ and $y = x + \ln x$ respectively. | \frac{3}{2} | math | 58 |
On an island, there are knights, who always tell the truth, and liars, who always lie. There are 15 islanders in a room. Each of the islanders in the room made two statements: "Among my acquaintances in this room, there are exactly six liars" and "Among my acquaintances in this room, there are no more than seven knight... | 9 \text{ knights} | math | 88 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.