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40.3k
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100
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|---|---|---|---|
21,100 | What is the coefficient of \(x^3y^5\) in the expansion of \(\left(\frac{2}{3}x - \frac{4}{5}y\right)^8\)? | -\frac{458752}{84375} | 42.1875 |
21,101 | Given the singing scores 9.4, 8.4, 9.4, 9.9, 9.6, 9.4, 9.7, calculate the average and variance of the remaining data after removing the highest and lowest scores. | 0.016 | 43.75 |
21,102 | What is the maximum value of $\frac{(3^t-2t)t}{9^t}$ for real values of $t$?
A) $\frac{1}{10}$
B) $\frac{1}{12}$
C) $\frac{1}{8}$
D) $\frac{1}{6}$
E) $\frac{1}{4}$ | \frac{1}{8} | 31.25 |
21,103 | What is the base $2$ representation of $125_{10}$? | 1111101_2 | 98.4375 |
21,104 | Given $f(x)= \sqrt {3}\sin \dfrac {x}{4}\cos \dfrac {x}{4}+ \cos ^{2} \dfrac {x}{4}+ \dfrac {1}{2}$.
(1) Find the period of $f(x)$;
(2) In $\triangle ABC$, sides $a$, $b$, and $c$ correspond to angles $A$, $B$, and $C$ respectively, and satisfy $(2a-c)\cos B=b\cos C$, find the value of $f(B)$. | \dfrac{\sqrt{3}}{2} + 1 | 80.46875 |
21,105 | In $\triangle ABC$, $\frac{1}{2}\cos 2A = \cos^2 A - \cos A$.
(I) Find the measure of angle $A$;
(II) If $a=3$, $\sin B = 2\sin C$, find the area of $\triangle ABC$. | \frac{3\sqrt{3}}{2} | 87.5 |
21,106 | A novice economist-cryptographer received a cryptogram from a ruler which contained a secret decree about implementing an itemized tax on a certain market. The cryptogram specified the amount of tax revenue that needed to be collected, emphasizing that a greater amount could not be collected in that market. Unfortunately, the economist-cryptographer made an error in decrypting the cryptogram—the digits of the tax revenue amount were identified in the wrong order. Based on erroneous data, a decision was made to introduce an itemized tax on producers of 90 monetary units per unit of goods. It is known that the market demand is represented by \( Q_d = 688 - 4P \), and the market supply is linear. When there are no taxes, the price elasticity of market supply at the equilibrium point is 1.5 times higher than the modulus of the price elasticity of the market demand function. After the tax was introduced, the producer price fell to 64 monetary units.
1) Restore the market supply function.
2) Determine the amount of tax revenue collected at the chosen rate.
3) Determine the itemized tax rate that would meet the ruler's decree.
4) What is the amount of tax revenue specified by the ruler to be collected? | 6480 | 2.34375 |
21,107 | Given that $α∈(0, \dfrac {π}{2})$ and $β∈(0, \dfrac {π}{2})$, with $cosα= \dfrac {1}{7}$ and $cos(α+β)=- \dfrac {11}{14}$, find the value of $sinβ$. | \dfrac{\sqrt{3}}{2} | 26.5625 |
21,108 | Given that the coefficient of the $x^3$ term in the expansion of $\left(x+a\right)\left(x-2\right)^5$ is $-60$, find the value of $a$. | \frac{1}{2} | 60.15625 |
21,109 | Given that point P(x, y) satisfies the equation (x-4 cos θ)^{2} + (y-4 sin θ)^{2} = 4, where θ ∈ R, find the area of the region that point P occupies. | 32 \pi | 92.1875 |
21,110 | If five squares of a $3 \times 3$ board initially colored white are chosen at random and blackened, what is the expected number of edges between two squares of the same color?
| \frac{16}{3} | 58.59375 |
21,111 | On the first day, 1 bee brings back 5 companions. On the second day, 6 bees (1 from the original + 5 brought back on the first day) fly out, each bringing back 5 companions. Determine the total number of bees in the hive after the 6th day. | 46656 | 67.1875 |
21,112 | Distribute 10 volunteer positions among 4 schools, with the requirement that each school receives at least one position. How many different ways can the positions be distributed? (Answer with a number.) | 84 | 97.65625 |
21,113 | Consider an isosceles right triangle with leg lengths of 1 each. Inscribed in this triangle is a square in such a way that one vertex of the square coincides with the right-angle vertex of the triangle. Another square with side length $y$ is inscribed in an identical isosceles right triangle where one side of the square lies on the hypotenuse of the triangle. What is $\dfrac{x}{y}$?
A) $\frac{1}{\sqrt{2}}$
B) $1$
C) $\sqrt{2}$
D) $\frac{\sqrt{2}}{2}$ | \sqrt{2} | 14.84375 |
21,114 | Derek is deciding between two different-sized pizzas at his favorite restaurant. The menu lists a 14-inch pizza and an 18-inch pizza. Calculate the percent increase in area if Derek chooses the 18-inch pizza over the 14-inch pizza. | 65.31\% | 44.53125 |
21,115 | Given that the center of the hyperbola is at the origin and one focus is F<sub>1</sub>(-$$\sqrt{5}$$, 0), if point P is on the hyperbola and the midpoint of segment PF<sub>1</sub> has coordinates (0, 2), then the equation of this hyperbola is _________ and its eccentricity is _________. | \sqrt{5} | 76.5625 |
21,116 | Peter borrows $2000$ dollars from John, who charges an interest of $6\%$ per month (which compounds monthly). What is the least integer number of months after which Peter will owe more than three times as much as he borrowed? | 19 | 2.34375 |
21,117 | The sum of the numbers 300, 2020, and 10001 is | 12321 | 67.96875 |
21,118 | If a computer executes the following program:
1. Initial values \( x = 3 \), \( S = 0 \)
2. \( x = x + 2 \)
3. \( S = S + x \)
4. If \( S \geq 10000 \), go to step 5; otherwise, repeat from step 2
5. Print \( x \)
6. Stop
What value will be printed by step 5? | 201 | 96.09375 |
21,119 | In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\left(a-c\right)\left(a+c\right)\sin C=c\left(b-c\right)\sin B$.
$(1)$ Find angle $A$;
$(2)$ If the area of $\triangle ABC$ is $\sqrt{3}$, $\sin B\sin C=\frac{1}{4}$, find the value of $a$. | 2\sqrt{3} | 67.1875 |
21,120 | Given a parabola $y=x^{2}-7$, find the length of the line segment $|AB|$ where $A$ and $B$ are two distinct points on it that are symmetric about the line $x+y=0$. | 5 \sqrt{2} | 22.65625 |
21,121 | Given the vertices of a pentagon at coordinates $(1, 1)$, $(4, 1)$, $(5, 3)$, $(3, 5)$, and $(1, 4)$, calculate the area of this pentagon. | 12 | 33.59375 |
21,122 | To popularize knowledge of fire safety, a certain school organized a competition on related knowledge. The competition is divided into two rounds, and each participant must participate in both rounds. If a participant wins in both rounds, they are considered to have won the competition. It is known that in the first round of the competition, the probabilities of participants A and B winning are $\frac{4}{5}$ and $\frac{3}{5}$, respectively; in the second round, the probabilities of A and B winning are $\frac{2}{3}$ and $\frac{3}{4}$, respectively. A and B's outcomes in each round are independent of each other.<br/>$(1)$ The probability of A winning exactly one round in the competition;<br/>$(2)$ If both A and B participate in the competition, find the probability that at least one of them wins the competition. | \frac{223}{300} | 24.21875 |
21,123 | Given that $\cos(\alpha+\pi)= \frac {3}{5}$ and $\pi \leq \alpha < 2\pi$, calculate the value of $\sin(-\alpha-2\pi)$. | \frac{4}{5} | 98.4375 |
21,124 | A region is bounded by semicircular arcs created on each side of an equilateral triangle, where each side measures \(1/\pi\). Calculate the total perimeter of this shaped region. | \frac{3}{2} | 75.78125 |
21,125 | The owner of an individual clothing store purchased 30 dresses for $32 each. The selling price of the 30 dresses varies for different customers. Using $47 as the standard price, any excess amount is recorded as positive and any shortfall is recorded as negative. The results are shown in the table below:
| Number Sold | 7 | 6 | 3 | 5 | 4 | 5 |
|-------------|---|---|---|---|---|---|
| Price/$ | +3 | +2 | +1 | 0 | -1 | -2 |
After selling these 30 dresses, how much money did the clothing store earn? | 472 | 17.96875 |
21,126 | Given the function $f(x)= \sqrt{2}\sin \left( 2x- \frac{\pi}{4} \right)$, where $x\in\mathbb{R}$, if the maximum and minimum values of $f(x)$ in the interval $\left[ \frac{\pi}{8}, \frac{3\pi}{4} \right]$ are $a$ and $b$ respectively, then the value of $a+b$ is ______. | \sqrt{2}-1 | 60.15625 |
21,127 | Five glass bottles can be recycled to make a new bottle. Additionally, for every 20 new bottles created, a bonus bottle can be made from residual materials. Starting with 625 glass bottles, how many total new bottles can eventually be made from recycling and bonuses? (Keep counting recycled and bonus bottles until no further bottles can be manufactured. Do not include the original 625 bottles in your count.) | 163 | 0 |
21,128 | Given a matrix $\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix}$ satisfies: $a_{11}$, $a_{12}$, $a_{21}$, $a_{22} \in \{0,1\}$, and $\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22}\end{vmatrix} =0$, determine the total number of distinct matrices. | 10 | 53.90625 |
21,129 | Read the following material and then answer the following questions: When simplifying algebraic expressions, sometimes we encounter expressions like $\frac{5}{\sqrt{3}}$, $\frac{2}{\sqrt{3}+1}$, in fact, we can further simplify them:<br/>$($1) $\frac{5}{\sqrt{3}}=\frac{5×\sqrt{3}}{\sqrt{3}×\sqrt{3}}=\frac{5}{3}\sqrt{3}$;<br/>$($2) $\frac{2}{\sqrt{3}+1}=\frac{2×(\sqrt{3}-1)}{(\sqrt{3}+1)(\sqrt{3}-1)}=\frac{2(\sqrt{3}-1)}{(\sqrt{3})^{2}-1}=\sqrt{3}-1$;<br/>$($3) $\frac{2}{\sqrt{3}+1}=\frac{3-1}{\sqrt{3}+1}=\frac{(\sqrt{3})^{2}-{1}^{2}}{\sqrt{3}+1}=\frac{(\sqrt{3}+1)(\sqrt{3}-1)}{\sqrt{3}+1}=\sqrt{3}-1$. The method of simplification mentioned above is called rationalizing the denominator.<br/>$(1)$ Simplify $\frac{2}{\sqrt{5}+\sqrt{3}}$ using different methods:<br/>① Refer to formula (2) to simplify $\frac{2}{\sqrt{5}+\sqrt{3}}=\_\_\_\_\_\_.$<br/>② Refer to formula (3) to simplify $\frac{2}{\sqrt{5}+\sqrt{3}}=\_\_\_\_\_\_.$<br/>$(2)$ Simplify: $\frac{1}{\sqrt{3}+1}+\frac{1}{\sqrt{5}+\sqrt{3}}+\frac{1}{\sqrt{7}+\sqrt{5}}+\ldots +\frac{1}{\sqrt{99}+\sqrt{97}}$. | \frac{3\sqrt{11}-1}{2} | 59.375 |
21,130 | Grandma Wang has 6 stools that need to be painted by a painter. Each stool needs to be painted twice. The first coat takes 2 minutes, but there must be a 10-minute wait before applying the second coat. How many minutes will it take to paint all 6 stools? | 24 | 8.59375 |
21,131 | Given the function $y = \cos\left(x+ \frac {\pi}{5}\right)$, where $x\in\mathbb{R}$, determine the horizontal shift required to obtain this function's graph from the graph of $y=\cos x$. | \frac {\pi}{5} | 39.84375 |
21,132 | Consider the ellipse $\frac{x^{2}}{6} + \frac{y^{2}}{2} = 1$ and the hyperbola $\frac{x^{2}}{3} - y^{2} = 1$ with common foci $F_{1}$ and $F_{2}$. Let $P$ be one of the intersection points of the two curves. Find the value of $\cos \angle F_{1}PF_{2}$. | \frac{1}{3} | 52.34375 |
21,133 | Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\diamondsuit$? | \frac{1}{52} | 45.3125 |
21,134 | Given four points on a sphere, $A$, $B$, $C$, $D$, with the center of the sphere being point $O$, and $O$ is on $CD$. If the maximum volume of the tetrahedron $A-BCD$ is $\frac{8}{3}$, then the surface area of sphere $O$ is ______. | 16\pi | 34.375 |
21,135 | Alex, Bonnie, and Chris each have $3$ blocks, colored red, blue, and green; and there are $3$ empty boxes. Each person independently places one of their blocks into each box. Each block placement by Bonnie and Chris is picked such that there is a 50% chance that the color matches the color previously placed by Alex or Bonnie respectively. Calculate the probability that at least one box receives $3$ blocks all of the same color.
A) $\frac{27}{64}$
B) $\frac{29}{64}$
C) $\frac{37}{64}$
D) $\frac{55}{64}$
E) $\frac{63}{64}$ | \frac{37}{64} | 5.46875 |
21,136 | Country $X$ has $40\%$ of the world's population and $60\%$ of the world's wealth. Country $Y$ has $20\%$ of the world's population but $30\%$ of its wealth. Country $X$'s top $50\%$ of the population owns $80\%$ of the wealth, and the wealth in Country $Y$ is equally shared among its citizens. Determine the ratio of the wealth of an average citizen in the top $50\%$ of Country $X$ to the wealth of an average citizen in Country $Y$. | 1.6 | 10.9375 |
21,137 | In acute triangle ABC, the sides opposite to angles A, B, C are a, b, c respectively, and a = 2b*sin(A).
(1) Find the measure of angle B.
(2) If a = $3\sqrt{3}$, c = 5, find b. | \sqrt{7} | 77.34375 |
21,138 | In response to the call for rural revitalization, Xiao Jiao, a college graduate who has successfully started a business in another place, resolutely returned to her hometown to become a new farmer and established a fruit and vegetable ecological planting base. Recently, in order to fertilize the vegetables in the base, she is preparing to purchase two types of organic fertilizers, A and B. It is known that the price per ton of organic fertilizer A is $100 more than the price per ton of organic fertilizer B. The total cost of purchasing 2 tons of organic fertilizer A and 1 ton of organic fertilizer B is $1700. What are the prices per ton of organic fertilizer A and B? | 500 | 59.375 |
21,139 | In triangle $ABC$, $BD$ is a median. $CF$ intersects $BD$ at $E$ so that $\overline{BE}=\overline{ED}$. Point $F$ is on $AB$. Then, if $\overline{BF}=5$, $\overline{BA}$ equals: | 15 | 24.21875 |
21,140 | Consider a city grid with intersections labeled A, B, C, and D. Assume a student walks from intersection A to intersection B every morning, always walking along the designated paths and only heading east or south. The student passes through intersections C and D along the way. The intersections are placed such that A to C involves 3 eastward moves and 2 southward moves, and C to D involves 2 eastward moves and 1 southward move, and finally from D to B requires 1 eastward move and 2 southward moves. Each morning, at each intersection where he has a choice, he randomly chooses whether to go east or south with probability $\frac{1}{2}$. Determine the probability that the student walks through C, and then D on any given morning.
A) $\frac{15}{77}$
B) $\frac{10}{462}$
C) $\frac{120}{462}$
D) $\frac{3}{10}$
E) $\frac{64}{462}$ | \frac{15}{77} | 13.28125 |
21,141 | A regular tetrahedron with four equilateral triangular faces has a sphere inscribed within it and another sphere circumscribed about it. Each of the four faces of the tetrahedron is tangent to a unique external sphere which is also tangent to the circumscribed sphere, but now these external spheres have radii larger than those in the original setup. Assume new radii are 50% larger than the radius of the inscribed sphere. A point $P$ is selected at random inside the circumscribed sphere. Compute the probability that $P$ lies inside one of these external spheres. | 0.5 | 39.0625 |
21,142 | Canadian currency has coins with values $2.00, 1.00, 0.25, 0.10,$ and $0.05. Barry has 12 coins including at least one of each of these coins. Find the smallest total amount of money that Barry could have. | 3.75 | 11.71875 |
21,143 | In $\triangle ABC$, let the sides opposite to angles $A$, $B$, and $C$ be $a$, $b$, and $c$ respectively. Given that $a\cos B=3$ and $b\sin A=4$.
(I) Find $\tan B$ and the value of side $a$;
(II) If the area of $\triangle ABC$ is $S=10$, find the perimeter $l$ of $\triangle ABC$. | 10 + 2\sqrt{5} | 20.3125 |
21,144 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $a > c$. Given that $\overrightarrow{BA} \cdot \overrightarrow{BC} = 2$ and $\cos B = \frac{1}{3}$, with $b=3$, find:
$(1)$ The values of $a$ and $c$; $(2)$ The value of $\cos(B - C)$. | \frac{23}{27} | 42.96875 |
21,145 | Point $P$ is a moving point on the curve $C_{1}$: $(x-2)^{2}+y^{2}=4$. Taking the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. Point $P$ is rotated counterclockwise by $90^{\circ}$ around the pole $O$ to obtain point $Q$. Suppose the trajectory equation of point $Q$ is the curve $C_{2}$.
$(1)$ Find the polar equations of curves $C_{1}$ and $C_{2}$;
$(2)$ The ray $\theta= \dfrac {\pi}{3}(\rho > 0)$ intersects curves $C_{1}$ and $C_{2}$ at points $A$ and $B$ respectively. Given the fixed point $M(2,0)$, find the area of $\triangle MAB$. | 3- \sqrt {3} | 0 |
21,146 | Celine has 240 feet of fencing. She needs to enclose a rectangular area such that the area is eight times the perimeter of the rectangle. If she uses up all her fencing material, how many feet is the largest side of the enclosure? | 101 | 1.5625 |
21,147 | In the diagram, \(p, q, r, s\), and \(t\) represent five consecutive integers, not necessarily in order. The two integers in the leftmost circle add to 63. The two integers in the rightmost circle add to 57. What is the value of \(r\)? | 30 | 18.75 |
21,148 | Given an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ with left and right foci $F_{1}(-c,0)$ and $F_{2}(c,0)$ respectively, and a point $P$ on the ellipse (different from the left and right vertices), the radius of the inscribed circle of $\triangle PF_{1}F_{2}$ is $r$. If the maximum value of $r$ is $\frac{c}{3}$, then the eccentricity of the ellipse is ______. | \frac{4}{5} | 66.40625 |
21,149 | In a circle with center $O$, $\overline{AB}$ and $\overline{CD}$ are diameters perpendicular to each other. Chord $\overline{DF}$ intersects $\overline{AB}$ at $E$. Given that $DE = 8$ and $EF = 4$, determine the area of the circle.
A) $24\pi$
B) $30\pi$
C) $32\pi$
D) $36\pi$
E) $40\pi$ | 32\pi | 9.375 |
21,150 | Construct a new sequence $\{a_n\}$ by extracting every 2nd, 4th, 8th, ..., $2^n$th, ... term from the sequence $\{3n+\log_2 n\}$ in the original order. Find the general term $a_n$ for the new sequence $\{a_n\}$ and calculate the sum of the first five terms, $S_5$. | 201 | 67.1875 |
21,151 | A point $Q$ is chosen in the interior of $\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\triangle DEF$, the resulting smaller triangles $u_{1}$, $u_{2}$, and $u_{3}$ have areas $9$, $16$, and $36$, respectively. Find the area of $\triangle DEF$. | 169 | 14.84375 |
21,152 | Given there are $50$ students in a class, a systematic sampling method is used to select $10$ students from these $50$. The $50$ students are randomly numbered from $1$ to $50$ and then evenly divided into $10$ groups in numerical order, with each group containing $5$ students. If the number drawn from the third group is $13$, determine the number drawn from the seventh group. | 33 | 91.40625 |
21,153 | A proper divisor of a number is one that is not equal to the number itself. What is the sum of the proper divisors of \( 450 \)? | 759 | 0 |
21,154 | Calculate the value of $\sin 135^{\circ}\cos (-15^{\circ}) + \cos 225^{\circ}\sin 15^{\circ}$. | \frac{1}{2} | 96.09375 |
21,155 | A large square has each side divided into four equal parts. A square is inscribed such that its vertices touch these division points, as illustrated below. Determine the ratio of the area of the inscribed square to the large square.
[asy]
draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);
draw((1,0)--(1,0.1)); draw((2,0)--(2,0.1)); draw((3,0)--(3,0.1));
draw((4,1)--(3.9,1)); draw((4,2)--(3.9,2)); draw((4,3)--(3.9,3));
draw((1,4)--(1,3.9)); draw((2,4)--(2,3.9)); draw((3,4)--(3,3.9));
draw((0,1)--(0.1,1)); draw((0,2)--(0.1,2)); draw((0,3)--(0.1,3));
draw((1,0)--(4,3)--(3,4)--(0,1)--cycle);
[/asy]
A) $\frac{1}{2}$
B) $\frac{1}{3}$
C) $\frac{5}{8}$
D) $\frac{3}{4}$ | \frac{5}{8} | 17.96875 |
21,156 | Evaluate \(\left(a^a - a(a-2)^a\right)^a\) when \( a = 4 \). | 1358954496 | 17.96875 |
21,157 | An ancient Greek was born on January 7, 40 B.C., and died on January 7, 40 A.D. Calculate the number of years he lived. | 79 | 62.5 |
21,158 | Given the curve E with the polar coordinate equation 4(ρ^2^-4)sin^2^θ=(16-ρ^2)cos^2^θ, establish a rectangular coordinate system with the non-negative semi-axis of the polar axis as the x-axis and the pole O as the coordinate origin.
(1) Write the rectangular coordinate equation of the curve E;
(2) If point P is a moving point on curve E, point M is the midpoint of segment OP, and the parameter equation of line l is $$\begin{cases} x=- \sqrt {2}+ \frac {2 \sqrt {5}}{5}t \\ y= \sqrt {2}+ \frac { \sqrt {5}}{5}t\end{cases}$$ (t is the parameter), find the maximum value of the distance from point M to line l. | \sqrt{10} | 1.5625 |
21,159 | Given that $a \in \{0,1,2\}$ and $b \in \{-1,1,3,5\}$, the probability that the function $f(x) = ax^2 - 2bx$ is an increasing function in the interval $(1, +\infty)$ is $(\quad\quad)$. | \frac{1}{3} | 5.46875 |
21,160 | In a square, points \(P\) and \(Q\) are the midpoints of the top and right sides, respectively. What fraction of the interior of the square is shaded when the region outside the triangle \(OPQ\) (assuming \(O\) is the bottom-left corner of the square) is shaded? Express your answer as a common fraction.
[asy]
filldraw((0,0)--(2,0)--(2,2)--(0,2)--cycle,gray,linewidth(1));
filldraw((0,2)--(2,1)--(2,2)--cycle,white,linewidth(1));
label("P",(0,2),N);
label("Q",(2,1),E);
[/asy] | \frac{1}{2} | 39.0625 |
21,161 | A robot colors natural numbers starting from 1 in ascending order according to the following rule: any natural number that can be expressed as the sum of two composite numbers is colored red, and those that do not meet the above criteria are colored yellow. If the numbers colored red are counted in ascending order, then the 1992nd number is $\boxed{2001}$. | 2001 | 91.40625 |
21,162 | A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answer as a common fraction. | \frac{9}{2} | 62.5 |
21,163 | Given $a^{x}=2$ and $a^{y}=3$, find the values of $a^{x+y}$ and $a^{2x-3y}$. | \frac{4}{27} | 99.21875 |
21,164 | In the set of four-digit numbers composed of the digits 0, 1, 2, 3, 4, 5 without any repetition, there are a total of numbers that are not divisible by 5. | 192 | 76.5625 |
21,165 | Twenty pairs of integers are formed using each of the integers \( 1, 2, 3, \ldots, 40 \) once. The positive difference between the integers in each pair is 1 or 3. If the resulting differences are added together, what is the greatest possible sum? | 58 | 31.25 |
21,166 | Car A and Car B are traveling from point A to point B. Car A departs 6 hours later than Car B. The speed ratio of Car A to Car B is 4:3. 6 hours after Car A departs, its speed doubles, and both cars arrive at point B simultaneously. How many hours in total did Car A take to travel from A to B? | 8.4 | 44.53125 |
21,167 | In the rectangular coordinate system, the line $l$ passes through the origin with an inclination angle of $\frac{\pi}{4}$; the parametric equations of the curve $C\_1$ are $\begin{cases} x = \frac{\sqrt{3}}{3} \cos \alpha \\ y = \sin \alpha \end{cases}$ (where $\alpha$ is the parameter); the parametric equations of the curve $C\_2$ are $\begin{cases} x = 3 + \sqrt{13} \cos \alpha \\ y = 2 + \sqrt{13} \sin \alpha \end{cases}$ (where $\alpha$ is the parameter).
(1) Find the polar coordinate equation of line $l$, and the Cartesian equations of curves $C\_1$ and $C\_2$;
(2) If the intersection points of line $l$ with curves $C\_1$ and $C\_2$ in the first quadrant are $M$ and $N$ respectively, find the distance between $M$ and $N$. | \frac{9\sqrt{2}}{2} | 92.1875 |
21,168 | If circle $$C_{1}: x^{2}+y^{2}+ax=0$$ and circle $$C_{2}: x^{2}+y^{2}+2ax+ytanθ=0$$ are both symmetric about the line $2x-y-1=0$, then $sinθcosθ=$ \_\_\_\_\_\_ . | -\frac{2}{5} | 3.125 |
21,169 | In an 11x11 grid making up a square, there are 121 uniformly spaced grid points including those on the edges. The point P is located in the very center of the square. A point Q is randomly chosen from the other 120 points. What is the probability that the line PQ is a line of symmetry for the square?
A) $\frac{1}{6}$
B) $\frac{1}{4}$
C) $\frac{1}{3}$
D) $\frac{1}{2}$
E) $\frac{2}{3}$ | \frac{1}{3} | 59.375 |
21,170 | There are three spheres and a cube. The first sphere is tangent to each face of the cube, the second sphere is tangent to each edge of the cube, and the third sphere passes through each vertex of the cube. What is the ratio of the surface areas of these three spheres? | 1:2:3 | 85.15625 |
21,171 | Convert $12012_3$ to a base 10 integer. | 140 | 100 |
21,172 | Given tetrahedron $P-ABC$, if one line is randomly selected from the lines connecting the midpoints of each edge, calculate the probability that this line intersects plane $ABC$. | \frac{3}{5} | 12.5 |
21,173 | Given $A_n^2 = 132$, calculate the value of $n$. | 12 | 76.5625 |
21,174 | Given that the function $f(x)=\ln (e^{x}+a+1)$ ($a$ is a constant) is an odd function on the set of real numbers $R$.
(1) Find the value of the real number $a$;
(2) If the equation $\frac{\ln x}{f(x)}=x^{2}-2ex+m$ has exactly one real root, find the value of $m$. | e^{2}+\frac{1}{e} | 0 |
21,175 | Three congruent cones, each with a radius of 8 cm and a height of 8 cm, are enclosed within a cylinder. The base of each cone is consecutively stacked and forms a part of the cylinder’s interior base, while the height of the cylinder is 24 cm. Calculate the volume of the cylinder that is not occupied by the cones, and express your answer in terms of $\pi$. | 1024\pi | 80.46875 |
21,176 | A three-wheeled vehicle travels 100 km. Two spare wheels are available. Each of the five wheels is used for the same distance during the trip. For how many kilometers is each wheel used? | 60 | 89.84375 |
21,177 | If $(1-2)^{9}=a_{9}x^{9}+a_{8}x^{8}+\ldots+a_{1}x+a_{0}$, then the sum of $a_1+a_2+\ldots+a$ is \_\_\_\_\_\_. | -2 | 29.6875 |
21,178 | Given the inequality $(|x|-1)^2+(|y|-1)^2<2$, determine the number of lattice points $(x, y)$ that satisfy it. | 16 | 82.03125 |
21,179 | In a speech competition, judges will score participants based on the content, delivery, and effectiveness of the speech, with weights of $4:4:2$ respectively. If a student receives scores of $91$, $94$, and $90$ in these three aspects, then the student's total score is ______ points. | 92 | 80.46875 |
21,180 | What is the largest four-digit number whose digits add up to 23? | 9950 | 90.625 |
21,181 | Given positive numbers $x$ and $y$ satisfying $x^2+y^2=1$, find the maximum value of $\frac {1}{x}+ \frac {1}{y}$. | 2\sqrt{2} | 91.40625 |
21,182 | Given the following system of equations: $$ \begin{cases} R I +G +SP = 50 R I +T + M = 63 G +T +SP = 25 SP + M = 13 M +R I = 48 N = 1 \end{cases} $$
Find the value of L that makes $LMT +SPR I NG = 2023$ true.
| \frac{341}{40} | 0 |
21,183 | Given a right triangular prism $ABC-A_{1}B_{1}C_{1}$, where $\angle BAC=90^{\circ}$, the area of the side face $BCC_{1}B_{1}$ is $16$. Find the minimum value of the radius of the circumscribed sphere of the right triangular prism $ABC-A_{1}B_{1}C_{1}$. | 2 \sqrt {2} | 0 |
21,184 | A certain school has a total of 1,000 students, including 380 first-year students and 180 male second-year students. It is known that the probability of drawing a female second-year student from all the students is 0.19. If stratified sampling (by grade) is used to draw 100 students from the entire school, then the number of third-year students to be drawn should be __________. | 25 | 96.09375 |
21,185 | What is $1010101_2 + 1001001_2$? Write your answer in base $10$. | 158 | 32.8125 |
21,186 | Point \( A \) lies on the line \( y = \frac{5}{12} x - 7 \), and point \( B \) lies on the parabola \( y = x^2 \). What is the minimum length of segment \( AB \)? | \frac{4007}{624} | 0 |
21,187 | The sequence $\{a\_n\}$ is a geometric sequence with the first term $a\_1=4$, and $S\_3$, $S\_2$, $S\_4$ form an arithmetic sequence.
(1) Find the general term formula of the sequence $\{a\_n\}$;
(2) If $b\_n=\log \_{2}|a\_n|$, let $T\_n$ be the sum of the first $n$ terms of the sequence $\{\frac{1}{b\_n b\_{n+1}}\}$. If $T\_n \leqslant \lambda b\_{n+1}$ holds for all $n \in \mathbb{N}^*$, find the minimum value of the real number $\lambda$. | \frac{1}{16} | 10.9375 |
21,188 | If $x + \frac{1}{x} = 3$, what is $x^6 + \frac{1}{x^6}$? | 322 | 93.75 |
21,189 | Add $91.234$ to $42.7689$ and round your answer to the nearest hundredth. | 134.00 | 96.875 |
21,190 | Given the function $f(x)=ax^{2}-2x+1$.
$(1)$ When $a\neq 0$, discuss the monotonicity of the function $f(x)$;
$(2)$ If $\frac {1}{3}\leqslant a\leqslant 1$, and the maximum value of $f(x)$ on $[1,3]$ is $M(a)$, the minimum value is $N(a)$, let $g(a)=M(a)-N(a)$, find the expression of $g(a)$;
$(3)$ Under the condition of $(2)$, find the minimum value of $g(a)$. | \frac {1}{2} | 13.28125 |
21,191 | You have four tiles marked X and three tiles marked O. The seven tiles are randomly arranged in a row. What is the probability that no two X tiles are adjacent to each other? | \frac{1}{35} | 73.4375 |
21,192 | Find the least integer value of $x$ for which $3|x| - 2 > 13$. | -6 | 94.53125 |
21,193 | Given four numbers $101010_{(2)}$, $111_{(5)}$, $32_{(8)}$, and $54_{(6)}$, the smallest among them is \_\_\_\_\_\_. | 32_{(8)} | 89.84375 |
21,194 | Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n$, and the sum of the first $n$ terms of the sequence ${b_n}$ is $T_n$. It is known that $a_1=2$, $3S_n=(n+m)a_n$, ($m\in R$), and $a_nb_n=\frac{1}{2}$. If for any $n\in N^*$, $\lambda>T_n$ always holds true, then the minimum value of the real number $\lambda$ is $\_\_\_\_\_\_$. | \frac{1}{2} | 24.21875 |
21,195 | Assume that the scores $X$ of 400,000 students in a math mock exam in Yunnan Province approximately follow a normal distribution $N(98,100)$. It is known that a certain student's score ranks among the top 9100 in the province. Then, the student's math score will not be lower than ______ points. (Reference data: $P(\mu -\sigma\ \ < X < \mu +\sigma )=0.6827, P(\mu -2\sigma\ \ < X < \mu +2\sigma )=0.9545$) | 118 | 73.4375 |
21,196 | An infinite geometric series has a first term of $15$ and a second term of $5$. A second infinite geometric series has the same first term of $15$, a second term of $5+n$, and a sum of three times that of the first series. Find the value of $n$. | 6.67 | 0 |
21,197 | For any positive integer \( n \), the value of \( n! \) is the product of the first \( n \) positive integers. Calculate the greatest common divisor of \( 8! \) and \( 10! \). | 40320 | 100 |
21,198 | If the graph of the function y = a^(x-2) (where a > 0 and a ≠ 1) always passes through a fixed point A, which lies on the graph of the linear function y = mx + 4n (where m, n > 0), find the minimum value of $\frac{1}{m} + \frac{2}{n}$. | 18 | 68.75 |
21,199 | In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given vectors $\overrightarrow{m} = (2\sin B, -\sqrt{3})$ and $\overrightarrow{n} = (\cos 2B, 2\cos^2 B - 1)$ and $\overrightarrow{m} \parallel \overrightarrow{n}$:
(1) Find the measure of acute angle $B$;
(2) If $b = 2$, find the maximum value of the area $S_{\triangle ABC}$ of triangle $ABC$. | \sqrt{3} | 70.3125 |
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