problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
If a line $l$ passes through the point $(-1,2)$ and is perpendicular to the line $x-3y+5=0$, then the equation of line $l$ is ________. | 3x+y+1=0 |
Given the complex number $z= \frac {(1+i)^{3}}{(1-i)^{2}}$ (where $i$ is the imaginary unit), the imaginary part of $z$ is ( )
A: -1
B: 1
C: -i
D: i | \text{A} |
## Task Condition
Calculate the volumes of bodies bounded by surfaces.
$$
z=x^{2}+5 y^{2}, z=5
$$ | \pi\cdot\frac{5\sqrt{5}}{2} |
Given a triangle $ABC$ with internal angles $A$, $B$, $C$ and opposite sides $a$, $b$, $c$ respectively, and area $S$. If $4S = a^2 - (b - c)^2$ and $b + c = 4$, find the maximum value of $S$. | 2 |
Given the function $$f(x)= \begin{cases} \log_{ \frac {1}{2}}x, & x\geq1 \\ 1-2^{x}, & x<1\end{cases}$$, then $f(f(2))= \_\_\_\_\_\_$. | \frac {1}{2} |
Given that line $l\_1$ passes through points $A(m,1)$ and $B(-3,4)$, and line $l\_2$ passes through points $C(1,m)$ and $D(-1,m+1)$, find the values of the real number $m$ when $l\_1$ is parallel to $l\_2$ or $l\_1$ is perpendicular to $l\_2$. | -\frac{9}{2} |
(50 points) Given a natural number $n \geqslant 5$. Try to find:
(1). In the $n$-element set $\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$, how many different numbers are produced by $a_{i}+a_{j}$ $(1<i<j \leqslant n)$ at least;
(2) Determine all $n$-element sets that achieve the above minimum value. | 2n-3 |
Given $x > 0$, $y > 0$, and $2x+y=2$, the minimum value of $\dfrac {1}{x}+ \dfrac {1}{y}$ is ( ).
A: $2$
B: $3$
C: $\dfrac{5}{2}$
D: $\dfrac {3}{2}+ \sqrt {2}$ | D |
5. On the sides of a square with length 1, isosceles trapezoids are constructed outward such that the vertices of all trapezoids are also vertices of a regular dodecagon. What is the perimeter of this dodecagon?
## City competition:2606, I. Faured,-B wkarijentan-: solutions to problems
Each correctly solved problem i... | 3 |
Let the sum of the first n terms of the sequence {an} be Sn, _____．
Choose one of the following three conditions to supplement the question below and answer it:
① The sequence {an} is a geometric sequence with a common ratio of 2, and a2, a3, a4-4 form an arithmetic sequence;
② Sn = 2an - 2;
③ Sn = 2n+1 - 2.
... | T_n = 3 - \frac{n+3}{2^n} |
Given the line $3x-4y+5=0$ and the circle $x^{2}+y^{2}=r^{2}$ $(r > 0)$, suppose they intersect at points $A$ and $B$, and $\angle AOB=120^{\circ}$ ($O$ is the coordinate origin), find the value of $r$. | 2 |
Given the function $f(x)= \frac {\cos x}{e^{x}}$, the equation of the tangent line to the graph of $f(x)$ at the point $(0,f(0))$ is __( )__.
A: $x+y+1=0$
B: $x+y-1=0$
C: $x-y+1=0$
D: $x-y-1=0$ | \text{B} |
On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell?
$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 35\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 112$ | 49 |
The central angle of a certain sector is $2$ radians, and its circumference is $4cm$. The area of this sector is _______ $cm^2$. | 1 |
The United Nations Educational, Scientific and Cultural Organization has designated March 14th as "International Mathematics Day" each year. A ninth-grade school conducted a reading activity with the theme of "Mathematics Culture" in March, and randomly sampled some students to read articles related to the activity.
... | 4470 |
10. (10 points) Taking 2 angles from the 4 interior angles of a quadrilateral to find their sum, there are 6 sum values in total, then the number of sums greater than $180^{\circ}$ is at most $\qquad$.
| 3 |
1. Find the sum of the digits of the number $A$, if $A=2^{63} \cdot 4^{25} \cdot 5^{106}-2^{22} \cdot 4^{44} \cdot 5^{105}-1$. Answer: 959. | 959 |
4. In $\triangle A B C$, the side lengths $a, b, c$ satisfy $(a+b+c)(a+b-c)=a b$, then the angle opposite to side $A B$ is ( ).
(A) $30^{\circ}$
(B) $60^{\circ}$
(C) $120^{\circ}$
(D) $150^{\circ}$ | C |
Example 3 Let $D$ be a point inside an acute $\triangle A B C$, $\angle A D B = \angle A C B + 90^{\circ}$, and $A C \cdot B D = A D \cdot B C$. Find the value of $\frac{A B \cdot C D}{A C \cdot B D}$. | \sqrt{2} |
V OM - I - Problem 7
In the plane, a line $ p $ and points $ A $ and $ B $ are given. Find a point $ M $ on the line $ p $ such that the sum of the squares $ AM^2 + BM^2 $ is minimized. | M |
Given real numbers $x$ and $y$ satisfying $x^2+y^2=1$, the minimum value of $$\frac {2xy}{x+y+1}$$ is ( )
A: $-1- \sqrt {2}$
B: $-1+ \sqrt {2}$
C: $1+ \sqrt {2}$
D: $1- \sqrt {2}$ | \text{A} |
6.4 Find the third term of an infinite geometric progression with a common ratio $|q|<1$, the sum of which is $\frac{8}{5}$, and the second term is $-\frac{1}{2}$. | \frac{1}{8} |
2. For the quadratic function $y=x^{2}+b x+c$, the vertex of its graph is $D$, and it intersects the positive x-axis at points $A$ and $B$ from left to right, and the positive y-axis at point $C$. If $\triangle A B D$ and $\triangle O B C$ are both isosceles right triangles (where $O$ is the origin), then $b+2 c=$ | 2 |
Given that $i$ is the imaginary unit, compute the product $(2+i)(1-3i)=$____. | 5-5i |
4.013. The sum of the first three terms of an increasing arithmetic progression is 15. If 1 is subtracted from the first two terms of this progression, and 1 is added to the third term, the resulting three numbers will form a geometric progression. Find the sum of the first 10 terms of the arithmetic progression. | 120 |
The condition "$x^{2}+5x-6 > 0$" is a __________ for "$x > 2$".
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D: Neither sufficient nor necessary condition | B |
## Problem Statement
Derive the equations of the tangent and normal lines to the curve at the point corresponding to the parameter value $t=t_{0}$.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=\sin t \\
y=a^{t}
\end{array}\right. \\
& t_{0}=0
\end{aligned}
$$ | x\cdot\ln+1-\frac{x}{\ln}+1 |
Given the function $f(x)=ax^{3}+x^{2}$ $(a\in\mathbb{R})$, the function takes an extreme value at $x=-\frac{4}{3}$. Find the value of $a$. | \frac{1}{2} |
If the graph of the function $y=\log_{2}(x-2)+3$ is translated by vector $\mathbf{a}$, resulting in the graph of the function $y=\log_{2}(x+1)-1$, then the coordinates of vector $\mathbf{a}$ are __________. | (-3, -4) |
Problem 10.3. On the side $AD$ of rectangle $ABCD$, a point $E$ is marked. On the segment $EC$, there is a point $M$ such that $AB = BM, AE = EM$. Find the length of side $BC$, given that $ED = 16, CD = 12$.
A: 6
B: 7
C: 8
D: 9 | C |
Given the function $f(x) = x\ln(x) - \frac{a}{2}x^{2}$, and the line $l: y = (k-2)x -k + 1$, where $k \in \mathbb{Z}$.
(1) If there exists an $x_0 \in [e,e^{2}]$ such that $f(x_0) > 0$ holds, find the range of values for the real number $a$.
(2) Assume $a = 0$, when $x > 1$, the graph of the function $f(x)$ is always a... | 4 |
Given the sets $M=\{x|x-2<0\}$, $N=\{x|x<a\}$. If $M \subseteq N$, then the range of the real number $a$ is ( )
A: $[2，+\infty)$
B: $(2，+\infty)$
C: $(-\infty，0)$
D: $(-\infty，0]$ | A |
When boarding the plane, a queue of p passengers forms, each with a ticket for one of $\mathrm{n}$ seats. The first in line is a crazy old lady. She rushes into the cabin and sits in a random seat (possibly even her own). Next, passengers take their seats in order, and if their seat is already occupied, they sit in a r... | \frac{1}{2} |
Given the function $f(x)=\log_{2}(x^{2}-2ax+3)$ is decreasing in the interval $\left( \frac{1}{2},1\right)$, the range of values for $a$ is \_\_\_\_\_\_. | [1,2] |
In the ancient Chinese mathematical masterpiece "The Mathematical Classic of Sunzi", there is a problem: "Three people share a car, and two cars are empty; two people share a car, and nine people walk. How many people and cars are there?" The essence of the problem is: there are a certain number of people and cars. If ... | D |
Problem 10.3. On the side $AD$ of rectangle $ABCD$, a point $E$ is marked. On the segment $EC$, there is a point $M$ such that $AB = BM, AE = EM$. Find the length of side $BC$, given that $ED = 16, CD = 12$.
^n \right) |
Three, (50 points) Find all positive integers $a, b, m, n$ such that $m$ and $n$ are coprime and $\left(a^{2}+b^{2}\right)^{m}=(a b)^{n}$.
| (,b,,n)=(2^{r},2^{r},2r,2r+1) |
11. Suppose that $\log _{2}\left[\log _{3}\left(\log _{4} a\right)\right]-\log _{3}\left[\log _{4}\left(\log _{2} b\right)\right]=\log _{4}\left[\log _{2}\left(\log _{3} c\right)\right]=0$. Find the value of $a+b+c$. | 89 |
10) A student of Fibonacci invented a sequence of numbers defined in this way: the first and second numbers of the sequence are 0 and 1, respectively; each number of the sequence, from the third onward, is equal to the sum of all the numbers that precede it (excluding itself!). What is the fifteenth number of the seque... | 4096 |
Given that $α$ is an angle in the fourth quadrant and $\cos ( \frac {π}{2}+α)= \frac {4}{5}$, find the value of $\tan α$. | -\frac{4}{3} |
Which of the following operations is correct?
A: $-2x+5x=-7x$
B: $yx-3xy=-2xy$
C: $-x^{2}-x^{2}=0$
D: $x^{2}-x=x$ | B |
The waiter at the restaurant U Šejdíře always adds the date to the bill: he increases the total amount spent by the paying customer by as many crowns as the day of the month it is.
In September, a trio of friends met at the restaurant twice. The first time, each of them paid separately, so the waiter added the date to... | 15 |
1. Two dice are thrown. Find the probability that the sum of the points on the faces that come up is even, and that a six appears on at least one of the dice. | \frac{5}{36} |
1. In a regular triangular prism $A B C-A_{1} B_{1} C_{1}$, $D$ is the midpoint of side $A C$.
(1) Prove: $A B_{1} / /$ plane $B D C_{1}$.
(2) When $\frac{A A_{1}}{A B}$ takes what value, $A B_{1} \perp B C_{1}$? | \frac{\sqrt{2}}{2} |
What is the smallest possible value of $\left|12^m-5^n\right|$, where $m$ and $n$ are positive integers? | 7 |
13. Given that $x+y=12$ and $x y=50$, find the exact value of $x^{2}+y^{2}$. | 44 |
In the Cartesian coordinate system, there is a point $P\left(x,y\right)$. If point $P$ is located in the second quadrant, and the distance from point $P$ to the $x$-axis and $y$-axis are $5$ and $2$ respectively, then the coordinates of point $P$ are ( ).
A: $(5,2)$
B: $(2,5)$
C: $(2,-5)$
D: $(-2,5)$ | D |
Let's find the largest number $A$ for which the following statement is true.
No matter how we choose seven real numbers between 1 and $A$, there will always be two of them for which the ratio $h$ satisfies $\frac{1}{2} \leqq h \leqq 2$. | 64 |
11. Tree person Groot is critically injured in a battle with a bounty hunter, and the Guardians of the Galaxy members need to inject it with 0.9% saline solution. Now there is 400 grams of 2.7% saline solution, and they need to add $\qquad$ grams of water to prepare the required saline solution. | 800 |
3. Determine all real numbers $x$ for which $\{x\}=\frac{x+[x]+\left[x+\frac{1}{2}\right]}{10}$, where $[x]$ is the greatest integer not greater than $x$ and $\{x\}=x-[x]$. | x_{1}=0,x_{2}=\frac{4}{3},x_{3}=\frac{25}{9} |
6. On an infinite tape, all natural numbers with the sum of digits equal to 2018 are written in ascending order. What number is written in the 225th position? (Method Commission) | 39..998(223nines) |
Problem 9.7. In triangle $ABC$, the bisector $AL$ is drawn. Points $E$ and $D$ are marked on segments $AB$ and $BL$ respectively such that $DL = LC$, $ED \parallel AC$. Find the length of segment $ED$, given that $AE = 15$, $AC = 12$.
.
(A) $\sqrt{2}$
(B) $\frac{\sqrt{2}}{2}$
(C) 2
(D) $\frac{1}{2}$ | D |
Given two distinct lines $l$ and $m$, and two distinct planes $α$ and $β$, which of the following propositions is correct? $(\,$ $)$
A: If $l$ is parallel to $α$ and $α∩β=m$, then $l$ is parallel to $m$.
B: If $l$ is perpendicular to $α$ and $l$ is parallel to $β$, then $α$ is perpendicular to $β$.
C: If $l$ is parall... | B |
The coordinates of point P (-2, -3) symmetric about the origin are ( )
A: (-2, 3)
B: (2, -3)
C: (2, 3)
D: (-3, -2) | \text{C} |
4. There is a basket of peaches, the number of which is a two-digit number. The first monkey takes away all $1 / 4$, the number of which is exactly the sum of the two digits of the total number. The second monkey takes away $1 / 4$ of the remainder, and the number is also exactly the sum of the two digits of the remain... | 48 |
1. At the intersection of roads $A$ and $B$ (straight lines) is a settlement $C$ (point). Sasha is walking along road $A$ towards point $C$, taking 45 steps per minute, with a step length of 60 cm. At the start of the movement, Sasha was 290 meters away from point $C$. Dan is walking towards $C$ along road $B$ at a spe... | d_{\}=57 |
Given $x>0, y>0$, and $2x+y=2$, find the minimum value of $\frac{1}{x}+\frac{1}{y}$. | \frac{3}{2} + \sqrt{2} |
12th Chinese 1997 Problem A1 The real numbers x 1 , x 2 , ... , x 1997 have sum -318 √3 and satisfy -1/√3 ≤ x i ≤ √3. What is the maximum possible value for the sum of their 12th powers? Solution | 1736/3^6+260\cdot3^6+2^{12}/3^6 |
# Problem 6. (3 points)
In an $8 \times 10$ table (8 rows, 10 columns), some cells are black, and the rest are white. In each white cell, the total number of black cells in the same row is written; nothing is written in the black cells. What is the maximum value that the sum of the numbers in the entire table can take... | 200 |
3. Let point $O$ be a point inside triangle $A B C$, and satisfy the relation
$$
\overrightarrow{O A}+2 \overrightarrow{O B}+3 \overrightarrow{O C}=3 \overrightarrow{A B}+2 \overrightarrow{B C}+\overrightarrow{C A}
$$
Then $\frac{S_{\triangle A O B}+2 S_{\triangle B O C}+3 S_{\triangle C O A}}{S_{\triangle A B C}}=$ $... | \frac{11}{6} |
Problem 10.8. Non-negative numbers $a, b, c$ sum up to 1. Find the maximum possible value of the quantity
$$
(a+3 b+5 c) \cdot\left(a+\frac{b}{3}+\frac{c}{5}\right)
$$ | \frac{9}{5} |
In the Cartesian coordinate system, the curve $C$ consists of all points whose product of distances to two fixed points $A(1,0)$ and $B(-1,0)$ equals $2$. For curve $C$, there are four statements:
1. The curve $C$ is symmetric about the axes;
2. The curve $C$ is symmetric about the origin;
3. All points on curve $C$ ar... | 1, 2 |
A tangent line is drawn at point $P(x_{0},y_{0})$ on the curve $y=e^{x}$. If the y-intercept of this tangent line is less than $0$, then the range of $x_{0}$ is ( ).
A: $(0,+∞)$
B: $( \dfrac {1}{e},+∞)$
C: $(1,+∞)$
D: $(2,+∞)$ | C |
1. Let $S=\{(x, y) \mid x y>0\}, T=\{(x, y) \mid$ $x>0$ and $y>0\}$. Then ().
(A) $S \cup T=S$
(B) $S \cup T=T$
(C) $S \cap T=S$
(D) $S \cap T=\varnothing$ | A |
9. Find all values of $a$ such that the roots $x_{1}, x_{2}, x_{3}$ of the polynomial $x^{3}-6 x^{2}+a x+a$ satisfy $\left(x_{1}-3\right)^{3}+\left(x_{2}-3\right)^{3}+\left(x_{3}-3\right)^{3}=0$. | -9 |
10. Let $p$ be a given odd prime. If the positive integer $k$ makes $\sqrt{k^{2}-p k}$ also a positive integer, then $k=$ $\qquad$ | \frac{(p+1)^2}{4} |
5. On the extension of side $A C$ of triangle $A B C$ beyond point $A$, point $T$ is marked such that $\angle B A C=2 \angle B T C$. Find the area of triangle $A B C$, given that $A B=A C, B T=42$, $A T=29$. | 420 |
2. Consider the natural number $a=1+3+3^{2}+3^{3}+\ldots+3^{2009}$.
a) Prove that the number $a$ is even.
b) Prove that the number $a$ is divisible by 13.
c) Find the remainder of the division of the number $b$ by 13, where $b=a+3^{2010}+3^{2011}$. | 4 |
The equation of the line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ can be represented as ( )
A: $$\frac {x-x_{1}}{x_{2}-x_{1}} = \frac {y-y_{1}}{y_{2}-y_{1}}$$
B: $$\frac {x-x_{2}}{x_{1}-x_{2}} = \frac {y-y_{2}}{y_{1}-y_{2}}$$
C: $(y-y_{1})(x_{2}-x_{1})=(x-x_{1})(y_{2}-y_{1})$
D: $y-y_{1}= \frac... | C |
Point $P$ is an arbitrary moving point on the line $3x-4y+2=0$. The minimum value of the distance from point $P$ to point $(3,-1)$ is ______. | 3 |
Let's determine $y$ such that the roots of the equation
$$
x^{2}-3 x y+y^{2}+2 x-9 y+1=0
$$
are equal ($x_{1}$ and $x_{2}$); for which values of $y$ are the roots of the equation real? Solve the equation for $y$ and determine $x$ such that the roots of the equation are real. | y_{1}=0,y_{2}=-\frac{24}{5},x\geqq-\frac{11}{5}orx\leqq-7 |
Sarah multiplied an integer by itself. Which of the following could be the result?
(A) 32
(B) 33
(C) 34
(D) 35
(E) 36 | 36 |
# 8.2. Condition:
A five-digit number is called a pit if the first three digits are in descending order, and the last three digits are in ascending order. For example, 73016 and 98346 are pits, while 88012 and 56821 are not pits. How many pits are there that are less than the number $22222?$ | 36 |
21. In triangle $P Q R$, the point $S$ is on $P Q$ so that the ratio of the length of $P S$ to the length of $S Q$ is $2: 3$. The point $T$ lies on $S R$ so that the area of triangle $P T R$ is 20 and the area of triangle SQT is 18 , as shown in the diagram.
What is the area of triangle $P Q R$ ?
A 100
B 90
C 80
D 70
E... | 80 |
Given that the central angle of a sector of a circle is 2 and its radius is 1, determine the area of the sector. | 1 |
Given circle $O: x^{2}+y^{2}=1$ intersects the negative half of the $x$-axis at point $A$ and the positive half of the $y$-axis at point $B$
$(1)$ If the line $l$ passing through point $C\left( \frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ cuts a chord of length $\sqrt{3}$ from circle $O$, find the equation of line $l$;
... | 1 |
The integer $m$ is a perfect cube exactly when it is equal to $n^{3}$ for some integer $n$. For example, 1000 is a perfect cube since $1000=10^{3}$. What is the smallest positive integer $k$ for which the integer $2^{4} \times 3^{2} \times 5^{5} \times k$ is a perfect cube?
(A) 12
(B) 30
(C) 60
(D) 480
(E) 1620 | 60 |
10. In a rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$ with edges $A B=4, A D=2$, $A A_{1}=3 \sqrt{2}$, a plane is drawn through the diagonal $B D_{1}$, intersecting the edge $A A_{1}$ such that the section of the parallelepiped by this plane has the smallest perimeter. Find the area of this section. | 8\sqrt{2} |
1. Let $\sin x+\cos x=\frac{1}{2}$. Then $\sin ^{3} x+\cos ^{3} x=$ $\qquad$ . | \frac{11}{16} |
7. For a positive integer $n$, let the product of its digits be denoted as $a(n)$. Then the positive integer solution to the equation $n^{2}-17 n+56=a(n)$ is
$\qquad$ . | 4 |
3. If $a, b, c$ are real numbers, $x=a^{2}-2 b+\frac{\pi}{3}, y=$ $b^{2}-2 c+\frac{\pi}{6}, z=c^{2}-2 a+\frac{\pi}{2}$, then among $x, y, z$
(A) at least one is greater than 0
(B) at least one is not less than 0
(C) at least one is not greater than 0
(D) at least one is less than 0 | A |
Given the equation of the line $$y-3= \sqrt {3}(x-4)$$, the fixed point through which this line passes and its angle of inclination are respectively ( )
A: (4, 3) and $$\frac {\pi}{3}$$
B: (-3, -4) and $$\frac {\pi}{6}$$
C: (4, 3) and $$\frac {\pi}{6}$$
D: (-4, -3) and $$\frac {\pi}{3}$$ | \text{A} |
Problem 3. The digit in the hundreds place of a given three-digit number is 7. If we move this digit to the units place, the digit in the tens place to the hundreds place, and the digit in the units place to the tens place, we will get a new three-digit number which is 567 less than the given number. Find the given num... | 714 |
The sales of a store in October were 5,000 yuan, and in December, they rose to 7,200 yuan. What is the average monthly growth rate? | 20\% |
Given points $M(4,0)$ and $N(1,0)$, any point $P$ on curve $C$ satisfies: $\overset{→}{MN} \cdot \overset{→}{MP} = 6|\overset{→}{PN}|$.
(I) Find the trajectory equation of point $P$;
(II) A line passing through point $N(1,0)$ intersects curve $C$ at points $A$ and $B$, and intersects the $y$-axis at point $H$. If $\o... | -\frac{8}{3} |
Given the sequence $\{a_n\}$ with the general term $a_n=2n-37$, when the sum of the first $n$ terms $S_n$ is minimized, the value of $n$ is ( )
A: $17$
B: $18$
C: $19$
D: $20$ | B |
Assume a university's female students' weight $y$ (unit: kg) has a linear correlation with their height $x$ (unit: cm). According to a set of sample data $(x_i, y_i) (i=1,2,…,n)$, the regression equation established by the least squares method is $\hat{y}=0.85x-85.71$. Among the following conclusions, the incorrect one... | D |
Given the function $f(x) = 2\left(x - \frac{1}{x}\right) - 2\ln x$, find the equation of the tangent line to the curve $y = f(x)$ at the point $(1, f(1))$. | 2x - y - 2 = 0 |
2. In an arithmetic progression, there are 12 terms, and their sum is 354. The sum of the terms with even indices is to the sum of the terms with odd indices as 32:27. Determine the first term and the common difference of the progression.
# | =2,=5 |
A line $L$ passes through the point $(\sqrt{2}, 0)$ and has exactly one common point with the hyperbola $x^2 - y^2 = 2$. The number of such lines is ( )
A: 1 line
B: 2 lines
C: 3 lines
D: 4 lines | C |
G. The number of non-overlapping triangles formed with the vertices of a regular $n$-sided polygon is equal to ( ).
(A) $\left[\frac{n^{2}}{10}\right]$
(B) $\left[\frac{n^{2}}{11}\right]$
(C) $\left[\frac{n^{2}}{12}\right]$
(D) None of the above | D |
$26 \cdot 3$ If a moving point $P(x, y)$ rotates counterclockwise on the unit circle with a constant angular velocity $\omega$, then the motion of point $Q\left(-2 x y, y^{2}-x^{2}\right)$ is
(A) rotating clockwise on the unit circle with angular velocity $\omega$.
(B) rotating counterclockwise on the unit circle with ... | C |
A rectangle with a length of 9 meters and a width of 6 meters is cut into the largest possible square. The area of the square is ____. | 36 |
1. The teacher cut a square sheet of paper with a side of 5 cm into two rectangles. The perimeter of one of these rectangles is $16 \mathrm{cm}$. What is the perimeter of the other? | 14 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.