problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
$$
\begin{array}{l}
\text { 9. For } \frac{1}{2} \leqslant x \leqslant 1 \text {, when } \\
(1+x)^{5}(1-x)(1-2 x)^{2}
\end{array}
$$
reaches its maximum value, $x=$ $\qquad$ . | x=\frac{7}{8} |
24. Define $\operatorname{gcd}(a, b)$ as the greatest common divisor of integers $a$ and $b$. Given that $n$ is the smallest positive integer greater than 1000 and satisfies:
$$
\begin{array}{l}
\operatorname{gcd}(63, n+120)=21, \\
\operatorname{gcd}(n+63,120)=60 .
\end{array}
$$
Then the sum of the digits of $n$ is (... | 18 |
10. A class has 48 students, of which 24 are interested in calligraphy, and 30 are interested in music. The number of students interested in both calligraphy and music is $n$. The sum of the maximum and minimum values of $n$ is $\qquad$ . | 30 |
8. Variant 1.
A line parallel to the leg $A C$ of the right triangle $A B C$ intersects the leg $B C$ at point $K$, and the hypotenuse $A B$ at point $N$. On the leg $A C$, a point $M$ is chosen such that $M K=M N$. Find the ratio $\frac{A M}{M C}$, if $\frac{B K}{B C}=14$. | 7 |
11. The floor plan of a castle is designed in this way: consider a circumference of 2019 meters with a regular polygon of 2019 vertices inscribed in it. Once the vertices of the polygon are numbered from 1 to 2019 in a clockwise direction, draw circles of length $2019 m$ centered at each point numbered with a perfect s... | 4038 |
In a triangle $ABC$, the median $AD$ (with $D$ on $BC$) and the angle bisector $BE$ (with $E$ on $AC$) are perpedicular to each other. If $AD = 7$ and $BE = 9$, find the integer nearest to the area of triangle $ABC$. | 47 |
10.4. Find the maximum value of the expression $a+b+c+d-ab-bc-cd-da$, if each of the numbers $a, b, c$ and $d$ belongs to the interval $[0 ; 1]$. | 2 |
Randomly select two numbers from the interval $[0,2]$. What is the probability that the larger of the two numbers is greater than $\frac{2}{3}$? | \frac{8}{9} |
8,9
Find the distance between the line passing through points $A(-3 ; 0 ; 1)$ and $B(2 ; 1 ;-1)$, and the line passing through points $C(-2 ; 2 ; 0)$ and $D(1 ; 3 ; 2)$. | \frac{5\sqrt{3}}{\sqrt{23}} |
Three. (25 points) Find all integer pairs $(a, b)$ that satisfy $a^{3}=7 \times 3^{b}-64$.
---
Please note that the translation retains the original formatting and structure of the text. | (-1,2),(5,3) |
2.297. $\sqrt[3]{26+15 \sqrt{3}} \cdot(2-\sqrt{3})=1$. | 1 |
Given that the area of a cross-section of sphere O is $\pi$, and the distance from the center O to this cross-section is 1, then the radius of this sphere is __________, and the volume of this sphere is __________. | \frac{8\sqrt{2}}{3}\pi |
Given $\alpha \in \left(\pi ,\frac{3\pi }{2}\right)$, and $\cos \alpha =-\frac{4}{5}$, then $\tan \left( \frac{\pi }{4}-\alpha \right)$ equals to ( ).
A: $7$
B: $\frac{1}{7}$
C: $-\frac{1}{7}$
D: $-7$ | B |
We call a natural number venerable if the sum of all its divisors, including $1$, but not including the number itself, is $1$ less than this number. Find all the venerable numbers, some exact degree of which is also venerable. | 2^k \text{ for } k \in \mathbb{Z}_{>0} |
3. For what values of $m$ and $n$, can the polynomial $x^{4}-5 x^{3}+11 x^{2}+$ $m x+n$ be divisible by $x^{2}-2 x+1$? | m=-11, n=4 |
## Task A-2.4. (4 points)
Three circles with a radius of $1 \mathrm{~cm}$ have exactly one common point, and their centers are the vertices of an equilateral triangle. Determine the area of the set of all points that belong to two of these circles.
A: 15
B: 16
C: 7
D: 8 | C |
2. (2007 Zhejiang Province High School Mathematics Competition B Paper) Let the non-constant function \( f(x) (x \in \mathbf{R}) \) be an even function, and its graph is symmetric about the line \( x = \frac{\sqrt{2}}{2} \). Then the function is
A. A non-periodic function
B. A periodic function with a period of \( \fra... | C |
The domain of the function $y= \dfrac {1}{ \sqrt {x+1}}$ is $\_\_\_\_\_\_$. | (-1,+\infty) |
Which of the following statements is correct?
A: The sum of two numbers is always greater than one of the addends.
B: A number whose absolute value is equal to its opposite is negative.
C: If the absolute values of two numbers are equal, then the two numbers are equal.
D: Integers and fractions are collectively ref... | D |
Which of the following real numbers is an irrational number?
A: $0.7$
B: $\frac{1}{2}$
C: $\sqrt{2}$
D: $\sqrt{9}$ | C |
Which of the following calculations is correct?
A: $a^{2}\cdot a^{6}=a^{8}$
B: $a^{6}\div a^{3}=a^{2}$
C: $2a^{2}+a^{6}=3a^{8}$
D: $(-3a)^{2}=-9a^{2}$ | A |
8、We know that the number of divisors of $2013$, $2014$, and $2015$ are the same. Therefore, for three consecutive natural numbers $n$, $n+1$, and $n+2$ that have the same property (the same number of divisors), the smallest value of $\mathbf{n}$ is | 33 |
11.1. Find all solutions of the system of equations in real numbers:
$$
\left\{\begin{array}{c}
x y+z+t=1 \\
y z+t+x=3 \\
z t+x+y=-1 \\
t x+y+z=1
\end{array}\right.
$$ | (1,0,-1,2) |
66. There is a class of three-digit numbers, the product of the digits in each place is 18. Among all such three-digit numbers, the difference between the largest and the smallest number is $\qquad$ . | 792 |
3. In a convex 12-sided polygon, all angles are equal. It is known that the lengths of some ten of its sides are 1, and the length of one more is 2. What can the area of this 12-sided polygon be? (M. A. Evdokimov) Solution. Consider the 12-sided polygon $A_{1} A_{2} \ldots A_{12}$, satisfying the condition of the probl... | 8+4\sqrt{3} |
Problem 8.3. In triangle $ABC$, the sides $AC=14$ and $AB=6$ are known. A circle with center $O$, constructed on side $AC$ as the diameter, intersects side $BC$ at point $K$. It turns out that $\angle BAK = \angle ACB$. Find the area of triangle $BOC$.
$? | 57 |
212. $\int \arctan x \, dx$. | x\operatorname{arctg}x-\frac{1}{2}\ln(1+x^{2})+C |
Given $x \gt 0$, $y \gt 0$, and $x+y=1$, if $\frac{1}{x}+\frac{{16}}{y} \gt {a^2}+24a$ always holds, then the range of real number $a$ is ( )
A: $\{a|{a≥\frac{1}{2}}\}$
B: $\{a|a\leqslant -3\}$
C: $\{a|-25 \lt a \lt 1\}$
D: $\{a|-1 \lt a \lt 25\}$ | C |
2. Given the function $f(x)=x^{2}-2 a x+2 a+4$ with the domain $R$ and the range $[1,+\infty]$. Then the value of $a$ is ).
(A) interval $[-1,3$ ]
(B) interval $(1-\sqrt{5}, 1+\sqrt{5})$
(C) interval $(-\infty,-1) \cup(3,+\infty)$
(D) set $\{-1,3\}$ | D |
3. The odd function $f(x)$ defined on $(-\infty,+\infty)$ is an increasing function, and the even function $g(x)$ has a graph that coincides with the graph of $f(x)$ on $[0,+\infty)$. Let $a>b>0$. Given the following inequalities:
(1) $f(b)-f(-a)>g(a)-g(-b)$
(2) $f(b)-f(-a)g(b)-g(-a)$
(3) $f(a)-f(-b)>g(b)-g(-a)$
(4) $f... | A |
Given that $m$ and $n$ are two different lines, and $\alpha$ and $\beta$ are two different planes, which of the following propositions is correct?
A: If $m \perp n$, $m \parallel \alpha$, and $n \parallel \beta$, then $\alpha \parallel \beta$
B: If $m \perp n$, $\alpha \cap \beta = m$, and $n \not\subset \alpha$, then... | \text{D} |
[Example 1.4.5] On the plane, there are finitely many polygons. If any two of these polygons are intersected by a line passing through the origin, then these polygons are said to be properly placed. Find the smallest natural number $m$, such that for any set of properly placed polygons, one can draw $m$ lines passing t... | 2 |
Solve the following system of equations:
$$
\begin{gathered}
\log _{2}(x y+4)-1=\log _{3}\left(x^{2}+y^{2}+2\right) \\
y=x-1
\end{gathered}
$$ | x_1=0,y_1=-1;x_2=1,y_2=0;x_3=-3,y_3=-4;x_4=4,y_4=3 |
Given a sequence $\{a_n\}$ satisfying $a_1=m$, $a_{n+1}=\begin{cases} 2a_n, & n=2k-1 \\ a_n+r, & n=2k \end{cases} (k\in\mathbb{N}^*, r\in\mathbb{R})$, and its sum of the first $n$ terms is $S_n$. If for any $n\in\mathbb{N}^*$, the sequence $\{a_n\}$ satisfies $a_{n+2}=a_n$, then the relationship between $m$ and $r$ is ... | m+r=0 |
An infinite sequence has the first three terms as 1, 2, 3. Which of the following cannot be its general term formula?
A: $a_n = n$
B: $a_n = n^3 - 6n^2 + 12n - 6$
C: $a_n = \frac{1}{2}n^2 - \frac{1}{2}n + 1$
D: $a_n = \frac{6}{n^2 - 6n + 11}$ | C |
A merchant sells a certain item for $48$ yuan, making a profit of $20\%$. What is the cost price of the item?
A: $30$ yuan
B: $40$ yuan
C: $50$ yuan
D: $60$ yuan | B |
Given the sets $A=\{x|x^{2}-x=0\}$ and $B=\{x|x^{2}+x=0\}$, find the set $A∪B=(\ \ \ )$.
A: $0$
B: $\{0\}$
C: $\varnothing$
D: $\{-1,0,1\}$ | D |
Among the following four 7-digit numbers, the one that is divisible by 8 is ( )
A: 23□□372
B: 53□□164
C: 5□□3416
D: 71□□172 | C |
Given the function $y=f(x)$, for any $x \in \mathbb{R}$, it satisfies $f(x+2)=\frac{1}{f(x)}$. When $x \in (0, 2]$, $f(x)=x$.
(1) Find the analytical expression of $f(x)$ when $x \in (2, 4]$;
(2) If $f(m)=1$, find the value of $m$;
(3) Calculate the sum: $f(1)+f(2)+f(3)+...+f(2015)$. | \frac{4535}{2} |
3. Given the ellipse $\Gamma_{1}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, $c=\sqrt{a^{2}-b^{2}}$, the parabola $\Gamma_{2}: y^{2}=2 c\left(x-\frac{c}{2}\right)$ intersects with $\Gamma_{1}$, and one of the intersection points is $A$. Let $O$ be the origin. The line $l: x=\frac{a^{2}}{c}$ intersects the $x$-a... | 0 |
Given the proposition $p$: For all $x \in \mathbb{R}$, $\cos x \leqslant 1$, then $\neg p$ is
A: There exists an $x \in \mathbb{R}$, $\cos x \geqslant 1$
B: For all $x \in \mathbb{R}$, $\cos x \geqslant 1$
C: There exists an $x \in \mathbb{R}$, $\cos x > 1$
D: For all $x \in \mathbb{R}$, $\cos x > 1$ | C |
16.1. [9.4 (15 points), 10.4 (15 points)] On the graph of the function $y=x+\frac{1}{x}$, where $x>0$, find the point closest to the origin. | (\frac{1}{\sqrt[4]{2}};\frac{1+\sqrt{2}}{\sqrt[4]{2}}) |
## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \sqrt{n+2}(\sqrt{n+3}-\sqrt{n-4})
$$ | \frac{7}{2} |
8. The sum of 9 fractions is 1, their numerators are all 1, five of them are $\frac{1}{3}, \frac{1}{7}, \frac{1}{9}, \frac{1}{11}, \frac{1}{33}$, the denominators of the remaining four numbers all end in 5, please write down these 4 fractions. | \frac{1}{5},\frac{1}{15},\frac{1}{45},\frac{1}{385} |
13. If the function $f$ satisfies $f\left(n^{2}\right)=f(n)+2, n \geqslant 2$ and $f(2)=1$, then, $f(256)=(\quad)$.
(A) 3
(B) 5
(C) 7
(D) 9
$(\mathrm{E}) *$ | C |
13. In $\triangle A B C, A B=8, B C=13$ and $C A=15$. Let $H, I, O$ be the orthocentre, incentre and circumcentre of $\triangle A B C$ respectively. Find $\sin \angle H I O$.
(2 marks)
在 $\triangle A B C$ 中, $A B=8 、 B C=13 、 C A=15$ 。設 $H 、 I 、 O$ 分別為 $\triangle A B C$ 的垂
心、內心和外心。求 $\sin \angle H I O$ 。
(2 分) | \frac{7\sqrt{3}}{26} |
A castle has infinitely many rooms labeled $1,2,3, \ldots$, which are divided into several halls. Suppose room $n$ is on the same hall as rooms $3 n+1$ and $n+10$ for every $n$. Determine the maximum possible number of different halls in the castle. | 3 |
3. (6 points) $(1234+2341+3412+4123) \div 11=$ | 1010 |
1. (8 points) The calculation result of the expression $3^{3}+4^{3}+5^{3}+6^{3}+7^{3}+8^{3}+9^{3}$ is | 2016 |
When a fair die is thrown twice, the probability that the sum of the numbers facing up is $6$ is ( ).
A: $ \dfrac {5}{36}$
B: $ \dfrac {1}{6}$
C: $ \dfrac {1}{12}$
D: $ \dfrac {1}{9}$ | A |
Given the line: \\(y=kx-k+1\\) and the curve \\(C\\): \\(x^{2}+2y^{2}=m\\) have common points, the range of \\(m\\) is \\((\\) \\()\\).
A: \\(m\\geqslant 3\\)
B: \\(m\\leqslant 3\\)
C: \\(m > 3\\)
D: \\(m < 3\\) | A |
2B. Solve the equation:
$$
\log _{\frac{1}{8}}(2 x)-4 \log _{\frac{1}{4}} x \cdot \log _{8} x=0
$$ | x_{1}=\frac{1}{\sqrt{2}},x_{2}=2 |
6. Given the sequence: $\frac{2}{3}, \frac{2}{9}, \frac{4}{9}, \frac{6}{9}, \frac{8}{9}, \frac{2}{27}, \frac{4}{27}, \cdots$, $\frac{26}{27}, \cdots, \frac{2}{3^{n}}, \frac{4}{3^{n}}, \cdots, \frac{3^{n}-1}{3^{n}}, \cdots$. Then $\frac{2018}{2187}$ is the $\qquad$th term of the sequence. | 1552 |
99. There are several people participating in an archery competition. In the first round, 18 people hit the target, in the second round, 15 people hit the target, and in the third round, 10 people hit the target. If 3 people hit the target in all three rounds, then what is the minimum number of people who participated ... | 20 |
10.4. How many solutions in integers $x, y$ does the equation $|3 x+2 y|+|2 x+y|=100$ have? | 400 |
Let $x \in \mathbb{R}$. Then "$|x-1|<2$" is a ( ) condition for "$x^2-4x-5<0$".
A: Sufficient but not necessary
B: Necessary but not sufficient
C: Necessary and sufficient
D: Neither sufficient nor necessary | \text{A} |
9. (14 points) Let the function $f(x)$ be defined on the closed interval
$[0,1]$, satisfying $f(0)=0, f(1)=1$, and for any $x, y \in[0,1](x \leqslant y)$, we have
$$
f\left(\frac{x+y}{2}\right)=\left(1-a^{2}\right) f(x)+a^{2} f(y),
$$
where the constant $a$ satisfies $0<a<1$. Find the value of $a$. | a=\frac{\sqrt{2}}{2} |
23. Find the sum of all the integral values of $x$ that satisfy
$$
\sqrt{x+3-4 \sqrt{x-1}}+\sqrt{x+8-6 \sqrt{x-1}}=1
$$ | 45 |
Given the complex numbers z1\=1+i, z2\=2+bi, if z1•z2 is a real number, find the value of b.
A: 2
B: 1
C: \-1
D: \-2 | D |
Problem 17. A vessel of cylindrical shape is filled with water. The height of the water layer is h. Water flows out of an opening (its dimensions can be neglected) located at height $z$ (Fig. 23). At what $z$ will the range of the water jet be the greatest? | \frac{}{2} |
$2+$
John had a full basket of trempons. First, he met Anna and gave her half of his trempons and another half-trempon. Then he met Banna and gave her half of the remaining trempons and another half-trempon. After meeting Vanna and giving her half of the trempons and another half-trempon, the basket was empty. How man... | 7 |
Example 4: On a circular road in a certain town, there are five primary schools in sequence: No.1 Primary School, No.2 Primary School, No.3 Primary School, No.4 Primary School, and No.5 Primary School. They have 15, 7, 11, 3, and 14 computers respectively. To make the number of computers in each school the same, how ma... | 12 |
Simplify and then evaluate the expression: $(x+2y)^{2}-(x+y)(x-y)$, where $x=-2$ and $y=\frac{1}{2}$. | -\frac{11}{4} |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and satisfy $\frac{c}{\cos C}=\frac{a+b}{\cos A+\cos B}$. Point $D$ is the midpoint of side $BC$.
$(1)$ Find the measure of angle $C$.
$(2)$ If $AC=2$ and $AD=\sqrt{7}$, find the length of side $AB$. | 2\sqrt{7} |
When $x \in [-2, 1]$, the range of the function $f(x) = x^2 + 2x - 2$ is ( )
A: $[1, 2]$
B: $[-2, 1]$
C: $[-3, 1]$
D: $[-3, +\infty)$ | C |
Given the sum of the first $n$ terms of an arithmetic sequence is $S_n$, where $S_{11} > 0$ and $S_{12} < 0$. Find the maximum value of $n$ for which $S_n$ is the largest. | n = 6 |
Given a parabola $C: x^2 = 2py$ ($p > 0$), a line $l$ with a slope of $1$ passes through its focus and intersects the parabola $C$ at points $M$ and $N$, and $|MN| = 16$.
(Ⅰ) Find the equation of the parabola $C$.
(Ⅱ) A moving circle $P$ with its center on the parabola $C$ passes through a fixed point $D(0, 4)$. If the... | \sqrt{2} - 1 |
3. (10 points) Two cyclists are training on different tracks. Cyclist $A$ uses a circular track, with a diameter of 1 kilometer; Cyclist $B$ uses a straight track, with a length of 5 kilometers. Cyclist $A$ completes 3 laps in 10 minutes, while Cyclist $B$ travels 2 round trips in 5 minutes. What is the ratio of the sp... | 3\pi:40 |
Given a quadratic equation in $x$, $mx^{2}+5x+m^{2}-2m=0$, with one root being $0$, find $m$. | m = 2 |
Among four people, A, B, C, and D, one person will be selected to participate in the garbage classification knowledge contest held by the district. After three rounds of preliminary competitions, their average scores are all $92.5$ points, with variances of $s_{A}^{2}=3.4$, $s_{B}^{2}=2.1$, $s_{C}^{2}=2.5$, and $s_{D}^... | B |
Given that the center of the ellipse is at the origin, its eccentricity $e= \frac{ \sqrt{3}}{2}$, and one of its foci coincides with the focus of the parabola ${x}^{2}=-4 \sqrt{3}y$, then the equation of this ellipse is ( ).
A: $ \frac{{x}^{2}}{4}+{y}^{2}=1 $
B: $ \frac{{x}^{2}}{4}+ \frac{{y}^{2}}{16}=1 $
C: $ {x}^{... | C |
In acute triangle $\triangle ABC$, $M$ is the midpoint of segment $BC$. Point $P$ lies in the interior of $\triangle ABC$ such that $AP$ bisects $\angle BAC$. Line $MP$ intersects the circumcircles of $\triangle ABP,\triangle ACP$ at $D,E$ respectively. Prove that if $DE=MP$, then $BC=2BP$. | BC = 2BP |
3. If the real number $x$ satisfies $\sin \left(x+20^{\circ}\right)=\cos \left(x+10^{\circ}\right)+\cos \left(x-10^{\circ}\right)$, then $\tan x=$ | \sqrt{3} |
7. Martha draws some pentagons, and counts the number of right-angles in each of her pentagons. No two of her pentagons have the same number of right-angles. Which of the following is the complete list of possible numbers of right-angles that could occur in Martha's pentagons?
A $1,2,3$
B $0,1,2,3,4$
C $0,1,2,3$
D $0,1... | C |
Problem 6. Suppose $T$ is the period of the function $y=f(x)$. From the condition, we get:
$$
\cos x=f(x)-3 f(x-\pi)=f(x-T)-3 f(x-T-\pi)=\cos (x-T)
$$
Since this equality holds for all $x$, the number $T$ is the period of the function $y=\cos x$, i.e., $T=2 \pi n$, where $n$ is some natural number. From the given equ... | f(x)=\frac{1}{4}\cosx |
21. For certain values of $x$, the list $x, x+6$ and $x^{2}$ contains just two different numbers. How many such values of $x$ are there?
A 1
В 2
C 3
D 4
E 5 | 4 |
My favorite numbers are those which, when multiplied by their own digit sum, become ten times larger. The product of three of my favorite numbers is 71668. Which are these numbers?
(Birdie) | 19,46,82 |
1. A die is rolled twice, and the sum of the points that come up is calculated and recorded. This procedure is repeated three times (a total of six rolls). Find the probability that only one of the three recorded sums is divisible by three. (12 points) | \frac{4}{9} |
7. As shown in the figure below, fold an A4 paper along the midpoints of its two longer sides, and you will get 2 smaller rectangles. The ratio of the length to width of the smaller rectangles is the same as that of the A4 paper. If the length of the A4 paper is 29.7 cm, then the area of the square with the width of th... | 441 |
The coordinates of the point $\left(-3,-4\right)$ with respect to the $x$-axis symmetry point are ____. | \left(-3,4\right) |
5. (3 points) Three circles with radii $\sqrt{13}, \sqrt{13}$, and $2 \sqrt{13-6 \sqrt{3}}$ are arranged such that the triangle formed by their centers is an equilateral triangle with side length $\sqrt{39}$. Find the radius of the circumscribed circle around the triangle, each vertex of which is the point of intersect... | 4\sqrt{39}-6\sqrt{13} |
5. Let $f(x)=a x^{5}+b x^{3}+c x+10$, and $f(3)$ $=3$. Then $f(-3)=$ $\qquad$ | 17 |
## Task 1 - 050831
Determine the number of all numbers between 10000 and 99999 that, like 35453, read the same sequence of digits forwards as backwards. | 900 |
6. A reel of film needs to be rewound from one spool to another. The diameters of the empty spools are the same and equal to $a$. Find the time required for rewinding if the length of the film is $L$, the thickness of the film is small and equal to $S$, and the receiving spool rotates at a constant angular velocity $\o... | \frac{\pi}{S\omega}(\sqrt{^{2}+\frac{4SL}{\pi}}-) |
35*. What is the greatest number of rays in space that form pairwise obtuse angles? | 4 |
Solve the following equations:
$(1) 6x-7=4x-5$
$(2) \frac{x+1}{2}-1=2+\frac{2-x}{4}$ | x = 4 |
Example 2 Note
$$
\begin{array}{l}
S=\frac{2}{1 \times 3}+\frac{2^{2}}{3 \times 5}+\cdots+\frac{2^{49}}{97 \times 99}, \\
T=\frac{1}{3}+\frac{2}{5}+\frac{2^{2}}{7}+\cdots+\frac{2^{48}}{99} .
\end{array}
$$
Then the value of $S-T$ is ( ).
(A) $\frac{2^{49}}{99}$
(B) $1-\frac{2^{49}}{99}$
(C) $\frac{2^{49}}{99}-1$
(D) $... | B |
Given the parabola $y=ax^2$ has the directrix $y=-2$, the value of the real number $a$ is \_\_\_\_\_\_. | \frac{1}{8} |
Find the solution set for the inequality $\frac{2x}{x+1} \leqslant 1$. | (-1, 1] |
Which of the following equations is correct after transformation?
A: If $a=b$, then $a-m=b+m$
B: From $-3x=2$, we get $x=-\frac{3}{2}$
C: If $|a|=|b|$, then $a=b$
D: If $\frac{a}{c}=\frac{b}{c}$, then $a=b$ | D |
Among the real numbers $\frac{{22}}{7}$, $\sqrt{5}$, $0$, $3.1415$, $\sqrt{16}$, $3\pi$, $3.1010010001\ldots$ (where the number of zeros between adjacent ones increases by $1$ each time), the number of irrational numbers is ( ).
A: $3$ numbers
B: $4$ numbers
C: $5$ numbers
D: $6$ numbers | A |
In the cartesian coordinate plane $(xoy)$, the left and right foci of the ellipse $(C)$: $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$ are $F\_1$ and $F\_2$ respectively. $P$ is a point on the ellipse $(C)$ such that $PF\_1 \perp PF\_2$. Find the area of $\triangle PF\_1F\_2$. | 9 |
(1) Given $f(x+1)=x^{2}-3 x+2, x \in\left(-\infty, \frac{3}{2}\right)$, find $f^{-1}(x)$. | f^{-1}(x)=\frac{5}{2}-\sqrt{x+\frac{1}{4}}(x>-\frac{1}{4}) |
5.3. A smooth sphere with a radius of 1 cm was dipped in red paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a red trail. During its mov... | 105.75 |
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