problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
4. Given $P_{1}\left(x_{1}, y_{1}\right), P_{2}\left(x_{2}, y_{2}\right), \cdots$, $P_{n}\left(x_{n}, y_{n}\right), \cdots$, where $x_{1}=1, y_{1}=0, x_{n+1}=$ $x_{n}-y_{n}, y_{n+1}=x_{n}+y_{n}\left(n \in \mathbf{Z}_{+}\right)$. If $a_{n}=$ $\overrightarrow{P_{n} P_{n+1}} \cdot \overrightarrow{P_{n+1} P_{n+2}}$, then t... | 10 |
Among the following groups, the one that contains like terms is ( )
A: $9a^{2}x$ and $9a^{2}$
B: $a^{2}$ and $2a$
C: $2a^{2}b$ and $3ab^{2}$
D: $4x^{2}y$ and $-yx^{2}$ | D |
10. (20 points) Point $A$ lies on the line $y=\frac{8}{15} x-6$, and point $B$ lies on the parabola $y=x^{2}$. What is the minimum length of segment $A B$? | \frac{1334}{255} |
3. Determine for which real number $p$ the equations
$$
\begin{array}{r}
x^{3}+x^{2}-36 x-p=0 \\
x^{3}-2 x^{2}-p x+2 p=0
\end{array}
$$
have a common root.
The school-based - closed part of the first round in category A takes place
## on Tuesday, December 3, 2002
so that it starts in the morning and the contestant... | 0,-60,36 |
Given vectors $\overrightarrow{m}=(2,\sin \alpha)$ and $\overrightarrow{n}=(\cos \alpha,-1)$, where $\alpha\in(0, \frac{\pi}{2})$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
$(1)$ Find the values of $\sin 2\alpha$ and $\cos 2\alpha$;
$(2)$ If $\sin (\alpha-\beta)= \frac{\sqrt{10}}{10}$, and $\beta\in(0, \frac{\... | \frac{\pi}{4} |
Simplify first, then evaluate: $(2x^{2}-2y^{2})-3(x^{2}y^{2}+x^{2})+3(x^{2}y^{2}+y^{2})$, where $x=-1$ and $y=2$. | 3 |
If we want to prove that the proposition "If $a \lt b$, then $a^{2} \lt b^{2}$" is false, the counterexample that can be given is ( ).
A: $a=2$, $b=3$
B: $a=-2$, $b=-3$
C: $a=-3$, $b=-2$
D: $a=-2$, $b=3$ | C |
If the complex number $(a^2-3a+2) + (a-2)i$ is a pure imaginary number, then the value of the real number $a$ is $\boxed{1}$. | 1 |
Given the line $ax - by = 1 (a > 0, b > 0)$ passes through the point $(1, -1)$, find the minimum value of $\frac{1}{a} + \frac{1}{b}$ $(\ )$.
A: $3$
B: $4$
C: $5$
D: $8$ | B |
A person flips a coin twice in succession. The mutually exclusive event to "getting heads at least once" is ( )
A: Getting heads at most once
B: Getting heads both times
C: Getting heads exactly once
D: Getting tails both times | D |
2. (5 points) There are three natural numbers, their sum is 2015, the sums of each pair are $m+1, m+2011$ and $m+2012$, then $m=$ $\qquad$ | 2 |
4. Let $P_{1}, P_{2}, \cdots, P_{7}$ be the seven vertices of a regular heptagon inscribed in the unit circle in sequence. Then
$$
\left|P_{1} P_{2}\right|^{2}+\left|P_{1} P_{3}\right|^{2}+\left|P_{1} P_{4}\right|^{2}=
$$
$\qquad$ | 7 |
If two balls are randomly selected from a pocket containing 2 red balls and 2 black balls, which of the following pairs of events are "mutually exclusive but not complementary"?
A: "At least one black ball" and "both are red balls"
B: "At least one black ball" and "both are black balls"
C: "At least one black ball" ... | \text{D} |
4.52 The number of integer solutions to the inequality $x-1<(x-1)^{2}<3 x+7$
(A) is equal to 4.
(B) is less than 4.
(C) is greater than 5.
(D) is equal to 5.
(China Junior High School Mathematics League, 1993) | A |
1. Let the function $y=\sin x \cdot \cos x(\sin x+\cos x)$, where $x \in\left[-\frac{\pi}{4}, \frac{\pi}{2}\right]$. Then the range of the function $y$ is $\qquad$ | [-\frac{\sqrt{3}}{9},\frac{\sqrt{2}}{2}] |
For a given positive integer $m$, the series
$$\sum_{k=1,k\neq m}^{\infty}\frac{1}{(k+m)(k-m)}$$
evaluates to $\frac{a}{bm^2}$, where $a$ and $b$ are positive integers. Compute $a+b$. | 7 |
5. In $\triangle A B C$, the lengths of the sides opposite to angles $A, B, C$ are $a, b, c$ respectively. If $c-a$ equals the height $h$ from vertex $A$ to side $AC$, then the value of $\sin \frac{C-A}{2}+\cos \frac{C+A}{2}$ is
(A) 1 ;
(B) $\frac{1}{2}$;
(C) $\frac{1}{3}$;
(D) -1 . | 1 |
Among the following numbers, the smallest one is:
A: $-2$
B: $0$
C: $\frac{1}{2}$
D: $2$ | A |
7.3. A flock of geese was flying over the lakes. On each lake, half of the geese and another half-goose landed, while the rest continued flying. All the geese landed on seven lakes. How many geese were in the flock? | 127 |
3. In the afternoon, 5 classes need to be scheduled: Physics, Chemistry, Biology, and two self-study periods. If the first class cannot be Biology and the last class cannot be Physics, then the number of different scheduling methods is ( ) kinds.
(A) 36
(B) 39
(C) 60
(D) 78 | B |
There are three rooms that need to be painted, and the painting scheme requires: each room is painted with only one color, and the three rooms must be painted with different colors. It is known that the painting areas of the three rooms (unit: m2) are x, y, z, respectively, and $x 2) are a, b, c, respectively, and $a <... | \text{B} |
Given the following statements: ① $\sqrt[4]{(-2)^{4}} = \pm2$; ② For $y=x^2+1$, where $x \in [-1, 2]$, the range of $y$ is $[2, 5]$; ③ The graph of a power function never passes through the fourth quadrant; ④ The graph of the function $f(x) = a^{x+1} - 2$ ($a > 0$, $a \neq 1$) passes through the fixed point $(-1, -1)$;... | B |
Show that if $t_1 , t_2, t_3, t_4, t_5$ are real numbers, then
$$\sum_{j=1}^{5} (1-t_j )\exp \left( \sum_{k=1}^{j} t_k \right) \leq e^{e^{e^{e}}}.$$ | e^{e^{e^{e}}} |
9. In tetrahedron $\mathrm{ABCD}$, the length of edge $\mathrm{AB}$ is $3 \mathrm{~cm}$, the area of face $\mathrm{ABC}$ is $15 \mathrm{~cm}^{2}$, and the area of face $\mathrm{ABD}$ is $12 \mathrm{~cm}^{2}$. The angle between these two faces is $30^{\circ}$. Find the volume of the tetrahedron (in $\mathrm{cm}^{3}$). | 20 |
3. The lateral sides $K L$ and $M N$ of trapezoid $K L M N$ are equal to 15 and 12, respectively, and the base $L M=3$. The bisector of angle $N K L$ passes through the midpoint of side $M N$. Find the area of the trapezoid. Answer: 80. | 80 |
The minimum value and the smallest positive period of the function $f(x)= \sqrt {3}\sin (2x- \frac {\pi}{6})-1$ are respectively $(\quad)$
A: $(- \sqrt {3}-1,\pi)$
B: $(- \sqrt {3}+1,\pi)$
C: $(- \sqrt {3},\pi)$
D: $(- \sqrt {3}-1,2\pi)$ | \text{A} |
A four-digit positive integer $m$, where all digits are not $0$, is called a "twin egg number" if the thousands digit is the same as the units digit, and the hundreds digit is the same as the tens digit. By exchanging the thousands and hundreds digits, and the tens and units digits, a new "twin egg number" $m'$ is obta... | 7117 |
5. Given that $[x]$ represents the greatest integer not exceeding $x$, if
$$
[x+0.1]+[x+0.2]+[x+0.3]+[x+0.4]+[x+0.5]+[x+0.6]+[x+0.7]+[x+0.8]+[x+0.9]=104
$$
then the minimum value of $x$ is ( ).
A: 11.5
B: 10.5
C: 9.5
D: 8.5 | 11.5 |
The solution set of the inequality $x-3>4x$ is. | x<-1 |
6. Given the sequence $\left\{a_{n}\right\}$
satisfies
$$
a_{n}=\sqrt{1+\frac{1}{n^{2}}+\frac{1}{(n+1)^{2}}}(n \geqslant 1),
$$
and its first $n$ terms sum is $S_{n}$. Then $\left[S_{n}\right]=$ $\qquad$ ( [ $x$ denotes the greatest integer not exceeding the real number $x$ ). | n |
7. There is a positive integer, its $\frac{1}{2}$ is the square of an integer, its $\frac{1}{3}$ is the cube of an integer, and its $\frac{1}{5}$ is the fifth power of an integer. Then the minimum value of this number is . $\qquad$ | 2^{15}\times3^{10}\times5^{6} |
The function $f(x) = x^2 + 2(a - 1)x + 2$ is decreasing on the interval $(-\infty, 4]$, then the range of the real number $a$ is | (-\infty, -3] |
Given a circle $C$ passes through points $A(-2,0)$, $B(0,2)$, and the center of the circle is on the line $y=x$. Also, the line $l:y=kx+1$ intersects circle $C$ at points $P$ and $Q$.
$(1)$ Find the equation of circle $C$;
$(2)$ If $\overrightarrow{OP} \cdot \overrightarrow{OQ} = -2$, find the value of the real numbe... | 0 |
3. Perica stands on one of the four floors of a building. In one move, he moves to an adjacent floor (the floor above or the floor below, if it exists). In how many ways can Perica make $n$ moves, where $n$ is a given non-negative integer, if he can start on any floor but must end on the last one? | B_{n}=F_{n+1} |
Given that $\log_{2}(\log_{3}x)=1$, find the value of the real number $x$. | 9 |
Among the following options, the one that does not represent a right-angled triangle $\triangle ABC$ is $(\ )$.
A: $b^{2}=a^{2}-c^{2}$
B: $a : b : c = 3 : 4 : 5$
C: $\angle C=\angle A-\angle B$
D: $\angle A : \angle B : \angle C=3 : 4 : 5$ | D |
Consider the following statements:
① The equation $x^{2}+y^{2}-2x+4y+6=0$ represents a circle;
② For $m > n > 0$, the equation $mx^{2}+ny^{2}=1$ represents an ellipse with foci on the $y$-axis;
③ Given points $M(-1,0)$ and $N(1,0)$, if $|PM|-|PN|=2$, then the locus of the moving point $P$ is the right branch of a hyper... | B |
9.4. In triangle $A B C$, the bisector $A M$ is perpendicular to the median $B K$. Find the ratios $B P: P K$ and $A P: P M$, where $P$ is the point of intersection of the bisector and the median. | BP:PK=1,AP:PM=3:1 |
In the Cartesian coordinate system $xOy$, the parabola $y=x^{2}+2bx+b^{2}-2\left(b \gt 0\right)$ is given.$(1)$ Express the coordinates of the vertex of the parabola in terms of $b$ as ______；$(2)$ If the parabola passes through the point $B\left(0,-1\right)$,① when $-2 \lt x \lt 3$, find the range of $y$；② if $k\leqsl... | -4 \leqslant k \leqslant -1 |
3. Find the real solutions of the equation: $x\left(\frac{6-x}{x+1}\right)\left(\frac{6-x}{x+1}+x\right)=8$.
## SOLUTION: | x_{1}=2-\sqrt{2},x_{2}=2+\sqrt{2} |
}[}{[}$
The median to the hypotenuse
Pythagorean Theorem (direct and inverse)
Mean proportionals in a right triangle
In a right triangle $A B C\left(\angle C=90^{\circ}\right)$, the height $C D$ and the median $C E$ are drawn. The areas of triangles $A B C$ and $C D E$ are 10 and 3, respectively. Find $A B$. | AB=\sqrt{50} |
Five, let $\overline{a b c d}$ be a four-digit number, where the first two digits and the last two digits form two-digit numbers $\overline{a b}$ and $\overline{c d}$, respectively. If the following conditions are met: $\overline{a b} \neq \overline{c d}$, $\overline{a b}$ divides $\overline{c d}$, and $\overline{c d}$... | 65 |
Given that $f(x)$ and $g(x)$ are functions defined on $\mathbb{R}$, and satisfy the following conditions:
1. $f(x) = a^x \cdot g(x)$ ($a>0$, $a \neq 1$);
2. $g(x) \neq 0$;
3. $f(x) \cdot g'(x) > f'(x) \cdot g(x)$;
If $\frac{f(1)}{g(1)} + \frac{f(-1)}{g(-1)} = \frac{5}{2}$, then $a = \_\_\_\_\_\_$. | \frac{1}{2} |
5. Satisfy $a>b>2$ and make $a b$, $\frac{b}{a}$, $a-b$, $a+b$ can be arranged in a certain order to form a geometric sequence of real number pairs $(a, b)(\quad)$.
(A) does not exist
(B) exactly 1
(C) exactly 2
(D) infinitely many | B |
5. The maximum value of the function $y=|\sin x|+|\sin 2 x|$ is $\qquad$ . | \frac{\sqrt{414+66 \sqrt{33}}}{16} |
55 Find all natural numbers $n(\geqslant 2)$, such that there exist real numbers $a_{1}, a_{2}, \cdots, a_{n}$, satisfying
$$
\left\{\left|a_{i}-a_{j}\right| \mid 1 \leqslant i<j \leqslant n\right\}=\left\{1,2, \cdots, \frac{n(n-1)}{2}\right\} .
$$ | 2,3,4 |
11. (20 points) Let $\alpha, \beta$ be any two interior angles of a triangle. If there exist $\alpha, \beta$ such that the inequality
$$
\cos \sqrt{\alpha}+\cos \sqrt{\beta}>a+\cos \sqrt{\alpha \beta}
$$
holds, find the range of the real number $a$.
保留了源文本的换行和格式。 | a \in (-\infty, 1) |
One. (25 points) Let the sequence $\left\{x_{n}\right\}$ satisfy
$$
x_{1}=1, x_{n}=\sqrt{x_{n-1}^{2}+x_{n-1}}+x_{n-1}(n \geqslant 2) \text {. }
$$
Find the general term formula for the sequence $\left\{x_{n}\right\}$.
(Zhang Lei, provided) | x_{n}=\frac{1}{2^{2^{1-n}}-1} |
Chord $ \left[AD\right]$ is perpendicular to the diameter $ \left[BC\right]$ of a circle. Let $ E$ and $ F$ be the midpoints of the arcs $ AC$ and $ CD$, respectively. If $ AD\bigcap BE\equal{}\left\{G\right\}$, $ AF\bigcap BC\equal{}\left\{H\right\}$ and $ m(AC)\equal{}\alpha$, find the measure of angle $ BHC$ in term... | 90^\circ - \frac{\alpha}{2} |
Let $\{a\_n\}$ be a geometric sequence with a common ratio of $q (q > 0$ and $q \neq 1)$, and the sequence $4a\_1$, $3a\_2$, $2a\_3$ forms an arithmetic sequence. The sum of the first 4 terms of this sequence is $S\_4 = 15$.
(1) Find the general term formula for $\{a\_n\}$;
(2) Let $b\_n = a\_n + 2n (n = 1, 2, 3, ...)$... | 2^{n} + n^{2} + n - 1 |
8. Let the ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ have an eccentricity $e=\frac{\sqrt{3}}{2}$, and its latus rectum (the focal diameter perpendicular to the major axis) $d=1, F_{1} 、 F_{2}$ be the two foci, $P$ be any point on the ellipse except the endpoints of the major axis, and the angle bise... | (-\frac{3}{2},\frac{3}{2}) |
25. (Northern) Let $x, y, z \in [0,1], |x-y| \leqslant \frac{1}{2}, |z-y| \leqslant \frac{1}{2}, |x-z| \leqslant \frac{1}{2}$, find
$$W(x, y, z)=x+y+z-xy-yz-zx$$
the maximum and minimum values. | \frac{5}{6} |
Compute the number of integers between $1$ and $100$, inclusive, that have an odd number of factors. Note that $1$ and $4$ are the first two such numbers. | 10 |
2.4. Find all complex numbers $z$ for which
$$
|z-2+i|=5, \quad \operatorname{Re}(z(1+i))=2
$$ | z_{1}=5+3i,z_{2}=-2-4i |
In the geometric sequence $\{a\_n\}$, it is known that $a\_2=4$ and $a\_6=6$. Find the value of $a\_{10}$ \_\_\_\_\_\_. | 9 |
5. Find the volume of the tetrahedron with vertices $(5,8,10),(10,10,17),(4,45,46),(2,5,4)$. | 0 |
Proizvoov V.v.
Ten consecutive natural numbers were written on the board. When one of them was erased, the sum of the nine remaining numbers turned out to be 2002.
What numbers remained on the board? | 218,219,220,221,222,224,225,226,227 |
2. Peppa Pig is fetching water from a well. At the beginning, the bucket is just on the water surface. If Peppa turns the handle 12 times, the bucket is still 2 meters away from the well's mouth; if Peppa turns the handle 16 times, the bucket just reaches the well's mouth. Then the distance from the water surface to th... | 8 |
5. Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ with left and right foci $F_{1}$ and $F_{2}$, respectively, and $P$ as any point on the ellipse not coinciding with the left or right vertices, points $I$ and $G$ are the incenter and centroid of $\triangle P F_{1} F_{2}$, respectively. When $I G$ ... | \frac{1}{3} |
In an arithmetic sequence $\{a_{n}\}$, $S_{n}$ represents the sum of its first $n$ terms. If $\frac{{S}_{2020}}{2020}-\frac{{S}_{20}}{20}=2000$, then the common difference $d$ of the sequence $\{a_{n}\}$ is $\_\_\_\_\_\_$. | 2 |
2. It is known that $n^{2012}+n^{2010}$ is divisible by 10 for some positive integer $n$. Which of the following numbers is not a possible value for $n$ ?
(A) 2 ;
(B) 13 ;
(C) 35 ;
(D) 47 ;
(E) 59 . | E |
7th Iberoamerican 1992 Problem B1 Let a n , b n be two sequences of integers such that: (1) a 0 = 0, b 0 = 8; (2) a n+2 = 2 a n+1 - a n + 2, b n+2 = 2 b n+1 - b n , (3) a n 2 + b n 2 is a square for n > 0. Find at least two possible values for (a 1992 , b 1992 ). | (1992\cdot1996,4\cdot1992+8),(1992\cdot1988,-4\cdot1992+8) |
Find all functions $f:\mathbb N\to\mathbb N$ such that
$$f(x)+f(y)\mid x^2-y^2$$holds for all $x,y\in\mathbb N$. | f(n) = n |
The solution to the equation $9^x + 3^x - 2 = 0$ is. | 0 |
Given the function $f(x) = (x^2 - 2x)\ln x + (a - \frac{1}{2})x^2 + 2(1 - a)x + a$.
1. Discuss the monotonicity of $f(x)$;
2. When $a < -2$, discuss the number of zeros of $f(x)$. | 3 |
We will call a three-digit natural number "even-loving" if it contains two even digits and the digit 1. We will call a three-digit natural number "odd-loving" if it contains two odd digits and the digit 2. Determine how many even-loving and how many odd-loving numbers there are.
(E. Novotná) | 65 |
Given the set $R_{n}=\{X|X=(x_{1},x_{2},\ldots,x_{n})\}$, where $x_{i}\in\{0,1\}$, $i=1, 2, \ldots, n$ $(n\geqslant 2)$. For $A=(a_{1},a_{2},\ldots,a_{n})\in R_{n}$ and $B=(b_{1},b_{2},\ldots,b_{n})\in R_{n}$, the distance between $A$ and $B$ is defined as $d(A,B)=|a_{1}-b_{1}|+|a_{2}-b_{2}|+\ldots+|a_{n}-b_{n}|= \sum\... | 4 |
There are 3 train services from location A to location B, and 2 ferry services from location B to location C. There are no other routes to travel from location A to C. How many different travel options are available to travel from location A to location C? | 6 |
If the function $f(x)=\begin{cases}3x-b & (x < 1), \\ 2^{x} & (x\geqslant 1)\end{cases}$ and $f(f(\frac{5}{6}))=4$, then $b=$ __________. | \frac{1}{2} |
5. The function $f(x)$ is defined on $(0, \infty)$, and satisfies
$$
f(x)-2 x f\left(\frac{1}{x}\right)+3 x^{2}=0 \text {. }
$$
Then the minimum value of $f(x)$ is $\qquad$ | 3 |
Example 26 Given $a_{i}, b_{i} \in \mathbf{R}(i=1,2, \cdots, n), \sum_{i=1}^{n} a_{i}^{2}=4, \sum_{i=1}^{n} b_{i}^{2}=9$, find the maximum value of $\sum_{i=1}^{n} a_{i} b_{i}$. | 6 |
10. On a plane, 2011 points are marked. We will call a pair of marked points $A$ and $B$ isolated if all other points are strictly outside the circle constructed on $A B$ as its diameter. What is the smallest number of isolated pairs that can exist? | 2010 |
Given the hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b < 0)$, its right vertex and right focus are $A$ and $F$ respectively, and the intersection point of its left directrix and the $x$-axis is $B$. If $A$ is the midpoint of segment $BF$, then the eccentricity of hyperbola $C$ is $\_\_\_\_\_\_$. | \sqrt{2} + 1 |
Let $m$ and $n$ be two different lines, and $\alpha$, $\beta$, $\gamma$ be three different planes. Then, among the following statements, the false one is ( )
A: If $m \perp \alpha$, $n \parallel \alpha$, then $m \perp n$
B: If $\alpha \parallel \beta$, $\beta \parallel \gamma$, $m \perp \alpha$, then $m \perp \gamma$
... | D |
Three of the vertices of a parallelogram are $(0,1),(1,2)$, and $(2,1)$. The area of the parallelogram is
(A) 1
(B) 2
(C) $\sqrt{2}$
(D) $2 \sqrt{2}$
(E) 4 | 2 |
[ Inequalities for elements of a triangle (other)]
In an acute-angled triangle $ABC$, the bisector $AD$, the median $BM$, and the altitude $CH$ intersect at one point. Within what limits can the measure of angle $A$ vary? | \angleA>\arccos(\frac{\sqrt{5}-1}{2})\approx5150' |
6. Let the complex number $z \neq 1, z^{13}=1$. Then the value of $z^{1}+z^{3}+z^{4}+z^{9}+z^{10}+z^{12}$ is ( ).
(A) Rational number
(B) Irrational number
(C) Imaginary number
(D) Pure imaginary number | B |
Let set $M = \{x | x < 2012\}$, and $N = \{x | 0 < x < 1\}$. Among the following relations, the correct one is ( ).
A: $M \cup N = \mathbb{R}$
B: $M \cap N = \{x | 0 < x < 1\}$
C: $N \in M$
D: $M \cap N = \varnothing$ | \text{B} |
4. Find all natural numbers $n \geq 2$, for which the equality $4 x_{n}+2 y_{n}=55 n^{2}+61 n-116$ holds, where $x_{n}=1 \cdot 2+2 \cdot 3+\cdots+(n-1) \cdot n, y_{n}=1^{2}+2^{2}+3^{2}+\cdots+(n-1)^{2}$.
(20 points) | 29 |
1. Simplify: $\left(\frac{2^{-1}+3^{-1}}{2^{-1}-3^{-1}}\right)^{-1}$. | \frac{1}{5} |
7.4. Winnie-the-Pooh and Tigger are climbing two identical fir trees. Winnie-the-Pooh climbs up at half the speed of Tigger, but he descends three times faster than Tigger. Winnie-the-Pooh and Tigger started and finished at the same time. How many times faster does Tigger climb up than he descends? | 1.5 |
Given the universal set $U=\{x\in\mathbb{N}^*|-1\leq x<6\}$, $A=\{1,3\}$, $B=\{3,5\}$, the complement of $A\cup B$ in $U$ is ( )
A: $\{-1,0,2,4,6\}$
B: $\{1,3,5\}$
C: $\{2,4\}$
D: $\{-1,0,2,4\}$ | \text{C} |
For which values of $k$ does the polynomial $X^{2017}-X^{2016}+X^{2}+k X+1$ have a rational root? | k=-2ork=0 |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0}\left(\tan\left(\frac{\pi}{4}-x\right)\right)^{\left(e^{x}-1\right) / x}$ | 1 |
Calculate: $(-\frac{1}{2})^{2022} \cdot 2^{2023}$ is ( )
A: $-1$
B: $1$
C: $-2$
D: $2$ | D |
Question 156, As shown in the figure, the edge length of the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ is $4, O$ is the center of the square $A D D_{1} A_{1}$, $E$ and $F$ are the midpoints of $B_{1} C_{1}$ and $C C_{1}$ respectively, $P$ is a moving point on the line segment $O B$, $M$ is a moving point on the base $A B ... | \sqrt{10}+1 |
The negation of the proposition "All numbers divisible by 2 are even" is ( )
A: All numbers that are not divisible by 2 are even
B: All numbers divisible by 2 are not even
C: There exists a number that is not divisible by 2 and is even
D: There exists a number divisible by 2 that is not even | \text{D} |
Evaluate
$$\frac{(2 + 2)^2}{2^2} \cdot \frac{(3 + 3 + 3 + 3)^3}{(3 + 3 + 3)^3} \cdot \frac{(6 + 6 + 6 + 6 + 6 + 6)^6}{(6 + 6 + 6 + 6)^6}$$ | 108 |
16. The store released two batches of notebooks. If the second batch had 5 more notebooks than it actually did, the product of the numbers expressing the quantities of notebooks in each batch would have increased by 1250, and if the first batch had 7 more notebooks, then this product would have increased by 3150. How m... | 700 |
6. Given a regular tetrahedron $ABCD$ with edge length 2, the sum of the areas of all sections obtained by planes equidistant from its four vertices is ( ).
(A) 4
(B) 3
(C) $\sqrt{3}$
(D) $3+\sqrt{3}$ | D |
Given that $\tan α= \dfrac {1}{2}$, find $\tan (α+ \dfrac {3π}{4})$ = ___. | -\dfrac{1}{3} |
When solving the equation $\frac{1}{2}-\frac{x-1}{3}=1$, the correct expression after removing the denominator is:
A: $1-\left(x-1\right)=1$
B: $3-2\left(x-1\right)=1$
C: $2-3\left(x-1\right)=6$
D: $3-2\left(x-1\right)=6$ | D |
[ Pascal's Triangle and Newton's Binomial ]
How many rational terms are contained in the expansion of
a) $(\sqrt{2}+\sqrt[4]{3})^{100}$
b) $(\sqrt{2}+\sqrt[8]{3})^{300}$? | 26 |
$$
\begin{array}{l}
A=\{2,0,1,7\}, \\
B=\{x \mid x=a b, a, b \in A\} .
\end{array}
$$
The number of elements in set $B$ is $\qquad$ | 7 |
6. Given that the hyperbola has asymptotes $2 x \pm y=0$, and it passes through the intersection point of the lines $x+y-3=0$ and $2 x-y+3 t=0$, where $-2 \leqslant t \leqslant 5$. Then the maximum possible value of the real axis length of the hyperbola is $\qquad$ . | 4 \sqrt{3} |
Evaluate the integral $$∫_{-1}^{0}(x-e^{x})dx=（\quad）$$
A: \-1- $$\frac {1}{e}$$
B: \-1
C: \- $$\frac {3}{2}$$+ $$\frac {1}{e}$$
D: \- $$\frac {3}{2}$$ | C |
## Problem 1
Find all continuous functions $f:[1,8] \rightarrow R$, such that
$$
\int_{1}^{2} f^{2}\left(t^{3}\right) d t+2 \int_{1}^{2} f\left(t^{3}\right) d t=\frac{2}{3} \int_{1}^{8} f(t) d t-\int_{1}^{2}\left(t^{2}-1\right)^{2} d t
$$ | f(x)=x^{2/3}-1,1\leqx\leq8 |
Given that the coefficient of the $x^{\frac{3}{2}}$ term in the expansion of $\left(\sqrt{x} - \frac{a}{\sqrt{x}}\right)^5$ is 30, find the real number $a$. | -6 |
1. Using the digits $4,5,6,7,8$ and 9, a six-digit number was written. Zoran, Darko, and Nikola guessed the number. Zoran: 574698, Darko: 786945, Nikola: 456789. It turned out that Zoran correctly guessed the positions of three digits. Darko also guessed the positions of three digits, and Nikola guessed the position of... | 576948 |
## Task Condition
Find the derivative.
$y=x^{\sin x^{3}}$ | x^{\sinx^{3}}\cdot(3x^{2}\cdot\lnx\cdot\cosx^{3}+\frac{\sinx^{3}}{x}) |
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