problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. It is known that in $S_n$, $S_{17} 0$. The smallest value of $S_n$ is ( )
A: $S_{10}$
B: $S_{9}$
C: $S_{8}$
D: $S_{7}$ | \text{C} |
Given that the random variable $X$ follows a normal distribution $N(1,\sigma^{2})$, and $P(x \leqslant 2) = 0.72$, then $P(x \leqslant 0) =$ \_\_\_\_\_\_. | 0.28 |
6. The minor axis of the ellipse $\rho=\frac{1}{2-\cos \theta}$ is equal to
保留了源文本的换行和格式。 | \frac{2}{3}\sqrt{3} |
4. If the equation $s x^{2}+t x+s-1=0$ with respect to $x$ has real roots for any real number $t$, then the range of the real number $s$ is ( ).
(A) $s<0$
(B) $s<0$ or $s \geqslant 1$
(C) $0 \leqslant s<1$
(D) $0<s \leqslant 1$ | D |
Arrange 3 volunteer teachers to teach in 6 schools, with no more than 2 teachers in each school. How many different distribution plans are there? (Answer with a number) | 210 |
Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two mutually perpendicular unit vectors, with $\overrightarrow{OA} = \overrightarrow{a}$, $\overrightarrow{OB} = \overrightarrow{b}$, and $\overrightarrow{OC} = m\overrightarrow{a} + n\overrightarrow{b}$. If $\triangle ABC$ is an isosceles right triangle with $A$ as ... | \text{B} |
We know that $R^3 = \{(x_1, x_2, x_3) | x_i \in R, i = 1, 2, 3\}$ is a vector space regarding the laws of composition
$(x_1, x_2, x_3) + (y_1, y_2, y_3) = (x_1 + y_1, x_2 + y_2, x_3 + y_3)$, $\lambda (x_1, x_2, x_3) = (\lambda x_1, \lambda x_2, \lambda x_3)$, $\lambda \in R$.
We consider the following subset of $R^3$ :... | (0, 0, 4) |
7. (2002 National High School Competition Question) Find the range of negative values of $a$ for which the inequality $\sin ^{2} x+a \cos x+a^{2} \geqslant 1+\cos x$ holds for all $x \in \mathbf{R}$. | \leqslant-2 |
Given that $\{a_n\}$ is an increasing sequence, and for any $n \in \mathbb{N}^*$, $a_n = n^2 + \lambda n$ always holds, then the range of the real number $\lambda$ is | (-3, +\infty) |
5 Let real numbers $a_{1}, a_{2}, \cdots, a_{100}$ satisfy:
(1) $a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{100} \geqslant 0$;
(2) $a_{1}+a_{2} \leqslant 100$;
(3) $a_{3}+a_{4}+\cdots+a_{100} \leqslant 100$.
Find the maximum value of $a_{1}^{2}+a_{2}^{2}+\cdots+a_{100}^{2}$. | 100^{2} |
Simplify first, then evaluate: $\left(x-2y\right)^{2}-\left(x-3y\right)\left(x+3y\right)-4y^{2}$, where $x=2$, $y=-1$. | 17 |
7. Let $D_{1}, D_{2}, \cdots, D_{2 n}$ be $2 n(n \geqslant 1)$ points on the hypotenuse $B C$ of the right triangle $\triangle A B C$. Denote
$$
\angle D_{i-1} A D_{i}=\alpha_{i}(i=1,2, \cdots, 2 n+1) \text {, }
$$
satisfying $D_{i-1} D_{i}=D_{i} D_{i+1}\left(i=1,2, \cdots, 2 n, D_{0}=B\right.$, $D_{2 n+1}=C$ ). Then
... | \frac{1}{2 n+1} |
The standard equation of a circle with center at point (-3, 4) and tangent to the y-axis is. | (x+3)^2+(y-4)^2=9 |
4. (1994 AIME Problem) For a real number $x$, $[x]$ denotes the greatest integer not exceeding $x$. Find the positive integer $n$ such that $\left[\log _{2} 1\right]+$ $\left[\log _{2} 2\right]+\left[\log _{2} 3\right]+\cdots+\left[\log _{2} n\right]=1994$. | 312 |
### 9.305 Find $a$, for which the inequality
$x^{2}-2^{a+2} \cdot x-2^{a+3}+12>0$ is true for any $x$. | \in(-\infty;0) |
Let $a$, $b$, and $c$ be three distinct lines, and $\alpha$, $\beta$, and $\gamma$ be three distinct planes. Now, consider the following four propositions:
1. $\left. \begin{matrix} & a\parallel \gamma \\ & b\parallel \gamma \\ \end{matrix} \right\}\Rightarrow a\parallel b$;
2. $\left. \begin{matrix} & \alpha \paralle... | \text{C} |
Task B-3.1. Solve the system of equations:
$$
\begin{aligned}
& 2^{x} \cdot 3^{y}=24 \\
& 2^{y} \cdot 3^{x}=54
\end{aligned}
$$ | 3,\quad1 |
Evokimov M.A.
A gold prospector has a pile of gold sand weighing 37 kg (and he has no more sand), a two-pan balance, and two weights of 1 and 2 kg. The prospector can perform actions of two types:
- balance the scales, i.e., if the scales are currently not in equilibrium, he can pour some sand from one pan to the oth... | 26 |
Given the function $f(x)=\sin x-x\cos x$.
(I) Find the equation of the tangent line to the curve $y=f(x)$ at the point $(\pi,f(\pi))$;
(II) Prove that when $x\in(0, \frac{\pi}{2})$, $f(x) kx-x\cos x$ holds for all $x\in(0, \frac{\pi}{2})$, find the maximum value of the real number $k$. | \frac{2}{\pi} |
## 10. Triomino
A triomino tile of shape $\square$ is placed on a board of dimensions $5 \times 5$ such that each covers three cells, and they do not overlap with each other. One cell remains uncovered. What is the sum of the numbers on all the cells of the board that can remain uncovered?
## Result: $\quad 234$ | 234 |
Find the values of $x$ in the following equations:
$(1) 4x^{2}-81=0$;
$(2) 8\left(x+1\right)^{3}=27$. | x = \frac{1}{2} |
2. Find the value of the expression $\cos ^{4} \frac{5 \pi}{24}+\cos ^{4} \frac{11 \pi}{24}+\sin ^{4} \frac{19 \pi}{24}+\sin ^{4} \frac{13 \pi}{24}$. | \frac{3}{2} |
40. (5 points)
Form a ten-digit number without repeating digits using numbers $0 \sim 9$, where the number can be divided by 11. There are $\qquad$ such numbers. | 285120 |
The condition for the equation $x^{2}+y^{2}+4x-2y+5m=0$ to represent a circle is ________ | m < 1 |
Given that $a$, $b$, and $c$ are the three side lengths of $\triangle ABC$, with $a=7$, $b=2$, $c$ being an odd number, the perimeter of $\triangle ABC$ is ______. | 16 |
57. Find the particular solution of the equation $y^{\prime}+y \operatorname{tg} x=0$, if $y=2$ when $x=0$. | 2\cosx |
If $x > 0$, then the minimum value of $f(x) = 4x + \frac{9}{x}$ is _______. | 12 |
Given real numbers $x$ and $y$ satisfying $(1-i)x + (1+i)y = 2$, find the value of $xy$. | 1 |
Athletes A, B, C, and D participated in the shooting pre-qualifying competition. The average scores and variances of their shooting results are shown in the table below. To select an athlete with good and stable performance to participate in the competition, which athlete should be chosen?
| Statistical | A | B | ... | C |
The unit vector that makes the same angle with vectors $\overrightarrow{a}=(3,-1)$ and $\overrightarrow{b}=(1,3)$ is
A: $(\frac{2\sqrt{5}}{5},\frac{\sqrt{5}}{5})$ or $(-\frac{2\sqrt{5}}{5},-\frac{\sqrt{5}}{5})$
B: $(\frac{\sqrt{5}}{5},\frac{2\sqrt{5}}{5})$ or $(-\frac{\sqrt{5}}{5},-\frac{2\sqrt{5}}{5})$
C: $(\frac{2... | \text{A} |
Given a function defined on $\mathbb{R}$, $f(x)=2\cos \omega x\sin (\omega x+ \dfrac {\pi}{6})- \dfrac {1}{2}(\omega > 0)$, with a period of $\pi$.
$(1)$ Find the value of $\omega$ and the intervals where $f(x)$ is increasing;
$(2)$ Let $g(x)=f(x)+\sin (x- \dfrac {\pi}{6})$, find the range of $g(x)$. | \left[-2, \dfrac {9}{8}\right] |
7.1 A cyclist planned to travel from point A to point B in 5 hours, moving at a constant speed. He moved at the planned speed until the midpoint of the journey, then decided to increase his speed by $25 \%$. He completed the journey to point B at the new speed. How long did the entire trip take? | 4.5 |
5. The organizers of a mathematics olympiad decided to photograph 60 participants. It is known that no more than 30 participants can fit in a single photograph, however, any two students must appear together in at least one photograph. What is the minimum number of photographs needed to achieve this? | 6 |
11.1. The price of a product was first increased by $x$ percent, and then decreased by $y$ percent. As a result, the price remained the same. Find all values that the difference $\frac{1}{x}-\frac{1}{y}$ can take. | -\frac{1}{100} |
Given that $A(-1,4)$ and point $B(a,2)$ lie on the same inverse proportion function graph, the value of $a$ is ( )
A: $-2$
B: $-1$
C: $-\frac{1}{2}$
D: $1$ | A |
24) A thief has seen Marco lock his bicycle using a padlock with a 4-digit combination (each digit ranges from 0 to 9). He was not able to see the combination but found out that at least two consecutive digits are the same. What is the maximum number of combinations the thief will have to try to steal Marco's bicycle?
... | 2710 |
Example 5 Given real numbers $x, y$ satisfy
$$
x+y=3, \frac{1}{x+y^{2}}+\frac{1}{x^{2}+y}=\frac{1}{2} \text {. }
$$
Find the value of $x^{5}+y^{5}$.
[3] | 123 |
Given $f(x) = x^5 - ax^3 + bx - 6$, and $f(-2) = 10$, find $f(2)$. | -22 |
Given are sheets and the numbers $00, 01, \ldots, 99$ are written on them. We must put them in boxes $000, 001, \ldots, 999$ so that the number on the sheet is the number on the box with one digit erased. What is the minimum number of boxes we need in order to put all the sheets? | 34 |
Given that \\(f(x)\\) is an odd function, and \\(g(x)\\) is an even function, and \\(f(x)+g(x)= \dfrac {1}{x-1}\\), then \\(f(3)=\\) \\()
A: \\(1\\)
B: \\(\dfrac {3}{4}\\)
C: \\(\dfrac {3}{8}\\)
D: \\(\dfrac {1}{8}\\) | C |
The graph of the function $f(x) = a^{x-1} + 4$ (where $a > 0$ and $a \neq 1$) always passes through a fixed point $P$. The coordinates of point $P$ are \_\_\_\_\_\_. | (1, 5) |
A7. The product of five different integers is 12 . What is the largest of the integers? | 3 |
3. At a rock festival, vocalists, guitarists, and drummers met, a total of 121 people. Each vocalist gave a slap on the back of each guitarist, each guitarist gave a slap on the back of each drummer, and each drummer gave a slap on the back of each vocalist. What is the maximum number of slaps on the back that the part... | 4880 |
9.195. $\log _{3} \log _{x^{2}} \log _{x^{2}} x^{4}>0$. | x\in(-\sqrt{2};-1)\cup(1;\sqrt{2}) |
The surface area of a sphere with a radius of 3 is ( )
A: $3\pi$
B: $9\pi$
C: $12\pi$
D: $36\pi$ | D |
Translate the graph of the function $y=\sin 2x$ along the $x$-axis to the left by $\dfrac{\pi }{6}$ units, and stretch the ordinate to twice its original length (the abscissa remains unchanged) to obtain the graph of the function $y=f(x)$. If the minimum value of the function $y=f(x)+a$ on the interval $\left[ 0,\dfrac... | 2 \sqrt{3} |
10. There are between 200 and 300 pencils that can be placed in red boxes of 10 or in blue boxes of 12 pencils each. When all the pencils were packed in red boxes, 7 pencils were left, and when packed in blue boxes, 3 pencils were short. How many pencils were intended for packing | 237,297 |
Given a sequence $\{a_n\}$ where each term is a positive integer and $a_1 < a_2 < \ldots < a_n$, let the set $A_k = \{x | x = \sum_{i=1}^k \lambda_i a_i, \lambda_i \in \{-1, 0, 1\}\}$ for $1 \leq k \leq n$.
Property 1: If for every $x \in A_k$ there exists a unique set of $\lambda_i$ $(i = 1, 2, \ldots, k)$ such that ... | a_n = 3^{n-1} |
8.1. Find the maximum value of the expression $\frac{\sin \left(x+\frac{\pi}{4}\right)}{2 \sqrt{2}(\sin x+\cos x) \cos 4 x-\cos 8 x-5}$. | 0.5 |
2. Find all solutions to the equation
$$
x^{4}=y^{2}+2 y+2
$$
where $x, y$ are integers. | (-1,-1)(1,-1) |
13. Angela has the polynomials $x-1,(x-1)(x-2),(x-1)(x-2)(x-3), \ldots$ up to $(x-1)(x-2) \cdots(x-2017)(x-2018)$, and she divides them into two groups. Let $p(x)$ be the product of the polynomials in the first group and $q(x)$ the product of the polynomials in the second group. Angela notices that the polynomial $p(x)... | 1009 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $a^{2}=b^{2}+ \frac{1}{4}c^{2}$, then $\frac{a\cos B}{c}=$ _____ . | \frac{5}{8} |
## Task 3 - 050713
The driver of a car registered in the GDR fled the scene after a traffic accident. After questioning several witnesses, the following information was obtained about the police registration number of the car:
a) The two letters of the license plate were AB or AD.
b) The two front digits were the sa... | 32 |
A number, after being increased by 5, decreased by 2, and then divided by 4, equals 7. This number is ( )
A: 35
B: 31
C: 25
D: 28 | C |
8.47 The number of real roots of the equation $x=\sin x+1993$ is .
(A) 0 .
(B) 1 .
(C) 2 .
(D) More than 2 .
(China Jiangsu Province Suzhou City High School Mathematics Competition, 1993) | 1 |
4. Ana chose the digits $1,2,3,4,5,6,7$ and 9. She decided to form groups of 4 prime numbers and use all the chosen digits for each group of prime numbers. What is the sum of the prime numbers in each group?
Naloge rešuj samostojno. Za reševanje imaš na voljo 210 minut.
Uporaba zapiskov, literature ali žepnega računa... | 190 |
3. As shown in the figure, there are two cubes of different sizes, the edge length of the larger cube is 6 times that of the smaller cube. The 6 faces of the larger cube are painted red, and the 6 faces of the smaller cube are painted yellow, then the two cubes are glued together. The red area on the surface of this so... | 43 |
6.130. In the equation $x^{2}-2 x+c=0$, determine the value of $c$ for which its roots $x_{1}$ and $x_{2}$ satisfy the condition $7 x_{2}-4 x_{1}=47$. | -15 |
Given that point $F$ is the focus of the parabola $y^{2}=4x$, and $A$, $B$, $C$ are three points on the parabola, and $O$ is the coordinate origin. If $F$ is the centroid of $\triangle ABC$, and the areas of $\triangle OFA$, $\triangle OFB$, and $\triangle OFC$ are respectively $S_{1}$, $S_{2}$, and $S_{3}$, then $S_{1... | 3 |
A math interest group at a certain middle school encountered some new mathematical symbols during an extracurricular study and exploration. They excerpted some materials as follows: For three real numbers $a$, $b$, $c$, $M\{a,b,c\}$ represents the average of these three numbers, and $\min \{a,b,c\}$ represents the smal... | a = 2 |
Two cards are dealt from a deck of four red cards labeled $A$, $B$, $C$, $D$ and four green cards labeled $A$, $B$, $C$, $D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair?
$\textbf{(A)}\ \frac{2}{7}\qquad\textbf{(B)}\ \frac{3}{8}\qquad\textbf{(C)}\... | \frac{4}{7} |
Which of the following calculations is correct?
A: $x^{2}+x^{3}=x^{5}$
B: $(x-y)^{2}=x^{2}-y^{2}$
C: $(x^{2})^{3}=x^{6}$
D: $6x^{6}\div 3x^{3}=2x^{2}$ | C |
To make the expression $x^2+kxy+81y^2$ a perfect square, the value of $k$ must be ( )
A: ±9
B: 18
C: 9
D: ±18 | D |
An arithmetic-geometric sequence $\{a\_n\}$ has a sum of its first $n$ terms denoted as $S\_n$. It is known that $S\_2$, $S\_4$, and $S\_3$ form an arithmetic sequence.
(1) Find the common ratio $q$ of the sequence $\{a\_n\}$.
(2) If $a\_1 - a\_3 = 3$, determine the number of terms for which the sum is $\frac{21}{8}$. | 6 |
24*. The case of Brown, Jones, and Smith is being investigated. One of them committed a crime. During the investigation, each of them made two statements.
Brown: "I didn't do it. Jones didn't do it."
Smith: "I didn't do it. Brown did it."
Jones: "Brown didn't do it. Smith did it."
It was later found out that one of... | Brown |
8. Find the smallest real number $m$, such that for any positive integers $a, b, c$ satisfying $a+b+c=1$, the inequality $m\left(a^{3}+b^{3}+c^{3}\right) \geqslant 6\left(a^{2}+b^{2}+c^{2}\right)+1$ holds.
(2006 China Southeast Mathematical Olympiad) | 27 |
Example 4 Solve the equation $\sqrt{12-\frac{12}{x^{2}}}+\sqrt{x^{2}-\frac{12}{x^{2}}}=x^{2}$. | x = \pm 2 |
\section*{Task 1 - 011241}
In 27,000 fertilization trials with phosphorus fertilizers, the following average potato yields were observed:
\[
\begin{array}{cc}
\text { Fertilizer application in terms of P2O5 (dt/ha) } & \text { Yield (dt/ha) } \\
0.0 & 237 \\
0.3 & 251 \\
0.9 & 269
\end{array}
\]
The relationship bet... | b=53.69,=290.69,k=0.44,y(0.6)=261.46\mathrm{dt}/\mathrm{ha},y(1.2)=274.77\mathrm{dt}/\mathrm{ha}, |
Let the function $f(x)=\log_{2}(2^{x}+1)$, $g(x)=\log_{2}(2^{x}-1)$. If the function $F(x)=g(x)-f(x)-m$ has a zero point in the interval $[1,2]$ for $x$, then the range of values for $m$ is \_\_\_\_\_\_\_\_. | \left[\log_{2}\frac{1}{3},\log_{2}\frac{3}{5}\right] |
Given that the function $f(x)$ satisfies the equation $$f( \frac {1}{x})+2f(x)=3x (x≠0)$$, determine the correct option from the following:
A: The function $f(x)$ is an even function and is monotonically decreasing on $(0, +\infty)$
B: The function $f(x)$ is an even function and is monotonically increasing on $(0, +\... | \text{D} |
A school conducted a survey test on 1400 high school science students and found that the total score $\xi$ (0 < $\xi$ < 750) of five subjects roughly follows a normal distribution $N(450, 130^2)$. If the probability of $\xi$ falling within (0, 280) is 0.107, then the number of students among the 1400 who scored above 6... | \text{C} |
$11 \cdot 1$ In the expansion of $\left(\frac{x^{2}}{2}-\frac{2}{x}\right)^{8}$, the coefficient of $x^{7}$ is
(A) 56 .
(B) -56 .
(C) 14 .
(D) -14 .
(E) 0 .
(11th American High School Mathematics Examination, 1960) | -14 |
6. Given $x, y, z \in \mathbf{R}_{+}$, and $\sqrt{x^{2}+y^{2}}+z=1$. Then the maximum value of $x y+2 x z$ is $\qquad$ | \frac{\sqrt{3}}{3} |
## problem statement
Calculate the area of the parallelogram constructed on vectors $a_{\text {and }} b$.
$a=3 p-4 q$
$b=p+3 q$
$|p|=2$
$|q|=3$
$(\widehat{p, q})=\frac{\pi}{4}$ | 39\sqrt{2} |
Given the function $f(x)=\ln \frac{x}{1-x}$, find the domain of the function $f(x+1)$. | (-1,0) |
20. From 1000 to 9999, the probability of randomly selecting a four-digit number where all digits are different and the unit digit is odd is ( ).
(A) $\frac{14}{75}$
(B) $\frac{56}{225}$
(C) $\frac{107}{400}$
(D) $\frac{7}{25}$
(E) $\frac{9}{25}$ | \frac{56}{225} |
7.3. In 60 chandeliers (each chandelier has 4 lampshades), lampshades need to be replaced. Each electrician spends 5 minutes replacing one lampshade. A total of 48 electricians will be working. Two lampshades in a chandelier cannot be replaced simultaneously. What is the minimum time required to replace all the lampsha... | 25 |
# 5. Problem 5
Consider all possible quadratic trinomials $x^{2}+p x+q$ with a positive discriminant, where the coefficients $p$ and $q$ are integers divisible by 5. Find the largest natural number $n$ such that for any trinomial with the described properties, the sum of the hundredth powers of the roots is an integer... | 50 |
(7) Let $S_{n}$ be the sum of the first $n$ terms of the arithmetic sequence $\left\{a_{n}\right\}$, $S_{17}=170, a_{2000}=2001$, then
$$
S_{2008}=
$$ | 1004\times2011 |
Example 6. Father's age is 48 years old, son's age is 20 years old. How many years later will the father's age be 3 times the son's age? | 6 |
Given a function $f(x)$ ($x \in \mathbb{R}$) that satisfies $f(-x) = 8 - f(4 + x)$, and a function $g(x) = \frac{4x + 3}{x - 2}$, if the graphs of functions $f(x)$ and $g(x)$ have 168 common points, denoted as $P_i(x_i, y_i)$ ($i = 1, 2, \ldots, 168$), then the value of $(x_1 + y_1) + (x_2 + y_2) + \ldots + (x_{168} + ... | D |
Given that the domain of $f(x+2)$ is $(-2,2)$, then the domain of $f(x-3)$ is \_\_\_\_\_\_. | (3,7) |
4. In $\triangle A B C$, it is known that $A B=7, A C=5$, the angle bisector $A D$ intersects the median $B M$ at point $P$. Then $\frac{A P}{P D}=$ | \frac{12}{7} |
The motion equation of an object is $s=2t\sin t + t$. Its velocity equation is ( )
A: $v=2\sin t + 2t\cos t + 1$
B: $v=2\sin t + 2t\cos t$
C: $v=2\sin t$
D: $v=2\sin t + 2\cos t + 1$ | A |
In 6 years, my father will be 3 times as old as I was when my father's age was equal to the sum of my age and my sister's age at that time. My current age is the same as my father's age was then. In 19 years, my father will be twice as old as my sister is now. How old are these three people today? | 69,47,44 |
Given the function $f\left(x\right)=2f'\left(0\right)x+e^{x}$, then $f\left(2\right)=\left(\ \ \right)$
A: $e^{2}-4$
B: $e^{2}-1$
C: $-1$
D: $e^{2}+4$ | A |
In the geometric sequence $\{a_n\}$, if $a_5=2$, then $a_1 \cdot a_9=$ ( )
A: $4$
B: $5$
C: $2$
D: $25$ | A |
Given vectors $a=(\sin x,-1)$ and $b=( \sqrt{3}\cos x,- \frac{1}{2})$, and the function $f(x)=(a+b)·a-2$.
(1) Find the smallest positive period $T$ of the function $f(x)$.
(2) Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ of $\triangle ABC$ respectively, where $A$ is an acute angle, ... | 2\sqrt{3} |
Given a unit cube $A B C D A_1 B_1 C_1 D_1$, $M$ is the midpoint of $B B_1$. Find the angle and the distance between the lines $A_1 B$ and $C M$. In what ratio does the common perpendicular of these lines divide the segments $C M$ and $A_1 B$? | \arccos\frac{1}{\sqrt{10}};\frac{1}{3},CY:YM=8:1;\frac{A_{1}X}{XB}=\frac{7}{2} |
19. (12 points) As shown in Figure 2, for the pyramid $P-ABCD$:
(1) Find a point $F$ on edge $PD$ such that $AF \parallel$ plane $PEC$;
(2) Find the cosine value of the dihedral angle $D-PE-A$. The base $ABCD$ is a rhombus with side length 2, $\angle ABC = 60^{\circ}$, $E$ is the midpoint of $AB$, $PA \perp$ plane $ABC... | \frac{4 \sqrt{31}}{31} |
The expression $21x^2 +ax +21$ is to be factored into two linear prime binomial factors with integer coefficients. This can be done if $a$ is:
$\textbf{(A)}\ \text{any odd number} \qquad\textbf{(B)}\ \text{some odd number} \qquad\textbf{(C)}\ \text{any even number}$
$\textbf{(D)}\ \text{some even number} \qquad\textbf{... | D |
In the $2013\cdot Jining$ test, if the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n = n^2 - 4n + 2$, then $|a_1| + |a_2| + \ldots + |a_{10}| = \_\_\_\_\_\_\_$. | 66 |
## Task A-3.1. (4 points)
Determine all $x \in[0,2 \pi]$ for which $3 \sin x$ and $\frac{4}{7} \cos (2 x)+\frac{5}{3} \sin x$ are integers. | x_{1}=\arcsin\frac{1}{3},\,x_{2}=\pi-\arcsin\frac{1}{3} |
The third chapter of "Jiuzhang Suanshu" (The Nine Chapters on the Mathematical Art) introduces the concept of "Shuai Fen," which refers to ratio-based decremental distribution. "Shuai Fen" literally means to distribute quantities in a decreasing manner according to certain proportions, often referred to as "Shuai Fen r... | D |
1. Given a sequence of positive numbers $\left\{a_{n}\right\}$ satisfying $a_{n+1} \geqslant 2 a_{n}+1$, and $a_{n}<2^{n+1}$ for $n \in \mathbf{Z}_{+}$. Then the range of $a_{1}$ is | 0<a_{1} \leqslant 3 |
2. Given that $f(x)$ is a periodic function on $\mathbf{R}$ with the smallest positive period of 2, and when $0 \leqslant x<2$, $f(x)=x^{3}-x$, then the number of intersections of the graph of the function $f(x)$ with the $x$-axis in the interval $[0,6]$ is $\qquad$ . | 7 |
A number $k$ is randomly taken from the interval $[-1, 2]$. The probability that the line $y=k(x-4)$ intersects with the circle $x^2+y^2=4$ is ( )
A: $$\frac{\sqrt{3}}{3}$$
B: $$\frac{\sqrt{3}}{2}$$
C: $$\frac{2\sqrt{3}}{9}$$
D: $$\frac{\sqrt{3}}{6}$$ | \text{C} |
In $\triangle ABC$, $AB= \sqrt {6}- \sqrt {2}$, $C=30^{\circ}$, find the maximum value of $AC+BC$. | 4 |
Calculate the following definite integrals.
(1) $$\int_{ -3 }^{ 2 }|x+1|dx$$
(2) Let $$f(x)= \begin{cases} x^{2} & (0\leq x<1) \\ 2-x & (1\leq x\leq 2) \end{cases}$$, then calculate $$\int_{ 0 }^{ 2 }f(x)dx$$. | \frac {5}{6} |
5. The distance $A B$ is 100 km. Cyclists start simultaneously from $A$ and $B$ towards each other with speeds of 20 and $30 \mathrm{km} / h$ respectively. At the same time, a fly with a speed of 50 km/h takes off from $A$, flies to meet the cyclist from $B$, then turns around and flies back to meet the cyclist from $A... | 70 |
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