problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given the pure imaginary number $z$ satisfies $(2-i)z=4+2ai$, where $i$ is the imaginary unit, find the value of the real number $a$. | 4 |
Define the determinant $$\begin{vmatrix} a & b \\ c & d \end{vmatrix}$$ as $ad - bc$. Compute the sum of the determinants: $$\begin{vmatrix} 2 & 4 \\ 6 & 8 \end{vmatrix} + \begin{vmatrix} 10 & 12 \\ 14 & 16 \end{vmatrix} + \ldots + \begin{vmatrix} 2010 & 2012 \\ 2014 & 2016 \end{vmatrix} = \_\_\_\_\_\_.$$ | -2016 |
If ¬(p∧q) is a false proposition, then ( )
A: p is true, q is false
B: p is false, q is false
C: p is true, q is true
D: p is false, q is true | \text{C} |
Which of the following statements about the function $f(x)=\frac{2-|x|}{1+|x|}$ are correct?
A: $f\left(x\right)$ is an even function.
B: $f\left(x\right)$ is monotonically decreasing on $\left(0,+\infty \right)$.
C: The range of $f\left(x\right)$ is $\left(-1,1\right]$.
D: The range of $f\left(x\right)$ is $\left(... | ABD |
Given \\(a > 0\\), the function \\(f(x) = x^3 - ax\\) is monotonically increasing on the interval \\([1, +\infty)\\). The maximum value of \\(a\\) is \_\_\_\_\_\_\_\_. | 3 |
Given a complex number $z$ that satisfies the following two conditions:
① $1 < z + \frac{2}{z} \leqslant 4$.
② The real part and the imaginary part of $z$ are both integers, and the corresponding point in the complex plane is located in the fourth quadrant.
(I) Find the complex number $z$;
(II) Calculate $|\overlin... | \frac{\sqrt{65}}{5} |
10. $[8]$ Evaluate the integral $\int_{0}^{1} \ln x \ln (1-x) d x$. | 2-\frac{\pi^{2}}{6} |
Let $S\_n$ be the sum of the first $n$ terms of the sequence ${a\_n}$, and $2a\_n + S\_n = An^2 + Bn + C$.
1. Find $a\_n$ when $A=B=0$, $C=1$.
2. If the sequence ${a\_n}$ is an arithmetic sequence and $A=1$, $C=-2$.
a. Find $a\_n$.
b. Let $b\_n = 2^n a\_n$, and let $T\_n$ be the sum of the first $n$ terms of the... | m \leq -3 |
Given the set $A = \{x | x > -1\}$, then ( )
A: $0 \subseteq A$
B: $\{0\} \in A$
C: $\emptyset \in A$
D: $\{0\} \subseteq A$ | \text{D} |
Given that the system of inequalities represents a planar region $\Omega$, where $k \geq 0$, find the value of $k$ when the area of $\Omega$ is minimized. | 1 |
Which of the following fractions is correctly simplified?
A: $\frac{y^{2}}{x^{2}}=\frac{y}{x}$
B: $\frac{a+c^{2}}{b+c^{2}}=\frac{a}{b}$
C: $\frac{a+b}{ma+mb}=\frac{1}{2}$
D: $\frac{a-b}{b-a}=-1$ | D |
A factory assigns five newly recruited employees, including A and B, to three different workshops. Each workshop must have at least one employee, and employees A and B must be assigned to the same workshop. The number of different assignment methods is ( )
A: 18
B: 24
C: 30
D: 36 | \text{D} |
Problem 3. Five different natural numbers are written in a circle on the board. Each of them, Petya divided by the next one in the clockwise direction, and then wrote down the 5 resulting numbers (not necessarily integers) on a piece of paper. Can the sum of the 5 numbers on the paper be an integer? | 18 |
Given the arithmetic sequence $\{a\_n\}$, the sum of its first $n$ terms is $S\_n$. If $S\_5 = 35$ and $S\_9 = 117$, find the value of $a\_4$. | 10 |
Given that $a\_1=2$, and the point $(a\_n,a_{n+1})$ is on the function $f(x)=x^{2}+2x$, where $n=1$, $2$, $3$, $…$
(1) Prove that the sequence ${\lg (1+a\_n)}$ is a geometric sequence, and find the general term formula for the sequence ${a\_n}$;
(2) Let $b\_n= \frac {1}{a\_n}+ \frac {1}{a\_n+2}$, find the sum of the fi... | 1- \frac {2}{3^{2^{n}}-1} |
Given vectors $\overrightarrow{a}=(1,2)$, $\overrightarrow{b}=(-2,-4)$, and $|\overrightarrow{c}|= \sqrt{5}$, if $(\overrightarrow{c}-\overrightarrow{b})\cdot\overrightarrow{a}=\frac{15}{2}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{c}$ is ( ).
A: $30^{\circ}$
B: $60^{\circ}$
C: $150^{\circ}$
D... | D |
In triangle $\triangle ABC$, given $a=1$, $b=2$, $C=\frac{{2π}}{3}$, then $c=\left(\ \ \right)$
A: $\sqrt{2}$
B: $\sqrt{3}$
C: $\sqrt{5}$
D: $\sqrt{7}$ | D |
8. In triangle $A B C$ with $\angle B=120^{\circ}$, the angle bisectors $A A_{1}, B B_{1}, C C_{1}$ are drawn. Find $\angle C_{1} B_{1} A_{1}$. | 90 |
2.1. Lisa wrote a quadratic equation. Artem erased its free term, so the equation now looks like $\operatorname{\operatorname {mak}} 2 x^{2}+20 x+\ldots=0$. Lisa doesn't remember what number Artem erased, but she remembers that the equation has exactly one real root. What is this root? | -5 |
Xiaoming and his father are making a cake at home during the holiday. Xiaoming made a cake in the shape of a right circular cone with a base radius of $10cm$ (the cross-section along the axis is an equilateral triangle). Now, they want to evenly sprinkle $1g$ of sesame seeds all over the cake's surface. Given that $1g$... | B |
Anička has 50 Kč, Anežka has 46 Kč, and they want to buy snacks for a family celebration with all their money. They are deciding between cupcakes and windmills: a windmill is 4 Kč more expensive than a cupcake, and with all their money, they could buy a third more cupcakes than windmills.
How much does each snack cost... | 12 |
From the sets $\{P, Q, R, S\}$ and $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$, select 2 elements from each set to form a sequence (both letters and numbers cannot be repeated), and in each sequence, the letter Q and the number 0 can appear at most once. The number of different sequences that can be formed under these condition... | 5832 |
7.5 There are 100 points distributed on a straight line. We mark the midpoints of all possible line segments with these points as endpoints. How many such midpoints can we get at least?
Will the above text be translated into English, please keep the original text's line breaks and format, and output the translation re... | 197 |
Given a line \\(l\\) passing through point \\(A(-4,0)\\) intersects with the parabola \\(G\\): \\(x^{2}=2py(p > 0)\\) at points \\(B\\) and \\(C\\). When the slope of the line is \\(\dfrac{1}{2}\\), \\(\overrightarrow{AC}=4\overrightarrow{AB}\\).
\\((1)\\) Find the equation of the parabola \\(G\\);
\\((2)\\) Let the pe... | (2,+\infty) |
Problem 6. The function $f$ is such that $f(x+2 y)-f(3 x-2 y)=2 y-x$ for all $x, y$. Find all possible values of the expression $\frac{f(5 t)-f(t)}{f(4 t)-f(3 t)}$. | 4 |
Given the function $f(x) = e^{x+1} - \ln(x+m)$, where $x=-1$ is a critical point of $f(x)$, then which of the following statements is true?
A: $f(x)$ is monotonically decreasing on $(-\infty, -1)$ and monotonically increasing on $(-1, +\infty)$.
B: $f(x)$ is monotonically increasing on $(-2, -1)$ and monotonically de... | D |
Given a circle of radius $\sqrt{13}$, let $A$ be a point at a distance $4 + \sqrt{13}$ from the center $O$ of the circle. Let $B$ be the point on the circle nearest to point $A$. A line passing through the point $A$ intersects the circle at points $K$ and $L$. The maximum possible area for $\triangle BKL$ can be writte... | 146 |
8. Given the function $f(x)=\mathrm{e}^{x}\left(x-a \mathrm{e}^{x}\right)$ has exactly two critical points $x_{1} 、 x_{2}\left(x_{1}<x_{2}\right)$. Then the range of values for $a$ is $\qquad$ | \left(0, \frac{1}{2}\right) |
Given four inequalities: ①$x \gt 1$, ②$x \gt 4$, ③$2-x \gt -1$, ④$x \lt 2$, choose two of them to form an inequality group. To make the integer solution of this inequality group be $2$, the two inequalities to be chosen are ______ (only fill in the numbers). | 1\text{ and }3 |
There are 5 people participating in a lottery drawing event, where each person sequentially draws one ticket from a box containing 5 tickets (among which 3 are winning tickets) without replacement, until all 3 winning tickets are drawn, ending the event. What is the probability that the event ends exactly after the 4th... | C |
If $a_1$, $a_2$, ..., $a_8$ form an arithmetic sequence with all terms greater than zero and the common difference $d \neq 0$, then ( )
A: $a_1a_8 > a_4a_5$
B: $a_1a_8 a_4 + a_5$
D: $a_1a_8 = a_4a_5$ | \text{B} |
If the variance of the sample $a_{1}$, $a_{2}$, $a_{3}$ is $2$, then the variance of the sample $2a_{1}+3$, $2a_{2}+3$, $2a_{3}+3$ is ______. | 8 |
For real constant numbers $ a,\ b,\ c,\ d,$ consider the function $ f(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d$ such that $ f( \minus{} 1) \equal{} 0,\ f(1) \equal{} 0,\ f(x)\geq 1 \minus{} |x|$ for $ |x|\leq 1.$
Find $ f(x)$ for which $ \int_{ \minus{} 1}^1 \{f'(x) \minus{} x\}^2\ dx$ is minimized. | f(x) = -x^2 + 1 |
Given $p:(x+2)(x-3) \lt 0$, $q:|x-1| \lt 2$, then $p$ is ( )
A: sufficient but not necessary condition
B: necessary but not sufficient condition
C: necessary and sufficient condition
D: neither necessary nor sufficient condition | B |
The imaginary unit is $i$, and the complex number $\frac {i}{2+i}$ equals ( )
A: $\frac {-1+2i}{3}$
B: $\frac {1+2i}{3}$
C: $\frac {1+2i}{5}$
D: $\frac {-1+2i}{5}$ | \text{C} |
Example 1 Solve the equation $\sqrt[4]{10+x}+\sqrt[4]{7-x}=3$.
| x_1=6, x_2=-9 |
20. (3 points) The PLA soldiers need to repair a dam while the flood continuously destroys it. If 10 people need 45 minutes, and 20 people need 20 minutes, then 14 people need $\qquad$ minutes to repair the dam. | 30 |
In tetrahedron $O\_ (ABC)$, point $M$ is on $OA$ with $OM$=$2MA$, and $N$ is the midpoint of $BC$. If $\overset{→}{OG} = \frac{1}{3} \overset{→}{OA} + \frac{x}{4} \overset{→}{OB} + \frac{x}{4} \overset{→}{OC}$, find the value of $x$ that makes $G$ collinear with $M$ and $N$. | x = 1 |
9. (This question is worth 16 points) Let $O$ be the circumcenter of acute $\triangle A B C$, and the areas of $\triangle B O C, \triangle C O A, \triangle A O B$ form an arithmetic sequence. Find the minimum value of $\tan A+2 \tan C$.
---
The translation is provided as requested, maintaining the original format and... | 2\sqrt{6} |
34. Find an integer $x$ that satisfies the equation
$$
x^{5}-101 x^{3}-999 x^{2}+100900=0
$$ | 10 |
Given that $(( \overline {z}-1+3i)(2-i)=4+3i$ (where $i$ is the imaginary unit, and $ \overline {z}$ is the complex conjugate of $z$), find the imaginary part of $z$.
A: $1$
B: $-1$
C: $i$
D: $-i$ | A |
2. A new math teacher came to the class. He conducted a survey among the students of this class, asking if they love math. It turned out that $50 \%$ love math, and $50 \%$ do not. The teacher conducted the same survey at the end of the school year. This time, $60 \%$ of the students answered "yes," and $40 \%$ answere... | 10;90 |
2. Let $x_{1}=\frac{1}{4}, x_{2}=\frac{3}{16}, x_{n+2}=\frac{3}{16}-\frac{1}{4}\left(x_{1}\right.$ $\left.+x_{2}+\cdots+x_{n}\right)(n=1,2, \cdots)$. Then $x_{n}=$ | \frac{n+1}{2^{n+2}} |
6.1.9 Find all real-coefficient polynomials $p(x)$ such that for any real numbers $a, b, c$ satisfying $a b + b c + c a = 0$, we have
$$
p(a-b) + p(b-c) + p(c-a) = 2 p(a+b+c).
$$ | p(x)=\alphax^{2}+\betax^{4},\alpha,\beta\in{R} |
8 In $\triangle A B C$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively, and $b \neq 1$. Also, $\frac{C}{A}, \frac{\sin B}{\sin A}$ are the roots of the equation $\log _{\sqrt{b}} x=\log _{b}(4 x-4)$. Then the shape of $\triangle A B C$ is $\qquad$ | A=\frac{\pi}{6},C=\frac{\pi}{3},B=\frac{\pi}{2} |
Given that the angle of inclination of the line $x=1$ is $\alpha$, determine the value of $\alpha$. | 90^\circ |
Given $$\overrightarrow {a} = (1, 2, -3)$$, and $$\overrightarrow {b} = (2, a-1, a^2 - \frac {1}{3})$$, then "a=1" is a condition for "$$\overrightarrow {a} \perp \overrightarrow {b}$$" that is ( )
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient cond... | \text{A} |
Calculate the area of the triangle formed by the line $x + y - 2 = 0$ and the coordinate axes. | S = 2 |
Given the following arrays, each composed of three numbers: $(1,2,3)$, $(2,4,6)$, $(3,8,11)$, $(4,16,20)$, $(5,32,37)$, ..., $(a\_n,b\_n,c\_n)$. If the sum of the first $n$ terms of the sequence $\{c\_n\}$ is denoted as $M\_n$, find the value of $M\_{10}$. | 2101 |
9. Let $-\frac{\pi}{2} \leqslant x \leqslant \frac{\pi}{2}$, and the equation $\cos 2x - 4a \cos x - a + 2 = 0$ has two distinct solutions. Find the range of values for $a$. | {\frac{1}{2}}\cup(\frac{3}{5},1] |
2.22 Given that $p$ and $q$ are rational numbers, and $x=\frac{\sqrt{5}-1}{2}$ satisfies $x^{3}+p x+q=0$. Then the value of $p+q$ is
(A) -1.
(B) 1.
(C) -3.
(D) 3.
(Anhui Province, China Junior High School Mathematics Competition, 1997) | -1 |
Given $a \in \mathbb{R}$, the function $f(x) = \sin x - |a|$, where $x \in \mathbb{R}$, is an odd function. Then, $a$ equals to ( )
A: 0
B: 1
C: -1
D: $\pm 1$ | A |
Given a sequence $\{a_{n}\}$ where each term is non-zero, and the sum of the first $n$ terms is $S_{n}$, with $a_{1}=1$, and $4S_{n}=a_{n+1}a_{n}+1$, find $a_{100}$. | 199 |
$33 \quad m$ and $n$ are two positive integers satisfying $1 \leq m \leq n \leq 40$. Find the number of pairs of $(m, n)$ such that their product $m n$ is divisible by 33 . | 64 |
Solve the equation: $\frac{2x-1}{3}-\frac{x-2}{6}=2$. | x = 4 |
## Task 20/83
We are looking for all triples of consecutive even or odd numbers, where the sum of their squares is a (in decimal notation truly) four-digit number with four identical digits. | (41;43;45) |
189. Find the integral curve of the differential equation $y^{\prime \prime}+2 y^{\prime}+2 y=0$, passing through the point $(0 ; 1)$ and tangent to the line $y=x+1$ at this point. | e^{-x}(\cosx+2\sinx) |
Let $p$ a prime number, $p\geq 5$. Find the number of polynomials of the form
\[ x^p + px^k + p x^l + 1, \quad k > l, \quad k, l \in \left\{1,2,\dots,p-1\right\}, \] which are irreducible in $\mathbb{Z}[X]$.
[i]Valentin Vornicu[/i] | 2 \binom{\frac{p-1}{2}}{2} |
The graph of ${(x^2 + y^2 - 1)}^3 = x^2 y^3$ is a heart-shaped curve, shown in the figure below.
[asy]
import graph;
unitsize(10);
real f(real x)
{
return sqrt(cbrt(x^4) - 4 x^2 + 4);
}
real g(real x)
{
return (cbrt(x^2) + f(x))/2;
}
real h(real x)
{
return (cbrt(x^2) - f(x)) / ... | 7 |
Let $T$ be an acute triangle. Inscribe a rectangle $R$ in $T$ with one side along a side of $T.$ Then inscribe a rectangle $S$ in the triangle formed by the side of $R$ opposite the side on the boundary of $T,$ and the other two sides of $T,$ with one side along the side of $R.$ For any polygon $X,$ let $A(X)$ denote t... | \frac{2}{3} |
Given the function $f(x)=4\sin (2x+\frac{\pi }{6})$ where $0\leqslant x\leqslant \frac{91\pi }{6}$, let ${x\_1}, {x\_2}, \cdots , {x\_n}$ denote all the zeros of the function $F(x)=f(x)-3$ in ascending order, i.e., ${x\_1} < {x\_2} < \cdots < {x\_n}$. Find the value of ${x\_1}+2{x\_2}+2{x\_3}+\cdots +2{x_{n-1}}+{x\_n}$... | 445\pi |
Among the following propositions about line $l$, line $m$, and planes $\alpha$, $\beta$, the true proposition is ( )
A: If $l \subset \beta$, and $\alpha \perp \beta$, then $l \perp \alpha$
B: If $l \perp \beta$, and $\alpha \parallel \beta$, then $l \perp \alpha$
C: If $\alpha \cap \beta = m$, and $l \perp m$, then $l... | \text{B} |
Two thieves stole an open chain with $2k$ white beads and $2m$ black beads. They want to share the loot equally, by cutting the chain to pieces in such a way that each one gets $k$ white beads and $m$ black beads. What is the minimal number of cuts that is always sufficient? | 2 |
Given the Earth's radius is $R$, if location A is at 45°N, 120°E, and location B is at 75°S, 120°E, then the surface distance between locations A and B is ( )
A: $\sqrt {3}R$
B: $\frac {\pi }{6}R$
C: $\frac {5\pi }{6}R$
D: $\frac {2\pi }{3}R$ | D |
Calculate: $\left(-0.25\right)^{2022}\times 4^{2023}$ is ( )
A: $-1$
B: $1$
C: $4$
D: $-4$ | C |
8.4. What is the maximum number of cells in an $n \times n$ square, drawn on graph paper, that can be colored so that no $2 \times 2$ square contains three colored cells?
## rectangular city | n^{2}/2 |
Given vectors $\overrightarrow{a}=(1,m)$ and $\overrightarrow{b}=(3,-2)$, and $(\overrightarrow{a}+ \overrightarrow{b}) \perp \overrightarrow{b}$, then $m=$______. | 8 |
Example 4. Find the derivative $y_{x}^{\prime}$ of the function $x=e^{t} \cos t$; $y=e^{t} \sin t$ at $t=0$. | 1 |
$$
\begin{array}{l}
\text { II. (16 points) Given } \\
\cos (\alpha+\beta)=\cos \alpha+\cos \beta .
\end{array}
$$
Find the maximum value of $\cos \alpha$. | \sqrt{3}-1 |
2. (10 points) The little rabbit and the little turtle start from location $A$ to the forest amusement park at the same time. The little rabbit jumps forward 36 meters per minute, and after every 3 minutes of jumping, it plays on the spot. The first time it plays for 0.5 minutes, the second time for 1 minute, the third... | 12 |
(2) If $\lg a, \lg b$ are the two real roots of the equation $2 x^{2}-4 x+1=0$, find the value of $\lg (a b) \cdot\left(\lg \frac{a}{b}\right)^{2}$. | 4 |
In triangle $\triangle ABC$, given $a=8$, $B=30^{\circ}$, $C=105^{\circ}$, then $b$ is equal to:
A: $\dfrac{32}{3}$
B: $4\sqrt{3}$
C: $4\sqrt{6}$
D: $4\sqrt{2}$ | D |
Ben starts with an integer greater than $9$ and subtracts the sum of its digits from it to get a new integer. He repeats this process with each new integer he gets until he gets a positive $1$-digit integer. Find all possible $1$-digit integers Ben can end with from this process. | 9 |
13.324. In the first week of his vacation, the engineer spent a few rubles less than $3 / 5$ of the amount of money he had taken with him; in the second week, $1 / 4$ of the remainder plus 30 rubles; in the third week, $2 / 5$ of the new remainder plus 12 rubles; after which $6 / 35$ of the amount of money taken remain... | 1160 |
Three. (20 points) Let the domain of the function $f(x)$ be $(0,+\infty)$, and for any positive real numbers $x, y$, the following always holds:
$$
f(x y)=f(x)+f(y)
$$
It is known that $f(2)=1$, and when $x>1$, $f(x)>0$.
(1) Find the value of $f\left(\frac{1}{2}\right)$;
(2) Determine the monotonicity of $y=f(x)$ on $... | S_{n}=\frac{n(n+1)}{2}, a_{n}=n |
2. Let the sum of the squares of the first 101 positive integers starting from a positive integer $k$ be equal to the sum of the squares of the next 100 positive integers. Then the value of $k$ is $\qquad$ . | 20100 |
Given the following functions: ①$y=-3x+2$; ②$y=\frac{3}{x}$; ③$y=2x^{2}$; ④$y=-5\left(x-1\right)^{2}$, the function that satisfies "when $x \gt 1$, the function value $y$ increases as the independent variable $x$ increases" is ( )
A: ①
B: ②
C: ③
D: ④ | C |
(I) Calculate: $$8^{ \frac {2}{3}}- \sqrt {( \sqrt {2}-1)^{2}}+2^{ \frac {1}{2}}+(\frac {1}{3})^{0}-\log_{10}100$$．
(II) Given $a>0$, and $a-a^{-1}=3$, find the value of $a^{2}+a^{-2}$． | 11 |
There are ten gardeners. Each of them can plant 30 pots of tulips or 20 pots of roses. It is known that the profit for each pot of tulip is $5, and for each pot of rose is $8. In order to ensure that the total profit is not less than $1560, the maximum number of gardeners that can be arranged to plant tulips is ______. | 4 |
Let $F_1$ and $F_2$ be the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) respectively. If there exists a point $P$ on the right branch of the hyperbola such that $\overrightarrow{PF_1} \cdot \overrightarrow{PF_2} = 0$, and the lengths of the sides of $\triangle F_1PF_2$... | \text{D} |
1. If $x+y=1$ and $x^{2}+y^{2}=2$, what is $x^{4}+y^{4}$? | \frac{7}{2} |
Solve the inequality system: $\left\{{\begin{array}{l}{5x-2＜3({x+2})}\\{\frac{{2x-1}}{3}-\frac{{5x+1}}{2}≤1}\end{array}}\right.$. | -1 \leqslant x < 4 |
Let $f(t)=\frac{t}{1-t}$, $t \not= 1$. If $y=f(x)$, then $x$ can be expressed as
$\textbf{(A)}\ f\left(\frac{1}{y}\right)\qquad \textbf{(B)}\ -f(y)\qquad \textbf{(C)}\ -f(-y)\qquad \textbf{(D)}\ f(-y)\qquad \textbf{(E)}\ f(y)$ | C |
1. There are 28 students in the class. On March 8th, each boy gave each girl one flower - a tulip, a rose, or a daffodil. How many roses were given if it is known that there were 4 times as many roses as daffodils, but 10 times fewer than tulips? (A. A. Tesler) | 16 |
Example 1. Compute the integral
$$
\int_{|z|=2} \frac{\operatorname{ch} i z}{z^{2}+4 z+3} d z
$$ | \pii\cos1 |
For the function $f(x)=\sin({2x+\frac{π}{3}})$, which of the following statements is correct?
A: The smallest positive period is $\pi$
B: $f({\frac{π}{2}})=\frac{{\sqrt{3}}}{2}$
C: The graph is symmetric about the point $({\frac{π}{3}，0})$
D: The maximum value on $[{0，\frac{π}{6}}]$ is $1$ | ACD |
7.3. Find the sum $\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{2020 \cdot 2021}$. | \frac{2020}{2021} |
Quality inspectors randomly select one of the five different products numbered $1$, $2$, $3$, $4$, and $5$ for quality testing. The probability of selecting a product with a number not less than $4$ is ( ).
A: $\frac{1}{5}$
B: $\frac{2}{5}$
C: $\frac{3}{5}$
D: $\frac{4}{5}$ | B |
Given that the graph of the quadratic function $y=ax^{2}-4ax+8 (a\neq 0)$ intersects the $x$-axis at points $A$ and $B$. If $AB=6$, then $a=\_\_\_\_\_\_$. | -\frac{8}{5} |
1. (3 points) Calculate: $100-99+98-97+96-95+94-93+93-92+91=$ | 96 |
3. Ben and David go to buy bagels. For every 1 yuan Ben spends, David can save 25 fen. This way, Ben spends 12.50 yuan more than David. Then the total amount spent by both is ( ) yuan.
(A) 37.50
(B) 50.00
( C) 87.50
(D) 90.00
(E) 92.50 | 87.50 |
Given a geometric sequence $\{a_n\}$, the sum of the first $n$ terms is denoted as $S_n$, and it is given that $S_2=1$, $S_4=3$. Find the value of $S_6$.
A: 5
B: 7
C: 9
D: 11 | B |
Given $a > b$ and $ab \neq 0$, which of the following inequalities always holds true?
A: $2^{a} > 2^{b}$
B: $\lg (a-b) > 0$
C: $ \dfrac {1}{a} < \dfrac {1}{b}$
D: $ \dfrac {b}{a} < 1$ | A |
8. Find the largest real number $a$ such that for all positive integers $n$ and all real numbers $x_{0}, x_{1}, \cdots, x_{n}\left(0=x_{0}<x_{1}<\cdots\right.$ $\left.<x_{n}\right)$, we have
$$
\sum_{i=1}^{n} \frac{1}{x_{i}-x_{i-1}} \geqslant a \sum_{i=1}^{n} \frac{i+1}{x_{i}} .
$$ | \frac{4}{9} |
Given sets $A=\{x|1\leq x<7\}$, $B=\{x|\log_{2}(x-2)<3\}$, $C=\{x|x<a\}$, with the universal set being the set of real numbers $R$.
(1) Find $A\cup B$;
(2) If $A\cap C\neq\emptyset$, and $B\cap C=\emptyset$, find the range of the real number $a$. | (1,2] |
Subject II
Consider eight distinct natural numbers. By summing any seven of these eight numbers, the results obtained are: $42, 47, 50, 52, 54, 55, 56, 57$. Determine the eight numbers.
Mathematical Gazette | 2,3,4,5,7,9,12,17 |
$10+10^{3}$ equals
(A) $2.0 \times 10^{3}$
(B) $8.0 \times 10^{3}$
(C) $4.0 \times 10^{1}$
(D) $1.0 \times 10^{4}$
(E) $1.01 \times 10^{3}$
## Part B: Each question is worth 6 credits. | 1.01\times10^{3} |
Compute the sum $S=\sum_{i=0}^{101} \frac{x_{i}^{3}}{1-3 x_{i}+3 x_{i}^{2}}$ for $x_{i}=\frac{i}{101}$. | 51 |
Let $\overrightarrow{a}=(m,-1,2)$ and $\overrightarrow{b}=(3,-4,n)$. If $\overrightarrow{a} \parallel \overrightarrow{b}$, then the values of $m$ and $n$ are respectively $(\quad)$.
A: $\frac{3}{4}$, $8$
B: $- \frac{3}{4}$, $-8$
C: $- \frac{3}{4}$, $8$
D: $\frac{3}{4}$, $-8$ | A |
2. Find the pattern and fill in the number:
2020, 2018, 2012, 2000, 1980, 1950, $\qquad$
$\qquad$ | 1908 |
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