problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
2. On a straight line, four distinct points in sequence are $A, B, C, D$. The point that minimizes the sum of the distances to $A, B, C, D$
(A) can be a point on the line $A D$;
(B) is only point $B$ or $C$;
(C) is only the midpoint of segment $A D$;
(D) has infinitely many.
| D |
13.253. For hydrodynamic studies, a small model of a channel has been made. Several pipes of the same cross-section, supplying water, and several pipes of another, but also the same cross-section, intended for removing water, are connected to this model. If four inlet and three outlet pipes are opened simultaneously, t... | 65 |
Given that the odd function $f(x)$ is a decreasing function on $\mathbb{R}$, the solution set of the inequality $f(a^2)+f(2a)>0$ with respect to $a$ is
A: $(-2,0)$
B: $(0,2)$
C: $(-2,0) \cup (0,2)$
D: $(-\infty,-2) \cup (0,+\infty)$ | \text{A} |
Ten fair coins are flipped. Given that there are at least nine heads, what is the probability that all the coins show heads? | \frac{1}{11} |
73. Find the equation of the line that forms an angle of $135^{\circ}$ with the $O x$ axis and intersects the $O y$ axis at the point $(0 ; 5)$. Determine whether this line passes through the points $A(2 ; 3)$ and $B(2 ;-3)$. Plot the line. | -x+5 |
Given that the coordinates of the direction vector of the line $l$ are $(2, 2\sqrt{3})$, find the inclination angle of the line $l$. | \frac{\pi}{3} |
The imaginary part of the complex number $(i^{-1}-i)^3$ is ( )
A: 8i
B: -8i
C: 8
D: -8 | C |
Given that $F(x)=(x^{3}-2x)f(x)(x\neq 0)$ is an even function, and $f(x)$ is not identically zero, then $f(x)$ is ($\quad$).
A: Odd function
B: Even function
C: Odd function or even function
D: Neither odd nor even function | A |
Example 3 Given that $x, y, z$ are non-negative real numbers, not all zero. Find
$$
u=\frac{\sqrt{x^{2}+y^{2}+x y}+\sqrt{y^{2}+z^{2}+y z}+\sqrt{z^{2}+x^{2}+z x}}{x+y+z}
$$
the minimum value. | \sqrt{3} |
Given sets A={1, 2, 3} and B={1, 2, 4}, find A∩B (the intersection of sets A and B).
A: {1, 2, 4}
B: {2, 3, 4}
C: {1, 2}
D: {1, 2, 3, 4} | \text{C} |
Given that the terminal sides of angles $\alpha$ and $\beta$ are symmetric with respect to the origin $O$, find $\cos(\alpha - \beta)$. | -1 |
Given the complex number $\dfrac{1+z}{1-z}=i$, find $|z|=$ ______. | 1 |
The number of intersection points between the parabola $y=3x^{2}+2x+1$ and the $x$-axis is ______. | 0 |
(5 points) In a geometric sequence $\{a_n\}$ where all terms are positive, if $a_6 = a_5 + 2a_4$, then the common ratio $q =$ ． | 2 |
1. The range of the function $f(x)=(\sin x-1)(\cos x-1)(x \in \mathbf{R})$ is $\qquad$ . | [0,\frac{3+2\sqrt{2}}{2}] |
3. A teacher and his students set off by bus to a Republican Mathematics Competition. Out of the 52 seats on the bus, they sat on seats whose numbers are consecutive natural numbers whose sum is 54. How many students did the teacher take with him if only one of the seat numbers on which they sat is a prime number? | 3 |
## Task Condition
Find the angle between the planes:
$3 x-2 y-2 z-16=0$
$x+y-3 z-7=0$ | \phi=\arccos\frac{7}{\sqrt{187}}\approx5912^{\}37^{\\} |
The length of the chord intercepted by the line $x - y - 5 = 0$ on the circle $x^{2} + y^{2} - 4x + 4y + 6 = 0$ is $\_\_\_\_\_\_\_\_$. | \sqrt{6} |
10. In the Cartesian coordinate system, let point $A(0,4)$, $B(3,8)$. If point $P(x, 0)$ makes $\angle A P B$ maximum, then $x=$ $\qquad$ | 5 \sqrt{2}-3 |
34. The sum of the infinite series $1-\frac{1}{4}+\frac{1}{16}-\frac{1}{64}+\cdots$ is ).
(A) 1
(B) $\frac{9}{10}$
(C) $\frac{4}{5}$
(D) $\frac{7}{10}$
(E) * | C |
4. As shown in Figure 1, in the tetrahedron $P-ABC$, the three lateral edges $PA$, $PB$, and $PC$ are mutually perpendicular. The dihedral angles formed by the lateral faces $PAB$, $PBC$, and $PCA$ with the base $ABC$ have plane angles of $\theta_{1}$, $\theta_{2}$, and $\theta_{3}$, respectively. The area of the base ... | C |
The real part of the complex number $z = (1-i)(2+i)$ is _______. | 3 |
If the sum of the binomial coefficients of the expansion of the binomial $(x^{2}- \frac {2}{x})^{n}$ is $8$, find the sum of the coefficients of this expansion. | -1 |
16. Among the following four numbers, which can be written as the sum of 100 consecutive natural numbers is ( ).
A. 1627384950
B. 2345678910
C. 3579111300
D. 4692581470 | A |
Given $a,b,c \in \mathbb{R}^+$ and $a+b+c=1$.
$(1)$ Find the minimum value of $\left(a+1\right)^{2}+4b^{2}+9c^{2}$.
$(2)$ Prove that: $\frac{1}{\sqrt{a}+\sqrt{b}}+\frac{1}{\sqrt{b}+\sqrt{c}}+\frac{1}{\sqrt{c}+\sqrt{a}}\geqslant \frac{3\sqrt{3}}{2}$. | \frac{3\sqrt{3}}{2} |
Given a sequence $\{a_{n}\}$ that satisfies $a_{n}=\left\{\begin{array}{l}{{t}^{n-3},n≤5,}&{n∈N*}\\{(t-2)n+7,n＞5,}&{n∈N*}\end{array}\right.$, and the sequence $\{a_{n}\}$ is a monotonically increasing sequence, then the range of $t$ is ______. | (2, 5) |
Calculate: $\lg 25 - 2\lg \frac{1}{2} = \_\_\_\_\_\_$. | 2 |
4. Given $1+2 \times 3+3 \times 3^{2}+\cdots+n \times 3^{n-1}$ $=3^{n}(n a-b)+c$, for all $n \in \mathbf{N}_{+}$. Then, the values of $a, b, c$ are (.
(A) $a=0, b=c=\frac{1}{4}$
(B) $a=b=c=\frac{1}{4}$
(C) $a=\frac{1}{2}, b=c=\frac{1}{4}$
(D) There do not exist such $a, b, c$ | C |
Given $\frac{\overline{z}}{1-i} = 2+i$, the imaginary part of the complex number $z$ is ______. | 1 |
In the Cartesian coordinate system, the equation
$$
x^{2}+2 x \sin x y+1=0
$$
represents the figure ( ).
(A) a straight line
(B) a parabola
(C) a point
(D) none of the above | D |
Convert the binary number $1101_{(2)}$ to a base-5 number.
A: $32_{(5)}$
B: $23_{(5)}$
C: $21_{(5)}$
D: $12_{(5)}$ | B |
Task 1. Sasha writes down the numbers 1, 2, 3, 4, 5 in some order, places the arithmetic operation signs "++", "-", "×" and parentheses, and looks at the result of the obtained expression. For example, he can get the number 8 using the expression $(4-3) \times (2+5)+1$. Can he get the number 123?
Forming numbers from ... | 123 |
Among the following functions, the one whose range is $(0,+\infty)$ is ( )
A: $y=2^{\frac{1}{x}}$
B: $y=\sin x+\cos x$
C: $y=|x|$
D: $y=\left(\frac{1}{2}\right)^{2-x}$ | D |
17. In a certain country, an aviation network has already been established, where any city is connected to no more than three other cities, and from any city to another city, it takes no more than one transfer. How many cities can this country have at most? | 10 |
Let D be a point in the plane of △ABC, with $$\overrightarrow {AD}$$\= - $$\frac {1}{3}$$ $$\overrightarrow {AB}$$ + $$\frac {4}{3}$$ $$\overrightarrow {AC}$$. If $$\overrightarrow {BC}$$\= λ $$\overrightarrow {DC}$$ (λ ∈ R), find the value of λ. | \lambda = -3 |
3. Given the ellipse $x^{2}+4(y-a)^{2}=4$ and the parabola $x^{2}=2 y$ have common points. Then the range of values for $a$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | \left[-1, \frac{17}{8}\right] |
## 47. The Return of Murdoch - Murdoch was thrown into the dungeon, - continued
the King, - but soon escaped and left the country, but then he infiltrated the country again, thoroughly disguised so that no one could recognize him. And this time he was arrested along with one knight and one liar. All three (let's denot... | B |
In $\triangle ABC$, $a$, $b$, $c$ are the side lengths corresponding to $\angle A$, $B$, $C$ respectively. If $\cos A + \sin A - \frac{2}{\cos B + \sin B} = 0$, find $\frac{a+b}{c}$ = \_\_\_\_\_\_. | \sqrt{2} |
$2 \cdot 9$ A town's population increased by 1200 people, and then this new population decreased by $11 \%$. The current number of people in the town is 32 fewer than before the increase of 1200 people. What was the original population?
(A) 1200.
(B) 11200.
(C) 9968.
(D) 10000.
(E) None of these.
(25th American High Sc... | 10000 |
In right triangle $ABC$, $\angle C=90^{\circ}$, $AB=6$, $AC=2$, then $\cos A=\_\_\_\_\_\_$. | \frac{1}{3} |
A box contains 36 balls of the same size, including 4 white balls. The probability of drawing four balls and having the event with probability $\dfrac{C_{32}^{1}C_{4}^{3}+C_{4}^{2}}{C_{36}^{4}}$ is ( )
A: No red balls
B: At most one red ball
C: At least one white ball
D: At least one red ball | \text{B} |
A factory produces three different models of products: A, B, and C. The ratio of their quantities is 2:3:7, respectively. If a stratified sampling method is used to draw a sample that contains 10 units of model A, what is the total number of units in this sample? | 60 |
Given integers $a$, $b$, $c$, and $d$ satisfy $abcd = 25$, and $a > b > c > d$, then $|a+b|+|c+d|$ equals to ( )
A: 0
B: 10
C: 2
D: 12 | D |
17. (3 points) From the sixteen natural numbers $1,2,3,4, \cdots, 15,16$, if $n$ numbers are randomly selected, there must be two numbers such that one is three times the other. The smallest $n$ is $\qquad$. | 13 |
$\left[\begin{array}{l}\text { B } \quad \text { Inscribed and Circumscribed Circles } \\ \text { [Trigonometric Ratios in a Right Triangle] }\end{array}\right.$
In triangle $ABC$, a circle with center $O$ is inscribed. The line $BO$ intersects this circle at points $M$ and $N$, and the segments $AO$ and $CO$ intersec... | \frac{\cos\frac{\gamma}{2}}{\cos\frac{\alpha}{2}} |
7. There are less than 100 cells in a row, and at the beginning, only the cells at both ends contain a chess piece each. Several students take turns placing chess pieces in the cells. Each person places one piece at a time, and it must be placed in the middle of two adjacent cells that already contain pieces (for examp... | 7 |
Given the sequence $\{a_n\}$ where $a_1=1$ and $a_n= \frac{1}{2}a_{n-1}+1$ for $n\geqslant 2$, then $a_n=$ ()
A: $2-(\frac{1}{2})^{n-1}$
B: $(\frac{1}{2})^{n-1}-2$
C: $2-2^{n-1}$
D: $2^{n}-1$ | A |
Given the equation $3x^{m-1} + 2 = -3$ is a linear equation in one variable, the solution to $3my + 2y = 3 + m$ is ( )
A: $y= \frac {8}{5}$
B: y=-3
C: y=3
D: $y= \frac {5}{8}$ | D |
Given a sequence $\{a_n\}$ with the general term $a_n = 2^n - \frac{1}{2}(n + 3)$, if the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n$, then $S_8$ equals \_\_\_\_\_. | S_8 = 480 |
Given that $|\vec{a}|=2$, $|\vec{b}|=5$, and $\vec{a} \cdot \vec{b}=-3$, find the magnitude of $\vec{a} + \vec{b}$. | \sqrt{23} |
What is the number of ways in which one can choose $60$ unit squares from a $11 \times 11$ chessboard such that no two chosen squares have a side in common? | 62 |
Given the function $f(x)=\sqrt{3}\sin(\omega x+\varphi)+2\sin^{2}\frac{\omega x+\varphi}{2}-1(\omega > 0,0 < \varphi < \pi)$, the distance between adjacent symmetry axes is $\frac{\pi}{2}$, and $f(0)=0$.
(1) Find the analytical expression of $f(x)$;
(2) Translate the graph of the function $y=f(x)$ to the right by $\fra... | [-2, \sqrt{3}] |
Given $a$, $b$, $c \in \mathbb{R}$, which of the following statements is correct?
A: If $a > b$, then $ac^{2} > bc^{2}$
B: If $\dfrac{a}{c} > \dfrac{b}{c}$, then $a > b$
C: If $a^{3} > b^{3}$ and $ab \dfrac{1}{b}$
D: If $a^{2} > b^{2}$ and $ab > 0$, then $\dfrac{1}{a} < \dfrac{1}{b}$ | C |
7.4. Can several rectangles, the sum of the perimeters of which is 100, be combined to form a square with a side length of 16? | Yes |
2. Compute $\frac{\tan ^{2}\left(20^{\circ}\right)-\sin ^{2}\left(20^{\circ}\right)}{\tan ^{2}\left(20^{\circ}\right) \sin ^{2}\left(20^{\circ}\right)}$. | 1 |
1. Given a right triangle $A B C$ with a right angle at vertex $C$. If the length of one leg is increased by $7 \mathrm{~cm}$, while the other leg remains the same, the hypotenuse of the new triangle would be $5 \mathrm{~cm}$ longer than the original hypotenuse. If the length of the same leg in the original triangle is... | 54\mathrm{~}^{2} |
Given set $A=\{x|x^{2}+x-6=0\}$, $B=\{2,3\}$, then $A$⋂$B=\left(\ \ \right)$
A: $\varnothing $
B: $\{2\}$
C: $\{3\}$
D: $\{2,3\}$ | B |
King Arthur's bodyguards held a knightly tournament. At the end of the tournament, it turned out that the king could find a third one for any two bodyguards who participated, who had defeated both of them. What is the minimum number of bodyguards who participated in the tournament? | 7 |
Problem 2. Calculate:
$$
\left(\frac{1+2}{3}+\frac{4+5}{6}+\frac{7+8}{9}+\ldots+\frac{2017+2018}{2019}\right)+\left(1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{673}\right)
$$ | 1346 |
## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{1+x \sin x-\cos 2 x}{\sin ^{2} x}
$$ | 3 |
Given the complex number $z= \frac {1}{i(i+1)}$, then $|z|=$ （ ）
A: $\frac { \sqrt {2}}{2}$
B: $\frac {1}{2}$
C: $\frac {1}{4}$
D: $\frac {1}{8}$ | A |
Given $\tan \alpha =2$, find $\frac{1}{\sin 2\alpha + \cos^{2}\alpha}$. | 1 |
[ Properties of medians. The centroid of a triangle.] [ The ratio of the areas of similar triangles $]$
The area of triangle $A B C$ is $S$. Find the area of the triangle whose sides are equal to the medians of triangle $A B C$. | \frac{3S}{4} |
5. Let $[t]$ denote the greatest integer not exceeding the real number $t$, and let $\{t\}=t-[t]$. Given that the real number $x$ satisfies $x^{3}+\frac{1}{x^{3}}=$
18. Then $\{x\}+\left\{\frac{1}{x}\right\}=(\quad)$.
(A) $\frac{1}{2}$
(B) $3-\sqrt{5}$
(C) $\frac{3-\sqrt{5}}{2}$
(D) 1 | D |
A polynomial subtracted by $x^{2}-3y^{2}$ equals $x^{2}+2y^{2}$, then this polynomial is ( )
A: $-2x^{2}+y^{2}$
B: $2x^{2}-y^{2}$
C: $x^{2}-2y^{2}$
D: $-2x^{2}-y^{2}$ | B |
The lateral area of a cone is $6\pi$, and the radius of the base circle is $2$. Find the length of the generatrix of the cone. | 3 |
9. (10 points) There are some identical square pieces of paper. Some of them are paired to form rectangles that are twice as long as they are wide. At this point, the total perimeter of all the newly formed rectangles is exactly equal to the total perimeter of the remaining squares, and it is known that the total perim... | 280 |
If $x=y$, then which of the following expressions is correct?
A: $2x=y+2$
B: $x+2a=y+a$
C: $x-2=2-y$
D: $-\frac{1}{3}x+1=-\frac{1}{3}y+1$ | D |
5. (3 points) Wang Lei was born on National Day. If three times his age minus 8 is exactly the total number of days in the month he was born, then Wang Lei is $\qquad$ years old this year. | 13 |
When two fair dice are rolled, the probability that the sum of the numbers facing up is $5$ equals \_\_\_\_\_\_． | \frac{1}{9} |
# Task № 2.4
## Condition:
Pillulkin and Nosnayka together weigh 35 kg more than Znayka, while Nosnayka and Znayka weigh 25 kg more than Pillulkin. How much does Nosnayka weigh? Give your answer in kilograms. | 30 |
Given two fixed points $F_1(-4,0)$, $F_2(4,0)$, and a moving point $P$ on the plane, such that $|PF_1|+|PF_2|=9$, the trajectory of point $P$ is ( )
A: Circle
B: Straight line
C: Ellipse
D: Line segment | \text{C} |
Among the real numbers $-5$, $0$, $3$, and $\frac{1}{3}$, the largest real number is:
A: $3$
B: $0$
C: $-5$
D: $\frac{1}{3}$ | A |
Given the function $f(x)= \dfrac {2}{2^{x}-1}$
(1) Determine the monotonicity of the function $f(x)$ on $(0,+\infty)$, and prove your conclusion;
(2) Find the maximum and minimum values of the function $f(x)$ on $[1,\log _{2}6]$. | \dfrac {2}{5} |
Given the function $f(x)= \frac {1}{2}ax^{2}-\ln x-2$, where $a\in\mathbb{R}$.
(I) Discuss the monotonicity of the function $f(x)$.
(II) If the function $f(x)$ has two zeros, find the range of values for the real number $a$. | (0,e^{3}) |
Task B-2.5. The ages of a father and his two children (they are not twins) are powers of the same prime number. A year ago, the ages of each of them were prime numbers. How old is the father, and how old is each of his two children? | 32,4,8 |
If we consider two skew lines as "a pair," then among the 12 edges of a hexagonal pyramid, the total number of pairs of skew lines is ( )
A: $12$
B: $24$
C: $36$
D: $48$ | B |
5. $\lim _{x \rightarrow 2} \frac{\ln \left(x^{2}-3\right)}{x-2}=$ | 4 |
13. (15 points) Person A and Person B start from points A and B simultaneously, walking towards each other. Person A walks 70 meters per minute, and Person B walks 60 meters per minute. They meet at a point 80 meters away from the midpoint. Find the distance between points A and B. | 2080 |
The one-variable quadratic equation about $x$ is $x^{2}-2x+2k-3=0$ with two distinct real roots.$(1)$ Find the range of $k$.$(2)$ If $k$ is a positive integer, find the roots of the equation. | x_{1} = 1 + \sqrt{2}, \quad x_{2} = 1 - \sqrt{2} |
We have an empty equilateral triangle with length of a side $l$. We put the triangle, horizontally, over a sphere of radius $r$. Clearly, if the triangle is small enough, the triangle is held by the sphere. Which is the distance between any vertex of the triangle and the centre of the sphere (as a function of $l$ and $... | \sqrt{r^2 + \frac{l^2}{4}} |
[
A sphere with a radius of 4 has a regular hexagonal pyramid with a height of 6 inscribed in it, and a second sphere is inscribed in the pyramid. Find the radius of the second sphere.
# | \frac{3(\sqrt{5}-1)}{2} |
14-17 Let $n$ be a natural number. How many solutions does the equation $x^{2}-\left[x^{2}\right]=(x-[x])^{2}$ have in the interval $1 \leqslant x \leqslant n$?
(Swedish Mathematical Competition, 1982) | n^2-n+1 |
Example 1 The number of proper subsets of the set $\left\{x \left\lvert\,-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2}\right., 1<\right.$ $x \in \mathbf{N}\}$ is $\qquad$
(1996, National High School Mathematics Competition) | 2^{90}-1 |
Example 2 Find the maximum value of
$$x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}$$ | 1 |
Calculate:$(1)2a-3b+a-5b$;$(2)(a^{2}-6a)-3(a^{2}-2a+1)+3$;$(3)4(x^{2}y-2xy^{2})-3(-xy^{2}+2x^{2}y)$. | -2x^{2}y - 5xy^{2} |
If $\theta$ is an angle in the second quadrant, what is the sign of the value of $\sin(\cos\theta) \cdot \cos(\sin 2\theta)$? | \text{negative} |
Given the function $f(x)= \frac{x}{\ln x}$, if there exists $x\in[e,e^{2}]$ such that $f(x)-mx- \frac{1}{2}+m\leqslant 0$ holds, then the range of values for $m$ is __________. | [\frac{1}{2},+\infty) |
1. Problem: The cost of 3 hamburgers, 5 milk shakes, and 1 order of fries at a certain fast food restaurant is $\$ 23.50$. At the same restaurant, the cost of 5 hamburgers, 9 milk shakes, and 1 order of fries is $\$ 39.50$. What is the cost of 2 hamburgers, 2 milk shakes ,and 2 orders of fries at this restaurant? | 15 |
10.2. Solve the equation:
$1+\frac{3}{x+3}\left(1+\frac{2}{x+2}\left(1+\frac{1}{x+1}\right)\right)=x$. | 2 |
3.18 The distance between points $A$ and $B$ is 270 m. A body moves uniformly from $A$ to $B$; upon reaching $B$, it immediately returns with the same speed. A second body, which leaves $B$ for $A$ 11 s after the first body leaves $A$, moves uniformly but more slowly. On its way from $\boldsymbol{B}$ to $\boldsymbol{A}... | 10 |
Given point $M(\sqrt{6}, \sqrt{2})$ is on the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$, and the eccentricity of the ellipse is $\frac{\sqrt{6}}{3}$.
$(1)$ Find the equation of ellipse $C$;
$(2)$ If a line $l$ with slope $1$ intersects ellipse $C$ at points $A$, $B$, and an isosceles triangle i... | \frac{9}{2} |
6. Pete came up with all the numbers that can be formed using the digits 2, 0, 1, 8 (each digit can be used no more than once). Find their sum. | 78331 |
Benefiting from multiple favorable factors such as the country's support for the development of new energy vehicles and the "Belt and Road" development strategy, the profit margin of a certain automobile parts manufacturing company has been increasing year by year. According to statistics, the profit was 2 billion yuan... | \text{No} |
A3. Which of the following expressions has an integer value?
(A) $(\sqrt{2}-1)^{2}$
(B) $(\sqrt{2}-1)(2 \sqrt{2}+1)$
(C) $(\sqrt{2}-1)(\sqrt{2}+1)$
(D) $\left(\sqrt{\frac{1}{2}}\right)^{2}$
(E) $(\sqrt[3]{2})^{2}$ | 1 |
Determine the number of intersection points between the circle $x^{2}+y^{2}+2x-6y-15=0$ and the line $(1+3m)x+(3-2m)y+4m-17=0$. | 2 |
Let $\sin \left( \frac{\pi}{2} + \theta \right) = \frac{1}{3}$. Then, $\cos 2\theta =$ ( )
A: $- \frac{7}{9}$
B: $- \frac{1}{9}$
C: $\frac{1}{9}$
D: $\frac{7}{9}$ | \text{A} |
Given vectors $\overrightarrow{a}=(1,0)$, $\overrightarrow{b}=(m,1)$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi}{4}$.
$(1)$ Find $|\overrightarrow{a}-2\overrightarrow{b}|$;
$(2)$ If $(\overrightarrow{a}+\lambda\overrightarrow{b})$ is perpendicular to $\overrightarrow{b}$, find th... | -\frac{1}{2} |
9 Given three sets, $A=\{x \mid x \in \mathbf{R}$ and $x>1\}, B=\left\{y \left\lvert\, y=\lg \frac{2}{x+1}\right., x \in \mathrm{A}\right\}$, $C=\left\{z \left\lvert\, z=\frac{m^{2} x-1}{m x+1}\right., x \in \mathrm{A}\right\}$. If $C \subseteq B$, then the range of values for $m$ is $\qquad$. | (-\infty,-1]\cup{0} |
9.1.4. Find the minimum value of the function $f(x)=\operatorname{tg}^{2} x+2 \operatorname{tg} x+6 \operatorname{ctg} x+9 \operatorname{ctg}^{2} x+4$ on the interval $\left(0 ; \frac{\pi}{2}\right)$. | 10+4\sqrt{3} |
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