CSCA Math Practice Test — Папа в Китае

Вопрос 1

If the sets $A = \{1, 2\}$ and $B = \{1, 2, 3\}$, then

$B \subseteq A$ $A = B$ $A \in B$ $A \subseteq B$

Вопрос 2

If the sets $A = (-1, 3)$ and $B = [2, 4)$, then $A \cap B$ equals

$[2, 3]$ $[2, 3)$ $(2, 3]$ $(2, 3)$

Вопрос 3

The solution set of the quadratic inequality $x^2 - 5x + 4 < 0$ is

$(-4, -1)$ $(-\infty, 1) \cup (4, +\infty)$ $(-\infty, -4) \cup (-1, +\infty)$ $(1, 4)$

Вопрос 4

The domain of the function $y = \dfrac{1}{\sqrt{x + 1}}$ is

$[-1, +\infty)$ $(-1, +\infty)$ $(0, +\infty)$ $(-\infty, -1)$

Вопрос 5

If $\{a_n\}$ is an arithmetic progression with first term $a_1 = 2$ and common difference $d = 3$, then $a_{100}$ equals

$302$ $305$ $299$ $298$

Вопрос 6

The point $(-1, 3 - \pi)$ lies in the

third quadrant first quadrant second quadrant fourth quadrant

Вопрос 7

Compute $\left(\sin \dfrac{\pi}{4} + \cos \dfrac{\pi}{4}\right) \cdot \tan \dfrac{\pi}{4} =$

$2\sqrt{2}$ $2$ $\sqrt{2}$ $1$

Вопрос 8

The following function is odd:

$y = x \sin x$ $y = x^4$ $y = \cos x$ $y = x + x^3$

Вопрос 9

The inverse function of $y = x^3$ is

$y = \dfrac{x}{3}$ $y = 3x$ $y = x^3$ $y = \sqrt[3]{x}$

Вопрос 10

The following function is monotonically increasing:

$y = \left(\dfrac{1}{3}\right)^x$ $y = \log_{0.5} x$ $y = 2x + 1$ $y = \sin x$

Вопрос 11

The following point lies in the third quadrant:

$(-1, -2)$ $(-1, 2)$ $(1, 2)$ $(1, -2)$

Вопрос 12

The solution set of the fractional inequality $\dfrac{x - 1}{x - 2} \leq 1$ is

$(2, +\infty)$ $[1, 2]$ $(-\infty, 2)$ $(-\infty, 1]$

Вопрос 13

It is known that $a, b, c$ form an arithmetic progression and satisfy $a + c = 20$. Then $b$ equals

$10$ $5$ $20$ $0$

Вопрос 14

The angle of inclination of the line $y = \sqrt{3}x + 10$ equals

$60°$ $30°$ $45°$ $120°$

Вопрос 15

The following inequality is true:

$0.7^{-0.2} < 0.7^{-0.4}$ $2.1^{-2} > 1.2^{-2}$ $3^{2.25} < 3^3$ $2.1^{2/3} < 1.2^{2/3}$

Вопрос 16

If the point $P(1, y)$ lies on the terminal side of an angle $\alpha$ and $\tan \alpha = 2$, then $y$ equals

$1$ $\dfrac{1}{2}$ $\sqrt{3}$ $2$

Вопрос 17

It is known that $\{a_n\}$ is an arithmetic progression and $a_1 = 1$, $a_2 = 2$. Then $a_{2025}$ equals

$2$ $2025$ $2024$ $20$

Вопрос 18

If the center of a circle has coordinates $(-3, 2)$ and its radius equals $4$, then the equation of the circle is

$(x + 3)^2 + (y - 2)^2 = 4$ $(x - 3)^2 + (y + 2)^2 = 4$ $(x - 3)^2 + (y + 2)^2 = 16$ $(x + 3)^2 + (y - 2)^2 = 16$

Вопрос 19

Find the distance between the two points $P(-1, -1)$ and $Q(1, 1)$.

$2$ $2\sqrt{2}$ $4$ $\sqrt{2}$

Вопрос 20

Find the equation of the line passing through the two points $P(1, 2)$ and $Q(2, 4)$.

$y = x + 1$ $y = 2x + 2$ $y = x + 2$ $y = 2x$

Вопрос 21

Let the equation of a circle be $x^2 + y^2 + 2x - 2y = 0$. Then the radius of this circle equals

$2$ $\dfrac{1}{2}$ $\sqrt{2}$ $1$

Вопрос 22

If $\alpha$ is an acute angle and $\sin \alpha = \dfrac{1}{2}$, then $\tan \alpha$ equals

$\dfrac{1}{2}$ $\dfrac{\sqrt{3}}{2}$ $\dfrac{1}{4}$ $\dfrac{\sqrt{3}}{3}$

Вопрос 23

It is known that the geometric progression $\{a_n\}$ has the first three terms $1, 2$, and $4$. Then the general term of this progression is

$a_n = 2n$ $a_n = 8$ $a_n = 2^{n-1}$ $a_n = 2^n$

Вопрос 24

It is known that $a > b$. Then

$\dfrac{1}{a} < \dfrac{1}{b}$ $a^2 > b^2$ $5^a > 5^b$ $\sin a > \sin b$

Вопрос 25

The distance from the point $P(-1, 2)$ to the point $Q(3, 1)$ equals

$\sqrt{17}$ $\sqrt{13}$ $\sqrt{5}$ $5$

Вопрос 26

The following statement about the function $y = \left(\dfrac{1}{2}\right)^x$ is true:

This function is symmetric with respect to the origin. This function is symmetric with respect to the $x$-axis. This function is monotonically increasing. This function is monotonically decreasing.

Вопрос 27

If $\sin \alpha = \dfrac{1}{4}$, then $\cos 2\alpha$ equals

$-\dfrac{7}{8}$ $\dfrac{1}{8}$ $\dfrac{7}{8}$ $-\dfrac{1}{8}$

Вопрос 28

The point of intersection of the line $l_1: y = 2x + 1$ and the line $l_2: x + y + 1 = 0$ is

$\left(\dfrac{1}{3}, \dfrac{2}{3}\right)$ $\left(\dfrac{2}{3}, \dfrac{1}{3}\right)$ $\left(-\dfrac{1}{3}, -\dfrac{2}{3}\right)$ $\left(-\dfrac{2}{3}, -\dfrac{1}{3}\right)$

Вопрос 29

The coordinates of the foci of the hyperbola $x^2 - y^2 = 1$ are

$(\pm\sqrt{2}, 0)$ $(0, \pm 2)$ $(0, \pm\sqrt{2})$ $(\pm 2, 0)$

Вопрос 30

The value of $\log_2 32 + \log_{0.5} 32$ equals

$-5$ $10$ $0$ $5$

Вопрос 31

The following statement about the function $y = \cos x$ is true:

$\cos(x + \pi) = \cos x$ This function is odd. This function is monotonically increasing on the interval $[0, \pi]$. This function is periodic with period $2\pi$.

Вопрос 32

The following statement about the parabola $y^2 = 4x$ is true:

It passes through the point $(-4, 4)$. Its axis of symmetry is $x = 0$. The coordinates of its focus are $(\pm 1, 0)$. The coordinates of its focus are $(1, 0)$.

Вопрос 33

The following line is perpendicular to the line $l: x + y + 3 = 0$:

$y = -2x + 1$ $y = x + 3$ $y = 1 - x$ $y = 2x + 1$

Вопрос 34

It is known that $\cos \alpha = -\dfrac{1}{2}$, $\alpha \in \left(\dfrac{\pi}{2}, \pi\right)$. Then the value of $\sin \dfrac{\alpha}{2}$ equals

$\dfrac{\sqrt{3}}{2}$ $-\dfrac{1}{2}$ $\dfrac{1}{2}$ $-\dfrac{\sqrt{3}}{2}$

Вопрос 35

It is known that $\sin \dfrac{\pi}{4} = \dfrac{\sqrt{2}}{2}$, $\sin \dfrac{\pi}{6} = \dfrac{1}{2}$. Then $\cos \dfrac{\pi}{12}$ equals

$\dfrac{\sqrt{6}}{4}$ $\dfrac{\sqrt{6} + \sqrt{2}}{4}$ $\dfrac{\sqrt{6} - \sqrt{2}}{4}$ $\dfrac{1 + \sqrt{3}}{4}$

Вопрос 36

The following function coincides with the function $y = |x|$:

$y = \sqrt[3]{x^3}$ $y = (\sqrt{x})^2$ $y = x$ $y = \sqrt{x^2}$

Вопрос 37

If $\alpha$ is an acute angle and $\cos \alpha = \dfrac{1}{2}$, then $\sin \dfrac{\alpha}{2}$ equals

$\pm\dfrac{\sqrt{3}}{2}$ $\pm\dfrac{1}{2}$ $\dfrac{\sqrt{3}}{2}$ $\dfrac{1}{2}$

Вопрос 38

Regarding the trigonometric reduction formulas, the following formula is incorrect:

$\cos(2\pi - \alpha) = \cos \alpha$ $\tan(\pi + \alpha) = -\tan \alpha$ $\sin(\pi - \alpha) = \sin \alpha$ $\sin\left(\dfrac{\pi}{2} - \alpha\right) = \cos \alpha$

Вопрос 39

The equation of the directrix of the parabola $y^2 = -x$ is

$x = \dfrac{1}{2}$ $x = -\dfrac{1}{2}$ $x = \dfrac{1}{4}$ $x = -\dfrac{1}{4}$

Вопрос 40

It is known that $\dfrac{\cos \alpha - \sin \alpha}{\cos \alpha + \sin \alpha} = \dfrac{1}{3}$. Then $\tan \alpha$ equals

$\dfrac{1}{2}$ $\dfrac{1}{4}$ $\dfrac{2}{3}$ $\dfrac{1}{3}$

Вопрос 41

If the sequence $\{a_n\}$ satisfies $a_1 = 1$ and $a_n = \dfrac{1}{1 + \dfrac{1}{a_{n-1}}}$, $n \geq 2$, then $a_{100}$ equals

$50$ $\dfrac{1}{100}$ $1$ $100$

Вопрос 42

The following statement about the logarithmic function $y = \log_a x$ is true:

The domain of this function is $(-\infty, +\infty)$. For $a > 1$ it is monotonically decreasing on its entire domain. For $a > 1$ and any $x > 0$ it holds that $\log_a x > 1$. For $0 < a < 1$ it is monotonically decreasing on its entire domain.

Вопрос 43

The following statement about the ellipse $\dfrac{x^2}{4} + \dfrac{y^2}{20} = 1$ is true:

The two foci of this ellipse are $(0, \pm 4)$. The two foci of this ellipse are $(\pm 4, 0)$. The vertices of this ellipse are $(\pm 4, 0)$ and $(0, \pm 20)$. The lengths of the major and minor axes are $20$ and $4$, respectively.

Вопрос 44

If a line is perpendicular to the line $l_1: x + 2y + 4 = 0$ and passes through the intersection point of the lines $l_2: x + y + 1 = 0$ and $l_3: 2x + y - 1 = 0$, then the equation of this line is

$x + 2y - 3 = 0$ $2x + y - 5 = 0$ $2x - y - 7 = 0$ $x - 2y - 1 = 0$

Вопрос 45

Let $O$ be the origin, $A$ and $B$ be two points in the plane, and $P$ be the midpoint of segment $AB$. Then the vector $\overrightarrow{OP}$ equals

$\dfrac{1}{2}(\overrightarrow{OA} + \overrightarrow{OB})$ $\overrightarrow{OA} + \overrightarrow{OB}$ $\overrightarrow{OB} - \overrightarrow{OA}$ $\dfrac{1}{2}(\overrightarrow{OB} - \overrightarrow{OA})$

Вопрос 46

If the complex number $z$ satisfies $z^3 = 1$ and $z \neq 1$, then $1 + z + z^2 + \cdots + z^{999} =$

$\dfrac{1 \pm \sqrt{3}i}{2}$ $-\dfrac{1 \pm \sqrt{3}i}{2}$ $0$ $1$

Вопрос 47

It is known that the arithmetic progression $\{a_n\}$ satisfies $3(a_2 + a_6) + 2(a_6 + a_{10} + a_{14}) = 24$. Then the sum $S_{13}$ equals

$104$ $52$ $39$ $26$

Вопрос 48

The twelve zodiac signs are twelve animals used in China to denote years. The probability that among three randomly chosen people at least two have the same zodiac sign equals

$\dfrac{17}{72}$ $\dfrac{1}{12}$ $\dfrac{55}{72}$ $\dfrac{143}{144}$