Let $A = \{ x \mid x = 3k + 1 , k \in Z \} , B = \{ x \mid x = 3k + 2 , k \in Z \} , U$ be the set of integers, then $C_{U}(A \bigcap B) =$ A. $\{ x \mid x = 3k , k \in Z \}$ B. $\{ x \mid x = 3k - 1 , k \in Z \}$ C. $\{ x \mid x = 3k - 1 , k \in \mathrm{Z} \}$ D. $\varnothing$
In the sequence $\left\{ a_{n} \right\}$ , let $S_{n}$ be the sum of the first $n$ terms of $\left\{ a_{n} \right\}$ , $S_{5} = 5S_{3} - 4$ , then $S_{4} =$ A. $7$ B. $9$ C. $15$ D. $20$
50 people registered for the soccer club, 60 people registered for the table tennis club, and 70 people registered for either the soccer or table tennis club. If a person is known to have registered for the soccer club, the probability that they also registered for the table tennis club is A. $0.8$ B. $0.4$ C. $0.2$ D. $0.1$
``$\sin^{2} \alpha + \sin^{2} \beta = 1$'' is ``$\cos \alpha + \cos \beta = 0$'' a A. sufficient but not necessary condition B. necessary but not sufficient condition C. necessary and sufficient condition D. neither sufficient nor necessary condition
The eccentricity of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \ (a > 0 , b > 0)$ is $\sqrt{5}$ . One of its asymptotes intersects the circle $(x - 2)^{2} + (y - 3)^{2} = 1$ at points $A , B$ , then $|AB| =$ A. $\frac{1}{5}$ B. $\frac{\sqrt{5}}{5}$ C. $\frac{2\sqrt{5}}{5}$ D. $\frac{4\sqrt{5}}{5}$
Five volunteers participate in community service over Saturday and Sunday. Each day, two people are randomly selected from them to participate. The number of ways to select such that exactly one person participates on both days is A. $120$ B. $60$ C. $40$ D. $30$
Let $f(x)$ be the function obtained by shifting $y = \cos\left(2x + \frac{\pi}{4}\right)$ to the left by $\frac{\pi}{6}$ units. The number of intersection points of $y = f(x)$ and $y = \frac{1}{2}x - \frac{1}{2}$ is A. $1$ B. $2$ C. $3$ D. $4$
In the quadrangular pyramid $P - ABCD$ , the base $ABCD$ is a square with $AB = 4$ , $PC = PD = 3$ , $\angle PCA = 45^{\circ}$ , then the area of $\triangle PBC$ is A. $2\sqrt{2}$ B. $3\sqrt{2}$ C. $4\sqrt{2}$ D. $5\sqrt{2}$
The ellipse $\frac{x^{2}}{9} + \frac{y^{2}}{6} = 1$ has foci $F_{1} , F_{2}$ and center $O$ . Let $P$ be a point on the ellipse. If $\cos \angle F_{1}PF_{2} = \frac{3}{5}$ , then $|OP| =$ A. $\frac{2}{5}$ B. $\frac{\sqrt{30}}{2}$ C. $\frac{3}{5}$ D. $\frac{\sqrt{35}}{2}$
Let $x , y$ satisfy the constraints $\left\{ \begin{array}{l} -2x + 3y \leqslant 3 \\ 3x - 2y \leqslant 3 \\ x + y = 1 \end{array} \right.$ . Let $z = 3x + 2y$ . The maximum value of $z$ is $\_\_\_\_$ .
In $\triangle ABC$ , $\angle BAC = 60^{\circ} , AB = 2 , BC = \sqrt{6}$ . $AD$ bisects $\angle BAC$ and intersects $BC$ at point $D$ . Then $AD =$ $\_\_\_\_$ .
In the sequence $\left\{ a_{n} \right\}$ , $a_{2} = 1$ . Let $S_{n}$ be the sum of the first $n$ terms of $\left\{ a_{n} \right\}$ . $2S_{n} = na_{n}$ . (1) Find the general term formula for $\left\{ a_{n} \right\}$ ; (2) Find the sum $T_{n}$ of the first $n$ terms of the sequence $\left\{ \frac{a_{n} + 1}{2^{n}} \right\}$ .
In the triangular prism $ABC - A_{1}B_{1}C_{1}$ , $AA_{1} = 2$ , $A_{1}C \perp$ base $ABC$ , $\angle ACB = 90^{\circ}$ , the distance from $A_{1}$ to plane $BCC_{1}B_{1}$ is 1 . (1) Prove: $AC = A_{1}C$ ; (2) If the distance between lines $AA_{1}$ and $BB_{1}$ is 2 , find the sine of the angle between $AB_{1}$ and plane $BCC_{1}B_{1}$ .
The line $x - 2y + 1 = 0$ intersects the parabola $y^{2} = 2px \ (p > 0)$ at points $A , B$ with $AB = 4\sqrt{15}$ . (1) Find the value of $p$ ; (2) Let $F$ be the focus of $y^{2} = 2px$ . Let $M , N$ be two points on the parabola such that $\overrightarrow{MF} \perp \overrightarrow{NF}$ . Find the minimum area of $\triangle MNF$ .
Given $f(x) = ax - \frac{\sin x}{\cos^{2} x} , \quad x \in \left(0 , \frac{\pi}{2}\right)$ , (1) When $a = 8$ , discuss the monotonicity of $f(x)$ ; (2) If $f(x) < \sin 2x$ , find the range of values for $a$ .
[Elective 4-4: Coordinate Systems and Parametric Equations] Given $P(2,1)$ and line $l : \left\{ \begin{array}{l} x = 2 + t\cos\alpha \\ y = 1 + t\sin\alpha \end{array} \right.$ ($t$ is a parameter). Line $l$ intersects the positive $x$-axis and positive $y$-axis at points $A , B$ respectively, with $|PA| \cdot |PB| = 4$ . (1) Find the value of $\alpha$ ; (2) With the origin as the pole and the positive $x$-axis as the polar axis, find the polar equation of line $l$ .
[Elective 4-5: Inequalities] Given $f(x) = 2|x - a| - a , \ a > 0$ . (1) Solve the inequality $f(x) < x$ ; (2) If the area enclosed by $y = f(x)$ and the coordinate axes is 2 , find $a$ .