Given set $A = \left\{ ( x , y ) \left| x ^ { 2 } + y ^ { 2 } \leqslant 3 , x \in \mathbf { Z } , y \in \mathbf { Z } \right. \right\}$, the number of elements in $A$ is A. 9 B. 8 C. 5 D. 4
The graph of function $f ( x ) = \frac { \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } } { x ^ { 2 } }$ is approximately A. [Graph A] B. [Graph B] C. [Graph C] D. [Graph D]
Given vectors $\boldsymbol { a } , \boldsymbol { b }$ satisfying $| \boldsymbol { a } | = 1 , \boldsymbol { a } \cdot \boldsymbol { b } = - 1$, then $\boldsymbol { a } \cdot ( 2 \boldsymbol { a } - \boldsymbol { b } ) =$ A. 4 B. 3 C. 2 D. 0
In $\triangle A B C$, $\cos \frac { C } { 2 } = \frac { \sqrt { 5 } } { 5 } , B C = 1 , A C = 5$, then $A B =$ A. $4 \sqrt { 2 }$ B. $\sqrt { 30 }$ C. $\sqrt { 29 }$ D. $2 \sqrt { 5 }$
Chinese mathematician Chen Jingrun achieved world-leading results in research on Goldbach's conjecture. Goldbach's conjecture states that ``every even number greater than 2 can be expressed as the sum of two prime numbers'', such as $30 = 7 + 23$. Among prime numbers not exceeding 30, if two different numbers are randomly selected, the probability that their sum equals 30 is A. $\frac { 1 } { 12 }$ B. $\frac { 1 } { 14 }$ C. $\frac { 1 } { 15 }$ D. $\frac { 1 } { 18 }$
In rectangular prism $A B C D - A _ { 1 } B C _ { 1 } D _ { 1 }$, $A B = B C = 1 , A A _ { 1 } = \sqrt { 3 }$, the cosine of the angle between skew lines $A D _ { 1 }$ and $D B _ { 1 }$ is A. $\frac { 1 } { 5 }$ B. $\frac { \sqrt { 5 } } { 6 }$ C. $\frac { \sqrt { 5 } } { 5 }$ D. $\frac { \sqrt { 2 } } { 2 }$
If $f ( x ) = \cos x - \sin x$ is an even function on $[ - a , a ]$, then the maximum value of $a$ is A. $\frac { \pi } { 4 }$ B. $\frac { \pi } { 2 }$ C. $\frac { 3 \pi } { 4 }$ D. $\pi$
Given that $f ( x )$ is an odd function with domain $( - \infty , + \infty )$ satisfying $f ( 1 - x ) = f ( 1 + x )$. If $f ( 1 ) = 2$, then $f ( 1 ) + f ( 2 ) + f ( 3 ) + \cdots + f ( 30 ) =$ A. $- 50$ B. $0$ C. $2$ D. $50$
Let $F _ { 1 } , F _ { 2 }$ be the left and right foci of ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$, $A$ is the left vertex of $C$. Point $P$ is on the line passing through $A$ with slope $\frac { \sqrt { 3 } } { 6 }$. $\triangle P F _ { 1 } F _ { 2 }$ is an isosceles triangle with $\angle F _ { 1 } F _ { 2 } P = 120 ^ { \circ }$, then the eccentricity of $C$ is A. $\frac { 2 } { 3 }$ B. $\frac { 1 } { 2 }$ C. $\frac { 1 } { 3 }$ D. $\frac { 1 } { 4 }$
If $x , y$ satisfy the constraints $\left\{ \begin{array} { l } x + 2 y - 5 \geqslant 0 , \\ x - 2 y + 3 \geqslant 0 , \\ x - 5 \leqslant 0 , \end{array} \right.$ then the maximum value of $z = x + y$ is $\_\_\_\_$.
The apex of a cone is $S$. The cosine of the angle between generatrices $S A$ and $S B$ is $\frac { 7 } { 8 }$. The angle between $S A$ and the base of the cone is $45 ^ { \circ }$. The area of $\triangle S A B$ is $5 \sqrt { 15 }$. Then the lateral surface area of the cone is $\_\_\_\_$.
(12 points) Let $S _ { n }$ be the sum of the first $n$ terms of arithmetic sequence $\{ a _ { n } \}$. Given $a _ { 1 } = - 7 , S _ { 3 } = - 15$. (1) Find the general term formula of $\{ a _ { n } \}$; (2) Find $S _ { n }$ and the minimum value of $S _ { n }$.
(12 points) The figure below is a line graph of environmental infrastructure investment $y$ (in units of 100 million yuan) from 2000 to 2016 in a certain region. To forecast the environmental infrastructure investment for 2018 in this region, two linear regression models were established for $y$ and time variable $t$. Based on data from 2000 to 2016 (time variable $t$ takes values $1, 2, \ldots, 17$ respectively), Model (1) is established: $\hat { y } = - 30.4 + 13.5 t$; based on data from 2010 to 2016 (time variable $t$ takes values $1, 2, \ldots, 7$ respectively), Model (2) is established: $\hat { y } = 99 + 17.5 t$. (1) Using each of these two models respectively, predict the environmental infrastructure investment for 2018 in this region; (2) Which model do you think gives a more reliable prediction? Explain your reasoning.
(12 points) Let the focus of parabola $C : y ^ { 2 } = 4 x$ be $F$. A line $l$ passing through $F$ with slope $k ( k > 0 )$ intersects $C$ at points $A$ and $B$. $| A B | = 8$. (1) Find the equation of line $l$; (2) Find the equation of the circle passing through points $A$ and $B$ and tangent to the directrix of $C$.
(12 points) As shown in the figure, in triangular pyramid $P - A B C$, $A B = B C = 2 \sqrt { 2 }$, $P A = P B = P C = A C = 4$, $O$ is the midpoint of $A C$. (1) Prove that $P O \perp$ plane $A B C$; (2) Point $M$ is on edge $B C$ such that the dihedral angle $M - P A - C$ is $30 ^ { \circ }$. Find the sine of the angle between $P C$ and plane $P A M$.