If $\frac { z } { z - 1 } = 1 + \mathrm { i }$ , then $z =$ A. $- 1 - \mathrm { i }$ B. $- 1 + \mathrm { i }$ C. $1 - \mathrm{i}$ D. $1 + \mathrm { i }$
Given vectors $\boldsymbol { a } = ( 0,1 ) , \boldsymbol { b } = ( 2 , x )$ , if $\boldsymbol { b } \perp ( \boldsymbol { b } - 4 \boldsymbol { a } )$ , then $x =$ A. $- 2$ B. $- 1$ C. $1$ D. $2$
Given function $f ( x ) = \left\{ \begin{array} { l l } - x ^ { 2 } - 2 a x - a , & x < 0 , \\ \mathrm { e } ^ { x } + \ln ( x + 1 ) , & x \geqslant 0 \end{array} \right.$ is monotonically increasing on $\mathbb { R }$ , then the range of $a$ is A. $( - \infty , 0 ]$ B. $[ - 1,0 ]$ C. $[ - 1,1 ]$ D. $[ 0 , + \infty )$
When $x \in [ 0,2 \pi ]$ , the number of intersection points of the curves $y = \sin x$ and $y = 2 \sin \left( 3 x - \frac { \pi } { 6 } \right)$ is A. $3$ B. $4$ C. $6$ D. $8$
Given that the domain of function $f ( x )$ is $\mathbb { R }$ , $f ( x ) > f ( x - 1 ) + f ( x - 2 )$ , and when $x < 3$ , $f ( x ) = x$ , then the following conclusion that must be correct is A. $f ( 10 ) > 100$ B. $f ( 20 ) > 1000$ C. $f ( 10 ) < 1000$ D. $f ( 20 ) < 10000$
To understand the per-acre income (in units of 10,000 yuan) after promoting exports, a sample was taken from the planting area. The sample mean of per-acre income after promoting exports is $\bar { x } = 2.1$ , and the sample variance is $s ^ { 2 } = 0.01$ . The historical per-acre income $X$ in the planting area follows a normal distribution $N \left( 1.8 , ~ 0.1 ^ { 2 } \right)$ . Assume that the per-acre income $Y$ after promoting exports follows a normal distribution $N \left( \bar { x } , s ^ { 2 } \right)$ . Then (if a random variable $Z$ follows a normal distribution $N \left( \mu , \sigma ^ { 2 } \right)$ , then $P ( Z < \mu + \sigma ) \approx 0.8413$ ) A. $P ( X > 2 ) > 0.2$ B. $P ( X > 2 ) < 0.5$ C. $P ( Y > 2 ) > 0.5$ D. $P ( Y > 2 ) < 0.8$
Let function $f ( x ) = ( x - 1 ) ^ { 2 } ( x - 4 )$ , then A. $x = 3$ is a local minimum point of $f ( x )$ B. When $0 < x < 1$ , $f ( x ) < f \left( x ^ { 2 } \right)$ C. When $1 < x < 2$ , $- 4 < f ( 2 x - 1 ) < 0$ D. When $- 1 < x < 0$ , $f ( 2 - x ) > f ( x )$
The shape ``$\varnothing$'' can be made into a beautiful ribbon. Consider it as part of the curve $C$ in the figure. It is known that $C$ passes through the origin $O$ , and points on $C$ satisfy: the abscissa is greater than $- 2$ , and the product of the distance to point $F ( 2,0 )$ and the distance to the line $x = a ( a < 0 )$ equals 4 . Then A. $a = - 2$ B. The point $( 2 \sqrt { 2 } , 0 )$ is on $C$ C. The maximum ordinate of points on $C$ in the first quadrant is 1 D. When point $\left( x _ { 0 } , y _ { 0 } \right)$ is on $C$ , $y _ { 0 } \leqslant \frac { 4 } { x _ { 0 } + 2 }$
Let the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ have left and right foci $F _ { 1 } , F _ { 2 }$ respectively. A line through $F _ { 2 }$ parallel to the $y$-axis intersects $C$ at points $A$ and $B$ . If $\left| F _ { 1 } A \right| = 13 , | A B | = 10$ , then the eccentricity of $C$ is $\_\_\_\_$ .
If the tangent line to the curve $y = \mathrm { e } ^ { x } + x$ at the point $( 0,1 )$ is also a tangent line to the curve $y = \ln ( x + 1 ) + a$ , then $a = $ $\_\_\_\_$ .
Person A and Person B each have four cards, with each card labeled with a number. Person A's cards are labeled with the numbers 1, 3, 5, 7, and Person B's cards are labeled with the numbers 2, 4, 6, 8. They play four rounds of competition. In each round, both players randomly select one card from their own cards and compare the numbers. The player with the larger number scores 1 point, and the player with the smaller number scores 0 points. Then each player discards the card used in that round (discarded cards cannot be used in subsequent rounds). The probability that Person A's total score after four rounds is at least 2 is $\_\_\_\_$ .
(13 points) Let the sides opposite to angles $A , B , C$ of $\triangle A B C$ be $a , b , c$ respectively. Given $\sin C = \sqrt { 2 } \cos B , a ^ { 2 } + b ^ { 2 } - c ^ { 2 } = \sqrt { 2 } a b$ . (1) Find $B$ ; (2) If the area of $\triangle A B C$ is $3 + \sqrt { 3 }$ , find $c$ .
(15 points) Given that $A ( 0,3 )$ and $P \left( 3 , \frac { 3 } { 2 } \right)$ are two points on the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ . (1) Find the eccentricity of $C$ ; (2) If a line $l$ through $P$ intersects $C$ at another point $B$ , and the area of $\triangle A B P$ is 9 , find the equation of $l$ .
(15 points) As shown in the figure, in the quadrangular pyramid $P - A B C D$ , $P A \perp$ base $A B C D , P A = A C = 2$ , $B C = 1 , A B = \sqrt { 3 }$ . (1) If $A D \perp P B$ , prove that $A D \|$ plane $P B C$ ; (2) If $A D \perp D C$ , and the sine of the dihedral angle $A - C P - D$ is $\frac { \sqrt { 42 } } { 7 }$ , find $A D$ .
(17 points) Given function $f ( x ) = \ln \frac { x } { 2 - x } + a x + b ( x - 1 ) ^ { 3 }$ . (1) If $b = 0$ and $f ^ { \prime } ( x ) \geqslant 0$ , find the minimum value of $a$ ; (2) Prove that the curve $y = f ( x )$ is centrally symmetric; (3) If $f ( x ) > - 2$ if and only if $1 < x < 2$ , find the range of $b$ .
(17 points) Let $m$ be a positive integer. The sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is an arithmetic sequence with nonzero common difference. If after removing two terms $a _ { i }$ and $a _ { j } ( i < j )$ , the remaining $4 m$ terms can be evenly divided into $m$ groups, and the 4 numbers in each group form an arithmetic sequence, then the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is called an $( i , j )$ -divisible sequence. (1) Write out all pairs $( i , j )$ with $1 \leqslant i < j \leqslant 6$ such that the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 6 }$ is an $( i , j )$ -divisible sequence; (2) When $m \geqslant 3$ , prove that the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is a $( 2,13 )$ -divisible sequence; (3) From $1,2 , \cdots , 4 m + 2$ , randomly select two numbers $i$ and $j$ ( $i < j$ ) at once. Let $P _ { m }$ denote the probability that the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is an $( i , j )$ -divisible sequence. Prove that $P _ { m } > \frac { 1 } { 8 }$ .