5. After the examination ends, submit both this test paper and the answer sheet together. I. Multiple Choice Questions: This section has 12 questions, each worth 5 points, for a total of 60 points. For each question, only one of the four options is correct. 1. Given sets $M = \{ x \mid - 4 < x < 2 \} , N = \left\{ x \mid x ^ { 2 } - x - 6 < 0 \right\}$, then $M \cap N =$ A. $\{ x \mid - 4 < x < 3 \}$ B. $\{ x \mid - 4 < x < - 2 \}$ C. $\{ x \mid - 2 < x < 2 \}$ D. $\{ x \mid 2 < x < 3 \}$ 2. Let complex number $z$ satisfy $| z - \mathrm { i } | = 1$, and the point corresponding to $z$ in the complex plane is $( x , y )$, then A. $( x + 1 ) ^ { 2 } + y ^ { 2 } = 1$ B. $( x - 1 ) ^ { 2 } + y ^ { 2 } = 1$ C. $x ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$ D. $x ^ { 2 } + ( y + 1 ) ^ { 2 } = 1$ 3. Given $a = \log _ { 2 } 0.2 , b = 2 ^ { 0.2 } , c = 0.2 ^ { 0.3 }$, then A. $a < b < c$ B. $a < c < b$ C. $c < a < b$ D. $b < c < a$ 4. In ancient Greece, people believed that the most beautiful human body has the ratio of the length from the top of the head to the navel to the length from the navel to the sole of the foot equal to $\frac { \sqrt { 5 } - 1 } { 2 } \left( \frac { \sqrt { 5 } - 1 } { 2 } \approx 0.618 \right.$, called the golden ratio), and the famous ``Venus de Milo'' exemplifies this. Furthermore, the ratio of the length from the top of the head to the throat to the length from the throat to the navel is also $\frac { \sqrt { 5 } - 1 } { 2 }$. If a person satisfies both golden ratio proportions, with a shoulder width of 105 cm and the length from the top of the head to the chin of 26 cm, then their height could be [Figure] A. 165 cm B. 175 cm C. 185 cm D. 190 cm 5. The graph of the function $f ( x ) = \frac { \sin x + x } { \cos x + x ^ { 2 } }$ on $[ - \pi , \pi ]$ is approximately A.[Figure] B.[Figure] C.[Figure] D.[Figure]
6. In ancient Chinese classics, the ``Book of Changes'' uses ``hexagrams'' to describe the changes of all things. Each ``hexagram'' consists of 6 lines arranged from bottom to top, with lines divided into yang lines ``—'' and yin lines ``- -''. The figure shows a hexagram. If a hexagram is randomly selected from all hexagrams, the probability that it has exactly 3 yang lines is A. $\frac { 5 } { 16 }$ B. $\frac { 11 } { 32 }$ C. $\frac { 21 } { 32 }$ D. $\frac { 11 } { 16 }$
7. Given non-zero vectors $a , b$ satisfying $| a | = 2 | b |$ and $( a - b ) \perp b$, the angle between $a$ and $b$ is A. $\frac { \pi } { 6 }$ B. $\frac { \pi } { 3 }$ C. $\frac { 2 \pi } { 3 }$ D. $\frac { 5 \pi } { 6 }$
9. Let $S _ { n }$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$. Given $S _ { 4 } = 0 , a _ { 5 } = 5$, then A. $a _ { n } = 2 n - 5$ B. $a _ { n } = 3 n - 10$ C. $S _ { n } = 2 n ^ { 2 } - 8 n$ D. $S _ { n } = \frac { 1 } { 2 } n ^ { 2 } - 2 n$
Mathematics (Science) Test Paper Page 2 (Total 5 Pages)
10. Given that the foci of ellipse $C$ are $F _ { 1 } ( - 1,0 ) , F _ { 2 } ( 1,0 )$, and a line through $F _ { 2 }$ intersects $C$ at points $A , B$. If $\left| A F _ { 2 } \right| = 2 \left| F _ { 2 } B \right| , | A B | = \left| B F _ { 1 } \right|$, then the equation of $C$ is A. $\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1$ B. $\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 2 } = 1$ C. $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$ D. $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 4 } = 1$
11. Regarding the function $f ( x ) = \sin | x | + | \sin x |$, there are four conclusions: (1) $f ( x )$ is an even function (2) $f ( x )$ is monotonically increasing on the interval $\left( \frac { \pi } { 2 } , \pi \right)$ (3) $f ( x )$ has 4 zeros on $[ - \pi , \pi ]$ (4) The maximum value of $f ( x )$ is 2 The numbers of all correct conclusions are A. (1)(2)(4) B. (2)(4) C. (1)(4) D. (1)(3)
12. Given that the four vertices of tetrahedron $P - A B C$ lie on the surface of sphere $O$, with $P A = P B = P C$, $\triangle A B C$ is an equilateral triangle with side length 2, $E , F$ are the midpoints of $P A , A B$ respectively, and $\angle C E F = 90 ^ { \circ }$, then the volume of sphere $O$ is A. $8 \sqrt { 6 } \pi$ B. $4 \sqrt { 6 } \pi$ C. $2 \sqrt { 6 } \pi$ D. $\sqrt { 6 } \pi$ II. Fill-in-the-Blank Questions: This section has 4 questions, each worth 5 points, for a total of 20 points.
14. Let $S _ { n }$ denote the sum of the first $n$ terms of a geometric sequence $\left\{ a _ { n } \right\}$. If $a _ { 1 } = \frac { 1 } { 3 } , a _ { 4 } ^ { 2 } = a _ { 6 }$, then $S _ { 3 } = \_\_\_\_$.
15. Teams A and B are playing a best-of-seven basketball series (the series ends when one team wins four games). Based on previous results, Team A's home and away arrangement is ``home, home, away, away, home, away, home'' in order. The probability that Team A wins at home is 0.6, and the probability that Team A wins away is 0.5. Each game is independent. The probability that Team A wins 4-1 is $\_\_\_\_$.
16. Given hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ with left and right foci $F _ { 1 } , F _ { 2 }$ respectively. A line through $F _ { 1 }$ intersects the two asymptotes of $C$ at points $A , B$ respectively. If $\overrightarrow { F _ { 1 } A } = \overrightarrow { A B } , \overrightarrow { F _ { 1 } B } \cdot \overrightarrow { F _ { 2 } B } = 0$, then the eccentricity of $C$ is $\_\_\_\_$. III. Solution Questions: Total 70 points. Solutions should include explanations, proofs, or calculation steps. Questions 17-21 are required for all students. Questions 22 and 23 are optional; students should choose one to answer. If more than one is answered, only the first one will be graded. (I) Required Questions: Total 60 points.
17. (12 points) In $\triangle A B C$, the angles $A , B , C$ have opposite sides $a , b , c$ respectively. Given $( \sin B - \sin C ) ^ { 2 } = \sin ^ { 2 } A - \sin B \sin C$. (1) Find $A$; (2) If $\sqrt { 2