Q5
Inequalities
Set Operations Using Inequality-Defined Sets
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5. Keep the answer sheet clean, do not fold it, do not tear or wrinkle it, and do not use correction fluid, correction tape, or scrapers.
I. Multiple Choice Questions: 12 questions in total, 5 points each, 60 points total. For each question, only one of the four options is correct.
1. Let $A = \left\{ x \mid x ^ { 2 } - 5 x + 6 > 0 \right\} , B = \{ x \mid x - 1 < 0 \}$, then $A \cap B =$
A. $( - \infty , 1 )$
B. $( - 2,1 )$
C. $( - 3 , - 1 )$
D. $( 3 , + \infty )$
2. Let $z = - 3 + 2 \mathrm { i }$. In the complex plane, the point corresponding to $\bar { z }$ is located in
A. the first quadrant
B. the second quadrant
C. the third quadrant
D. the fourth quadrant
3. Given $\overrightarrow { A B } = ( 2,3 ) , \overrightarrow { A C } = ( 3 , t ) , | \overrightarrow { B C } | = 1$, then $\overrightarrow { A B } \cdot \overrightarrow { B C } =$
A. $- 3$
B. $- 2$
C. $2$
D. $3$
4. On January 3, 2019, the Chang'e-4 probe successfully achieved humanity's first soft landing on the far side of the moon, marking another major achievement in China's space program. A key technical challenge in achieving soft landing on the far side of the moon is maintaining communication between the ground and the probe. To solve this problem, the Queqiao relay satellite was launched, which orbits around the Earth-Moon Lagrange point $L _ { 2 }$. The $L _ { 2 }$ point is an equilibrium point located on the extension of the Earth-Moon line. Let the Earth's mass be $M _ { 1 }$, the Moon's mass be $M _ { 2 }$, the Earth-Moon distance be $R$, and the distance from the $L _ { 2 }$ point to the Moon be $r$. According to Newton's laws of motion and the law of universal gravitation, $r$ satisfies the equation: $\frac { M _ { 1 } } { ( R + r ) ^ { 2 } } + \frac { M _ { 2 } } { r ^ { 2 } } = ( R + r ) \frac { M _ { 1 } } { R ^ { 3 } }$. Let $\alpha = \frac { r } { R }$. Since $\alpha$ is very small, in approximate calculations $\frac { 3 \alpha ^ { 3 } + 3 \alpha ^ { 4 } + \alpha ^ { 5 } } { ( 1 + \alpha ) ^ { 2 } } \approx 3 \alpha ^ { 3 }$. Then the approximate value of $r$ is
A. $\sqrt { \frac { M _ { 2 } } { M _ { 1 } } } R$
B. $\sqrt { \frac { M _ { 2 } } { 2 M _ { 1 } } } R$
C. $\sqrt [ 3 ] { \frac { 3 M _ { 2 } } { M _ { 1 } } } R$
D. $\sqrt [ 3 ] { \frac { M _ { 2 } } { 3 M _ { 1 } } } R$
5. In a speech competition, 9 judges each give an original score to a contestant. When determining the contestant's final score, 1 highest score and 1 lowest score are removed from the 9 original scores, leaving 7 valid scores. Compared with the 9 original scores, the numerical characteristic that remains unchanged for the 7 valid scores is
A. median
B. mean
C. variance
D. range