1. Given the universal set $U = \{ 1,2,3,4,5,6 \}$, set $A = \{ 2,3,4 \}$, set $B = \{ 1,3,4,6 \}$, then set $A \cap C _ { U } B =$ (A) $\{ 3 \}$ (B) $\{ 2,5 \}$ (C) $\{ 1,4,6 \}$ (D) $\{ 2,3,5 \}$
Given the universal set $U = \{1,2,3,4,5,6,7,8\}$, set $A = \{2,3,5,6\}$, set $\mathrm{B} = \{1,3,4,6,7\}$, then $\mathrm{A} \cap \mathrm{C}_{U}\mathrm{B} =$ (A) $\{2,5\}$ (B) $\{3,6\}$ (C) $\{2,5,6\}$ (D) $\{2,3,5,6,8\}$
2. Let $A$ and $B$ be two sets. Then ``$A \cap B = A$'' is ``$A \subseteq B$'' a A. sufficient but not necessary condition B. necessary but not sufficient condition C. necessary and sufficient condition D. neither sufficient nor necessary condition
5. A bag contains 4 balls of identical shape and size, including 1 white ball, 1 red ball, and 2 yellow balls. If 2 balls are randomly drawn at once, then the probability that the 2 balls have different colors is $\_\_\_\_$ .
10. Given the set $A = \left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 1 , x , y \in Z \right\} , A = \{ ( x , y ) \| x | \leq 2 , | y | \leq 2 , x , y \in Z \}$, define the set $A \oplus B = \left\{ \left( x _ { 1 } + x _ { 2 } , y _ { 1 } + y _ { 2 } \right) \mid \left( x _ { 1 } , y _ { 1 } \right) \in A , \left( x _ { 2 } , y _ { 2 } \right) \in B \right.$, then the number of elements in $A \oplus B$ is A. $ 77$ B. $ 49$ C. $ 45$ D. $ 30$
11. Given the set $\mathrm { U } = \{ 1,2,3,4 \} , \mathrm { A } = \{ 1,3 \} , \mathrm { B } = \{ 1,3,4 \}$, then $\mathrm { A } \cup ( C \cup B ) =$ $\_\_\_\_$
16. (This question is worth 13 points) Groups $A$ and $B$ each have 7 patients. Their recovery time (in days) after taking a certain drug is recorded as follows: Group A: $10,11,12,13,14,15,16$ Group B: $12,13,15,16,17,14 , a$ Assume that the recovery times of all patients are mutually independent. Randomly select 1 person from each of groups A and B. The person selected from group A is denoted as patient 甲, and the person selected from group B is denoted as patient 乙. (I) Find the probability that the recovery time of patient 甲 is at least 14 days; (II) If $a = 25$, find the probability that the recovery time of patient 甲 is longer than that of patient 乙; (III) For what value of $a$ are the variances of recovery times for groups A and B equal? (Proof of the conclusion is not required)
16. A bank stipulates that if a bank card has 3 incorrect password attempts in one day, the card will be locked. Xiaowang went to the bank to withdraw money and found that he forgot his bank card password, but he is certain that the correct password is one of his 6 commonly used passwords. Xiaowang decides to randomly select one without replacement to try. If the password is correct, he stops trying; otherwise, he continues trying until the card is locked. (1) Find the probability that Xiaowang's bank card is locked that day; (2) Let $X$ denote the number of password attempts Xiaowang makes that day. Find the probability distribution of $X$ and its mathematical expectation.
16. (This question is worth 12 points) A shopping mall is holding a promotional lottery activity. After customers purchase goods of a certain amount, they can participate in the lottery. The lottery method is as follows: randomly draw 1 ball each from box A containing 2 red balls $\mathrm { A } _ { 1 } , \mathrm { A } _ { 2 }$ and 1 white ball B, and from box B containing 2 red balls $\mathrm { a } _ { 1 } , \mathrm { a } _ { 2 }$ and 2 white balls $\mathrm { b } _ { 1 } , \mathrm { b } _ { 2 }$. If both balls drawn are red, the customer wins; otherwise, the customer does not win. (I) List all possible outcomes of drawing balls using the ball labels. (II) Someone claims: Since there are more red balls than white balls in both boxes, the probability of winning is greater than the probability of not winning. Do you agree? Please explain your reasoning.
To promote the development of table tennis, a certain table tennis competition allows athletes from different associations to form teams. There are 3 athletes from Association A, of which 2 are seeded players, and 5 athletes from Association B, of which 3 are seeded players. Randomly select 4 people from these 8 athletes to participate in the competition. (I) Let A be the event ``exactly 2 seeded players are selected, and these 2 seeded players are from the same association''. Find the probability of this event. (II) Let X be the number of seeded players among the 4 selected people. Find the probability distribution and mathematical expectation of the random variable X.
Given sets $A = \{ x \mid x < 1 \} , B = \left\{ x \mid 3 ^ { x } < 1 \right\}$, then A. $A \cap B = \{ x \mid x < 0 \}$ B. $A \cup B = \mathbf { R }$ C. $A \cup B = \{ x \mid x > 1 \}$ D. $A \cap B = \varnothing$
Five cards numbered $1, 2, 3, 4, 5$ are shuffled and three are drawn in order. The probability that the number on the first card is greater than the number on the third card is A. $\dfrac{1}{10}$ B. $\dfrac{1}{5}$ C. $\dfrac{3}{10}$ D. $\dfrac{2}{5}$
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
Given sets $A = \{ - 1,0,1,2 \} , B = \left\{ x \mid x ^ { 2 } \leqslant 1 \right\}$ , then $A \cap B =$ A. $\{ - 1,0,1 \}$ B. $\{ 0,1 \}$ C. $\{ - 1,1 \}$ D. $\{ 0,1,2 \}$
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]
15. Teams A and B are in a basketball championship series using a best-of-seven format (the series ends when one team wins four games). Based on previous results, Team A's home/away schedule is ``home, home, away, away, home, away, home'' in order. Team A's probability of winning at home is 0.6, and away is 0.5. Each game is independent. The probability that Team A wins 4-1 is $\_\_\_\_$ .
Given the sets $A = \left\{ ( x , y ) \mid x , y \in \mathbf { N } ^ { * } , y \geqslant x \right\} , B = \{ ( x , y ) \mid x + y = 8 \}$ , the number of elements in $A \cap B$ is A. 2 B. 3 C. 4 D. 6
Let $O$ be the center of square $A B C D$. If we randomly select 3 points from $O , A , B , C , D$, the probability that the 3 points are collinear is A. $\frac { 1 } { 5 }$ B. $\frac { 2 } { 5 }$ C. $\frac { 1 } { 2 }$ D. $\frac { 4 } { 5 }$