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$

17. (This question is worth 12 points)

A minibus has 5 seats numbered $1,2,3,4,5$. Passengers $P _ { 1 } , P _ { 2 } , P _ { 3 } , P _ { 4 } , P _ { 5 }$ have assigned seat numbers $1,2,3,4,5$ respectively. They board in order of increasing seat numbers. Passenger $P _ { 1 }$ did not sit in seat 1 due to health reasons. The driver then requires the remaining passengers to be seated according to the following rule: if their own seat is empty, they must sit in their own seat; if their own seat is occupied, they can choose any of the remaining empty seats. (1) If passenger $P _ { 1 }$ sits in seat 3 and other passengers are seated according to the rule, there are 4 possible seating arrangements. The table below shows two of them. Please fill in the remaining two arrangements (enter the seat numbers where passengers sit in the blank cells);
(2) If passenger

Passenger	$P _ { 1 }$	$P _ { 2 }$	$P _ { 3 }$	$P _ { 4 }$	$P _ { 5 }$
\multirow{3}{*}{Seat Number}	3	2	1	4	5
	3	2	4	5	1

$P _ { 1 }$ sits in seat 2, and other passengers are seated according to the rule, find the probability that passenger $P _ { 5 }$ sits in seat 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

We consider a general balanced urn with parameters $a_{0}, b_{0}, a, b, c, d \in \mathbb{N}$ satisfying $a + b = c + d = s$. For $n \geqslant 1$, an outcome resulting from $n$ successive draws is modeled by the $n$-tuple indicating the color and number of the balls successively obtained. We denote by $\Omega_{n}$ the set of possible outcomes of these $n$ draws.
By examining the number of balls in the urn just before each draw, justify that, for $n \geqslant 1$, $$\operatorname{card}(\Omega_{n}) = (a_{0} + b_{0}) \times \cdots \times (a_{0} + b_{0} + s(n-1)) = s^{n} L_{n}\left(\frac{a_{0} + b_{0}}{s}\right).$$

In a win-or-lose game, the winner gets 2 points whereas the loser gets 0. Six players A, B, C, D, E and F play each other in a preliminary round from which the top three players move to the final round. After each player has played four games, A has 6 points, B has 8 points and C has 4 points. It is also known that E won against F. In the next set of games D, E and F win their games against A, B and C respectively. If A, B and D move to the final round, the final scores of E and F are, respectively,
(A) 4 and 2
(B) 2 and 4
(C) 2 and 2
(D) 4 and 4

In a win-or-lose game, the winner gets 2 points whereas the loser gets 0. Six players A, B, C, D, E and F play each other in a preliminary round from which the top three players move to the final round. After each player has played four games, A has 6 points, B has 8 points and C has 4 points. It is also known that E won against F. In the next set of games D, E and F win their games against A, B and C respectively. If A, B and D move to the final round, the final scores of E and F are, respectively,
(A) 4 and 2
(B) 2 and 4
(C) 2 and 2
(D) 4 and 4

A party is attended by twenty people. In any subset of four people, there is at least one person who knows the other three (we assume that if $X$ knows $Y$, then $Y$ knows $X$). Suppose there are three people in the party who do not know each other. How many people in the party know everyone?
(A) 16
(B) 17
(C) 18
(D) Cannot be determined from the given data.

3. In a college of 300 students, every student reads 5 newspapers and every newspaper is read by 60 students. The number of newspaper is:
(A) at least 30
(B) at most 20
(C) exactly 25
(D) none of these

Let $|X|$ denote the number of elements in a set $X$. Let $S = \{1,2,3,4,5,6\}$ be a sample space, where each element is equally likely to occur. If $A$ and $B$ are independent events associated with $S$, then the number of ordered pairs $(A, B)$ such that $1 \leq |B| < |A|$, equals\_\_\_\_

Let $S = \left\{ E _ { 1 } , E _ { 2 } \ldots E _ { 8 } \right\}$ be a sample space of a random experiment such that $P \left( E _ { n } \right) = \frac { n } { 36 }$ for every $n = 1,2 \ldots 8$. Then the number of elements in the set $\left\{ A \subset S : P ( A ) \geq \frac { 4 } { 5 } \right\}$ is $\_\_\_\_$.

Let $A = \{ 0 , 3 , 4 , 6 , 7 , 8 , 9 , 10 \}$ and $R$ be the relation defined on $A$ such that $R = \{ ( x , y ) \in A \times A : x - y$ is odd positive integer or $x - y = 2 \}$. The minimum number of elements that must be added to the relation $R$, so that it is a symmetric relation, is equal to $\_\_\_\_$

The number of symmetric relations defined on the set $\{1, 2, 3, 4\}$ which are not reflexive is $\underline{\hspace{1cm}}$.

Let $A = \{ 1,2,3,4,5 \}$. Let R be a relation on A defined by $x \mathrm { R } y$ if and only if $4 x \leq 5 \mathrm { y }$. Let m be the number of elements in R and n be the minimum number of elements from $\mathrm { A } \times \mathrm { A }$ that are required to be added to R to make it a symmetric relation. Then $\mathrm { m } + \mathrm { n }$ is equal to:
(1) 25
(2) 24
(3) 26
(4) 23

Let $\mathbf { R } = \{ ( 1,2 ) , ( 2,3 ) , ( 3,3 ) \}$ be a relation defined on the set $\{ 1,2,3,4 \}$. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:
(1) 10
(2) 7
(3) 8
(4) 9

Q70. Let the relations $R _ { 1 }$ and $R _ { 2 }$ on the set $X = \{ 1,2,3 , \ldots , 20 \}$ be given by $R _ { 1 } = \{ ( x , y ) : 2 x - 3 y = 2 \}$ and $R _ { 2 } = \{ ( x , y ) : - 5 x + 4 y = 0 \}$. If $M$ and $N$ be the minimum number of elements required to be added in $R _ { 1 }$ and $R _ { 2 }$, respectively, in order to make the relations symmetric, then $M + N$ equals
(1) 12
(2) 16
(3) 8
(4) 10
Q71. For $\alpha , \beta \in \mathbb { R }$ and a natural number $n$, let $A _ { r } = \left| \begin{array} { c c c } r & 1 & \frac { n ^ { 2 } } { 2 } + \alpha \\ 2 r & 2 & n ^ { 2 } - \beta \\ 3 r - 2 & 3 & \frac { n ( 3 n - 1 ) } { 2 } \end{array} \right|$. Then
(1) 0
(2) $4 \alpha + 2 \beta$
(3) $2 \alpha + 4 \beta$
(4) $2 n$

Q69. Let $A = \{ 1,2,3,4,5 \}$. Let R be a relation on A defined by $x \mathrm { R } y$ if and only if $4 x \leq 5 \mathrm { y }$. Let m be the number of elements in R and n be the minimum number of elements from $\mathrm { A } \times \mathrm { A }$ that are required to be added to R to make it a symmetric relation. Then $\mathrm { m } + \mathrm { n }$ is equal to :
(1) 25
(2) 24
(3) 26
(4) 23

Q85. Let $A = \{ 2,3,6,7 \}$ and $B = \{ 4,5,6,8 \}$. Let $R$ be a relation defined on $A \times B$ by ( $\left. a _ { 1 } , b _ { 1 } \right) R \left( a _ { 2 } , b _ { 2 } \right)$ if and only if $a _ { 1 } + a _ { 2 } = b _ { 1 } + b _ { 2 }$. Then the number of elements in $R$ is $\_\_\_\_$

Let $A = \{2, 3, 5, 7, 9\}$. Consider a relation defined as
$R = \{(x, y) : 2x \leq 3y, x \in A, y \in A\}$
$l =$ total number of elements in relation R
$m =$ Number of elements required in R to make it symmetric.
Find $l + m$.
(A) 18
(B) 25
(C) 27
(D) 30

Let $\mathrm { A } = \{ - 2 , - 1,0,1,2,3,4 \}$ and R be a relation R , such that $\mathbf { R } = \{ ( \mathbf { x } , \mathbf { y } ) : ( \mathbf { 2 x } + \mathbf { y } ) \leq - \mathbf { 2 } , \mathbf { x } \in \mathbf { A } , \mathbf { y } \in \mathbf { A } \}$.\
Let $\boldsymbol { l } =$ number of elements in $\mathbf { R }$\ $\mathrm { m } =$ minimum number of elements to be added in R to make it reflexive.\ $\mathrm { n } =$ minimum number of elements to be added in R to make it symmetric, then $( 1 + m + n )$ is\ (A) 17\ (B) 10\ (C) 11\ (D) 14

$$\begin{aligned} & A = \{ a , b , e \} \\ & B = \{ a , b , c , d \} \end{aligned}$$
Given this, how many sets $K$ satisfy the condition $( A \cap B ) \subseteq K \subseteq ( A \cup B )$?
A) 3
B) 4
C) 5
D) 8
E) 9

Let $\mathbf { A } = \{ \mathbf { a } , \mathbf { b } , \mathbf { c } , \mathbf { d } \}$. For non-empty subsets $X , Y$ of A
$$\begin{aligned} & X \cap Y = \emptyset \\ & X \cup Y = A \end{aligned}$$
How many ordered pairs (X, Y) are there such that these conditions hold?
A) 6
B) 8
C) 10
D) 12
E) 14

The figure below shows a lamp and the appearance of a string that operates this lamp. The lamp;

• when closed, if the string is pulled and released, it gives dim light,
• when giving dim light, if the string is pulled and released, it gives daylight,
• when giving daylight, if the string is pulled and released, it gives bright light,
• when giving bright light, if the string is pulled and released, it turns off.

Initially, this lamp was closed. The string was pulled and released A times and the lamp was observed to give bright light. Then, the lamp's string was pulled and released B more times and the lamp was observed to give daylight. Later, the lamp's string was pulled and released C more times and the lamp was observed to turn off.
Accordingly, which of the following is an even number?
A) $A \cdot B + C$ B) $B \cdot C + A$ C) $A \cdot (B + C)$ D) $B \cdot (A + C)$ E) $C \cdot (A + B)$

At the entrance of a hotel, there are three digital wall clocks showing the local times of cities $\mathrm{A}, \mathrm{B}$ and C. A customer looking at these clocks observed that the local time difference between cities A and B is 4 hours, and the local time difference between cities B and C is 3 hours.
When the clock showing the local time of city A reads 14.00, which of the following cannot be the time shown on the clock for city C?
A) 07.00 B) 13.00 C) 15.00 D) 17.00 E) 21.00

The number of different digits in a natural number N is defined as shown in the figure.
Example: $4202 = 3$
Let A be a digit, and the equality shown in the figure holds.
What is the sum of different A values that satisfy the equality?
A) 8 B) 9 C) 10 D) 11 E) 12

Among the three-digit natural numbers $ABB$ and $BAB$, one is divisible by 11 and the other is divisible by 12.
Accordingly, what is the sum $\mathrm{A} + \mathrm{B}$?
A) 7 B) 8 C) 10 D) 11 E) 13