One die has two faces marked 1 , two faces marked 2 , one face marked 3 and one face marked 4 . Another die has one face marked 1 , two faces marked 2 , two faces marked 3 and one face marked 4 . The probability of getting the sum of numbers to be 4 or 5 , when both the dice are thrown together, is
(1) $\frac { 2 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 4 } { 9 }$
(4) $\frac { 3 } { 5 }$

Define a relation R on the interval $\left[0, \frac{\pi}{2}\right)$ by $x\mathrm{R}y$ if and only if $\sec^2 x - \tan^2 y = 1$. Then R is:
(1) both reflexive and transitive but not symmetric
(2) an equivalence relation
(3) reflexive but neither symmetric nor transitive
(4) both reflexive and symmetric but not transitive

Two numbers $\mathrm { k } _ { 1 }$ and $\mathrm { k } _ { 2 }$ are randomly chosen from the set of natural numbers. Then, the probability that the value of $\mathrm { i } ^ { \mathrm { k } _ { 1 } } + \mathrm { i } ^ { \mathrm { k } _ { 2 } } , ( \mathrm { i } = \sqrt { - 1 } )$ is non-zero, equals
(1) $\frac { 1 } { 2 }$
(2) $\frac { 3 } { 4 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 2 } { 3 }$

Let $A = \left[ a_{ij} \right]$ be a square matrix of order 2 with entries either 0 or 1. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $\mathrm{P}(\mathrm{E})$ is:
(1) $\frac{3}{16}$
(2) $\frac{5}{8}$
(3) $\frac{3}{8}$
(4) $\frac{1}{8}$

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

$A$ and $B$ alternately throw a pair of dice. $A$ wins if he throws a sum of 5 before $B$ throws a sum of 8, and $B$ wins if he throws a sum of 8 before $A$ throws a sum of 5. The probability that $A$ wins if $A$ makes the first throw, is
(1) $\frac{8}{17}$
(2) $\frac{9}{19}$
(3) $\frac{9}{17}$
(4) $\frac{8}{19}$

The number of non-empty equivalence relations on the set $\{ 1,2,3 \}$ is:
(1) 6
(2) 5
(3) 7
(4) 4

A board has 16 squares as shown in the figure (a $4 \times 4$ grid of squares). Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is :
(1) $7/10$
(2) $4/5$
(3) $23/30$
(4) $3/5$

Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains $n$ white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is $29 / 45$, then $n$ is equal to :
(1) 6
(2) 3
(3) 5
(4) 4

The relation $R = \{ ( x , y ) : x , y \in \mathbb { Z }$ and $x + y$ is even $\}$ is:
(1) reflexive and symmetric but not transitive
(2) an equivalence relation
(3) symmetric and transitive but not reflexive
(4) reflexive and transitive but not symmetric

Q68. Let $A = \{ 2,3,6,8,9,11 \}$ and $B = \{ 1,4,5,10,15 \}$. Let $R$ be a relation on $A \times B$ defined by ( $a , b$ ) $R ( c , d )$ if and only if $3 a d - 7 b c$ is an even integer. Then the relation $R$ is
(1) an equivalence relation.
(2) reflexive and symmetric but not transitive.
(3) transitive but not symmetric.
(4) reflexive but not symmetric.

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

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) $2 n$

Q80. The coefficients $a , b , c$ in the quadratic equation $a x ^ { 2 } + b x + c = 0$ are from the set $\{ 1,2,3,4,5,6 \}$. If the probability of this equation having one real root bigger than the other is $p$, then 216 p equals :
(1) 57
(2) 76
(3) 38
(4) 19

Q80. If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is:
(1) $\frac { 18 } { 25 }$
(2) $\frac { 12 } { 25 }$
(3) $\frac { 6 } { 25 }$
(4) $\frac { 4 } { 25 }$

Q80. Let the sum of two positive integers be 24 . If the probability, that their product is not less than $\frac { 3 } { 4 }$ times their greatest possible product, is $\frac { m } { n }$, where $\operatorname { gcd } ( m , n ) = 1$, then $n - m$ equals
(1) 10
(2) 9
(3) 11
(4) 8

Q80. If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $i ^ { \text {th } }$ roll than the number obtained in the $( i - 1 ) ^ { \text {th } }$ roll, $i = 2,3$, is equal to
(1) $3 / 54$
(2) $2 / 54$
(3) $1 / 54$
(4) $5 / 54$

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

There are nine cards on which the integers from 1 to 9 are written in a box. Two cards are taken simultaneously from this box. Let $S$ denote the sum of the numbers written on the two cards.
(1) The probability that $S$ is 5 or less is $\frac { \mathbf { A } } { \mathbf { B } }$. Let us assign a score to the result $S$.
When $S$ is 5 or less the score is $10 - S$, and when it is greater than 5 the score is 2. Then the expected value of the score is $\frac { \mathbf { C D } } { \mathbf { E F } }$.
(2) Let us perform the above trial twice, returning the two cards to the box before the second trial.
(i) The probability that $S$ is 5 or less in both trials is $\frac { \mathbf { G } } { \mathbf { H } }$.
(ii) The probability that $S$ is 5 or less in at least one trial is $\frac { \mathbf { J K } } { \mathbf { L M } }$.

Let P be a point in a plane with a coordinate system that is initially located at the origin $(0,0)$ and moves in the plane according to the following rule:
One dice is thrown. When the number on the dice is a multiple of three, point P moves 1 unit in the positive direction of the $x$-axis, and when the number on the dice is not a multiple of three, point P moves 1 unit in the positive direction of the $y$-axis.
Assume that the dice is thrown four times.
(1) The probability that P reaches point $(3,1)$ is $\frac{\mathbf{A}}{\mathbf{BC}}$.
(2) Altogether, the number of the points which P can reach is $\mathbf{D}$, and the coordinates of these points can be expressed in terms of an integer $k$ as
$$(k,\, \mathbf{E} - k) \quad (\mathbf{F} \leq k \leq \mathbf{G}).$$
Let us denote the probability that P can reach a given point $(k, \mathbf{E} - k)$ by $p_k$. Then the maximum value of $p_k$ is $\frac{\mathbf{HI}}{\mathbf{HI}}$, and the minimum value of $p_k$ is $\frac{\mathbf{J}}{\mathbf{BC}}$.
(3) The probability that $P$ passes through point $(1,1)$ and reaches point $(2,2)$ is $\frac{\mathbf{KL}}{\mathbf{BC}}$.

In a bag there are a total of nine balls: one white, three red and five black. The white ball is worth five points, a red ball is worth three points, and a black ball is worth one point. Two balls are taken from the bag together, and the trial is scored by the sum of the values of the two balls.
(1) The highest possible score is $\mathbf{A}$, and the probability that it happens is $\dfrac{\mathbf{B}}{\mathbf{CD}}$.
(2) The probability that the score is 6 is $\dfrac{\mathbf{E}}{\mathbf{F}}$.
(3) The expected value of the score is $\dfrac{\mathbf{GH}}{\mathbf{I}}$.

Consider the sets $A = \{ 4 m \mid m$ is a natural number $\}$ and $B = \{ 6 m \mid m$ is a natural number $\}$.
(1) For each of the following $\mathbf { L } \sim \mathbf{O}$, choose the correct answer from among (0) $\sim$ (3) below.
Let $n$ be a natural number.
(i) $n \in A$ is $\mathbf { L }$ for $n$ to be divisible by 2 .
(ii) $n \in B$ is $\mathbf { M }$ for $n$ to be divisible by 24 .
(iii) $n \in A \cup B$ is $\mathbf { N }$ for $n$ to be divisible by 3 .
(iv) $n \in A \cap B$ is $\square\mathbf{O}$ for $n$ to be divisible by 12 . (0) a necessary and sufficient condition
(1) a necessary condition but not a sufficient condition
(2) a sufficient condition but not a necessary condition
(3) neither a necessary condition nor a sufficient condition
(2) Let $C = \{ m \mid m$ is a natural number satisfying $1 \leqq m \leqq 100 \}$.
The number of elements which belong to $( \bar { A } \cup \bar { B } ) \cap C$ is $\mathbf { P Q }$, and the number of elements which belong to $\bar { A } \cap \bar { B } \cap C$ is $\mathbf { R S }$. Note that $\bar { A }$ and $\bar { B }$ denote the complements of $A$ and $B$, where the universal set is the set of all natural numbers.

There are two bags, A and B. Bag A contains four white balls and one red ball, and bag B contains two white balls and three red balls. Two balls are taken simultaneously out of bag A, then two balls are taken simultaneously out of bag B.
(1) The probability that two white balls are taken out of A, and one white ball and one red ball are taken out of B is $\frac{\mathbf{J}}{\mathbf{J}}$.
(2) The probability that the four balls taken out consist of three white balls and one red ball is $\frac{\mathbf{M}}{\mathbf{M}}$.
(3) The probability that the four balls taken out all have the same color is $\square$ PQ
(4) The probability that of the four balls taken out, two or fewer are white balls is $\frac{\mathbf{RS}}{\mathbf{TU}}$.