Consider the statement "The match will be played only if the weather is good and ground is not wet". Select the correct negation from the following:
(1) The match will not be played and weather is not good and ground is wet.
(2) If the match will not be played, then either weather is not good or ground is wet.
(3) The match will be played and weather is not good or ground is wet.
(4) The match will not be played or weather is good and ground is not wet.

The number of choices for $\Delta \in \{ \wedge , \vee , \Rightarrow , \Leftrightarrow \}$, such that $( p \Delta q ) \Rightarrow ( ( p \Delta \sim q ) \vee ( ( \sim p ) \Delta q ) )$ is a tautology, is
(1) 1
(2) 2
(3) 3
(4) 4

Let a set $A = A _ { 1 } \cup A _ { 2 } \cup \ldots \cup A _ { k }$, where $A _ { i } \cap A _ { j } = \phi$ for $i \neq j ; 1 \leq i , j \leq k$. Define the relation $R$ from $A$ to $A$ by $R = \left\{ ( x , y ) : y \in A _ { i } \right.$ if and only if $\left. x \in A _ { i } , 1 \leq i \leq k \right\}$. Then, $R$ is:
(1) reflexive, symmetric but not transitive
(2) reflexive, transitive but not symmetric
(3) reflexive but not symmetric and transitive
(4) an equivalence relation

Consider the following statements: $P$: Ramu is intelligent. $Q$: Ramu is rich. $R$: Ramu is not honest. The negation of the statement ``Ramu is intelligent and honest if and only if Ramu is not rich'' can be expressed as:
(1) $((P \wedge (\sim R)) \wedge Q) \wedge ((\sim Q) \wedge ((\sim P) \vee R))$
(2) $((P \wedge R) \wedge Q) \vee ((\sim Q) \wedge ((\sim P) \vee (\sim R)))$
(3) $((P \wedge R) \wedge Q) \wedge ((\sim Q) \wedge ((\sim P) \vee (\sim R)))$
(4) $((P \wedge (\sim R)) \wedge Q) \vee ((\sim Q) \wedge ((\sim P) \wedge R))$

Negation of the Boolean expression $p \leftrightarrow (q \rightarrow p)$ is
(1) $\sim p \wedge q$
(2) $p \wedge \sim q$
(3) $\sim p \vee \sim q$
(4) $\sim p \wedge \sim q$

Let the operations $*, \odot \in \{\wedge, \vee\}$. If $p * q \odot p \odot {\sim}q$ is a tautology, then the ordered pair $(*, \odot)$ is
(1) $(\vee, \wedge)$
(2) $(\vee, \vee)$
(3) $(\wedge, \wedge)$
(4) $(\wedge, \vee)$

The relation $R = a , b : \operatorname { gcd} a , b = 1 , \quad 2 a \neq b , \quad a , \quad b \in \mathbb { Z }$ is:
(1) transitive but not reflexive
(2) symmetric but not transitive
(3) reflexive but not symmetric
(4) neither symmetric nor transitive

The number of values of $r \in \{p, q, \sim p, \sim q\}$ for which $((p \wedge q) \Rightarrow (r \vee q)) \wedge ((p \wedge r) \Rightarrow q)$ is a tautology, is:
(1) 1
(2) 2
(3) 4
(4) 3

Among the relations $S = \left\{(a,b) : a, b \in R - \{0\},\ 2 + \frac{a}{b} > 0\right\}$ and $T = \left\{(a,b) : a, b \in R,\ a^2 - b^2 \in Z\right\}$,
(1) $S$ is transitive but $T$ is not
(2) both $S$ and $T$ are symmetric
(3) neither $S$ nor $T$ is transitive
(4) $T$ is symmetric but $S$ is not

Among the statements: $(S1): 2023^{2022} - 1999^{2022}$ is divisible by 8. $(S2): 13(13)^{n} - 11n - 13$ is divisible by 144 for infinitely many $n \in \mathbb{N}$
(1) Only $(S2)$ is correct
(2) Only $(S1)$ is correct
(3) Both $(S1)$ and $(S2)$ are correct
(4) Both $(S1)$ and $(S2)$ are incorrect

Among the statements $(S1): (p \Rightarrow q) \vee ((\sim p) \wedge q)$ is a tautology $(S2): (q \Rightarrow p) \Rightarrow ((\sim p) \wedge q)$ is a contradiction
(1) Neither $(S1)$ and $(S2)$ is True
(2) Both $(S1)$ and $(S2)$ are True
(3) Only $(S2)$ is True
(4) Only $(S1)$ is True

Negation of $p \wedge ( q \wedge \sim ( p \wedge q ) )$ is
(1) $( \sim ( p \wedge q ) ) \vee p$
(2) $p \vee q$
(3) $\sim ( p \vee q )$
(4) $( \sim ( p \wedge q ) ) \wedge q$

Let $x = (8\sqrt{3} + 13)^{13}$ and $y = (7\sqrt{2} + 9)^{9}$. If $[t]$ denotes the greatest integer $\leq t$, then
(1) $[x] + [y]$ is even
(2) $[x]$ is odd but $[y]$ is even
(3) $[x]$ is even but $[y]$ is odd
(4) $[x]$ and $[y]$ are both odd

Consider the relations $R_1$ and $R_2$ defined as $a R_1 b \Leftrightarrow a^2 + b^2 = 1$ for all $a, b \in \mathbb{R}$ and $(a,b) R_2 (c,d) \Leftrightarrow a + d = b + c$ for all $a, b, c, d \in \mathbb{N} \times \mathbb{N}$. Then
(1) Only $R_1$ is an equivalence relation
(2) Only $R_2$ is an equivalence relation
(3) $R_1$ and $R_2$ both are equivalence relations
(4) Neither $R_1$ nor $R_2$ is an equivalence relation

Let $X = \mathbf { R } \times \mathbf { R }$. Define a relation $R$ on $X$ as : $\left( a _ { 1 } , b _ { 1 } \right) R \left( a _ { 2 } , b _ { 2 } \right) \Leftrightarrow b _ { 1 } = b _ { 2 }$
Statement I : $\quad \mathrm { R }$ is an equivalence relation.
Statement II : For some $( a , b ) \in X$, the set $S = \{ ( x , y ) \in X : ( x , y ) R ( a , b ) \}$ represents a line parallel to $y = x$.
In the light of the above statements, choose the correct answer from the options given below :
(1) Both Statement I and Statement II are false
(2) Statement I is true but Statement II is false
(3) Both Statement I and Statement II are true
(4) Statement I is false but Statement II is true

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

Statement I: $25 ^ { 13 } + 20 ^ { 13 } + 8 ^ { 13 } + 3 ^ { 13 }$ is divisible $b - 7$. Statement II: The integral value of $( 7 + 4 \sqrt { 3 } ) \sqrt { 25 } )$ is an odd number (A) Neither statements are correct (B) Only statement I is correct (C) Only statement II is correct (D) Both the statements are correct

Is the set $\{ 1,2,4,5,6,9,10,11 \}$ nice? Justify your answer.

For $n = 3$, explain why the list $( 2,1,1 )$ is good, but the list $( 2,2,2 )$ is not good.

11. Decomposing the positive integer 18 into a product of two positive integers gives
$$1 \times 18, 2 \times 9, 3 \times 6$$
three ways. Among these three decompositions, $3 \times 6$ has the smallest difference between the two numbers, so we call $3 \times 6$ the optimal decomposition of 18. When $p \times q ( p \leq q )$ is the optimal decomposition of a positive integer $n$, we define the function $F ( n ) = \frac { p } { q }$. For example, $F ( 18 ) = \frac { 3 } { 6 } = \frac { 1 } { 2 }$. Which of the following statements about the function $F ( n )$ are correct?
(1) $F ( 4 ) = 1$.
(2) $F ( 24 ) = \frac { 3 } { 8 }$.
(3) $F ( 27 ) = \frac { 1 } { 3 }$.
(4) If $n$ is a prime number, then $F ( n ) = \frac { 1 } { n }$.
(5) If $n$ is a perfect square, then $F ( n ) = 1$.

Part Two: Fill-in-the-Blank Questions (45 points)

Instructions: 1. For questions A through I, mark your answers on the "Answer Section" of the answer sheet at the indicated row numbers (12–32).
2. Each completely correct answer is worth 5 points. Wrong answers are not penalized. Incomplete answers receive no points.
A. A sample survey of 1000 families with two children in a certain region obtained the following data, where (boy, girl) represents a family where the first child is a boy and the second child is a girl, and so on.

Family Type	Number of Families
(boy, boy)	261
(boy, girl)	249
(girl, boy)	255
(girl, girl)	235

From this data, the estimated ratio of boys to girls in families with two children in this region is approximately (rounded to the nearest integer).
B

8. Consider the following statement about the positive integer $n$ :
Statement (*): The sum of the four consecutive integers, the smallest of which is $n$, is a multiple of 6 .
Which one of the following is true?
A Statement () is true for all values of $n$.
B Statement () is true for all values of $n$ which are odd, but not for any other values of $n$.
C Statement (*) is true for all values of $n$ which are multiples of 3 , but not for any other values of $n$.
D Statement (*) is true for all values of $n$ which are multiples of 6 , but not for any other values of $n$.
E Statement (\textit{) is not true for any value of $n$.

In this question, $a , b$, and $c$ are positive integers.
The following is an attempted proof of the false statement:
If $a$ divides $b c$, then $a$ divides $b$ or $a$ divides $c$.
['$a$ divides $b c$' means '$a$ is a factor of $b c$']
Which line contains the error in this proof?
1. The statement is equivalent to if $a$ does not divide $b$ and $a$ does not divide $c$ then $a$ does not divide $b c$'.
2. Suppose $a$ does not divide $b$ and $a$ does not divide $c$. Then the remainder when dividing $b$ by $a$ is $r$, where $0 < r < a$, and the remainder when dividing $c$ by $a$ is $s$, where $0 < s < a$.
3. So $b = a x + r$ and $c = a y + s$ for some integers $x$ and $y$.
4. Thus $b c = a ( a x y + x s + y r ) + r s$.
5. So the remainder when dividing $b c$ by $a$ is $r s$.
6. Since $r > 0$ and $s > 0$, it follows that $r s > 0$.
7. Hence $a$ does not divide $b c$.

Consider the following three statements:
$1 \quad 10 p ^ { 2 } + 1$ and $10 p ^ { 2 } - 1$ are both prime when $p$ is an odd prime.
2 Every prime greater than 5 is of the form $6 n + 1$ for some integer $n$.
3 No multiple of 7 greater than 7 is prime.
The result $91 = 7 \times 13$ can be used to provide a counterexample to which of the above statements?
A none of them
B 1 only
C 2 only
D 3 only
E 1 and 2 only
F 1 and 3 only
G 2 and 3 only
H 1, 2 and 3

Consider the following attempt to prove this true theorem:
Theorem: $a ^ { 3 } + b ^ { 3 } = c ^ { 3 }$ has no solutions with $a , b$ and $c$ positive integers.
Attempted proof:
Suppose that there are positive integers $a , b$ and $c$ such that $a ^ { 3 } + b ^ { 3 } = c ^ { 3 }$.
I We have $a ^ { 3 } = c ^ { 3 } - b ^ { 3 }$.
II $\quad$ Hence $a ^ { 3 } = ( c - b ) \left( c ^ { 2 } + c b + b ^ { 2 } \right)$.
III It follows that $a = c - b$ and $a ^ { 2 } = c ^ { 2 } + c b + b ^ { 2 }$, since $a \leqslant a ^ { 2 }$ and $c - b \leqslant c ^ { 2 } + c b + b ^ { 2 }$.
IV Eliminating $a$, we have $( c - b ) ^ { 2 } = c ^ { 2 } + c b + b ^ { 2 }$.
V Multiplying out, we have $c ^ { 2 } - 2 c b + b ^ { 2 } = c ^ { 2 } + c b + b ^ { 2 }$.
VI Hence $3 c b = 0$ so one of $b$ and $c$ is zero.
But this is a contradiction to the original assumption that all of $a , b$ and $c$ are positive. It follows that the equation has no solutions.
Comment on this proof by choosing one of the following options:
A The proof is correct
B The proof is incorrect and the first mistake occurs on line I.
C The proof is incorrect and the first mistake occurs on line II.
D The proof is incorrect and the first mistake occurs on line III.
E The proof is incorrect and the first mistake occurs on line IV.
F The proof is incorrect and the first mistake occurs on line V.
G The proof is incorrect and the first mistake occurs on line VI.

Consider the following statement:
A car journey consists of two parts. In the first part, the average speed is $u \mathrm {~km} / \mathrm { h }$. In the second part, the average speed is $v \mathrm {~km} / \mathrm { h }$. Hence the average speed for the whole journey is $\frac { 1 } { 2 } ( u + v ) \mathrm { km } / \mathrm { h }$.
Which of the following examples of car journeys provide(s) a counterexample to the statement?
I In the first part of the journey, the car travels at a constant speed of $50 \mathrm {~km} / \mathrm { h }$ for 100 km . In the second part of the journey, the car travels at a constant speed of $40 \mathrm {~km} / \mathrm { h }$ for 100 km .
II In the first part of the journey, the car travels at a constant speed of $50 \mathrm {~km} / \mathrm { h }$ for one hour. In the second part of the journey, the car travels at a constant speed of $40 \mathrm {~km} / \mathrm { h }$ for one hour.
III In the first part of the journey, the car travels at a constant speed of $50 \mathrm {~km} / \mathrm { h }$ for 80 km . In the second part of the journey, the car travels at a constant speed of $40 \mathrm {~km} / \mathrm { h }$ for 100 km .