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

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

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

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) $( \sim ( p \wedge q ) ) \wedge q$

The statement $\sim p \vee ( \sim p \wedge q )$ is equivalent to
(1) $\sim p \wedge q$
(2) $p \wedge q \wedge \sim p$
(3) $\sim p \wedge q \wedge q$
(4) $\sim p \vee q$

The statement $( p \wedge ( \sim q ) ) \vee ( ( \sim p ) \wedge q ) \vee ( ( \sim p ) \wedge ( \sim q ) )$ is equivalent to
(1) $\sim p \vee q$
(2) $\sim p \vee \sim q$
(3) $p \vee \sim q$
(4) $p \vee q$

Let $p$ and $q$ be two statements. Then $\sim ( p \wedge ( p \rightarrow \sim q ) )$ is equivalent to
(1) $p \vee ( p \wedge ( \sim q ) )$
(2) $p \vee ( ( \sim p ) \wedge q )$
(3) $( \sim p ) \vee q$
(4) $p \wedge q$

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 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

The negation of $( p \wedge ( - q ) ) \vee ( - p )$ is equivalent to
(1) $p \wedge ( - q )$
(2) $p \wedge q$
(3) $p \vee ( q \vee ( - p ) )$
(4) $p \wedge ( q \wedge ( - p ) )$

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

If $R$ is the smallest equivalence relation on the set $\{ 1,2,3,4 \}$ such that $\{ ( 1,2 ) , ( 1,3 ) \} \subset R$, then the number of elements in $R$ is
(1) 10
(2) 12
(3) 8
(4) 15

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 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

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

Let $A = \{ 1,2,3 \}$. The number of relations on $A$, containing $( 1,2 )$ and $( 2,3 )$, which are reflexive and transitive but not symmetric, is $\_\_\_\_$

Consider the integral expression
$$P = ( x - 1 ) ^ { 2 } ( y + 5 ) + ( 2 x - 3 ) ( y + 4 ) - ( x - 1 ) ^ { 2 } .$$
(1) $P$ can be transformed into
$$P = \left( x ^ { 2 } - \mathbf { K } \right) ( y + \mathbf { L } ) .$$
(2) The pairs $( x , y )$ of integers $x$ and $y$ which give $P = 7$ are
$$( \pm \mathbf { M } , \mathbf { N O P } ) , \quad ( \pm \mathbf { Q } , \mathbf { R S } ) .$$
(3) Let $a$ be a rational number. If $x = \sqrt { 2 } + 2 \sqrt { 3 }$ and $y = a + \sqrt { 6 }$, then the value of $a$ such that the value of $P$ is a rational number is $\mathbf { T U }$.

A natural number $n$ is said to be a perfect square when there exists a natural number $x$ satisfying $n = x ^ { 2 }$. Similarly, $n$ is said to be a perfect cube when there exists a natural number $x$ satisfying $n = x ^ { 3 }$.
In the following two cases, find the natural number $n$ that satisfies the conditions.
(i) $n$ is a perfect square. Furthermore, the number obtained by adding 13 to $n$ is also a perfect square.
(ii) $n$ is a perfect cube. Furthermore, the number obtained by adding 61 to $n$ is also a perfect cube.
First, consider (i). From the definition of a perfect square number, $n$ can be expressed as $n = x ^ { 2 }$, where $x$ is a natural number. In addition, there exists a natural number $y$ such that
$$x ^ { 2 } + 13 = y ^ { 2 } .$$
Since $x < y$, $y - x = \square \mathbf { J }$ and $y + x = \mathbf { K L }$. It follows that
$$x = \mathbf { M } , \quad y = \mathbf { N } ,$$
and finally that $n = \mathbf { O P }$.
Next, consider (ii). Similar to (i), in (ii), there exists a natural number $x$ such that $n = x ^ { 3 }$, and there also exists a natural number $y$ such that
$$x ^ { 3 } + 61 = y ^ { 3 } .$$
When we solve this equation, we obtain
$$x = \mathbf { Q } , \quad y = \mathbf { R } ,$$
and hence the perfect cube $n = \mathbf{ST}$.

Consider four natural numbers $a , b , c$ and $d$ satisfying $1 < a < b < c < d$. Suppose that two sets using these numbers, $A = \{ a , b , c , d \}$ and $B = \left\{ a ^ { 2 } , b ^ { 2 } , c ^ { 2 } , d ^ { 2 } \right\}$, satisfy the following two conditions:
(i) Just two elements belong to the intersection $A \cap B$, and the sum of these two elements is greater than or equal to 15, and less than or equal to 25.
(ii) The sum of all the elements belonging to the union $A \cup B$, is less than or equal to 300.
We are to find the values of $a , b , c$ and $d$.
First, set $A \cap B = \{ x , y \}$, where $x < y$. Since $x \in B$ and $y \in B$, it follows from (i) that $y = \mathbf { A B }$ and that $x$ is either $\mathbf{C}$ or $\mathbf { D }$. (Write the answers in the order $\mathbf { C } < \mathbf { D }$.) Here, when we consider (ii), we see that $x = \mathbf { E }$. Hence $A$ includes the elements $\mathbf { F }$, $\mathbf{F}$ and $\mathbf{F}$.
Furthermore, when we denote the remaining element of $A$ by $z$, from (ii) we see that $z$ satisfies
$$z ^ { 2 } + z \leqq \mathbf { G H } .$$
Hence we have $z = \mathbf { I }$. From the above we obtain
$$a = \mathbf { J } , \quad b = \mathbf { K } , \quad c = \mathbf { L } \text { and } d = \mathbf { M N } .$$

Consider integers $a$ and $b$ satisfying the equation $$14a + 9b = 147. \tag{1}$$
(1) We are to find the positive integers $a$ and $b$ satisfying equation (1).
Since $$14a = \mathbf{A}(\mathbf{BC} - \mathbf{D}b) \text{ and } 9b = \mathbf{E}(\mathbf{FG} - \mathbf{H}a),$$ $a$ is a multiple of $\mathbf{A}$, and $b$ is a multiple of $\mathbf{E}$.
So, when we set $a = \mathbf{A}m$ and $b = \mathbf{E}n$, where $m$ and $n$ are integers, from (1) we have $$\mathbf{I}m + \mathbf{J}n = \mathbf{K}.$$
Since the positive integers $m$ and $n$ satisfying this are $$m = \mathbf{L} \text{ and } n = \mathbf{M},$$ we obtain $$a = \mathbf{N} \text{ and } b = \mathbf{O}.$$
(2) We are to find the solutions $a$ and $b$ of equation (1) satisfying $0 < a + b < 5$.
Since $14 \times \mathbf{N} + 9 \times \mathbf{O} = 147$, from this equality and (1) we have $$14(a - \mathbf{N}) = 9(\mathbf{O} - b).$$
Since 14 and 9 are relatively prime, $a$ and $b$ can be expressed in terms of an integer $k$ as $$a = \mathbf{P} + \mathbf{Q}k, \quad b = -\mathbf{RS}k + \mathbf{T}.$$
Since $0 < a + b < 5$, we have $k = \mathbf{U}$, and we obtain $$a = \mathbf{VW}, \quad b = -\mathbf{XY}.$$

As shown in the figure, consider a rectangular stone block with a vertex $A$ and a face containing point $A$. Let the midpoints of the edges of this face be $B , E , F , D$ respectively. Another face of the rectangular block containing point $B$ has its edge midpoints as $B , C , H , G$ respectively. Given that $\overline { B C } = 8$ and $\overline { B D } = \overline { D C } = 9$. The stone block is now cut to remove eight corners, such that the cutting plane for each corner passes through the midpoints of the three adjacent edges of that corner.
How many faces does the stone block have after cutting the corners? (Single choice question, 3 points)
(1) Octahedron
(2) Decahedron
(3) Dodecahedron
(4) Tetradecahedron
(5) Hexadecahedron

Let $\mathbf { p } , \mathbf { q }$ and $\mathbf { r }$ be propositions with their negations denoted by $\mathbf { p } ^ { \prime } , \mathbf { q } ^ { \prime } , \mathbf { r } ^ { \prime }$ respectively. Which of the following is equivalent to the proposition
$$p \vee q \Rightarrow q \wedge r$$
?
A) $\mathrm { p } ^ { \prime } \wedge \mathrm { q } ^ { \prime } \Rightarrow \mathrm { q } ^ { \prime } \vee \mathrm { r } ^ { \prime }$
B) $\mathrm { p } ^ { \prime } \wedge \mathrm { q } ^ { \prime } \Rightarrow \mathrm { q } ^ { \prime } \wedge \mathrm { r } ^ { \prime }$
C) $p ^ { \prime } \vee q ^ { \prime } \Rightarrow q ^ { \prime } \wedge r ^ { \prime }$
D) $q ^ { \prime } \wedge r ^ { \prime } \Rightarrow p ^ { \prime } \vee q ^ { \prime }$
E) $q ^ { \prime } \vee r ^ { \prime } \Rightarrow p ^ { \prime } \wedge q ^ { \prime }$

A rectangular piece of paper ABCD shown below is folded so that vertices B and D coincide. Let E be the folding point on side [AB] such that $|AE| = 1$ unit.
As a result of the folding, the overlapping parts of the paper form a dark-colored equilateral triangular region DEF.
Accordingly, what is the area of the paper in square units?
A) $6\sqrt{2}$ B) $2\sqrt{2}$ C) $4\sqrt{3}$ D) $3\sqrt{3}$ E) $4\sqrt{2}$

A student made an error while proving the following claim that he thought was true.
Claim: For any sets $A$, $B$, $C$, we have $A \backslash ( B \cap C ) \subseteq ( A \backslash B ) \cap ( A \backslash C )$.
The student's proof:
If I show that every element of the set $A \backslash ( B \cap C )$ is in the set $( A \backslash B ) \cap ( A \backslash C )$, the proof is complete.
Now, let $x \in A \backslash ( B \cap C )$. (I) From this, $x \in A$ and $x \notin ( B \cap C )$. (II) From this, $x \in A$ and $( x \notin B$ and $x \notin C )$. (III) From this, $( x \in A$ and $x \notin B )$ and $( x \in A$ and $x \notin C )$. (IV) From this, $x \in A \backslash B$ and $x \in A \backslash C$. (V) From this, $x \in [ ( A \backslash B ) \cap ( A \backslash C ) ]$.
In which of the numbered steps did this student make an error?
A) I
B) II
C) III
D) IV
E) V