An organization awarded 48 medals in event 'A', 25 in event 'B' and 18 in event 'C'. If these medals went to total 60 men and only five men got medals in all the three events, then, how many received medals in exactly two of three events?

(a) 10                           (b) 9                                 (c) 21                               (d) 15

Let $\Omega$ be the sample space and $A\subseteq \Omega$ be an event. Given below are two statements

(S1) : If P(A) = 0, then A = $\mathrm{\varphi }$

(S2) : If P(A) = 1, then A = $\Omega$

Then

(a) both (S1) and (S2) are true                          (b) only (S1) is true

(c) only (S2) is true                                                    (d) both (S1) and (S2) are false

The number of elements in the set $\{n\in \mathrm{\mathbb{Z}}:|n^2-10n+19|<6\}$ is ___________ .

(A) 3                        (B) 0                      (C) 4                        (D) 6

The number of elements in the set {$n\in \mathrm{\mathbb{N}}:10\le n\le 100$ and $3^n-3$ is a multiple of 7} is __________ .

(A) 15                           (B) 12                           (C) 10                          (D) 18

Let $\lambda \in R$ and let the equation E be $\left|x^2\right|-2\left|x\right|+|\lambda -3|=0.$ Then the largest element in the set $S=\{x+\lambda :x$ is an integer solution of E$\}$ is _________ .

(A) 1                           (B) 5                        (C) 6                      (D) 4

Let A = {1, 2, 3, 4, 5, 6, 7}. Then the relation $R=\left\{\right(x,y)\in A\times A:x+y=7\}$ is

(a) an equivalence relation

(b) reflexive but neither symmetric nor transitive

(c) transitive but neither symmetric nor reflexive

(d) symmetric but neither reflexive nor transitive

Let A = {2, 3, 4} and B = {8, 9, 12}. Then the number of elements in the relation

$R=\left\{\right((a_1,b_1),(a_2,b_2))\in (A\times B,A\times B):a_1$ divides $b_2$ and $a_2$ divides $b_1\}$ is

(a) 24                            (b) 12                                (c) 36                                (d) 18

Let A = {1, 3, 4, 6, 9} and B = {2, 4, 5, 8, 10}. Let R be a relation defined on A x B such that $R=\left\{\right((a_1,b_1),(a_2,b_2)):a_1\le b_2$ and $b_1\le a_2\}.$ Then the number of elements in the set R is

(a) 52                        (b) 180                           (c) 26                        (d) 160

Let R be a relation on R, given by

R = $\left\{\right(a,b):3a-3b+\sqrt{7}$ is an irrational number$\}.$

Then R is

(a) reflexive and symmetric but not transitive

(b) reflexive and transitive but not symmetric

(c) reflexive but neither symmetric nor transitive

(d) an equivalence relation

Let P(S) denote the power set of S = {1, 2, 3, ...., 10}. Define the relations $R_1$ and $R_2$ on P(S) as $A{R}_{1}B$ if $(A\cap B^C)\cup (B\cap A^C)=\varphi$ and $A{R}_{2}B$ if $A\cup B^C=B\cup A^C,\forall A,B\in P\left(S\right)$. Then

(a) both $R_1$ and $R_2$ are not equivalence relations

(b) only $R_2$ is an equivalence relation

(c) only $R_1$ is an equivalence relation

(d) both $R_1$ and $R_2$ are equivalence relations

The relation $R=\left\{\right(a,b):gcd(a,b)=1,2a\ne b,a,b,\in Z\}$ is

(a) reflexive but not symmetric                       (b) transitive but not reflexive

(c) symmetric but not transitive                       (d) neither symmetric nor transitive

Let R be a relation defined on $\mathrm{\mathbb{N}}$ as $a R b$ if $2a+3b$ is a multiple of 5, $a,b\in \mathrm{\mathbb{N}}$.

(a) symmetric but not transitive                   (b) not reflexive

(c) an equivalence relation                                 (d) transitive but not symmetric

The minimum number of elements that must be added to the relation R = {(a, b), (b, c)} on the set {a, b, c} so that it becomes symmetric and transitive is

(a) 3                               (b) 7                                 (c) 4                                (d) 5

Let R be a relation on N x N defined by (a, b) R(c, d) if and only if $ad(b-c)=bc(a-d)$. Then R is

(a) transitive but neither reflexive nor symmetric

(b) symmetric but neither reflexive nor transitive

(c) symmetric and transitive but not reflexive

(d) reflexive and symmetric but not transitive

Among the relations

$S=\left\{\right(a,b):a,b\in \mathit{R}-\{0\},2+\frac{a}{b}>0\}$ and 

$T=\left\{\right(a,b):a,b\in \mathit{R}\mathbf{,}{a}^{\mathbf{2}}-b^2\in Z\}$

(a) S is transitive but T is not                     (b) both S and T are symmetric

(c) neither S nor T is transitive                    (d)  T is symmetric but S is not

Let A = {1, 2, 3, 4, .... ,10} and B = {0, 1, 2, 3, 4}. The number of elements in the relation $R=\left\{\right(a,b)\in A\times A:2{(a-b)}^2+3(a-b)\in B\}$ is __________ .

(A) 14                            (B) 12                         (C) 16                         (D) 18

Let A = {0, 3, 4, 6, 7, 8, 9, 10} and R be the relation defined on A such that $R=\left\{\right(x,y)\in A\times A:x-y$ 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 __________ .

(A) 19                          (B) 18                              (C) 16                        (D) 17

The number of relations, on the set {1, 2, 3} containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _________ .

(A) 5                                 (B) 5                          (C) 3                         (D) 7

Let A = {-4, -3, -2, 0, 1, 3, 4} and $R=\left\{\right(a,b)\in A\times A:b=|a|or b^2=a+1\}$ be a relation on A.

Then the minimum number of elements, that must be added to the relation R so that it becomes reflexive and symmetric, is ________ .

(A) 7                                   (B) 5                             (C) 3                           (D) 1

Let A = {1, 2, 3, 4} and R be a relation on the set A x A defined by $R=\left\{\right((a,b),(c,d)):2a+3b=4c+5d\}.$

Then the number of elements in R is ___________ .

(A) 4                            (B) 6                         (C) 8                            (D) 10

The minimum number of elements that must be added to the relation R = {(a, b), (b, c), (b, d)} on the set {a, b, c, d} so that it is an equivalence relation is _____________ .

(A) 15                        (B) 17                        (C) 13                        (D) 11

Let $A=\{x\in \mathrm{ℝ}:[x+3]+[x+4]\le 3\},$

$B=\{x\in \mathrm{ℝ}:3^x{\left(\sum _{r=1}^{\infty }\frac{3}{{10}^r}\right)}^{x-3}<3^{-3x}\},$

where [t] denotes greatest integer function. Then,

(a) $A\cap B=\varphi$                  (b) $B\subset C,A\ne B$                      (c) $A\subset B,A\ne B$                     (d) $A=B$

Let the sets A and B denote the domain and range respectively of the function $f\left(x\right)=\frac{1}{\sqrt{\left[x\right]-x}}$, where $\left[x\right]$ denotes the smallest integer greater than or equal to x. Then among the statements

$\left(S1\right):A\cap B=(1,\infty )-N$ and

$\left(S2\right): A\cup B=(1,\infty )$

(a) only (S1) is true                                        (b) both (S1) and (S2) are true

(c) only (S2) is true                                         (d) neither (S1) nor (S2) is true

If $f\left(x\right)=\frac{\left(\tan 1^o\right)x+{\log }_e\left(123\right)}{x{\log }_e\left(1234\right)-\left(\tan 1^o\right)},x>0,$ then the least value of $f\left(f\right(x\left)\right)+f\left(f\right(\frac{4}{x}\left)\right)$ is

(a) 4                              (b) 2                       (c) 0                          (d) 8

The number of integral solutions x of ${\log }_{(x+\frac{7}{2})}{\left(\frac{x-7}{2x-3}\right)}^2\ge 0$ is

(a) 6                          (b) 8                           (c) 5                      (d) 7

The domain of the function $f\left(x\right)=\frac{1}{\sqrt{{\left[x\right]}^2-3\left[x\right]-10}}$ is (where [x] denotes the greatest integer less than or equal to x).

(a) $(-\infty ,-3]\cup (5,\infty )$                       (b) $(-\infty ,-2)\cup [6,\infty )$

(c) $(-\infty ,-3]\cup [6,\infty )$                        (d) $(-\infty ,-2)\cup (5,\infty )$

For $x\in \mathrm{ℝ}$, two real valued functions $f\left(x\right)$ and $g\left(x\right)$ are such that, $g\left(x\right)=\sqrt{x}+1$ and $fog\left(x\right)=x+3-\sqrt{x}.$ Then $f\left(0\right)$ is equal to

(a) 5                           (b) 1                             (c) 0                             (d) - 3

Let $f:R-\{0,1\}\to \mathrm{ℝ}$ be a function such that $f\left(x\right)+f\left(\frac{1}{1-x}\right)=1+x.$ Then $f\left(2\right)$ is equal to

(a) $\frac{7}{3}$                              (b) $\frac{9}{2}$                               (c) $\frac{9}{4}$                            (d) $\frac{7}{4}$

The equation $x^2-4x+\left[x\right]+3=x\left[x\right],$ where [x] denotes the greatest integer function, has

(a) no solution                                                 (b) exactly two solutions in $(-\infty ,\infty )$

(c) a unique solution in $(-\infty ,1)$                      (d) a unique solution in $(-\infty ,\infty )$

If $f\left(x\right)=\frac{2^{2x}}{2^{2x}+2},x\in R,$ then

$f\left(\frac{1}{2023}\right)+f\left(\frac{2}{2023}\right)+....+f\left(\frac{2022}{2023}\right)$ is equal to

(a) 1011                            (b) 2010                              (c) 1010                              (d) 2011

Let $f\left(x\right)$ be a function such that $f(x+y)=f\left(x\right)·f\left(y\right)$ for all $x,y\in N.$ If $f\left(1\right)=3$ and $\sum _{k=1}^{n}f\left(k\right)=3279,$ then the value of n is

(a) 8                             (b) 6                                (c) 7                             (d) 9

The number of functions $f:\{1,2,3,4\}\to \{a\in :Z|a|\le 8\}$ satisfying $f\left(n\right)+\frac{1}{n}f(n+1)=1, \forall n\in \{1,2,3\}$ is

(a) 2                             (b) 1                           (c) 4                      (d) 3

Let $f\left(x\right)=2x^n+\lambda ,\lambda \in \mathit{R},n\in \mathit{N}, and f\left(4\right)=133,f\left(5\right)=255.$ Then the sum of all the positive integer divisors of $\left(f\right(3)-f(2\left)\right)$ is

(a) 59                              (b) 60                            (c) 61                              (d) 58

Let $f:R\to R$ be a function defined by $f\left(x\right)={\log }_{\sqrt{m}}\left\{\sqrt{2}\right(\sin x-\cos x)+m-2\},$ for some $m$, such that the range of $f$ is [0, 2]. Then the value of $m$ is

(a) 3                               (b) 5                                  (c) 4                                 (d) 2

Let $f:\mathit{R}\to \mathit{R}$ be a function such that $f\left(x\right)=\frac{x^2+2x+1}{x^2+1}.$ Then

(a) $f\left(x\right)$ is many-one in $(-\infty ,-1)$

(b) $f\left(x\right)$ is one-one in $(-\infty ,\infty )$

(c) $f\left(x\right)$ is many-one in $(1,\infty )$

(d) $f\left(x\right)$ is one-one in $[1,\infty )$ but not in $(-\infty ,\infty )$

The domain of $f\left(x\right)=\frac{{\log }_{(x+1)}(x-2)}{e^{2{\log }_{e}x}-(2x+3)},x\in \mathrm{ℝ}$ is

(a) $(2,\infty )-\left\{3\right\}$                           (b) $\mathrm{ℝ}-\left\{3\right\}$                          (c) $\mathrm{ℝ}-\{-1,3\}$                       (d) $(-1,\infty )-\left\{3\right\}$

Consider a function $f:\mathrm{\mathbb{N}}\to \mathrm{ℝ},$ satisfying $f\left(1\right)+2f\left(2\right)+3f\left(3\right)+...+xf\left(x\right)=x(x+1)f\left(x\right);x\ge 2$

with $f\left(1\right)=1.$ Then $\frac{1}{f\left(2022\right)}+\frac{1}{f\left(2028\right)}$ is equal to

(a) 8400                            (b) 8200                               (c) 8100                                (d) 8000

The range of the function $f\left(x\right)=\sqrt{3-x}+\sqrt{2+x}$ is

(a) $[2\sqrt{2},\sqrt{11}]$                        (b) $[\sqrt{5},\sqrt{10}]$                           (c) $[\sqrt{5},\sqrt{13}]$                      (d) $[\sqrt{2},\sqrt{7}]$

If the domain of the function $f\left(x\right)=\frac{\left[x\right]}{1+x^2},$ where [x] is greatest integer $\le x,$ is [2,6), then its range is

(a) $(\frac{5}{37},\frac{2}{5}]$

(b) $(\frac{5}{26},\frac{2}{5}]-\{\frac{9}{29},\frac{27}{109},\frac{18}{89},\frac{9}{53}\}$

(c) $(\frac{5}{26},\frac{2}{5}]$

(d) $(\frac{5}{37},\frac{2}{5}]-\{\frac{9}{29},\frac{27}{109},\frac{18}{89},\frac{9}{53}\}$

Let $f:\mathit{R}-\{2,6\}\to \mathit{R}$ be real valued function defined as $f\left(x\right)=\frac{x^2+2x+1}{x^2-8x+12}.$ Then range of $f$ is

(a) $(-\infty ,-\frac{21}{4}]\cup [\frac{21}{4},\infty )$

(b) $(-\infty ,-\frac{21}{4}]\cup [0,\infty )$

(c) $(-\infty ,-\frac{21}{4})\cup (0,\infty )$

(d) $(-\infty ,-\frac{21}{4}]\cup [1,\infty )$

If domain of the function

${\log }_e\left(\frac{6x^2+5x+1}{2x-1}\right)+{\cos }^{-1}\left(\frac{2x^2-3x+4}{3x-5}\right)$ is $(\alpha ,\beta )\cup (\gamma ,\delta ],$ then 18$({\alpha }^2+{\beta }^2+{\gamma }^2+{\delta }^2)$ is equal to ____________.

(A) 20                         (B) 30                         (C) 40                        (D) 60

Let R = {a, b, c, d, e} and S = {1, 2, 3, 4}. Total number of onto functions $f:R\to S$ such that $f\left(a\right)\ne 1,$ is equal to ________ .

(A) 150                        (B) 180                    (C) 160                      (D) 140

Let a, b, c be three distinct positive real numbers such that ${\left(2a\right)}^{{\log }_{e}a}={\left(bc\right)}^{{\log }_{e}b}$ and $b^{{\log }_{e}2}=a^{{\log }_{e}c}$.

Then 6a + 5bc is equal to _____________ .

(A) 5                        (B) 4                      (C) 8                         (D) 6

Let A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6}. Then the number of functions $f:A\to B$ satisfying $f\left(1\right)+f\left(2\right)=f\left(4\right)-1$ is equal to __________ .

Let $\left[\alpha \right]$ denote the greatest integer $\le \alpha$. Then $\left[\sqrt{1}\right]+\left[\sqrt{2}\right]+\left[\sqrt{3}\right]+...+\left[\sqrt{120}\right]$ is equal to __________ .

For some a, b, c$\in$N, let $f\left(x\right)=ax-3$ and $g\left(x\right)=x^b+c,x\in R.$ If ${\left(fog\right)}^{-1}\left(x\right)={\left(\frac{x-7}{2}\right)}^{1/3},$ then (fog)(ac) + (gof)(b) is equal to ___________ .

Suppose $f$ is a function satisfying $f(x+y)=f\left(x\right)+f\left(y\right)$ for all $x,y\in \mathrm{\mathbb{N}}$ and $f\left(1\right)=\frac{1}{5}.$ If $\sum _{n=1}^m\frac{f\left(n\right)}{n(n+1)(n+2)}=\frac{1}{12},$ then $m$ is equal to ___________ .

Let S = {1, 2, 3, 4, 5, 6}. Then the number of one-one functions $f:S\to P\left(S\right),$ where $P\left(S\right)$ denote the power set of S, such that $f\left(n\right)\subset f\left(m\right)$ where $n<m$ is __________ .

Let $f^1\left(x\right)=\frac{3x+2}{2x+3},x\in R-\left\{\frac{-3}{2}\right\}$

For $n\ge 2,$ define $f^n\left(x\right)=f^{1}o{f}^{n-1}\left(x\right).$

If $f^5\left(x\right)=\frac{ax+b}{bx+a},gcd(a,b)=1,$ then a + b is equal to _________ .

Let A = {1, 2, 3, 5, 8, 9}. Then the number of possible functions $f:A\to A$ such that $f(m·n)=f\left(m\right)·f\left(n\right)$ for every $m,n\in A$ with $m·n\in A$ is equal to __________________ .