Sets Relations and Functions — JEE Main PYQs

Let $D$ be the domain of the function $f(x)={\sin }^{-1}\left({\log }_{3x}\left(\frac{6+2{\log }_{3}x}{-5x}\right)\right)$. If the range of the function $g:D\to \mathbb{R}$ defined by $g(x)=x-[x]$, ([x] is the greatest integer function), is $(\alpha ,\beta )$, t …

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 ______.

Let $\mathrm{A}=\{-4,-3,-2,0,1,3,4\}$ and $\mathrm{R}=\{(\mathrm{a},\mathrm{b})\in \mathrm{A}\times \mathrm{A}:\mathrm{b}=|\mathrm{a}|$ or ${\mathrm{b}}^2=\mathrm{a}+1\}$ be a relation on A. Then the minimum number of elements, that must be added to the relation $R$ s …

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

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

If the domain of the function
$f(x)={\log }_e\left(4x^2+11x+6\right)+{\sin }^{-1}(4\mathrm{x}+3)+{\cos }^{-1}\left(\frac{10\mathrm{x}+6}{3}\right)$ is $(\alpha ,\beta ]$,
Then $36|\alpha +\beta |$ is equal to :

Let $\mathrm{A}=\{1,2,3,4\}$ and R be a relation on the set $A\times A$ defined by $\mathrm{R}=\{((\mathrm{a},\mathrm{b}),(\mathrm{c},\mathrm{d})):2\mathrm{a}+3\mathrm{b}=4\mathrm{c}+5\mathrm{d}\}$. Then the number of elements in R is:

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

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

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

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?

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

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

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

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

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

For $n\ge 2$, define $f^n(x)=f^1\circ f^{n-1}(x)$.
If $f^5(x)=\frac{ax+b}{bx+a}$, $\gcd (a,b)=1$, then $a+b$ is equal to ____.

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

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

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

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

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

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

Among the relations
$\mathrm{S}=\left\{(\mathrm{a},\mathrm{b}):\mathrm{a},\mathrm{b}\in \mathbb{R}-\{0\},2+\frac{\mathrm{a}}{\mathrm{b}}>0\right\}$
And …

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

Let $\mathrm{A}=\{1,2,3,5,8,9\}$. Then the number of possible functions $\mathrm{f}:\mathrm{A}\to \mathrm{A}$ such that $\mathrm{f}(\mathrm{m}\cdot \mathrm{n})=\mathrm{f}(\mathrm{m})\cdot \mathrm{f}(\mathrm{n})$ for every …

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

Let $R$ be a relation on $\mathbb{R}$, given by $\mathrm{R}=\{(\mathrm{a},\mathrm{b}):3\mathrm{a}-3\mathrm{b}+\sqrt{7}$ is an irrational number}. Then $R$ is

For some $\mathrm{a},\mathrm{b},\mathrm{c}\in \mathbb{N}$, let $\mathrm{f}(\mathrm{x})=\mathrm{ax}-3$ and
$g(x)=x^b+c,x\in \mathbb{R}$. If $(f\circ g{)}^{-1}(x)={\left(\frac{x-7}{2}\right)}^{1/3}$, then $(\mathrm{fg})(\mathrm{ac})+(\mathrm{gof})(\mathrm{b})$ is eq …

Let
$\mathrm{S}=\left\{(\mathrm{x},\mathrm{y})\in \mathbb{N}\times \mathbb{N}:9(\mathrm{x}-3{)}^2+16(\mathrm{y}-4{)}^2\le 144\right\}$
and $\mathrm{T}=\left\{(\mathrm{x},\mathrm{y})\in \mathbb{R}\times \mathbb{R}:(\mathrm{x}-7{)}^2+(\mathrm{y}-4{)}^2\le 36\right\}$.
Th …

For $\alpha \in \mathrm{N}$, consider a relation R on N given by $\mathrm{R}=\{(\mathrm{x},\mathrm{y}):3\mathrm{x}+\alpha \mathrm{y}$ is a multiple of 7}. The relation R is an equivalence relation if and only if :

Let $\alpha ,\beta$ and $\gamma$ be three positive real numbers. Let $\mathrm{f}(\mathrm{x})=\alpha {\mathrm{x}}^5+\beta {\mathrm{x}}^3+\gamma \mathrm{x},\mathrm{x}\in \mathrm{R}$ and $\mathrm{g}:\mathrm{R}\to \mathrm{R}$ be such that …

Let $R_1$ and $R_2$ be two relations defined on $\mathbb{R}$ by $a{\mathrm{R}}_{1}b\iff ab\ge 0$ and $a{R}_{2}b\iff a\ge b$, then

Let $f,g:\mathbb{N}-\{1\}\to \mathbb{N}$ be functions defined by $f(\mathrm{a})=\alpha$, where $\alpha$ is the maximum of the powers of those primes p such that $p^{\alpha }$ divides $a$, and $g(a)=a+1$, for all $a\in \mathbb{N}-\{1\}$. Then, the function …

The total number of functions, f : $\{1,2,3,4\}\to \{1,2,3,4,5,6\}$ such that $\mathrm{f}(1)+\mathrm{f}(2)=\mathrm{f}(3)$, is equal to :

Let $A=\{1,2,3,4,5,6,7\}$ and $B=\{3,6,7,9\}$. Then the number of elements in the set $\{C\subseteq A:C\cap B\ne \varphi \}$ is

The domain of the function
$f(x)={\sin }^{-1}\left[2x^2-3\right]+{\log }_2\left({\log }_{\frac{1}{2}}\left(x^2-5x+5\right)\right)$,
where $[t]$ is the greatest integer function, is :

The number of functions f, from the set $A=\left\{x\in N:x^2-10x+9\le 0\right\}$ to the set $B=\left\{n^2:n\in N\right\}$ such that $f(x)\le (x-3{)}^2+1$, for every $x\in A$, is ______.

Let $\mathrm{A}:\{1,2,3,4,5,6,7\}$. Define $\mathrm{B}=\{\mathrm{T}\subseteq \mathrm{A}$ : either $1\notin \mathrm{T}$ or $2\in \mathrm{T}\}$ and $\mathrm{C}=\mathrm{T}\subseteq \mathrm{A}:\mathrm{T}$ the sum of all the elements of T is a prime number}. Then the …

Let $\mathrm{f}:\mathrm{R}\to \mathrm{R}$ be a continuous function such that $\mathrm{f}(3\mathrm{x})-\mathrm{f}(\mathrm{x})=\mathrm{x}$. If $\mathrm{f}(8)=7$, then $\mathrm{f}(14)$ is equal to :

Let ${\mathrm{R}}_1=\{(\mathrm{a},\mathrm{b})\in \mathrm{N}\times \mathrm{N}:|\mathrm{a}-\mathrm{b}|\le 13\}$ and

Let $S=\{1,2,3,4,5,6,7,8,9,10\}$. Define $f:S\to S$ as $f(n)=\{\begin{array}{ll}2n,& \text{if }n=1,2,3,4,5\\ 2n-11& \text{if }n=6,7,8,9,10\end{array}$.

Let $g:S\to S$ be a function such that …

Let $[\mathrm{t}]$ denote the greatest integer $\le \mathrm{t}$ and $\{\mathrm{t}\}$ denote the fractional part of t. Then integral value of $\alpha$ for which the left hand limit of the function …

Let a function $\mathrm{f}:\mathbb{N}\to \mathbb{N}$ be defined by …

Let ${\mathrm{R}}_1$ and ${\mathrm{R}}_2$ be relations on the set $\{1,2,\dots ,50\}$ such that
${\mathrm{R}}_1=\{\left(\mathrm{p},{\mathrm{p}}^{\mathrm{n}}\right):\mathrm{p}$ is a prime and $\mathrm{n}\ge 0$ is an integer $\}$
and …

The probability that a relation R from $\{\mathrm{x},\mathrm{y}\}$ to $\{\mathrm{x},\mathrm{y}\}$ is both symmetric and transitive, is equal to:

Let $\mathrm{f}(\mathrm{x})$ and $\mathrm{g}(\mathrm{x})$ be two real polynomials of degree 2 and 1 respectively. If $\mathrm{f}(\mathrm{g}(\mathrm{x}))=8{\mathrm{x}}^2-2\mathrm{x}$, and $g(f(x))=4x^2+6x+1$, then the value of $f(2)+g(2)$ is ______.

Let $f:\mathbb{R}\to \mathbb{R}$ be defined as $f(x)=x-1$ and $\mathrm{g}:\mathbb{R}-\{1,-1\}\to \mathbb{R}$ be defined as $\mathrm{g}(\mathrm{x})=\frac{{\mathrm{x}}^2}{{\mathrm{x}}^2-1}$.

Then the function fog is :

Let $\mathrm{f}:\mathbb{R}\to \mathbb{R}$ satisfy $\mathrm{f}(\mathrm{x}+\mathrm{y})=2^{\mathrm{x}}\mathrm{f}(\mathrm{y})+4^{\mathrm{y}}\mathrm{f}(\mathrm{x}),\forall \mathrm{x}$, $y\in \mathbb{R}$. If $f(2)=3$, then …

Let a set $\mathrm{A}={\mathrm{A}}_1\cup {\mathrm{A}}_2\cup \dots \cup {\mathrm{A}}_k$, where $A_i\cap A_j=\varphi$ for $i\ne j,1\le i,j\le k$. Define the relation $R$ from $A$ to $A$ by …

Let $\mathrm{c},\mathrm{k}\in \mathrm{R}$. If $\mathrm{f}(\mathrm{x})=(\mathrm{c}+1){\mathrm{x}}^2+\left(1-{\mathrm{c}}^2\right)\mathrm{x}+2\mathrm{k}$ and $\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x})+\mathrm{f}(\mathrm{y})-\mathrm{xy}$, for all …