Sets Relations and Functions — JEE Main PYQs

Let $f(x)=\frac{x-1}{x+1},x\in R-\{0,-1,1)$. If $f^{n+1}(x)=f\left(f^n(x)\right)$ for all $\mathrm{n}\in \mathrm{N}$, then ${\mathrm{f}}^6(6)+{\mathrm{f}}^7(7)$ is equal to:

Let $A=\{n\in \mathbb{N}:\text{H.C.F.}(n,45)=1\}$ and $B=\{2k:k\in \{1,2,\dots ,100\}\}$. Then the sum of all the elements of $A\cap B$ is ____.

Let R be a relation from the set $\{1,2,3\dots \dots \dots ,60\}$ to itself such that $\mathrm{R}=\{(\mathrm{a},\mathrm{b}):\mathrm{b}=\mathrm{pq}$, where $\mathrm{p},\mathrm{q}\ge 3$ are prime numbers $\}$. Then, the number of elements in R is :

Let $\mathrm{f}:\mathrm{N}\to \mathrm{R}$ be a function such that $\mathrm{f}(\mathrm{x}+\mathrm{y})=2\mathrm{f}(\mathrm{x})\mathrm{f}(\mathrm{y})$ for natural numbers x and y . If $f(1)=2$, then the value of $\alpha$ for which …

Out of all the patients in a hospital $89\%$ are found to be suffering from heart ailment and $98\%$ are suffering from lungs infection. If $\mathrm{K}\%$ of them are suffering from both ailments, then K can not belong to the set:

Let $f:R\to R$ be defined as $f(x)=2x-1$ and $g:R-\{1\}\to R$ be defined as $g(x)=\frac{x-\frac{1}{2}}{x-1}$. Then the composition function $\mathrm{f}(\mathrm{g}(x))$ is :

Let $A=\{n\in N:n\text{ is a 3-digit number}\}$
$B=\{9k+2:k\in N\}$
And $C=\{9k+\ell :k\in N\}$ for some $\ell (0<\ell <9)$
If the sum of all the elements of the set $A\cap (B\cup C)$ is $274\times 400$, then $\ell$ is equal to ______.

If $a+\alpha =1,b+\beta =2$ and $af(x)+\alpha f\left(\frac{1}{x}\right)=bx+\frac{\beta }{x},x\ne 0$, then the value of the equation $\frac{f(x)+f\left(\frac{1}{x}\right)}{x+\frac{1}{x}}$ is. ____

Let $f:R-\left\{\frac{\alpha }{6}\right\}\to R$ be defined by $f(x)=\frac{5x+3}{6x-\alpha }$. Then the value of $\alpha$ for which (fof) $(x)=x$, for all $x\in R-\left\{\frac{\alpha }{6}\right\}$, is :

Let $Z$, be the set of all integers,
$A=\left\{(x,y)\in Z\times Z:(x-2{)}^2+y^2\le 4\right\}$
$B=\left\{(x,y)\in Z\times Z:x^2+y^2\le 4\right\}$
$C=\left\{(x,y)\in Z\times Z:(x-2{)}^2+(y-2{)}^2\le 4\right\}$
If the total number of relations from $A\cap B$ t …

Let $f,g:\mathbb{N}\to \mathbb{N}$ such that $f(n+1)=f(n)+f(1)\forall n\in \mathbb{N}$ and $g$ be any arbitrary function. Which of the following statements is NOT true ?

Let $[x]$ denote the greatest integer less than or equal to $x$. Then, the values of $x\in R$ satisfying the equation ${\left[e^x\right]}^2+\left[e^x+1\right]-3=0$ lie in the interval :

Let $A=\{0,1,2,3,4,5,6,7\}$. Then the number of bijective functions $f:A\to A$ such that $f(1)+f(2)=3-f(3)$ is equal to ____

Let $R=\{(P,Q)\mid P$ and $Q$ are at the same distance from the origin} be a relation, then the equivalence class of $(1,-1)$ is the set :

If $A=\{x\in R;|x-2|>1\},B=\left\{x\in R:\sqrt{x^2-3}>1\right\},C=\{x\in R;|x-4|\ge 2\}$ and $Z$ is the set of all integers, then the number of subsets of the set $(A\cap B\cap C{)}^C\cap Z$ is

Let $f(x)={\sin }^{-1}x$ and $g(x)=\frac{x^2-x-2}{2x^2-x-6}$. If $g(2)={\lim }_{x\to 2}g(x)$, then the domain of the function fog is:

Let $A=\{1,2,3,\dots ,10\}$ and $f:A\to A$ be defined as
$f(k)=\{\begin{array}{cc}k+1& \text{ if }k\text{ is odd }\\ k& \text{ if }k\text{ is even }\end{array}$
Then the number of possible functions $g:A\to A$ such that $g$ o …

Let $f:N\to N$ be a function such that $f(m+n)=f(m)+f(n)$ for every $m,n\in N$. If $f(6)=18$, then $f(2).f(3)$ is equal to :

Which of the following is not correct for relation on the set of real numbers?

Let $S=\{1,2,3,4,5,6,7\}$. Then the number of possible function $f:S\to S$ such that $f(m\cdot n)=f(m)\cdot f(n)$ for every $m,n\in S$ and $m\cdot n\in S$ is equal to ____.

The inverse of $y=5^{\log x}$ is:

In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement?

Let $\mathrm{g}:\mathrm{N}\to \mathrm{N}$ be defined as


Then which of the following statements is true?

If $[x]$ be the greatest integer less than or equal to $x$, then ${\sum }_{n=8}^{100}\left[\frac{(-1{)}^{n}n}{2}\right]$ is equal to :

Consider functions $f:A\to B$ and $g:B\to C(A,B,C\subseteq R)$ such that $(g\circ f{)}^{-1}$ exists, then :

Let $N$ be the set of natural numbers and a relation $R$ on $N$ be defined by $R=\left\{(x,y)\in N\times N:x^3-3x^{2}y-x{y}^2+3y^3=0\right\}$. Then the relation $R$ is :

Let $f:R\to R$ be defined as $f(x+y)+f(x-y)=2f(x)f(y),f\left(\frac{1}{2}\right)=-1$. Then, the value of ${\sum }_{k=1}^{20}\frac{1}{\sin (k)\sin (k+f(k))}$ is equal to :

Let $A=\left\{n\in N\mid n^2\le n+10,000\right\},B=\{3k+1\mid k\in N\}$ and $C=\{2k\mid k\in N\}$, then the sum of all the elements of the set $A\cap (B-C)$ is equal to ______

The range of the function
$f(x)={\log }_{\sqrt{5}}\left(3+\cos \left(\frac{3\pi }{4}+x\right)+\cos \left(\frac{\pi }{4}+x\right)+\cos \left(\frac{\pi }{4}-x\right)-\cos \left(\frac{3\pi }{4}-x\right)\right)$ is:

Let $A=\{2,3,4,5,\dots ,30\}$ and ' $\sim$ ' be an equivalence relation on $A\times A$, defined by $(a,b)\sim (c,d)$, if and only if $ad=bc$. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $(4,3)$ is equal to :

Define a relation $R$ over a class of $n\times n$ real matrices $A$ and $B$ as "ARB iff there exists a nonsingular matrix $P$ such that $PA{P}^{-1}=B$". Then which of the following is true?

Let $f:R-\{3\}\to R-\{1\}$ be defined by $f(x)=\frac{x-2}{x-3}$.
Let $g:R\to R$ be given as $g(x)=2x-3$. Then, the sum of all the values of $x$ for which $f^{-1}(x)+g^{-1}(x)=\frac{13}{2}$ is equal to :

The real function $f(x)=\frac{{\mathrm{cosec}}^{-1}x}{\sqrt{x-[x]}}$, where $[x]$ denotes the greatest integer less than or equal to $x$, is defined for all $x$ belonging to :

If the functions are defined as $\mathrm{f}(\mathrm{x})=\sqrt{\mathrm{x}}$ and $\mathrm{g}(\mathrm{x})=\sqrt{1-\mathrm{x}}$, then what is the common domain of the following functions: $f+g,f-g,f/g,g/f,g-f$ where …

If $g(x)=x^2+x-1$ and $(g\circ f)(x)=4x^2-10x+5$, then $f\left(\frac{5}{4}\right)$ is equal to

Let $X=\{n\in N:1\le n\le 50\}$. If $A=\{n\in X:n$ is a multiple of 2$\}$ and $B=\{n\in X,n$ is a multiple of 7$\}$, then the number of elements in the smallest subset of $X$ containing both $A$ and $B$ is ____

The inverse function of $f(x)=\frac{8^{2x}-8^{-2x}}{8^{2x}+8^{-2x}},x\in (-1,1)$, is

Let $f:(1,3)\to R$ be a function defined by $f(x)=\frac{x[x]}{1+x^2}$, where $[x]$ denotes the greatest integer $\le \mathrm{X}$. Then the range of f is:

Let $A=(a,b,c)$ and $B=(1,2,3,4)$. Then the number of elements in the set $C=\{f:A\to B\mid 2\in f(A)$ and $f$ is not one-one$\}$ is ____.

Let ${\mathrm{R}}_1$ and ${\mathrm{R}}_2$ be two relations defined as follows:
${\mathrm{R}}_1=\left\{(\mathrm{a},\mathrm{b})\in {\mathrm{R}}^2:{\mathrm{a}}^2+{\mathrm{b}}^2\in \mathrm{Q}\right\}$ and
$R_2=\left\{(a,b)\in R^2:a^2+b^2\notin Q\right\}$, wher …

For a suitably chosen real constant $a$, let a function, $f:\mathbb{R}-\{-a\}\to \mathbb{R}$ be defined by $f(x)=\frac{a-x}{a+x}$. Further suppose that for any real number $x\ne -a$ and $f(x)\ne -a$, $(f\circ f)(x)=x$. Then $f\left(-\frac{1}{2}\right)$ is equal to:

A survey shows that $73\%$ of the persons working in an office like coffee, whereas $65\%$ like tea. If $x$ denotes the percentage of them, who like both coffee and tea, then $x$ cannot be:

Set A has m elements and Set B has n elements. If the total number of subsets of A is 112 more than the total number of subsets of $B$, then the value of $m.n$ is ______.

Let $[t]$ denote the greatest integer $\le t$. Then the equation in $x,[x{]}^2+2[x+2]-7=0$ has:

A survey shows that $63\%$ of the people in a city read newspaper A whereas $76\%$ read newspaper B. if $\mathrm{x}\%$ of the people read both the newspapers, then a possible value of x can be:

Let ${\bigcup }_{i=1}^{50}{X}_i={\bigcup }_{i=1}^n{Y}_i=T$, where each $X_i$ contains 10 elements and each $Yi$ contains 5 elements. If each element of the set T is an element of exactly 20 of sets ${\mathrm{X}}_{\mathrm{i}}$'s and exactly 6 of sets $Y_i$'s then n is e …

If $f(x)={\log }_e\left(\frac{1-x}{1+x}\right),|x|<1$, then $f\left(\frac{2x}{1+x^2}\right)$ is equal to:

Let $f(x)=a^x(a>0)$ be written as $f(x)=f_1(x)+f_2(x)$, where $f_1(x)$ is an even function and $f_2(x)$ is an odd function. Then $f_1(x+y)+f_1(x-y)$ equals

If the function $f:R-\{1,-1\}\to A$ defined by $f(x)=\frac{x^2}{1-x^2}$, is surjective, then $A$ is equal to:

In a class of 140 students numbered 1 to 140 , all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of th …