Sets, Relations & Functions · JEE Main previous year questions

If the function $f:(-\infty ,-1]\to (a,b]$ defined by $f(x)=e^{x^3-3x+1}$ is one - one and onto, then the distance of the point $P(2b+4,a+2)$ from the line $x+e^{-3}y=4$ is :

1. A.$2\sqrt{1+e^6}$
2. B.$\sqrt{1+e^6}$
3. C.$3\sqrt{1+e^6}$
4. D.$4\sqrt{1+e^6}$

Let $a,b,c$ be the lengths of three sides of a triangle satisfying the condition $\left(a^2+b^2\right)x^2-2b(a+c)x+\left(b^2+c^2\right)=0$. If the set of all possible values of $x$ is the interval $(\alpha ,\beta )$, then $12\left({\alpha }^2+{\beta }^2\right)$ is equal to ____.

Let $A=\{1,2,3,\dots \dots \dots \dots ,100\}$. Let $R$ be a relation on $\mathrm{A}$ defined by $(x,y)\in R$ if and only if $2x=3y$. Let $R_1$ be a symmetric relation on $A$ such that $R\subset R_1$ and the number of elements in $R_1$ is $\mathrm{n}$. Then, the minimum value of $\mathrm{n}$ is ____.

Let $g:\mathbb{R}\to \mathbb{R}$ be a non constant twice differentiable function such that $g^{\prime }\left(\frac{1}{2}\right)=g^{\prime }\left(\frac{3}{2}\right)$. If a real valued function $f$ is defined as $f(x)=\frac{1}{2}[g(x)+g(2-x)]$, then

1. A.$f^{\prime \prime }(x)=0$ for at least two $x$ in $(0,2)$
2. B.$f^{\prime }\left(\frac{3}{2}\right)+f^{\prime }\left(\frac{1}{2}\right)=1$
3. C.$f^{\prime \prime }(x)=0$ for no $x$ in $(0,1)$
4. D.$f^{\prime \prime }(x)=0$ for exactly one $x$ in $(0,1)$

If the domain of the function $f(x)={\cos }^{-1}\left(\frac{2-|x|}{4}\right)+\{{\log }_e(3-x){\}}^{-1}$ is $[-\alpha ,\beta )-\{\gamma \}$, then $\alpha +\beta +\gamma$ is equal to :

1. A.11
2. B.12
3. C.9
4. D.8

Let $A=\{1,2,3,\dots ,7\}$ and let $P(A)\text{ denote the power set of A. If the number of functions }f:A\to P(A)$ such that $a\in f(a),\forall a\in A$ is $m^n,m$ and $n\in \mathbf{N}$ and $m$ is least, then $m+n$ is equal to _________.

If the function $$f(x)=\{\begin{array}{ll}\frac{1}{|x|},& |x|\ge 2\\ a{x}^2+2b,& |x|<2\end{array}$$ is differentiable on $\mathbf{R}$, then $48(a+b)$ is equal to _________.

Let $R$ be a relation on $Z\times Z$ defined by $(a,b)R(c,d)$ if and only if $ad-bc$ is divisible by 5. Then $R$ is

1. A.Reflexive and transitive but not symmetric
2. B.Reflexive and symmetric but not transitive
3. C.Reflexive but neither symmetric nor transitive
4. D.Reflexive, symmetric and transitive

If $f(x)=\{\begin{array}{ll}2+2x,& -1\le x<0\\ 1-\frac{x}{3},& 0\le x\le 3\end{array}$; $g(x)=\{\begin{array}{ll}-x,& -3\le x\le 0\\ x,& 0<x\le 1\end{array}$, then range of $(f\circ g)(x)$ is

1. A.$[0,1)$
2. B.$[0,3)$
3. C.$(0,1]$
4. D.$[0,1]$

A function $y=f(x)$ satisfies $f(x)\sin 2x+\sin x-(1+{\cos }^{2}x)f^{\prime }(x)=0$ with condition $f(0)=0$. Then, $f\left(\frac{\pi }{2}\right)$ is equal to

1. A.$2$
2. B.$1$
3. C.$-1$
4. D.$0$

The function $f(x)=\frac{x}{x^2-6x-16},x\in \mathbb{R}-\{-2,8\}$

1. A.decreases in $(-\infty ,-2)\cup (-2,8)\cup (8,\infty )$
2. B.increases in $(-\infty ,-2)\cup (-2,8)\cup (8,\infty )$
3. C.decreases in $(-2,8)$ and increases in $(-\infty ,-2)\cup (8,\infty )$
4. D.decreases in $(-\infty ,-2)$ and increases in $(8,\infty )$

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

1. A.15
2. B.10
3. C.12
4. D.8

The function $f(x)=2x+3(x{)}^{\frac{2}{3}},x\in \mathbb{R}$, has

1. A.exactly one point of local minima and no point of local maxima
2. B.exactly one point of local maxima and exactly one point of local minima
3. C.exactly two points of local maxima and exactly one point of local minima
4. D.exactly one point of local maxima and no point of local minima

Let the set $C=\{(x,y)\mid x^2-2^y=2023,x,y\in \mathbb{N}\}$. Then ${\sum }_{(x,y)\in C}(x+y)$ is equal to _________.

Let $S=\{1,2,3,\dots ,10\}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $\mathrm{R}=\{(\mathrm{A},\mathrm{B}):\mathrm{A}\cap \mathrm{B}\ne \varphi ;\mathrm{A},\mathrm{B}\in \mathrm{M}\}$ is :

1. A.symmetric only
2. B.reflexive only
3. C.symmetric and reflexive only
4. D.symmetric and transitive only

Consider the function. $$f(x)=\{\begin{array}{ll}\frac{\mathrm{a}\left(7x-12-x^2\right)}{\mathrm{b}|x^2-7x+12|}& ,x<3\\ 2^{\frac{\sin (x-3)}{x-[x]}}& ,x>3\\ \mathrm{b}& ,x=3,\end{array}$$ where $[x]$ denotes the greatest integer less than or equal to $x$. If $\mathrm{S}$ denotes the set of all ordered pairs $(\mathrm{a},\mathrm{b})$ such that $f(x)$ is continuous at $x=3$, then the number of elements in $\mathrm{S}$ is :

1. A.Infinitely many
2. B.4
3. C.2
4. D.1

The function $f:\mathbf{N}-\{1\}\to \mathbf{N}$; defined by $f(\mathrm{n})=$ the highest prime factor of $\mathrm{n}$, is :

1. A.one-one only
2. B.neither one-one nor onto
3. C.onto only
4. D.both one-one and onto

Let for a differentiable function $f:(0,\infty )\to \mathbf{R}$, $f(x)-f(y)\ge {\log }_e\left(\frac{x}{y}\right)+x-y,\forall x,y\in (0,\infty )$. Then $\sum _{n=1}^{20}{f}^{\prime }\left(\frac{1}{n^2}\right)$ is equal to ____.

Consider the function $f:(0,2)\to \mathbf{R}$ defined by $f(x)=\frac{x}{2}+\frac{2}{x}$ and the function $g(x)$ defined by $g(x)=\{\begin{array}{ll}\min \{f(t)\},& 0<t\le x\text{ and }0<x\le 1\\ \frac{3}{2}+x,& 1<x<2\end{array}$. Then,

1. A.$g$ is continuous but not differentiable at $x=1$
2. B.$g$ is continuous and differentiable for all $x\in (0,2)$
3. C.$g$ is not continuous for all $x\in (0,2)$
4. D.$g$ is neither continuous nor differentiable at $x=1$

Let $A$ and $B$ be two finite sets with $m$ and $n$ elements respectively. The total number of subsets of the set $A$ is 56 more than the total number of subsets of $B$. Then the distance of the point $P(m,n)$ from the point $Q(-2,-3)$ is :

1. A.$8$
2. B.$10$
3. C.$4$
4. D.$6$

Let $f:\mathbf{R}-\{-\frac{1}{2}\}\to \mathbf{R}$ and $g:\mathbf{R}-\{-\frac{5}{2}\}\to \mathbf{R}$ be defined as $f(x)=\frac{2x+3}{2x+1}$ and $g(x)=\frac{|x|+1}{2x+5}$. Then, the domain of the function $f\circ g$ is :

1. A.$\mathbf{R}-\{-\frac{7}{4}\}$
2. B.$\mathbf{R}$
3. C.$\mathbf{R}-\{-\frac{5}{2},-\frac{7}{4}\}$
4. D.$\mathbf{R}-\{-\frac{5}{2}\}$

Let $\text{R}$ be the interior region between the lines $3x-y+1=0$ and $x+2y-5=0$ containing the origin. The set of all values of $a$, for which the points $(a^2,a+1)$ lie in $R$, is :

1. A.$(-3,0)\cup (\frac{2}{3},1)$
2. B.$(-3,0)\cup (\frac{1}{3},1)$
3. C.$(-3,-1)\cup (\frac{1}{3},1)$
4. D.$(-3,-1)\cup (-\frac{1}{3},1)$

Let $f(x)={\int }_0^{x}g(t){\log }_e\left(\frac{1-t}{1+t}\right)\mathrm{d}t$, where $g$ is a continuous odd function. If ${\int }_{-\pi /2}^{\pi /2}\left(f(x)+\frac{x^2\cos x}{1+e^x}\right)\mathrm{d}x={\left(\frac{\pi }{\alpha }\right)}^2-\alpha$, then $\alpha$ is equal to _______.

The set of all $a\in \mathbb{R}$ for which the equation $x|x-1|+|x+2|+a=0$ has exactly one real root, is :

1. A.$(-\infty ,\infty )$
2. B.$(-6,\infty )$
3. C.$(-\infty ,-3)$
4. D.$(-6,-3)$

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

1. A.$5$
2. B.$0$
3. C.$-3$
4. D.$1$