Sets, Relations & Functions · JEE Main previous year questions

The number of elements in the set $\{n\in \{1,2,3,\dots ,100\}\mid (11{)}^n>(10{)}^n+(9{)}^n\}$ is ________.

Let [ x ] denote the greatest integer $\le x$, where $x\in \mathrm{R}$. If the domain of the real valued function $f(x)=\sqrt{\frac{|[x]|-2}{|[x]|-3}}$ is $(-\infty ,a)\cup [b,c)\cup [4,\infty )$, $a<b<c$, then the value of a + b + c is :

1. A.8
2. B.1
3. C.$-2$
4. D.$-3$

Let 'a' be a real number such that the function $f(x)=a{x}^2+6x-15,x\in \mathrm{R}$ is increasing in $\left(-\infty ,\frac{3}{4}\right)$ and decreasing in $\left(\frac{3}{4},\infty \right)$. Then the function $g(x)=a{x}^2-6x+15,x\in \mathrm{R}$ has a :

1. A.local maximum at $x=-\frac{3}{4}$
2. B.local minimum at $x=-\frac{3}{4}$
3. C.local maximum at $x=\frac{3}{4}$
4. D.local minimum at $x=\frac{3}{4}$

The real valued 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 :

1. A.all real except integers
2. B.all non-integers except the interval $[-1,1]$
3. C.all integers except $0,-1,1$
4. D.all real except the interval $[-1,1]$

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

1. A.$0\le x\le 1$
2. B.$0\le x<1$
3. C.$0<x<1$
4. D.$0<x\le 1$

The number of elements in the set $\{x\in \mathbb{R}:(|x|-3)|x+4|=6\}$ is equal to :

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

Let the functions $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ be defined as : $f(x)=\{\begin{array}{ll}x+2,& x<0\\ x^2,& x\ge 0\end{array}$ and $g(x)=\{\begin{array}{ll}{x}^3,& x<1\\ 3x-2,& x\ge 1\end{array}$ Then, the number of points in $\mathbb{R}$ where $(f\circ g)(x)$ is NOT differentiable is equal to :

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

The range of $a\in \mathbb{R}$ for which the function $f(x)=(4a-3)(x+{\log }_{e}5)+2(a-7)\cot \left(\frac{x}{2}\right){\sin }^2\left(\frac{x}{2}\right)$, $x\ne 2n\pi ,n\in \mathbb{N}$ has critical points, is :

1. A.$[1,\infty )$
2. B.$(-3,1)$
3. C.$\left[-\frac{4}{3},2\right]$
4. D.$(-\infty ,-1]$

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

1. A.$S=\{(x,y)\mid x^2+y^2=\sqrt{2}\}$
2. B.$S=\{(x,y)\mid x^2+y^2=2\}$
3. C.$S=\{(x,y)\mid x^2+y^2=1\}$
4. D.$S=\{(x,y)\mid x^2+y^2=4\}$

Let f be any function defined on R and let it satisfy the condition : $|f(x)-f(y)|\le |(x-y{)}^2|,\forall (x,y)\in R$ If $f(0)=1$, then :

1. A.f(x) can take any value in R
2. B.$f(x)<0,\forall x\in R$
3. C.$f(x)>0,\forall x\in R$
4. D.$f(x)=0,\forall x\in R$

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

1. A.$5^5$
2. B.$10^5$
3. C.$5!$
4. D.${}^{10}{\mathrm{C}}_5$

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 :

1. A.$(-\infty ,-2]\cup \left[-\frac{4}{3},\infty \right)$
2. B.$(-\infty ,-2]\cup [-1,\infty )$
3. C.$(-\infty ,-2]\cup \left[-\frac{3}{2},\infty \right)$
4. D.$(-\infty ,-1]\cup [2,\infty )$

Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one-one functions form the set A to the set $A\times B$. Then :

1. A.$2y=273x$
2. B.$y=91x$
3. C.$2y=91x$
4. D.$y=273x$

A function f is defined on $[-3,3]$ as $$f(x)=\{\begin{array}{ll}\min \{|x|,2-x^2\},& -2\le x\le 2\\ [|x|],& 2<|x|\le 3\end{array}$$ where [x] denotes the greatest integer $\le$ x. The number of points, where f is not differentiable in $(-3,3)$ is ______ .

If $f(x+y)=f(x)f(y)$ and ${\sum }_{x=1}^{\infty }f(x)=2$, $x,y\in \mathbb{N}$, where $\mathbb{N}$ is the set of all natural number, then the value of $\frac{f(4)}{f(2)}$ is :

1. A.$\frac{2}{3}$
2. B.$\frac{1}{9}$
3. C.$\frac{1}{3}$
4. D.$\frac{4}{9}$

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 $f:\mathbb{R}\to \mathbb{R}$ be a function defined by $f(x)=\max \{x,x^2\}$. Let S denote the set of all points in $\mathbb{R}$, where f is not differentiable. Then S is equal to :

1. A.$\{0,1\}$
2. B.$\{0\}$
3. C.$\varphi$(an empty set)
4. D.$\{1\}$

The set of all real values of $\lambda$ for which the function $$f(x)=(1-{\cos }^{2}x)(\lambda +\sin x),x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$$ has exactly one maxima and exactly one minima, is :

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

For all twice differentiable functions $f:\mathbb{R}\to \mathbb{R}$, with $f(0)=f(1)=f^{\prime }(0)=0$

1. A.$f^{\prime \prime }(x)\ne 0$, at every point $x\in (0,1)$
2. B.$f^{\prime \prime }(x)=0$, for some $x\in (0,1)$
3. C.$f^{\prime \prime }(0)=0$
4. D.$f^{\prime \prime }(x)=0$, at every point $x\in (0,1)$

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 :

1. A.$\frac{1}{3}$
2. B.$-3$
3. C.$-\frac{1}{3}$
4. D.3

Suppose that a function $f:\mathbb{R}\to \mathbb{R}$ satisfies $f(x+y)=f(x)f(y)$ for all $x,y\in \mathbb{R}$ and $f(1)=3$. If ${\sum }_{i=1}^{n}f(i)=363$ then n is equal to ______ .

Consider the two sets : $A=\{m\in \mathbb{R}:\text{both the roots of }{x}^2-(m+1)x+m+4=0\text{ are real}\}$ and $B=[-3,5)$. Which of the following is not true?

1. A.$A\cap B=\{-3\}$
2. B.$B-A=(-3,5)$
3. C.$A\cup B=\mathbb{R}$
4. D.$A-B=(-\infty ,-3)\cup (5,\infty )$

The function, $f(x)=(3x-7)x^{2/3},x\in \mathbb{R}$, is increasing for all $x$ lying in :

1. A.$(-\infty ,0)\cup \left(\frac{3}{7},\infty \right)$
2. B.$(-\infty ,0)\cup \left(\frac{14}{15},\infty \right)$
3. C.$\left(-\infty ,\frac{14}{15}\right)$
4. D.$\left(-\infty ,\frac{14}{15}\right)\cup (0,\infty )$

Let ${\mathrm{R}}_1$ and ${\mathrm{R}}_2$ be two relation defined as follows : ${\mathrm{R}}_1=\{(a,b)\in {\mathbb{R}}^2:a^2+b^2\in \mathbb{Q}\}$ and ${\mathrm{R}}_2=\{(a,b)\in {\mathbb{R}}^2:a^2+b^2\notin \mathbb{Q}\}$, where $\mathbb{Q}$ is the set of all rational numbers. Then :

1. A.Neither ${\mathrm{R}}_1$ nor ${\mathrm{R}}_2$ is transitive.
2. B.${\mathrm{R}}_2$ is transitive but ${\mathrm{R}}_1$ is not transitive.
3. C.${\mathrm{R}}_1$ and ${\mathrm{R}}_2$ are both transitive.
4. D.${\mathrm{R}}_1$ is transitive but ${\mathrm{R}}_2$ is not transitive.

Let S be the set of all $\lambda \in \mathbb{R}$ for which the system of linear equations $2x-y+2z=2$ $x-2y+\lambda z=-4$ $x+\lambda y+z=4$ has no solution. Then the set S :

1. A.contains more than two elements.
2. B.contains exactly two elements.
3. C.is a singleton.
4. D.is an empty set.