Sets, Relations & Functions · JEE Main previous year questions

The domain of the definition of the function $f(x)=\frac{1}{4-x^2}+{\log }_{10}(x^3-x)$ is

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

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

1. A.$2f_1(x)f_1(y)$
2. B.$2f_1(x+y)f_1(x-y)$
3. C.$2f_1(x)f_2(y)$
4. D.$2f_1(x+y)f_2(x-y)$

Let $f(x)={\int }_0^{x}g(t)dt$ where g is a non-zero even function. If $f(x+5)=g(x)$, then ${\int }_0^{x}f(t)dt$ equals-

1. A.$5{\int }_{x+5}^{5}g(t)dt$
2. B.${\int }_{x+5}^{5}g(t)dt$
3. C.${\int }_5^{x+5}g(t)dt$
4. D.$2{\int }_5^{x+5}g(t)dt$

Let $S=\{1,2,3,\dots ,100\}$. The number of non-empty subsets A of S such that the product of elements in A is even is :

1. A.$2^{50}-1$
2. B.$2^{50}(2^{50}-1)$
3. C.$2^{100}-1$
4. D.$2^{50}+1$

Considering only the principal values of inverse functions, the set $A=\{x\ge 0:{\tan }^{-1}(2x)+{\tan }^{-1}(3x)=\frac{\pi }{4}\}$

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

Let S be the set of all points in $(-\pi ,\pi )$ at which the function, $f(x)=\min \{\sin x,\cos x\}$ is not differentiable. Then S is a subset of which of the following ?

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

Let Z be the set of integers. If $A=\{x\in Z:2^{(x+2)(x^2-5x+6)}=1\}$ and $B=\{x\in Z:-3<2x-1<9\}$, then the number of subsets of the set $A\times B$, is

1. A.$2^{12}$
2. B.$2^{18}$
3. C.$2^{10}$
4. D.$2^{15}$

If the function f given by $f(x)=x^3-3(a-2)x^2+3ax+7$, for some $a\in \mathbb{R}$ is increasing in $(0,1]$ and decreasing in $[1,5)$, then a root of the equation, $\frac{f(x)-14}{(x-1{)}^2}=0(x\ne 1)$ is :

1. A.$-7$
2. B.5
3. C.7
4. D.6

Let f be a differentiable function such that $f(1)=2$ and $f^{\prime }(x)=f(x)$ for all $x\in \mathbb{R}$. If $h(x)=f(f(x))$, then $h^{\prime }(1)$ is equal to :

1. A.$4e$
2. B.$2e^2$
3. C.$4e^2$
4. D.$2e$

The set of all values of $\lambda$ for which the system of linear equations $x-2y-2z=\lambda x$ $x+2y+z=\lambda y$ $-x-y=\lambda z$ has a non-trivial solution :

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

Let K be the set of all real values of x where the function $f(x)=\sin |x|-|x|+2(x-\pi )\cos |x|$ is not differentiable. Then the set K is equal to :

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

The number of functions f from $\{1,2,3,\dots ,20\}$ onto $\{1,2,3,\dots ,20\}$ such that $f(k)$ is a multiple of 3, whenever k is a multiple of 4, is :

1. A.$6^5\times (15)!$
2. B.$5^6\times 15$
3. C.$(15)!\times 6!$
4. D.$5!\times 6!$

Let a function $f:(0,\infty )\to (0,\infty )$ be defined by $f(x)=\left|1-\frac{1}{x}\right|$. Then f is :

1. A.not injective but it is surjective
2. B.neither injective nor surjective
3. C.injective only
4. D.both injective as well as surjective

Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined as $$f(x)=\{\begin{array}{ll}5& ;x\le 1\\ a+bx& ;1<x<3\\ b+5x& ;3\le x<5\\ 30& ;x\ge 5\end{array}$$ Then, f is

1. A.continuous if $a=0$ and $b=5$
2. B.continuous if $a=-5$ and $b=10$
3. C.continuous if $a=5$ and $b=5$
4. D.not continuous for any values of a and b

For $x\in \mathbb{R}-\{0,1\}$, Let $f_1(x)=\frac{1}{x}$, $f_2(x)=1-x$ and $f_3(x)=\frac{1}{1-x}$ be three given functions. If a function, $J(x)$ satisfies $(f_2\circ J\circ f_1)(x)=f_3(x)$ then $J(x)$ is equal to :

1. A.$f_1(x)$
2. B.$\frac{1}{x}{f}_3(x)$
3. C.$f_2(x)$
4. D.$f_3(x)$

Let $\mathbf{N}$ denote the set of all natural numbers. Define two binary relations on $\mathbf{N}$ as $R_1=\{(x,y)\in \mathbf{N}\times \mathbf{N}:2x+y=10\}$ and $R_2=\{(x,y)\in \mathbf{N}\times \mathbf{N}:x+2y=10\}$. Then :

1. A.Range of $R_1$ is $\{2,4,8\}$.
2. B.Range of $R_2$ is $\{1,2,3,4\}$.
3. C.Both $R_1$ and $R_2$ are symmetric relations.
4. D.Both $R_1$ and $R_2$ are transitive relations.

Let M and m be respectively the absolute maximum and the absolute minimum values of the function, $f(x)=2x^3-9x^2+12x+5$ in the interval [0, 3]. Then M $-$ m is equal to :

1. A.5
2. B.9
3. C.4
4. D.1

Consider the following two binary relations on the set $A=\{a,b,c\}$ : $R_1=\{(c,a),(b,b),(a,c),(c,c),(b,c),(a,a)\}$ and $R_2=\{(a,b),(b,a),(c,c),(c,a),(a,a),(b,b),(a,c)\}$. Then :

1. A.both $R_1$ and $R_2$ are not symmetric.
2. B.$R_1$ is not symmetric but it is transitive.
3. C.$R_2$ is symmetric but it is not transitive.
4. D.both $R_1$ and $R_2$ are transitive.

The set of all $\alpha \in \mathbf{R}$, for which $w=\frac{1+(1-8\alpha )z}{1-z}$ is purely imaginary number, for all $z\in \mathbf{C}$ satisfying $|z|=1$ and $\text{Re }z\ne 1$, is :

1. A.an empty set
2. B.$\{0\}$
3. C.$\left\{0,\frac{1}{4},-\frac{1}{4}\right\}$
4. D.equal to $\mathbf{R}$

Let S be the set of all real values of k for which the system of linear equations $x+y+z=2$ $2x+y-z=3$ $3x+2y+kz=4$ has a unique solution. Then S is :

1. A.an empty set
2. B.equal to $\{0\}$
3. C.equal to $\mathbf{R}$
4. D.equal to $\mathbf{R}-\{0\}$

Let $S=\{(\lambda ,\mu )\in \mathbf{R}\times \mathbf{R}:f(t)=(|\lambda |e^{|t|}-\mu ).\sin (2|t|),t\in \mathbf{R}\text{, is a differentiable function}\}$. Then S is a subset of :

1. A.$\mathbf{R}\times [0,\infty )$
2. B.$[0,\infty )\times \mathbf{R}$
3. C.$\mathbf{R}\times (-\infty ,0)$
4. D.$(-\infty ,0)\times \mathbf{R}$

Two sets A and B are as under : $\mathrm{A}=\{(a,b)\in \mathrm{R}\times \mathrm{R}:|a-5|<1\text{ and }|b-5|<1\}$; $\mathrm{B}=\{(a,b)\in \mathrm{R}\times \mathrm{R}:4(a-6{)}^2+9(b-5{)}^2\le 36\}$; Then

1. A.neither $\mathrm{A}\subset \mathrm{B}$ nor $\mathrm{B}\subset \mathrm{A}$
2. B.$\mathrm{B}\subset \mathrm{A}$
3. C.$\mathrm{A}\subset \mathrm{B}$
4. D.$\mathrm{A}\cap \mathrm{B}=\varphi$ ( an empty set )

The function $f:\mathbf{N}\to \mathbf{N}$ defined by $f(x)=x-5\left[\frac{x}{5}\right]$, Where $\mathbf{N}$ is the set of natural numbers and $[x]$ denotes the greatest integer less than or equal to x, is :

1. A.one-one and onto
2. B.one-one but not onto.
3. C.onto but not one-one.
4. D.neither one-one nor onto.

The function f defined by $f(x)=x^3-3x^2+5x+7$ , is :

1. A.increasing in $\mathbf{R}$.
2. B.decreasing in $\mathbf{R}$.
3. C.decreasing in $(0,\infty )$ and increasing in $(-\infty ,0)$
4. D.increasing in $(0,\infty )$ and decreasing in $(-\infty ,0)$

Let f be a polynomial function such that $f(3x)=f{}^{\prime }(x)\cdot f{}^{\prime \prime }(x)$, for all $x\in \mathbf{R}$. Then :

1. A.$f(2)+f{}^{\prime }(2)=28$
2. B.$f{}^{\prime \prime }(2)-f{}^{\prime }(2)=0$
3. C.$f{}^{\prime \prime }(2)-f(2)=4$
4. D.$f(2)-f{}^{\prime }(2)+f{}^{\prime \prime }(2)=10$