Sets, Relations & Functions · JEE Main previous year questions

If $\mathrm{R}=\{(x,y):x,y\in \mathbb{Z},x^2+3y^2\le 8\}$ is a relation on the set of integers $\mathbb{Z}$, then the domain of ${\mathrm{R}}^{-1}$ is :

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

The domain of the function $f(x)={\sin }^{-1}\left(\frac{|x|+5}{x^2+1}\right)$ is $(-\infty ,-a]\cup [a,\infty )$. Then a is equal to :

1. A.$\frac{\sqrt{17}-1}{2}$
2. B.$\frac{1+\sqrt{17}}{2}$
3. C.$\frac{\sqrt{17}}{2}+1$
4. D.$\frac{\sqrt{17}}{2}$

Let $f:\mathbb{R}\to \mathbb{R}$ be a function which satisfies $f(x+y)=f(x)+f(y)\forall x,y\in \mathbb{R}$. If $f(1)=2$ and $g(n)={\sum }_{k=1}^{(n-1)}f(k),n\in \mathbb{N}$ then the value of $n$, for which $g(n)=20$, is :

1. A.20
2. B.9
3. C.5
4. D.4

Let $f:(-1,\infty )\to \mathbb{R}$ be defined by $f(0)=1$ and $f(x)=\frac{1}{x}{\log }_e(1+x),x\ne 0$. Then the function $f$ :

1. A.decreases in $(-1,\infty )$
2. B.decreases in $(-1,0)$ and increases in $(0,\infty )$
3. C.increases in $(-1,\infty )$
4. D.increases in $(-1,0)$ and decreases in $(0,\infty )$

Let $A=\{X=(x,y,z{)}^T:PX=0\text{ and }{x}^2+y^2+z^2=1\}$ where $$P=\left[\begin{array}{ccc}1& 2& 1\\ -2& 3& -4\\ 1& 9& -1\end{array}\right],$$ then the set $A$ :

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

Let f and g be differentiable functions on $\mathbb{R}$ such that $f\circ g$ is the identity function. If for some $a,b\in \mathbb{R}$, $g^{\prime }(a)=5$ and $g(a)=b$, then $f^{\prime }(b)$ is equal to :

1. A.1
2. B.5
3. C.$\frac{2}{5}$
4. D.$\frac{1}{5}$

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

1. A.$\frac{1}{4}{\log }_e\left(\frac{1-x}{1+x}\right)$
2. B.$\frac{1}{4}({\log }_{8}e){\log }_e\left(\frac{1-x}{1+x}\right)$
3. C.$\frac{1}{4}({\log }_{8}e){\log }_e\left(\frac{1+x}{1-x}\right)$
4. D.$\frac{1}{4}{\log }_e\left(\frac{1+x}{1-x}\right)$

If c is a point at which Rolle's theorem holds for the function, $f(x)={\log }_e\left(\frac{x^2+\alpha }{7x}\right)$ in the interval $[3,4]$, where $a\in \mathrm{R}$, then $f^{\prime \prime }(c)$ is equal to

1. A.$\frac{1}{12}$
2. B.$\frac{\sqrt{3}}{7}$
3. C.$-\frac{1}{12}$
4. D.$-\frac{1}{24}$

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

1. A.$\left(\frac{2}{5},\frac{1}{2}\right)\cup \left(\frac{3}{5},\frac{4}{5}\right]$
2. B.$\left(\frac{3}{5},\frac{4}{5}\right)$
3. C.$\left(\frac{2}{5},\frac{4}{5}\right]$
4. D.$\left(\frac{2}{5},\frac{3}{5}\right]\cup \left(\frac{3}{4},\frac{4}{5}\right)$

Let S be the set of all real roots of the equation, $3^x(3^x-1)+2=|3^x-1|+|3^x-2|$. Then S :

1. A.contains exactly two elements.
2. B.is an empty set.
3. C.is a singleton.
4. D.contains at least four elements.

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:

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

Let S be the set of points where the function, $f(x)=|2-|x-3||$, $x\in \mathbb{R}$ is not differentiable. Then ${\sum }_{x\in S}f(f(x))$ is equal to _____.

Let $y=y(x)$ be a function of $x$ satisfying $y\sqrt{1-x^2}=k-x\sqrt{1-y^2}$ where $k$ is a constant and $y\left(\frac{1}{2}\right)=-\frac{1}{4}$. Then $\frac{dy}{dx}$ at $x=\frac{1}{2}$, is equal to :

1. A.$\frac{2}{\sqrt{5}}$
2. B.$-\frac{\sqrt{5}}{2}$
3. C.$\frac{\sqrt{5}}{2}$
4. D.$-\frac{\sqrt{5}}{4}$

The value of c in the Lagrange's mean value theorem for the function $f(x)=x^3-4x^2+8x+11$, when $x\in [0,1]$ is:

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

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

Let S be the set of all $\alpha \in \mathbb{R}$ such that the equation, $\cos 2x+\alpha \sin x=2\alpha -7$ has a solution. Then S is equal to :

1. A.$[2,6]$
2. B.$[3,7]$
3. C.$[1,4]$
4. D.$\mathbb{R}$

Let $f(x)=e^x-x$ and $g(x)=x^2-x$, $\forall x\in \mathrm{R}$. Then the set of all $x\in \mathrm{R}$, where the function $h(x)=(f\circ g)(x)$ is increasing, is :

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

Let $f(x)={\log }_e(\sin x)$, $(0<x<\pi )$ and $g(x)={\sin }^{-1}(e^{-x})$, $(x\ge 0)$. If $\alpha$ is a positive real number such that a = (fog)'($\alpha$) and b = (fog)($\alpha$), then :

1. A.$a{\alpha }^2+b\alpha -a=-2{\alpha }^2$
2. B.$a{\alpha }^2+b\alpha +a=0$
3. C.$a{\alpha }^2-b\alpha -a=0$
4. D.$a{\alpha }^2-b\alpha -a=1$

If f(x) is a non-zero polynomial of degree four, having local extreme points at x = –1, 0, 1; then the set $S=\{x\in R:f(x)=f(0)\}$ Contains exactly :

1. A.four rational numbers.
2. B.four irrational numbers.
3. C.two irrational and one rational number.
4. D.two irrational and two rational numbes.

All the points in the set $S=\left\{\frac{\alpha +i}{\alpha -i}:\alpha \in R\right\}(i=\sqrt{-1})$ lie on a :

1. A.straight line whose slope is –1
2. B.straight line whose slope is 1.
3. C.circle whose radius is 1.
4. D.circle whose radius is $\sqrt{2}$.

Let $f(x)=15-|x-10|$; $x\in R$. Then the set of all values of x, at which the function, $g(x)=f(f(x))$ is not differentiable, is :

1. A.{10,15}
2. B.{5,10,15,20}
3. C.{10}
4. D.{5,10,15}

Let $\sum _{k=1}^{10}f(a+k)=16(2^{10}-1)$ where the function f satisfies $f(x+y)=f(x)f(y)$ for all natural numbers x, y and f(1) = 2. then the natural number 'a' is

1. A.2
2. B.16
3. C.4
4. D.3

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

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

Let S be the set of all values of x for which the tangent to the curve $y=f(x)=x^3-x^2-2x$ at (x, y) is parallel to the line segment joining the points (1, f(1)) and (−1, f(−1)), then S is equal to :

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

If $f(x)=[x]-\left[\frac{x}{4}\right],x\in 4$, where $[x]$ denotes the greatest integer function, then

1. A.Both ${\lim }_{x\to 4-}f(x)$ and ${\lim }_{x\to 4+}f(x)$ exist but are not equal
2. B.f is continuous at x = 4
3. C.${\lim }_{x\to 4+}f(x)$ exists but ${\lim }_{x\to 4-}f(x)$ does not exist
4. D.${\lim }_{x\to 4-}f(x)$ exists but ${\lim }_{x\to 4+}f(x)$ does not exist