Sets, Relations & Functions · JEE Main previous year questions

Let S be the set of all the natural numbers, for which the line $\frac{x}{a}+\frac{y}{b}=2$ is a tangent to the curve ${\left(\frac{x}{a}\right)}^n+{\left(\frac{y}{b}\right)}^n=2$ at the point (a, b), $ab\ne 0$. Then :

1. A.$S=\varphi$
2. B.$n(S)=1$
3. C.$S=\{2k:k\in \mathrm{N}\}$
4. D.$S=\mathrm{N}$

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

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

Let $f:\mathbb{N}\to \mathbb{R}$ be a function such that $f(x+y)=2f(x)f(y)$ for natural numbers x and y. If $f(1)=2$, then the value of $\alpha$ for which $\sum _{k=1}^{10}f(\alpha +k)=\frac{512}{3}(2^{20}-1)$ holds, is :

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

Let $f:\mathbb{R}\to \mathbb{R}$ be defined as $f(x)=x^3+x-5$. If $g(x)$ is a function such that $f(g(x))=x$, $\forall x\in \mathbb{R}$, then $g^{\prime }(63)$ is equal to $\underline{}$.

1. A.$\frac{1}{49}$
2. B.$\frac{3}{49}$
3. C.$\frac{43}{49}$
4. D.$\frac{91}{49}$

Let $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ be two functions defined by $f(x)={\log }_e(x^2+1)-e^{-x}+1$ and $g(x)=\frac{1-2e^{2x}}{e^x}$. Then, for which of the following range of $\alpha$, the inequality $f\left(g\left(\frac{(\alpha -1{)}^2}{3}\right)\right)>f\left(g\left(\alpha -\frac{5}{3}\right)\right)$ holds ?

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

Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined by $f(x)={\left(2\left(1-\frac{x^{25}}{2}\right)(2+x^{25})\right)}^{\frac{1}{50}}$. If the function $g(x)=f(f(f(x)))+f(f(x))$, then the greatest integer less than or equal to $g(1)$ is $\underline{}$.

Let $A=\left(\begin{array}{cc}2& -2\\ 1& -1\end{array}\right)$ and $B=\left(\begin{array}{cc}-1& 2\\ -1& 2\end{array}\right)$. Then the number of elements in the set $\{(n,m):n,m\in \{1,2,\dots ,10\}\text{ and }n{A}^n+m{B}^m=I\}$ is $\underline{}$.

Let ${\lambda }^{\ast }$ be the largest value of $\lambda$ for which the function $f_{\lambda }(x)=4\lambda x^3-36\lambda x^2+36x+48$ is increasing for all $x\in \mathrm{R}$. Then $f_{{\lambda }^{\ast }}(1)+f_{{\lambda }^{\ast }}(-1)$ is equal to :

1. A.36
2. B.48
3. C.64
4. D.72

The sum of all the elements of the set $\{\alpha \in \{1,2,\dots ,100\}:HCF(\alpha ,24)=1\}$ is ____.

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 :

1. A.$(0,\sqrt{5})$
2. B.$[-2,2]$
3. C.$\left[\frac{1}{\sqrt{5}},\sqrt{5}\right]$
4. D.$[0,2]$

The function f(x), that satisfies the condition $f(x)=x+{\int }_0^{\pi /2}\sin x\cos yf(y)dy$, is :

1. A.$x+\frac{2}{3}(\pi -2)\sin x$
2. B.$x+(\pi +2)\sin x$
3. C.$x+\frac{\pi }{2}\sin x$
4. D.$x+(\pi -2)\sin x$

The local maximum value of the function $f(x)={\left(\frac{2}{x}\right)}^{x^2}$, $x>0$, is

1. A.$(2\sqrt{e}{)}^{\frac{1}{e}}$
2. B.${\left(\frac{4}{\sqrt{e}}\right)}^{\frac{e}{4}}$
3. C.$(e{)}^{\frac{2}{e}}$
4. D.1

The domain of the function ${\mathrm{cosec}}^{-1}\left(\frac{1+x}{x}\right)$ is :

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

Let a and b respectively be the points of local maximum and local minimum of the function $f(x)=2x^3-3x^2-12x$. If A is the total area of the region bounded by $y=f(x)$, the x-axis and the lines $x=a$ and $x=b$, then 4A is equal to ________.

Let $f:[0,\infty )\to [0,3]$ be a function defined by $$f(x)=\{\begin{array}{ll}\max \{\sin t:0\le t\le x\},& 0\le x\le \pi \\ 2+\cos x,& x>\pi \end{array}$$ Then which of the following is true?

1. A.f is continuous everywhere but not differentiable exactly at one point in $(0,\infty )$
2. B.f is differentiable everywhere in $(0,\infty )$
3. C.f is not continuous exactly at two points in $(0,\infty )$
4. D.f is continuous everywhere but not differentiable exactly at two points in $(0,\infty )$

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

1. A.symmetric but neither reflexive nor transitive
2. B.reflexive but neither symmetric nor transitive
3. C.reflexive and symmetric, but not transitive
4. D.an equivalence relation

Let $A=\{n\in N\mid n^2\le n+10,000\}$, $B=\{3k+1\mid k\in N\}$ 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 area (in sq. units) of the region, given by the set $\{(x,y)\in R\times R\mid x\ge 0,2x^2\le y\le 4-2x\}$ is :

1. A.$\frac{8}{3}$
2. B.$\frac{17}{3}$
3. C.$\frac{13}{3}$
4. D.$\frac{7}{3}$

Let $S=\left\{n\in N|{\left(\begin{array}{cc}0& i\\ 1& 0\end{array}\right)}^n\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\forall a,b,c,d\in R\right\}$, where $i=\sqrt{-1}$. Then the number of 2-digit numbers in the set S is ____.

Let $M=\left\{A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right):a,b,c,d\in \{\pm 3,\pm 2,\pm 1,0\}\right\}$. Define f : M $\to$ Z, as f(A) = det(A), for all $A\in M$, where z is set of all integers. Then the number of $A\in M$ such that f(A) = 15 is equal to ____.

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

1. A.f and g both are one-one
2. B.f and g both are onto
3. C.f is one-one and g is onto
4. D.f is onto and g is one-one

Let $f:\mathbb{R}\to \mathbb{R}$ be defined as $$f(x)=\{\begin{array}{ll}-\frac{4}{3}{x}^3+2x^2+3x,& x>0\\ 3x{e}^x,& x\le 0\end{array}.$$ Then f is increasing function in the interval

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

If the domain of the function $f(x)=\frac{{\cos }^{-1}\sqrt{x^2-x+1}}{\sqrt{{\sin }^{-1}\left(\frac{2x-1}{2}\right)}}$ is the interval $(\alpha ,\beta ]$, then $\alpha +\beta$ is equal to :

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

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

The sum of all the elements in the set $\{n\in \{1,2,\dots ,100\}\mid \text{H.C.F. of }n\text{ and }2040\text{ is }1\}$ is equal to ________.