Sets, Relations & Functions · JEE Main previous year questions

Let $f$ and $g$ be the twice differentiable functions on $\mathbb{R}$ such that $f^{\prime \prime }(x)=g^{\prime \prime }(x)+6x$ $f^{\prime }(1)=4g^{\prime }(1)-3=9$ $f(2)=3g(2)=12$. Then which of the following is NOT true?

1. A.$g(-2)-f(-2)=20$
2. B.There exists $x_0\in (1,3/2)$ such that $f(x_0)=g(x_0)$
3. C.$|f^{\prime }(x)-g^{\prime }(x)|<6\Rightarrow -1<x<1$
4. D.If $-1<x<2$, then $|f(x)-g(x)|<8$

The minimum value of the function $f(x)={\int }_0^2{e}^{|x-t|}dt$ is :

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

Let $x=2$ be a local minima of the function $f(x)=2x^4-18x^2+8x+12$, $x\in (-4,4)$. If $\mathrm{M}$ is local maximum value of the function $f$ in $(-4,4)$, then $\mathrm{M}=$

1. A.$18\sqrt{6}-\frac{33}{2}$
2. B.$18\sqrt{6}-\frac{31}{2}$
3. C.$12\sqrt{6}-\frac{33}{2}$
4. D.$12\sqrt{6}-\frac{31}{2}$

Let $z_1=2+3i$ and $z_2=3+4i$. The set $\mathrm{S}=\left\{z\in \mathbb{C}:|z-z_1{|}^2-|z-z_2{|}^2=|z_1-z_2{|}^2\right\}$ represents a

1. A.hyperbola with the length of the transverse axis 7
2. B.hyperbola with eccentricity 2
3. C.straight line with the sum of its intercepts on the coordinate axes equals $-18$
4. D.straight line with the sum of its intercepts on the coordinate axes equals 14

Let $f:(0,1)\to \mathbb{R}$ be a function defined $f(x)=\frac{1}{1-e^{-x}}$, and $g(x)=(f(-x)-f(x))$. Consider two statements (I) g is an increasing function in $(0,1)$ (II) g is one-one in $(0,1)$ Then,

1. A.Both (I) and (II) are true
2. B.Neither (I) nor (II) is true
3. C.Only (II) is true
4. D.Only (I) is true

Let S = {1, 2, 3, 5, 7, 10, 11}. The number of non-empty subsets of S that have the sum of all elements a multiple of 3, is _____________.

For some $\mathrm{a},\mathrm{b},\mathrm{c}\in \mathbb{N}$, let $f(x)=ax-3$ and $\mathrm{g}(x)=x^{\mathrm{b}}+\mathrm{c},x\in \mathbb{R}$. If $(fog{)}^{-1}(x)={\left(\frac{x-7}{2}\right)}^{1/3}$, then $(fog)(\mathrm{ac})+(gof)(\mathrm{b})$ is equal to ____.

The number of functions $f:\{1,2,3,4\}\to \{a\in \mathbb{Z}:|a|\le 8\}$ satisfying $f(n)+\frac{1}{n}f(n+1)=1,\forall n\in \{1,2,3\}$ is

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

Let the function $f(x)=2x^3+(2p-7)x^2+3(2p-9)x-6$ have a maxima for some value of $x<0$ and a minima for some value of $x>0$. Then, the set of all values of p is

1. A.$\left(-\frac{9}{2},\frac{9}{2}\right)$
2. B.$\left(\frac{9}{2},\infty \right)$
3. C.$\left(0,\frac{9}{2}\right)$
4. D.$\left(-\infty ,\frac{9}{2}\right)$

Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined by $f(x)={\log }_{\sqrt{m}}\{\sqrt{2}(\sin x-\cos x)+m-2\}$, for some $m$, such that the range of $f$ is [0, 2]. Then the value of $m$ is ____

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

The relation $R=\{(a,b):\gcd (a,b)=1,2a\ne b,a,b\in \mathbb{Z}\}$ is :

1. A.reflexive but not symmetric
2. B.transitive but not reflexive
3. C.symmetric but not transitive
4. D.neither symmetric nor transitive

The equation $x^2-4x+[x]+3=x[x]$, where $[x]$ denotes the greatest integer function, has :

1. A.exactly two solutions in $(-\infty ,\infty )$
2. B.no solution
3. C.a unique solution in $(-\infty ,\infty )$
4. D.a unique solution in $(-\infty ,1)$

Let $\lambda \in \mathbb{R}$ and let the equation E be $|x{|}^2-2|x|+|\lambda -3|=0$. Then the largest element in the set $S=\{x+\lambda :x\text{ is an integer solution of E}\}$ is ____

Let $f(x)$ be a function such that $f(x+y)=f(x).f(y)$ for all $x,y\in \mathbb{N}$. If $f(1)=3$ and ${\sum }_{k=1}^{n}f(k)=3279$, then the value of n is

1. A.9
2. B.7
3. C.6
4. D.8

The minimum number of elements that must be added to the relation R = {(a, b), (b, c), (b, d)} on the set {a, b, c, d} so that it is an equivalence relation, is __________.

Let $\mathrm{R}$ be a relation from the set $\{1,2,3,\dots ,60\}$ to itself such that $R=\{(a,b):b=pq$, where $p,q⩾3$ are prime numbers$\}$. Then, the number of elements in $\mathrm{R}$ is :

1. A.600
2. B.660
3. C.540
4. D.720

The number of points, where the function $f:\mathbf{R}\to \mathbf{R}$, $f(x)=|x-1|\cos |x-2|\sin |x-1|+(x-3)\left|x^2-5x+4\right|$, is NOT differentiable, is :

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

Let $S=\{4,6,9\}$ and $T=\{9,10,11,\dots ,1000\}$. If $A=\left\{a_1+a_2+\dots +a_k:k\in \mathbf{N},a_1,a_2,a_3,\dots ,a_k\in S\right\}$, then the sum of all the elements in the set $T-A$ is equal to ____.

The domain of the function $f(x)={\sin }^{-1}\left(\frac{x^2-3x+2}{x^2+2x+7}\right)$ is :

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

The number of elements in the set $S=\left\{x\in \mathbb{R}:2\cos \left(\frac{x^2+x}{6}\right)=4^x+4^{-x}\right\}$ is :

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

Considering only the principal values of the inverse trigonometric functions, the domain of the function $f(x)={\cos }^{-1}\left(\frac{x^2-4x+2}{x^2+3}\right)$ is :

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

For $\alpha \in \mathbf{N}$, consider a relation $\mathrm{R}$ on $\mathbf{N}$ given by $\mathrm{R}=\{(x,y):3x+\alpha y\text{ is a multiple of }7\}$. The relation $R$ is an equivalence relation if and only if :

1. A.$\alpha =14$
2. B.$\alpha$ is a multiple of 4
3. C.4 is the remainder when $\alpha$ is divided by 10
4. D.4 is the remainder when $\alpha$ is divided by 7

Let $S$ be the set of all passwords which are six to eight characters long, where each character is either an alphabet from $\{A,B,C,D,E\}$ or a number from $\{1,2,3,4,5\}$ with the repetition of characters allowed. If the number of passwords in $S$ whose at least one character is a number from $\{1,2,3,4,5\}$ is $\alpha \times 5^6$, then $\alpha$ is equal to ____.

Let $f:[0,1]\to \mathbf{R}$ be a twice differentiable function in $(0,1)$ such that $f(0)=3$ and $f(1)=5$. If the line $y=2x+3$ intersects the graph of $f$ at only two distinct points in $(0,1)$, then the least number of points $x\in (0,1)$, at which $f^{\prime \prime }(x)=0$, is ____.

The function $f(x)=x{e}^{x(1-x)}$, $x\in \mathbb{R}$, is :

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