Sets, Relations & Functions · JEE Main previous year questions

The relation $R=\{(x,y):x,y\in \mathbb{Z}\text{ and }x+y\text{ is even }\}$ is:

1. A.reflexive and transitive but not symmetric
2. B.reflexive and symmetric but not transitive
3. C.an equivalence relation
4. D.symmetric and transitive but not reflexive

The sum of all local minimum values of the function $$f(x)=\{\begin{array}{ll}1-2x,& x<-1\\ \frac{1}{3}(7+2|x|),& -1\le x\le 2\\ \frac{11}{18}(x-4)(x-5),& x>2\end{array}$$ is

1. A.$\frac{167}{72}$
2. B.$\frac{157}{72}$
3. C.$\frac{171}{72}$
4. D.$\frac{131}{72}$

Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined by $f(x)=(2+3a)x^2+\left(\frac{a+2}{a-1}\right)x+b,a\ne 1$. If $f(x+y)=f(x)+f(y)+1-\frac{2}{7}xy$, then the value of $28{\sum }_{i=1}^5|f(i)|$ is

1. A.735
2. B.675
3. C.715
4. D.545

Let $S=\left\{p_1,p_2\dots ,p_{10}\right\}$ be the set of first ten prime numbers. Let $A=S\cup P$, where $P$ is the set of all possible products of distinct elements of $S$. Then the number of all ordered pairs $(x,y),x\in S$, $y\in A$, such that $x$ divides $y$, is ____ .

Let $(2,3)$ be the largest open interval in which the function $f(x)=2{\log }_{\mathrm{e}}(x-2)-x^2+ax+1$ is strictly increasing and $(b,c)$ be the largest open interval, in which the function $g(x)=(x-1{)}^3(x+2-a{)}^2$ is strictly decreasing. Then $100(a+b-c)$ is equal to :

1. A.360
2. B.420
3. C.160
4. D.280

Let $f:(0,\infty )\to \mathbf{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2{f}^{\prime }(x)=2xf(x)+3$, with $f(1)=4$. Then $2f(2)$ is equal to :

1. A.19
2. B.23
3. C.29
4. D.39

The function $f:(-\infty ,\infty )\to (-\infty ,1)$, defined by $f(x)=\frac{2^x-2^{-x}}{2^x+2^{-x}}$ 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

Number of functions $f:\{1,2,\dots ,100\}\to \{0,1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98 , is equal to ________.

Let $\mathrm{X}=\mathbf{R}\times \mathbf{R}$. Define a relation $\mathrm{R}$ on $\mathrm{X}$ as : $(a_1,b_1)\mathrm{R}(a_2,b_2)\iff b_1=b_2$ Statement I: $\mathrm{R}$ is an equivalence relation. Statement II : For some $(a,b)\in \mathrm{X}$, the set $\mathrm{S}=\{(x,y)\in \mathrm{X}:(x,y)\mathrm{R}(a,b)\}$ represents a line parallel to $y=x$. In the light of the above statements, choose the correct answer from the options given below :

1. A.Both Statement I and Statement II are true
2. B.Statement I is true but Statement II is false
3. C.Both Statement I and Statement II are false
4. D.Statement I is false but Statement II is true

Let the range of the function $f(x)=6+16\cos x\cdot \cos \left(\frac{\pi }{3}-x\right)\cdot \cos \left(\frac{\pi }{3}+x\right)\cdot \sin 3x\cdot \cos 6x,x\in \mathbf{R}$ be $[\alpha ,\beta ]$. Then the distance of the point $(\alpha ,\beta )$ from the line $3x+4y+12=0$ is :

1. A.11
2. B.10
3. C.8
4. D.9

Let $f(x)$ be a real differentiable function such that $f(0)=1$ and $f(x+y)=f(x)f^{\prime }(y)+f^{\prime }(x)f(y)$ for all $x,y\in \mathbf{R}$. Then ${\sum }_{n=1}^{100}{\log }_{e}f(n)$ is equal to :

1. A.2406
2. B.5220
3. C.2525
4. D.2384

The number of non-empty equivalence relations on the set $\{1,2,3\}$ is :

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

Let $f:\mathbf{R}\to \mathbf{R}$ be a twice differentiable function such that $f(x+y)=f(x)f(y)$ for all $x,y\in \mathbf{R}$. If $f^{\prime }(0)=4a$ and $f$ satisfies $f^{\prime \prime }(x)-3a{f}^{\prime }(x)-f(x)=0,a>0$, then the area of the region $R=\{(x,y)\mid 0\le y\le f(ax),0\le x\le 2\}$ is :

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

Let the function, $$f(x)=\{\begin{array}{ll}-3\mathrm{a}{x}^2-2,& x<1\\ {\mathrm{a}}^2+bx,& x⩾1\end{array}$$ be differentiable for all $x\in \mathbf{R}$, where $\mathrm{a}>1$, $\mathrm{b}\in \mathbf{R}$. If the area of the region enclosed by $y=f(x)$ and the line $y=-20$ is $\alpha +\beta \sqrt{3}$, $\alpha ,\beta \in Z$, then the value of $\alpha +\beta$ is ____ .

If the domain of the function $f(x)={\sin }^{-1}\left(\frac{x-1}{2x+3}\right)$ is $\mathbf{R}-(\alpha ,\beta )$, then $12\alpha \beta$ is equal to :

1. A.40
2. B.36
3. C.24
4. D.32

Let $A=\{2,3,6,7\}$ and $B=\{4,5,6,8\}$. Let $R$ be a relation defined on $A\times B$ by $(a_1,b_1)R(a_2,b_2)$ if and only if $a_1+a_2=b_1+b_2$. Then the number of elements in $R$ is ______.

If a function $f$ satisfies $f(m+n)=f(m)+f(n)$ for all $m,n\in \mathbf{N}$ and $f(1)=1$, then the largest natural number $\lambda$ such that ${\sum }_{k=1}^{2022}f(\lambda +k)\le (2022{)}^2$ is equal to ______.

Let the set of all positive values of $\lambda$, for which the point of local minimum of the function $(1+x({\lambda }^2-x^2))$ satisfies $\frac{x^2+x+2}{x^2+5x+6}<0$, be $(\alpha ,\beta )$. Then ${\alpha }^2+{\beta }^2$ is equal to ______.

Let the range of the function $f(x)=\frac{1}{2+\sin 3x+\cos 3x}$, $x\in \mathbb{R}$ be $[a,b]$. If $\alpha$ and $\beta$ are respectively the A.M. and the G.M. of $a$ and $b$, then $\frac{\alpha }{\beta }$ is equal to

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

Let $A=\{(x,y):2x+3y=23,x,y\in \mathbb{N}\}$ and $B=\{x:(x,y)\in A\}$. Then the number of one-one functions from $A$ to $B$ is equal to ____.

The number of critical points of the function $f(x)=(x-2{)}^{2/3}(2x+1)$ is

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

Let $[t]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of all prime factors of 2310 and $f:A\to \mathbb{Z}$ be the function $f(x)=\left[{\log }_2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]$. The number of one-to-one functions from $A$ to the range of $f$ is

1. A.$20$
2. B.$120$
3. C.$25$
4. D.$24$

If the set $R=\{(a,b):a+5b=42,a,b\in \mathbb{N}\}$ has $m$ elements and ${\sum }_{n=1}^m(1-i^{n!})=x+iy$, where $i=\sqrt{-1}$, then the value of $m+x+y$ is

1. A.$12$
2. B.$4$
3. C.$8$
4. D.$5$

Let $f(x)=\{\begin{array}{ll}-a& \text{if }-a\le x\le 0\\ x+a& \text{if }0<x\le a\end{array}$ where $a>0$ and $g(x)=(f(|x|)-|f(x)|)/2$. Then the function $g:[-a,a]\to [-a,a]$ is

1. A.neither one-one nor onto.
2. B.both one-one and onto.
3. C.one-one.
4. D.onto

Let $A=\{2,3,6,8,9,11\}$ and $B=\{1,4,5,10,15\}$. Let $R$ be a relation on $A\times B$ defined by $(a,b)R(c,d)$ if and only if $3ad-7bc$ is an even integer. Then the relation $R$ is

1. A.reflexive but not symmetric.
2. B.an equivalence relation.
3. C.reflexive and symmetric but not transitive.
4. D.transitive but not symmetric.