Sets, Relations & Functions · JEE Main previous year questions

2024 59 questions

1. Q32024 · 27 Jan 2024 · Shift 1

   Let $S=\{1,2,3,\dots ,10\}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $\mathrm{R}=\{(\mathrm{A},\mathrm{B}):\mathrm{A}\cap \mathrm{B}\ne \varphi ;\mathrm{A},\mathrm{B}\in \mathrm{M}\}$ is :

   1. A.symmetric only
   2. B.reflexive only
   3. C.symmetric and reflexive only
   4. D.symmetric and transitive only
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   Answer: A

2. Q92024 · 27 Jan 2024 · Shift 1

   Consider the function. $$f(x)=\{\begin{array}{ll}\frac{\mathrm{a}\left(7x-12-x^2\right)}{\mathrm{b}|x^2-7x+12|}& ,x<3\\ 2^{\frac{\sin (x-3)}{x-[x]}}& ,x>3\\ \mathrm{b}& ,x=3,\end{array}$$ where $[x]$ denotes the greatest integer less than or equal to $x$. If $\mathrm{S}$ denotes the set of all ordered pairs $(\mathrm{a},\mathrm{b})$ such that $f(x)$ is continuous at $x=3$, then the number of elements in $\mathrm{S}$ is :

   1. A.Infinitely many
   2. B.4
   3. C.2
   4. D.1
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   Answer: D

3. Q202024 · 27 Jan 2024 · Shift 1

   The function $f:\mathbf{N}-\{1\}\to \mathbf{N}$; defined by $f(\mathrm{n})=$ the highest prime factor of $\mathrm{n}$, is :

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

4. Q252024 · 27 Jan 2024 · Shift 1· Numerical answer

   Let for a differentiable function $f:(0,\infty )\to \mathbf{R}$, $f(x)-f(y)\ge {\log }_e\left(\frac{x}{y}\right)+x-y,\forall x,y\in (0,\infty )$. Then $\sum _{n=1}^{20}{f}^{\prime }\left(\frac{1}{n^2}\right)$ is equal to ____.

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   Answer: 2890

5. Q32024 · 27 Jan 2024 · Shift 2

   Consider the function $f:(0,2)\to \mathbf{R}$ defined by $f(x)=\frac{x}{2}+\frac{2}{x}$ and the function $g(x)$ defined by $g(x)=\{\begin{array}{ll}\min \{f(t)\},& 0<t\le x\text{ and }0<x\le 1\\ \frac{3}{2}+x,& 1<x<2\end{array}$. Then,

   1. A.$g$ is continuous but not differentiable at $x=1$
   2. B.$g$ is continuous and differentiable for all $x\in (0,2)$
   3. C.$g$ is not continuous for all $x\in (0,2)$
   4. D.$g$ is neither continuous nor differentiable at $x=1$
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   Answer: A

6. Q62024 · 27 Jan 2024 · Shift 2

   Let $A$ and $B$ be two finite sets with $m$ and $n$ elements respectively. The total number of subsets of the set $A$ is 56 more than the total number of subsets of $B$. Then the distance of the point $P(m,n)$ from the point $Q(-2,-3)$ is :

   1. A.$8$
   2. B.$10$
   3. C.$4$
   4. D.$6$
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   Answer: B

7. Q102024 · 27 Jan 2024 · Shift 2

   Let $f:\mathbf{R}-\{-\frac{1}{2}\}\to \mathbf{R}$ and $g:\mathbf{R}-\{-\frac{5}{2}\}\to \mathbf{R}$ be defined as $f(x)=\frac{2x+3}{2x+1}$ and $g(x)=\frac{|x|+1}{2x+5}$. Then, the domain of the function $f\circ g$ is :

   1. A.$\mathbf{R}-\{-\frac{7}{4}\}$
   2. B.$\mathbf{R}$
   3. C.$\mathbf{R}-\{-\frac{5}{2},-\frac{7}{4}\}$
   4. D.$\mathbf{R}-\{-\frac{5}{2}\}$
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   Answer: D

8. Q152024 · 27 Jan 2024 · Shift 2

   Let $\text{R}$ be the interior region between the lines $3x-y+1=0$ and $x+2y-5=0$ containing the origin. The set of all values of $a$, for which the points $(a^2,a+1)$ lie in $R$, is :

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

9. Q242024 · 27 Jan 2024 · Shift 2· Numerical answer

   Let $f(x)={\int }_0^{x}g(t){\log }_e\left(\frac{1-t}{1+t}\right)\mathrm{d}t$, where $g$ is a continuous odd function. If ${\int }_{-\pi /2}^{\pi /2}\left(f(x)+\frac{x^2\cos x}{1+e^x}\right)\mathrm{d}x={\left(\frac{\pi }{\alpha }\right)}^2-\alpha$, then $\alpha$ is equal to _______.

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   Answer: 2