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Algebra

Sets, Relations and Function: JEE Main & Advanced PYQs

214 previous year questions on Sets, Relations and Function from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 1-50 of 214

  1. If $A=\{x\in\mathbf{R}:|x|<2\}$ and $B=\{x\in\mathbf{R}:|x-2|\ge3\}$; then :
  2. Let $R$ be a relation on $\mathbb{Z}\times\mathbb{Z}$ defined by $(a,b)R(c,d)$ if and only if $ad-bc$ is divisible by $5$. Then $R$ is
  3. Let the relations $R_1$ and $R_2$ on the set $X=\{1,2,3,\ldots,20\}$ be given by $R_1=\{(x,y):2x-3y=2\}$ and $R_2=\{(x,y):-5x+4y=0\}$. If $M$ and $N$ be the minimum …
  4. Define a relation $R$ over a class of $n\times n$ real matrices $A$ and $B$ as "$ARB$ iff there exists a non-singular matrix $P$ such that $PAP^{-1}=B$". Then which of …
  5. Let $A=\{x\in\mathbb{R}:[x+3]+[x+4]\le3\}$, $B=\left\{x\in\mathbb{R}:3^{x}\left(\sum_{r=1}^{\infty}\frac{3}{10^{r}}\right)^{x-3}<3^{-3x}\right\}$, where $[t]$ denotes …
  6. Let $A=\{1,3,4,6,9\}$ and $B=\{2,4,5,8,10\}$. Let $R$ be a relation defined on $A\times B$ such that $R=\{((a_1,b_1),(a_2,b_2)):a_1\le b_2\text{ and }b_1\le a_2\}$. Then …
  7. Let $A=\{-2,-1,0,1,2,3\}$. Let $R$ be a relation on $A$ defined by $xRy$ if and only if $y=\max\{x,1\}$. Let $l$ be the number of elements in $R$. Let $m$ and $n$ be the …
  8. Let $A=\{1,2,3,\ldots,100\}$ and $R$ be a relation on $A$ such that $R=\{(a,b):a=2b+1\}$. Let $(a_1,a_2),(a_2,a_3),(a_3,a_4),\ldots,(a_k,a_{k+1})$ be a sequence of $k$ …
  9. Let $A=\{0,1,2,3,4,5\}$. Let $R$ be a relation on $A$ defined by $(x,y)\in R$ if and only if $\max\{x,y\}\in\{3,4\}$. Then among the statements $(S_1)$ : The number of …
  10. Consider the relations $R_1$ and $R_2$ defined as $aR_1b\Leftrightarrow a^{2}+b^{2}=1$ for all $a,b\in\mathbf{R}$ and $(a,b)R_2(c,d)\Leftrightarrow a+d=b+c$ for all …
  11. Let $A=\{(\alpha,\beta)\in\mathbb{R}\times\mathbb{R}:|\alpha-1|\le4\text{ and }|\beta-5|\le6\}$ and …
  12. Let $f,g:\mathbb{N}-\{1\}\to\mathbb{N}$ be functions defined by $f(a)=\alpha$, where $\alpha$ is the maximum of the powers of those primes $p$ such that $p^\alpha$ …
  13. Let $R_1=\{(a,b)\in\mathbf{N}\times\mathbf{N}:|a-b|\le13\}$ and $R_2=\{(a,b)\in\mathbf{N}\times\mathbf{N}:|a-b|\neq13\}$. Then on $\mathbf{N}$ :
  14. The total number of functions, $f:\{1,2,3,4\}\to\{1,2,3,4,5,6\}$ such that $f(1)+f(2)=f(3)$, is equal to :
  15. Let $\alpha,\beta$ and $\gamma$ be three positive real numbers. Let $f(x)=\alpha x^5+\beta x^3+\gamma x$, $x\in\mathbf{R}$ and $g:\mathbf{R}\to\mathbf{R}$ be such that …
  16. Let $R$ be a relation from the set $\{1,2,3,\ldots,60\}$ to itself such that $R=\{(a,b):b=pq$, where $p,q\ge3$ are prime numbers$\}$. Then, the number of elements in $R$ …
  17. Let $A=\{x\in\mathbf{R}:|x+1|<2\}$ and $B=\{x\in\mathbf{R}:|x-1|\ge2\}$. Then which one of the following statements is NOT true ?
  18. 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 …
  19. Let $A=\{2,3,4\}$ and $B=\{8,9,12\}$. Then the number of elements in the relation …
  20. The probability that a relation $R$ from $\{x,y\}$ to $\{x,y\}$ is both symmetric and transitive, is equal to
  21. Let $\mathbb{Z}$ be the set of all integers, $A=\{(x,y)\in\mathbb{Z}\times\mathbb{Z}:(x-2)^{2}+y^{2}\le4\}$, $B=\{(x,y)\in\mathbb{Z}\times\mathbb{Z}:x^{2}+y^{2}\le4\}$ …
  22. Which of the following is not correct for relation $R$ on the set of real numbers ?
  23. Let $A=\{2,3,4,5,\ldots,30\}$ and '$\simeq$' be an equivalence relation on $A\times A$, defined by $(a,b)\simeq(c,d)$, if and only if $ad=bc$. Then the number of ordered …
  24. An organization awarded $48$ medals in event 'A', $25$ in event 'B' and $18$ in event 'C'. If these medals went to total $60$ men and only five men got medals in all the …
  25. Let $R=\{(P,Q)\mid P$ and $Q$ are at the same distance from the origin$\}$ be a relation, then the equivalence class of $(1,-1)$ is the set :
  26. Let a set $A=A_1\cup A_2\cup\ldots\cup A_k$, where $A_i\cap A_j=\phi$ for $i\neq j$, $1\le i,j\le k$. Define the relation $R$ from $A$ to $A$ by $R=\{(x,y):y\in A_i$ if …
  27. Let $\mathbf{N}$ be the set of natural numbers and a relation $R$ on $\mathbf{N}$ be defined by $R=\{(x,y)\in\mathbf{N}\times\mathbf{N}:x^3-3x^2y-xy^2+3y^3=0\}$. Then …
  28. Let $A=\{1,2,3,4,5,6,7\}$. Then the relation $R=\{(x,y)\in A\times A:x+y=7\}$ is
  29. The domain of the function $f(x)=\sin^{-1}[2x^2-3]+\log_2\left(\log_{\frac12}(x^2-5x+5)\right)$, where $[t]$ is the greatest integer function, is :
  30. Let $f:\mathbb{R}-\{5\}\to\mathbb{R}$ be defined as $f(x)=\frac{2x^2+3x-2}{x-5}$. Then $f$ is
  31. Consider the relation $R$ on the set $\{-2,-1,0,1,2\}$ defined by $(a,b)\in R$ if and only if $1+ab>0$. Then, among the statements : I. The number of elements in $R$ is …
  32. Let $A=\{1,2,3,4,5\}$. Let $R$ be a relation on $A$ defined by $xRy$ if and only if $4x\le5y$. Let $m$ be the number of elements in $R$ and $n$ be the minimum number of …
  33. Let $f:(1,3)\to\mathbf{R}$ 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 :
  34. Let $A=\{-3,-2,-1,0,1,2,3\}$. Let $R$ be a relation on $A$ defined by $xRy$ if and only if $0\le x^{2}+2y\le4$. Let $l$ be the number of elements in $R$ and $m$ be the …
  35. Let $x*y=x^{2}+y^{3}$ and $(x*1)*1=x*(1*1)$. Then a value of $2\sin^{-1}\left(\frac{x^{4}+x^{2}-2}{x^{4}+x^{2}+2}\right)$ is
  36. Let $\mathbb{R}$ be the set of real numbers and '$*$' be a binary operation on $\mathbb{R}-\{0\}$ defined by $a*b=a+2b+\frac{a}{b}$. Then the operation '$*$' is
  37. If $R=\{(x,y):x,y\in\mathbf{Z},x^2+3y^2\le8\}$ is a relation on the set of integers $\mathbf{Z}$, then the domain of $R^{-1}$ is :
  38. Let $R_1$ and $R_2$ be two relations defined on $\mathbb{R}$ by $aR_1b\Leftrightarrow ab\ge0$ and $aR_2b\Leftrightarrow a\ge b$. Then,
  39. Let $S=\left\{x\in[-6,3]-\{-2,2\}:\frac{|x+3|-1}{|x|-2}\ge0\right\}$ and $T=\{x\in\mathbb{Z}:x^2-7|x|+9\le0\}$. Then the number of elements in $S\cap T$ is :
  40. For $\alpha\in\mathbf{N}$, consider a relation $R$ on $\mathbf{N}$ given by $R=\{(x,y):3x+\alpha y\text{ is a multiple of }7\}$. The relation $R$ is an equivalence …
  41. Consider the sets $A=\{(x,y)\in\mathbb{R}\times\mathbb{R}:x^{2}+y^{2}=25\}$, $B=\{(x,y)\in\mathbb{R}\times\mathbb{R}:x^{2}+9y^{2}=144\}$ …
  42. Let $A=\{-3,-2,-1,0,1,2,3\}$ and $R$ be a relation on $A$ defined by $xRy$ if and only if $2x-y\in\{0,1\}$. Let $l$ be the number of elements in $R$. Let $m$ and $n$ be …
  43. Let $[t]$ denote the greatest integer $\le t$ and $\lim_{x\to0}x\left[\frac4x\right]=A$. Then the function, $f(x)=[x^2]\sin(\pi x)$ is discontinuous, when $x$ is equal …
  44. Let $A=\{1,6,11,16,\ldots\}$ and $B=\{9,16,23,30,\ldots\}$ be the sets consisting of the first $2025$ terms of two arithmetic progressions. Then $n(A\cup B)$ is
  45. Let $f:\mathbf{R}\to\mathbf{R}$ be such that for all $x\in\mathbf{R}$, $(2^{1+x}+2^{1-x})$, $f(x)$ and $(3^x+3^{-x})$ are in A.P., then the minimum value of $f(x)$ is :
  46. Let $A$ be the set of all functions $f:\mathbf{Z}\to\mathbf{Z}$ and $R$ be a relation on $A$ such that $R=\{(f,g):f(0)=g(1)\text{ and }f(1)=g(0)\}$. Then $R$ is :
  47. Out of all the patients in a hospital $89\%$ are found to be suffering from heart ailment and $98\%$ are suffering from lungs infection. If $K\%$ of them are suffering …
  48. 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 …
  49. Let $S=\{1,2,3,\ldots,10\}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $R=\{(A,B):A\cap B\neq\phi;\ A,B\in M\}$ is :
  50. A group of $40$ students appeared in an examination of $3$ subjects - Mathematics, Physics and Chemistry. It was found that all students passed in atleast one of the …

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