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Yukti Vault

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 51-100 of 214

  1. If $[t]$ denotes the greatest integer $\le t$, then the number of points, at which the function $f(x)=4|2x+3|+9\left[x+\frac12\right]-12[x+20]$ is not differentiable in …
  2. The number of relations on the set $A=\{1,2,3\}$, containing at most $6$ elements including $(1,2)$, which are reflexive and transitive but not symmetric, is …
  3. Let $A=\{1,2,3,\ldots,7\}$ and let $P(A)$ denote the power set of $A$. If the number of functions $f:A\to P(A)$ such that $a\in f(a)$, $\forall a\in A$ is $m^{n}$, $m$ …
  4. The number of symmetric relations defined on the set $\{1,2,3,4\}$ which are not reflexive is $\_\_\_\_\_\_$ .
  5. The sum of the maximum and minimum values of the function $f(x)=|5x-7|+[x^2+2x]$ in the interval $\left[\frac54,2\right]$, where $[t]$ is the greatest integer $\le t$ …
  6. Let $A=\{1,2,3,\ldots,20\}$. Let $R_1$ and $R_2$ two relation on $A$ such that $R_1=\{(a,b):b$ is divisible by $a\}$, $R_2=\{(a,b):a$ is an integral multiple of $b\}$. …
  7. Let $R_1$ and $R_2$ be two relations defined as follows : $R_1=\{(a,b)\in\mathbf{R}^2:a^2+b^2\in\mathbf{Q}\}$ and $R_2=\{(a,b)\in\mathbf{R}^2:a^2+b^2\notin\mathbf{Q}\}$ …
  8. The number of functions $f$, from the set $A=\{x\in\mathbf{N}:x^2-10x+9\le0\}$ to the set $B=\{n^2:n\in\mathbf{N}\}$ such that $f(x)\le(x-3)^2+1$, for every $x\in A$, is …
  9. Let $A=\{-4,-3,-2,0,1,3,4\}$ and $R=\{(a,b)\in A\times A:b=|a|\text{ or }b^{2}=a+1\}$ be a relation on $A$. Then the minimum number of elements, that must be added to …
  10. Let $f:\mathbf{R}\to\mathbf{R}$ be a function which satisfies $f(x+y)=f(x)+f(y)$ $\forall x,y\in\mathbf{R}$. If $f(1)=2$ and $g(n)=\sum_{k=1}^{(n-1)}f(k)$ …
  11. Let $A=\{n\in\mathbf{N}\mid n^2\le n+10{,}000\}$, $B=\{3k+1\mid k\in\mathbf{N}\}$ and $C=\{2k\mid k\in\mathbf{N}\}$, then the sum of all the elements of the set …
  12. If $\log_32$, $\log_3(2^x-5)$, $\log_3\left(2^x-\frac72\right)$ are in an arithmetic progression, then the value of $x$ is equal to ________.
  13. Let $A=\{1,2,3,4,5,6,7\}$. Define $B=\{T\subseteq A:$ either $1\notin T$ or $2\in T\}$ and $C=\{T\subseteq A:$ the sum of all the elements of $T$ is a prime number$\}$. …
  14. Let $R=\{(x,y)\in\mathbf{N}\times\mathbf{N}:\log_e(x+y)\le2\}$. Then the minimum number of elements, required to be added in $R$ to make it a transitive relation, is …
  15. Let $S=\{1,2,3,4,5,6,7\}$. Then the number of possible functions $f:S\to S$ such that $f(m\cdot n)=f(m)\cdot f(n)$ for every $m,n\in S$ and $m\cdot n\in S$ is equal to …
  16. Let $A=\{1,2,3,4\}$ and $R$ be a relation on the set $A\times A$ defined by $R=\{((a,b),(c,d)):2a+3b=4c+5d\}$. Then the number of elements in $R$ is $\_\_\_\_\_\_$ .
  17. Let $A=\{1,4,7\}$ and $B=\{2,3,8\}$. Then the number of elements, in the relation $R=\{((a_1,b_1),(a_2,b_2))\in((A\times B)\times(A\times B)):a_1+b_2$ divides …
  18. Let $\bigcup_{i=1}^{50}X_i=\bigcup_{i=1}^{n}Y_i=T$, where each $X_i$ contains $10$ elements and each $Y_i$ contains $5$ elements. If each element of the set $T$ is an …
  19. Let $[t]$ denote the greatest integer $\le t$. Then the equation in $x$, $[x]^2+2[x+2]-7=0$ has :
  20. The number of relations, on the set $\{1,2,3\}$ containing $(1,2)$ and $(2,3)$, which are reflexive and transitive but not symmetric, is $\_\_\_\_\_\_$ .
  21. Consider the two sets : $A=\{m\in\mathbf{R}$ : both the roots of $x^2-(m+1)x+m+4=0$ are real$\}$ and $B=[-3,5)$. Which of the following is not true ?
  22. Let $A=\{1,2,3,\ldots,100\}$. Let $R$ be a relation on $A$ defined by $(x,y)\in R$ if and only if $2x=3y$. Let $R_1$ be a symmetric relation on $A$ such that …
  23. Let $S=\{4,6,9\}$ and $T=\{9,10,11,\ldots,1000\}$. If $A=\{a_1+a_2+\ldots+a_k:k\in\mathbf{N},a_1,a_2,a_3,\ldots,a_k\in S\}$, then the sum of all the elements in the set …
  24. Let $R_1$ and $R_2$ be relations on the set $\{1,2,\ldots,50\}$ such that $R_1=\{(p,p^{n}):p$ is a prime and $n\ge0$ is an integer$\}$ and $R_2=\{(p,p^{n}):p$ is a prime …
  25. The number of solutions of the equation $|x^2+3x+2|+|x+5|-2=0$, $x\in\mathbb{R}$, is ________.
  26. 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 …
  27. Let $A=\{1,2,3,4,\ldots,10\}$ and $B=\{0,1,2,3,4\}$. The number of elements in the relation $R=\{(a,b)\in A\times A:2(a-b)^{2}+3(a-b)\in B\}$ is $\_\_\_\_\_\_$ .
  28. If $A=\{x\in\mathbf{R}:|x-2|>1\}$, $B=\left\{x\in\mathbf{R}:\sqrt{x^{2}-3}>1\right\}$, $C=\{x\in\mathbf{R}:|x-4|\ge2\}$ and $\mathbf{Z}$ is the set of all integers, then …
  29. For a suitably chosen real constant $a$, let a function, $f:\mathbf{R}-\{-a\}\to\mathbf{R}$ be defined by $f(x)=\frac{a-x}{a+x}$. Further suppose that for any real …
  30. A survey shows that $63\%$ of the people in a city read newspaper $A$ whereas $76\%$ read newspaper $B$. If $x\%$ of the people read both the newspapers, then a possible …
  31. Let $A=\{2,3,4,5,6\}$. Let $R$ be a relation on the set $A\times A$ given by $(x,y)R(z,w)$ if and only if $x$ divides $z$ and $y\le w$. Then the number of elements in …
  32. Let $A=\{1,2,3,4,5,6,7\}$ and $B=\{3,6,7,9\}$. Then the number of elements in the set $\{C\subseteq A:C\cap B\ne\phi\}$ is ________.
  33. 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 $\_\_\_\_\_\_$ .
  34. Let $A=\{0,3,4,6,7,8,9,10\}$ and $R$ be the relation defined on $A$ such that $R=\{(x,y)\in A\times A:x-y\text{ is odd positive integer or }x-y=2\}$. The minimum number …
  35. Let $f(x)=x\cdot\left[\frac x2\right]$, for $-10<x<10$, where $[t]$ denotes the greatest integer function. Then the number of points of discontinuity of $f$ is equal to …
  36. Let $A=\{n\in\mathbf{N}:\text{H.C.F.}(n,45)=1\}$ and let $B=\{2k:k\in\{1,2,\ldots,100\}\}$. Then the sum of all the elements of $A\cap B$ is $\_\_\_\_\_\_$ .
  37. A survey shows that $73\%$ of the persons working in an office like coffee, whereas $65\%$ like tea. If $x$ denotes the percentage of them, who like both coffee and tea …
  38. Let $A=\{1,2,3,4\}$ and $R=\{(1,2),(2,3),(1,4)\}$ be a relation on $A$. Let $S$ be the equivalence relation on $A$ such that $R\subset S$ and the number of elements in …
  39. Let $R=\{(3,3),(6,6),(9,9),(12,12),(6,12),(3,9),(3,12),(3,6)\}$ be a relation on the set $A=\{3,6,9,12\}$. The relation is :
  40. The statement $p\to(q\to p)$ is equivalent to :
  41. Which of the following is equivalent to the Boolean expression $p\wedge\sim q$ ?
  42. Let $p$ and $q$ be any two logical statements and $r:p\to(\sim p\vee q)$. If $r$ has a truth value $F$, then the truth values of $p$ and $q$ are respectively :
  43. Consider the following statements: P: Suman is brilliant Q: Suman is rich R: Suman is honest The negation of the statement, "Suman is brilliant and dishonest if and only …
  44. 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 …
  45. For integers $m$ and $n$, both greater than $1$, consider the following three statements : $P$ : $m$ divides $n$ $Q$ : $m$ divides $n^{2}$ $R$ : $m$ is prime, then
  46. In a certain town, $25\%$ of the families own a phone and $15\%$ own a car; $65\%$ families own neither a phone nor a car and $2000$ families own both a car and a phone. …
  47. The proposition $(\sim p)\vee(p\wedge\sim q)$ is equivalent to :
  48. Statement-1: The statement $A\to(B\to A)$ is equivalent to $A\to(A\vee B)$. Statement-2: The statement $\sim\left[(A\wedge B)\to(\sim A\vee B)\right]$ is a Tautology.
  49. The proposition $\sim(p\vee\sim q)\vee\sim(p\vee q)$ is logically equivalent to :
  50. Let $p,q,r$ denote arbitrary statements. Then the logically equivalent of the statement $p\Rightarrow(q\vee r)$ is :

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