Sets, Relations and Functions — JEE Main PYQs

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 $y \leq w$. Then the number of elements in R is __________.

Let $[\cdot]$ denote the greatest integer function. If the domain of the function $f(x)=\cos^{-1}\left(\frac{4x+2[x]}{3}\right)$ is $[\alpha, \beta]$, then $12(\alpha+\beta)$ is equal to:

For the function $f:[1,\infty) \to [1,\infty)$ defined by $f(x) = (x-1)^4 + 1$, among the two statements:

(I) The set $S = \{x \in [1,\infty) : f(x) = f^{-1}(x)\}$ contains exactly two elements, and

(II) The set $S = \{x \in [1,\infty) : f(x) = f^{-1}(x+1)\}$ is an empty set,

Let for some $\alpha \in \mathbb{R}$, $f : \mathbb{R} \to \mathbb{R}$ be a function satisfying $f(x+y) = f(x) + 2y^2 + y + \alpha xy$ for all $x, y \in \mathbb{R}$. If $f(0) = -1$ and $f(1) = 2$, then the value of $\displaystyle\sum_{n=1}^{5}\left(\alpha + f(n)\right)$ is:

Let $A = \{1, 4, 7\}$ and $B = \{2, 3, 8\}$. Then the number of elements, in the relation $R = \left\{\left((a_1, b_1), (a_2, b_2)\right) \in ((A \times B) \times (A \times B)) : a_1 + b_2 \text{ divides } a_2 + b_1\right\}$ is __________.

Let $[\cdot]$ denote the greatest integer function. If the domain of the function $f(x)=\sin^{-1}\left(\frac{x+[x]}{3}\right)$ is $[\alpha, \beta)$, then $\alpha^2+\beta^2$ is equal to:

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \frac{2x^2 - 3x + 2}{3x^2 + x + 3}$. Then $f$ is :

Let $R = \left\{(x, y) \in \mathbb{N} \times \mathbb{N} : \log_e (x + y) \le 2\right\}$. Then the minimum number of elements, required to be added in R to make it a transitive relation, is ________.

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 17

II. $R$ is an equivalence relation

Let A be the set of all functions $f: \mathbf{Z} \to \mathbf{Z}$ and R be a relation on A such that $\mathrm{R} = \{(\mathrm{f}, \mathrm{g}) : f(0) = g(1) \text{ and } f(1) = g(0)\}$. Then R is :

If the domain of the function $f(x)=\frac{1}{\sqrt{10+3x-x^2}}+\frac{1}{\sqrt{x+|x|}}$ is $(a, b)$, then $(1+a)^2+b^2$ is equal to :

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 $k$ elements of $R$ such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer $k$, for which such a sequence exists, is equal to :

Let $A = \{-3, -2, -1, 0, 1, 2, 3\}$. Let R be a relation on A defined by $x\mathrm{R}y$ if and only if $0 \le x^2 + 2y \le 4$. Let $l$ be the number of elements in R and $m$ be the minimum number of elements required to be added in R to make it a reflexive relation. Then $l + m$ is equal to

If the domain of the function $f(x) = \log_e\left(\frac{2x-3}{5+4x}\right) + \sin^{-1}\left(\frac{4+3x}{2-x}\right)$ is $[\alpha, \beta)$, then $\alpha^2 + 4\beta$ is equal to

Let $A = \{-2, -1, 0, 1, 2, 3\}$. Let R be a relation on $A$ defined by $x\mathrm{R}y$ if and only if $y = \max\{x, 1\}$. Let $l$ be the number of elements in R. Let $m$ and $n$ be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then $l + m + n$ is equal to

If the domain of the function $f(x) = \log_7 (1 - \log_4 (x^2 - 9x + 18))$ is $(\alpha, \beta) \cup (\gamma, \delta)$, then $\alpha + \beta + \gamma + \delta$ is equal to

Let $f$ be a function such that $f(x) + 3f\left(\dfrac{24}{x}\right) = 4x,\ x \neq 0$. Then $f(3) + f(8)$ is equal to

Consider the sets $A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + y^2 = 25\}$, $B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + 9y^2 = 144\}$, $C = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} : x^2 + y^2 \le 4\}$ and $D = A \cap B$. The total number of one-one functions from the set $D$ to the set $C$ is:

Let $f, g : (1, \infty) \to \mathbb{R}$ be defined as $f(x) = \frac{2x+3}{5x+2}$ and $g(x) = \frac{2-3x}{1-x}$. If the range of the function $fog : [2, 4] \to \mathbb{R}$ is $[\alpha, \beta]$, then $\frac{1}{\beta - \alpha}$ is equal to

Let the domains of the functions $f(x)=\log_4\log_3\log_7(8-\log_2(x^2+4x+5))$ and $g(x)=\sin^{-1}\left(\dfrac{7x+10}{x-2}\right)$ be $(\alpha,\beta)$ and $[\gamma,\delta]$, respectively. Then $\alpha^2+\beta^2+\gamma^2+\delta^2$ is equal to :

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 the minimum number of elements required to be added in $R$ to make it reflexive and symmetric relations, respectively. Then $l+m+n$ is equal to :

The number of relations on the set $\mathrm{A} = \{1, 2, 3\}$, containing at most 6 elements including $(1, 2)$, which are reflexive and transitive but not symmetric, is __________

Let $A = \{(\alpha, \beta) \in \mathbf{R} \times \mathbf{R} : |\alpha - 1| \leq 4 \text{ and } |\beta - 5| \leq 6\}$ and $B = \{(\alpha, \beta) \in \mathbf{R} \times \mathbf{R} : 16(\alpha - 2)^2 + 9(\beta - 6)^2 \leq 144\}$. Then

If the range of the function $f(x) = \dfrac{5 - x}{x^2 - 3x + 2}$, $x \neq 1, 2$, is $(-\infty, \alpha] \cup [\beta, \infty)$, then $\alpha^2 + \beta^2$ is equal to :

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

$(\mathrm{S}_1)$ : The number of elements in $R$ is 18, and

$(\mathrm{S}_2)$ : The relation $R$ is symmetric but neither reflexive nor transitive

Let the domain of the function $f(x) = \cos^{-1}\left(\frac{4x+5}{3x-7}\right)$ be $[\alpha, \beta]$ and the domain of $g(x) = \log_2\left(2 - 6\log_{27}(2x+5)\right)$ be $(\gamma, \delta)$.

Then $|7(\alpha + \beta) + 4(\gamma + \delta)|$ is equal to __________

Let $A=\{1,2,3,\ldots,10\}$ and $B=\left\{\frac{m}{n}: m, n \in A, m<n \text{ and } \gcd(m, n)=1\right\}$. Then $n(B)$ is equal to :

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

Let $A = \{1, 2, 3, 4\}$ and $B = \{1, 4, 9, 16\}$. Then the number of many-one functions $f : A \to B$ such that $1 \in f(A)$ is equal to :

Let $A = \{1, 2, 3\}$. The number of relations on $A$, containing $(1, 2)$ and $(2, 3)$, which are reflexive and transitive but not symmetric, is ________.

Let $f(x) = \log_e x$ and $g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1}$. Then the domain of $fog$ is

Let $R = \{(1, 2), (2, 3), (3, 3)\}$ be a relation defined on the set $\{1, 2, 3, 4\}$. Then the minimum number of elements, needed to be added in $R$ so that $R$ becomes an equivalence relation, is:

Let $A = \{(x, y) \in \mathbf{R} \times \mathbf{R} : |x+y| \geq 3\}$ and $B = \{(x, y) \in \mathbf{R} \times \mathbf{R} : |x| + |y| \leq 3\}$. If $C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\}$, then $\sum_{(x, y) \in C} |x+y|$ is :

Let $X = \mathbf{R} \times \mathbf{R}$. Define a relation R on X as :

$(a_1, b_1) \text{ R } (a_2, b_2) \Leftrightarrow b_1 = b_2$.

Statement I : R is an equivalence relation.

Statement II : For some $(a, b) \in X$, the set $S = \{(x, y) \in X : (x, y) \text{ R } (a, b)\}$ represents a line parallel to $y = x$.

In the light of the above statements, choose the correct answer from the options given below :

Let $f(x)=\frac{2^{x+2}+16}{2^{2x+1}+2^{x+4}+32}$. Then the value of $8\left(f\left(\frac{1}{15}\right)+f\left(\frac{2}{15}\right)+\ldots+f\left(\frac{59}{15}\right)\right)$ is equal to

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

Let $A = \left\{x \in (0, \pi) - \left\{\dfrac{\pi}{2}\right\} : \log_{(2/\pi)}|\sin x| + \log_{(2/\pi)}|\cos x| = 2\right\}$ and

$B = \left\{x \geq 0 : \sqrt{x}\left(\sqrt{x} - 4\right) - 3\left|\sqrt{x} - 2\right| + 6 = 0\right\}.$ Then $\mathrm{n(A \cup B)}$ is equal to :

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

If $f(x) = \frac{2^x}{2^x + \sqrt{2}}$, $x \in \mathbb{R}$, then $\sum_{k=1}^{81} f\left(\frac{k}{82}\right)$ is equal to

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

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Let $f : [0,\ 3] \to \mathrm{A}$ be defined by $f(x) = 2x^3 - 15x^2 + 36x + 7$ and $g : [0,\ \infty) \to \mathrm{B}$ be defined by $g(x) = \frac{x^{2025}}{x^{2025}+1}$. If both the functions are onto and $S = \{x \in \mathbf{Z} : x \in \mathrm{A} \text{ or } x \in \mathrm{B}\}$, then $n(S)$ is equal to :

If the domain of the function $\sin^{-1}\left(\frac{3x-22}{2x-19}\right)+\log_{e}\left(\frac{3x^{2}-8x+5}{x^{2}-3x-10}\right)$ is $(\alpha, \beta]$, then $3\alpha+10\beta$ is equal to :

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $m$ and $n$ respectively be the least and the most number of students who studied all the three subjects. Then $m+n$ is equal to ________.

Let a relation R on $\mathbb{N} \times \mathbb{N}$ be defined as: $(x_1, y_1)$ R $(x_2, y_2)$ if and only if $x_1 \le x_2$ or $y_1 \le y_2$. Consider the two statements: (I) R is reflexive but not symmetric. (II) R is transitive Then which one of the following is true?

Consider the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \dfrac{2x}{\sqrt{1 + 9x^2}}$. If the composition of $f$, $\underbrace{(f \circ f \circ f \circ \cdots \circ f)}_{10 \text{ times}}(x) = \dfrac{2^{10} x}{\sqrt{1 + 9\alpha x^2}}$, then the value of $\sqrt{3\alpha + 1}$ is equal to ______

Let $A = \{1, 3, 7, 9, 11\}$ and $B = \{2, 4, 5, 7, 8, 10, 12\}$. Then the total number of one-one maps $f : A \to B$, such that $f(1) + f(3) = 14$, is :

If $S = \{a \in \mathbb{R} : |2a - 1| = 3[a] + 2\{a\}\}$, where $[t]$ denotes the greatest integer less than or equal to $t$ and $\{t\}$ represents the fractional part of $t$, then $72\displaystyle\sum_{a \in S} a$ is equal to ________.

Let $f, g : \mathbf{R} \rightarrow \mathbf{R}$ be defined as :

$f(x) = |x - 1|$ and $g(x) = \begin{cases} \mathrm{e}^{x}, & x \geq 0 \\ x + 1, & x \leq 0. \end{cases}$

Then the function $f(g(x))$ is

Let the relations $R_1$ and $R_2$ on the set $X = \{1, 2, 3, ...., 20\}$ 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 number of elements required to be added in $R_1$ and $R_2$, respectively, in order to make the relations symmetric, then $M+N$ equals

The function $f(x) = \dfrac{x^2 + 2x - 15}{x^2 - 4x + 9}$, $x \in \mathbb{R}$ is

Let $A = \{n \in [100, 700] \cap \mathbb{N}: n$ is neither a multiple of 3 nor a multiple of 4$\}$. Then the number of elements in $A$ is

Let $f(x)=\frac{1}{7-\sin 5x}$ be a function defined on $\mathbf{R}$. Then the range of the function $f(x)$ is equal to :

Let $A=\{1, 2, 3, 4, 5\}$. Let $R$ be a relation on $A$ defined by $xRy$ if and only if $4x \le 5y$. Let $m$ be the number of elements in $R$ and $n$ be the minimum number of elements from $A \times A$ that are required to be added to $R$ to make it a symmetric relation. Then $m+n$ is equal to :

Let $[t]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of all prime factors of 2310 and $f : A \rightarrow \mathbb{Z}$ 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

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

Let $f(x) = \begin{cases} -a & \text{if } -a \le x \le 0 \\ x + a & \text{if } 0 < x \le a \end{cases}$ where $a > 0$ and $g(x) = (f(|x|) - |f(x)|)/2$.

Then the function $g : [-a, a] \to [-a, a]$ 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 :

Let $\mathrm{A}=\{2, 3, 6, 7\}$ and $\mathrm{B}=\{4, 5, 6, 8\}$. Let R be a relation defined on $\mathrm{A}\times\mathrm{B}$ by $(\mathrm{a}_1, \mathrm{b}_1)\,\mathrm{R}\,(\mathrm{a}_2, \mathrm{b}_2)$ if and only if $\mathrm{a}_1+\mathrm{a}_2=\mathrm{b}_1+\mathrm{b}_2$. Then the number of elements in R is ________.

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 __________.

Let $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \begin{cases} \log_e x, & x > 0 \\ e^{-x}, & x \le 0 \end{cases}$ and $g(x) = \begin{cases} x, & x \ge 0 \\ e^{x}, & x < 0 \end{cases}$. Then, $gof: \mathbb{R} \to \mathbb{R}$ is :

Let $A = \{1, 2, 3, \ldots, 20\}$. Let $R_1$ and $R_2$ two relation on $A$ such that $R_1 = \{(a, b) : b \text{ is divisible by } a\}$ $R_2 = \{(a, b) : a \text{ is an integral multiple of } b\}$. Then, number of elements in $R_1 - R_2$ is equal to ________.

If the domain of the function $f(x)=\frac{\sqrt{x^2-25}}{(4-x^2)}+\log_{10}(x^2+2x-15)$ is $(-\infty, \alpha) \cup [\beta, \infty)$, then $\alpha^2+\beta^3$ is equal to :

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 $(a, b), (c, d) \in \mathbf{N} \times \mathbf{N}$. Then

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 :

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

If $f(x)=\begin{cases} 2+2x, & -1 \le x < 0 \\ 1-\dfrac{x}{3}, & 0 \le x \le 3 \end{cases}$ ; $g(x)=\begin{cases} -x, & -3 \le x \le 0 \\ x, & 0 < x \le 1 \end{cases}$, then range of $(fog)(x)$ is

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

If the domain of the function $f(x)=\cos^{-1}\left(\frac{2-|x|}{4}\right)+\{\log_e(3-x)\}^{-1}$ is $[-\alpha, \beta)-\{\gamma\}$, then $\alpha+\beta+\gamma$ is equal to :

Let $\mathrm{A}=\{1, 2, 3, ...., 7\}$ and let $\mathrm{P(A)}$ denote the power set of A. If the number of functions $f:\mathrm{A}\to\mathrm{P(A)}$ such that $a\in f(a)$, $\forall\ a\in\mathrm{A}$ is $m^n$, $m$ and $n\in\mathbf{N}$ and $m$ is least, then $m+n$ is equal to ________.

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 subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed in Chemistry, atmost 11 students passed in both Mathematics and Physics, atmost 15 students passed in both Physics and Chemistry, atmost 15 students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is ________.

If the domain of the function $f(x)=\log_e\left(\frac{2x+3}{4x^2+x-3}\right)+\cos^{-1}\left(\frac{2x-1}{x+2}\right)$ is $(\alpha, \beta]$, then the value of $5\beta-4\alpha$ is equal to

The number of symmetric relations defined on the set $\{1, 2, 3, 4\}$ which are not reflexive is ______.

If $f(x)=\frac{4x+3}{6x-4}, x \neq \frac{2}{3}$ and $(fof)(x)=g(x)$, where $g: \mathbb{R}-\left\{\frac{2}{3}\right\} \rightarrow \mathbb{R}-\left\{\frac{2}{3}\right\}$, then $(gogog)(4)$ is equal to

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 S is n. Then, the minimum value of n is ______