Sets Relations and Functions — JEE Main PYQs

233 previous year questions on Sets Relations and Functions (Mathematics), newest first. Open any question for its complete step-by-step solution.

Let a function $f : (0, \infty) \to (0, \infty)$ be defined by $f (x) = 1 - \frac{1}{x}$ . Then $f$ is:
JEE Main 2019View solution →
For $x \in (0, \frac{3}{2})$ , let $f (x) = x, g (x) = tan x$ and $h (x) = \frac{1 - x ^{2}}{1 + x ^{2}}$ . If $ϕ (x) = ((h of) og) (x)$ , then $ϕ (\frac{π}{3})$ is equal to :
JEE Main 2019View solution →
For $x \in R$ , let $[x]$ denote the greatest integer $\leq x$ , then the sum of the series …
JEE Main 2019View solution →
Let $f : R \to R$ be defined by $f (x) = \frac{x}{1 + x ^{2}}, x \in R$ . Then the range of $f$ is:
JEE Main 2019View solution →
Let $A, B$ and C be sets such that $ϕ \neq = A \cap B \subseteq C$ . Then which of the following statements is not true?
JEE Main 2019View solution →
Let Z be the set of integers. If
$A = {x \in Z : 2^{(x + 2) (x^{2} - 5 x + 6)} = 1}$ and $B = {x \in Z : - 3 < 2 x - 1 < 9}$ , then the number of subsets of the set $A \times B$ , is :
JEE Main 2019View solution →
The domain of the definition of the function $f (x) = \frac{1}{4 - x ^{2}} + lo g (x^{3} - x)$ is
JEE Main 2019View solution →
Let $A = {x \in R : x$ is not a positive integer $}$ Define a function $f : A \to R$ as $f (x) = \frac{2 x}{x - 1}$ then $f$ is
JEE Main 2019View solution →
For $x \in R - [0, 1]$ , let $f_{1} (x) = \frac{1}{x}, f_{2} (x) = 1 - x$ and $f_{3} (x) = \frac{1}{1 - x}$ be three given functions. If a function, $J (x)$ satisfies …
JEE Main 2019View solution →
Let $f (x) = x^{2}, x \in R$ . For any $A \subseteq R$ , define $g (A) = {x \in R : f (x) \in A}$ . If $S = [0, 4]$ , then which one of the following statements is not true?
JEE Main 2019View solution →
Two sets A and B are as under : $A = {(a, b) \in R \times R : ∣ a - 5∣ < 1$ and $∣ b - 5∣ < 1}; B = {(a, b) \in R \times R : 4 (a - 6)^{2} + 9 (b - 5)^{2} \leq 36}$ . Then :
JEE Main 2018View solution →
The function $f : R \to [- \frac{1}{2}, \frac{1}{2}]$ defined as $f (x) = \frac{x}{1 + x ^{2}}$ , is :
JEE Main 2017View solution →
If $f (x) + 2 f (\frac{1}{x}) = 3 x, x \neq = 0$ , and $S = {x \in R : f (x) = f (- x)}$ ; then S :
JEE Main 2016View solution →
Let A and B be two sets containing four and two elements respectively. Then the number of subsets of the set $A \times B$ , each having at least three elements is :
JEE Main 2015View solution →
If $a \in R$ and the equation $- 3 (x - [x])^{2} + 2 (x - [x]) + a^{2} = 0$ (where $[x]$ denotes the greatest integer $\leq x$ ) has no integral solution, then all possible values of $a$ lie in the interval
JEE Main 2014View solution →
If $g$ is the inverse of a function $f$ and $f^{'} (x) = \frac{1}{1 + x ^{5}}$ , then $g^{'} (x)$ is equal to
JEE Main 2014View solution →
Let $A$ and $B$ be two sets containing 2 elements and 4 elements respectively. The number of subsets of $A \times B$ having 3 or more elements is
JEE Main 2013View solution →
Let $X = {1, 2, 3, 4, 5}$ . The number of different ordered pairs $(Y, Z)$ that can be formed such that $Y \subseteq X, Z \subseteq X$ and $Y \cap Z$ is empty, is
JEE Main 2012View solution →
Let R be the set of real numbers

Statement-1 : $A = {(x, y) \in R \times R : y - x$ is an integer $}$ is an equivalence relation on $R$ .
Statement-2: $B = {(x, y) \in R \times R : x = α y$ for some rational number $α}$ is an equivalence relation on R .
JEE Main 2011View solution →
The domain of the function $f (x) = \frac{1}{∣ x ∣ - x}$ is
JEE Main 2011View solution →
Consider the following relations:
$R = {(x, y) ∣ x, y$ are real numbers and $x = w y$ for some rational number $w}$ ; …
JEE Main 2010View solution →
If $A, B$ and $C$ are three sets such that $A \cap B = A \cap C$ and $A \cup B = A \cup C$ , then
JEE Main 2009View solution →
For real $x$ , let $f (x) = x^{3} + 5 x + 1$ , then
JEE Main 2009View solution →
Let $f (x) = (x + 1)^{2} - 1, x \geq - 1$

Statement-1: The set ${x : f (x) = f^{- 1} (x)} = {0, - 1}$
Statement-2: $f$ is a bijection.
JEE Main 2009View solution →
Let $f : N \to Y$ be a function defined as $f (x) = 4 x + 3$ , where $Y = {y \in N : y = 4 x + 3$ for some $x \in N}$ . Show that $f$ is invertible and its inverse is
JEE Main 2008View solution →
Let R be the real line. Consider the following subsets of the plane $R \times R$ .
$S = {(x, y) : y = x + 1$ and $0 < x < 2}, T = {(x, y) : x - y$ is an integer $}$ . Wh …
JEE Main 2008View solution →
The largest interval lying in $(- \frac{π}{2}, \frac{π}{2})$ for which the function $[f (x) = 4^{- x^{2}} + cos^{- 1} (\frac{x}{2} - 1) + lo g (cos x)]$ is defined, is
JEE Main 2007View solution →
Let W denote the words in the English dictionary. Define the relation R by:
$R = {(x, y) \in W \times W ∣$ the words x and y have at least one letter in common $}$ . Then R is
JEE Main 2006View solution →
Let $R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)}$ be a relation on the set $A = {3, 6, 9, 12}$ be a relation on the set $A = {3, 6, 9, 12}$ . The relation is
JEE Main 2005View solution →
A real valued function $f (x)$ satisfies the functional equation $f (x - y) = f (x) f (y) - f (a - x) f (a + y)$ where $a$ is a given constant and $f (0) = 1$ , $f (2 a - x)$ is equal to
JEE Main 2005View solution →
Let $R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)}$ be a relation on the set $A = {1, 2, 3, 4}$ . The relation $R$ is
JEE Main 2004View solution →
If $f : R \to S$ , defined by $f (x) = sin x - 3 cos x + 1$ , is onto, then the interval of $S$ is
JEE Main 2004View solution →
The graph of the function $y = f (x)$ is symmetrical about the line $x = 2$ , then
JEE Main 2004View solution →