Relations & Functions Questions - JEE

Let $f: \mathbb{R} \to \mathbb{R}$, $f(x) = \dfrac{2x^2-5x+3}{8x^2+9x+11}$. Determine the nature of $f$.

Piecewise functions $f$ and $g$ given. Which are true?(A) $(f+g)(1)=9$ (B) $(f-g)(3.5)=1$ (C) $(fg)(0)=24$ (D) $(f/g)(5)=8/3$

Let $f : \mathbb{R} \to \mathbb{R}$ is defined by $f(x) = \begin{cases} (x+1)^3 & ; x \leq 1 \\ \ln x + (b^2 - 3b + 10) & ; x > 1 \end{cases}$. If $f(x)$ is invertible, then the set of all values of $b$ is:

Let $f(x)=x^2+3x+2$. Which are correct?(A) $f(|x|)=2$ has 1 solution (B) 3 solutions (C) $|f(x)|=0.125$ has 4 solutions (D) $|f(|x|)|=0.125$ has 8 solutions

Find the number of functions from set A = {1, 2, 3} to set B = {1, 2, 3, 4} such that the co-domain contains exactly the element 2.

Which is the simplified representation of $(A' \cap B' \cap C) \cup (B \cap C) \cup (A \cap C)$ where A, B and C are subsets of set X?

Which of the following is not logically equivalent to the following proposition? 'A real number is either rational or irrational'

Let $A$ and $B$ be two sets containing four and two elements respectively. Then the number of subsets for the set $A \times B$, each having at least three elements is

If $(p \wedge \sim q) \wedge (p \wedge r) \rightarrow \sim p \vee q$ is false, then the truth values of p, q and r are, respectively,

The function $f:[0,3] \to [1,29]$, defined by $f(x) = 2x^3 - 15x^2 + 36x + 1$, is:

Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be defined as $f(x)=\begin{cases}\log_e x,&x>0\\e^{-x},&x\leq0\end{cases}$ and $g(x)=\begin{cases}x,&x\geq0\\e^x,&x<0\end{cases}$. Then $g\circ f:\mathbb{R}\to\mathbb{R}$ is:

Consider the following statements:Statement-1: The relation A on the set of integers defined by $x\, A\, y \Leftrightarrow x - y$ is an integer, is an equivalence relation.Statement-2: The relation B on the set of real numbers defined by $x\, B\, y \Leftrightarrow \frac{x}{y}$ is a rational number, is an equivalence relation.

Let A = {1, 2, 3, 4, 5, 6, 7}. The relation R = {(x, y) ∈A × A : x + y = 7} is:

Let $S = \{1, 2, 3, 4\}$. The total number of unordered pairs of disjoint subsets of S is equal to

Determine which of the following is NOT a tautology:(a) (p ∨ q) ∨ (p ∨ (¬q))(b) (p ∧ q) ∨ (p ∧ (¬q))(c) (p ∨ q) ∧ (p ∨ (¬q))(d) (p ∨ q) ∧ ((¬p) ∨ (¬q))

The functions $f(x) = \cos^{-1}\sqrt{1-x^2}$ and $g(x) = \sin^{-1}x$ are identical for $x$ belonging to

ConsiderStatement-1: $(p \wedge \sim q) \wedge (\sim p \wedge q)$ is a fallacy.Statement-2: $(p \rightarrow q) \leftrightarrow (\sim q \rightarrow \sim p)$ is a tautology.

The domain of definition of the function $f(x) = \sqrt{\log_{10}\left(\dfrac{5x - x^2}{4}\right)}$ is

In the truth table for the statement $(\sim p \Rightarrow \sim q) \land (\sim q \Rightarrow \sim p)$, the last column has the truth value in the following order

Suppose $p$: A natural number $n$ is odd and $q$: natural number $n$ is not divisible by 2, then the biconditional statement $p \Leftrightarrow q$ is

The relation R = {(a, b) : gcd(a, b) = 1, 2a ̸= b, a, b ∈Z} is: 11 JEE Main 2019–2024 | Relations Complete Solutions Booklet

$(A \cup B) - (A \cap B)$ is equal to

Let $f(x)=\dfrac{x-2}{x-3}$, $g(x)=2x-3$. Find sum of all $x$ where $f^{-1}(x)+g^{-1}(x)=\frac{13}{2}$.

If $q$ is false and $p \wedge q \leftrightarrow r$ is true, then which one of the following statements is a tautology?

If A, B and C are three sets such that $A \cap B = A \cap C$ and $A \cup B = A \cup C$, then

Which of the following is not a statement?

The contrapositive of the statement "If you are born in India, then you are a citizen of India", is:

Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(x+2) = f(2-x)$ and $f(7+x) = f(7-x)$ for all real numbers $x$. If $f(0) = 0$ and there are at least $m$ number of integer solutions for $f(x) = 0$ in the interval $[-2010, 2010]$, then $m$ can be __________.

In a linear programming problem, the objective function is $z = 4x + 6y$. The corner points are (0, 2), (3, 0), (6, 0), (6, 8), and (0, 5). The minimum value of $z$ occurs at:

●Ex. 17 If A = {2, 3}, B = {4, 5} and C = {5, 6}, then n{(A × B) ∪ (B × C)} is

●15. A ∪ B equals

If $f(x) = \log(x + \sqrt{1 + x^2})$, then $f(x)$ is

The Boolean expression $\sim(p \Rightarrow (\sim q))$ is equivalent to

A function $f$ from the set of natural numbers to integers defined by\[f(n) = \begin{cases} \dfrac{n-1}{2}, & \text{when } n \text{ is odd} \\ -\dfrac{n}{2}, & \text{when } n \text{ is even} \end{cases}\]is

Let A, B and C be finite sets such that $A \cap B \cap C = \phi$ and each one of the sets $A \Delta B$, $B \Delta C$ and $C \Delta A$ has 200 elements. The number of elements in $A \cup B \cup C$ is __________.

If $S$ is the set of all real numbers. A relation $R$ has been defined on $S$ by $aRb \Longleftrightarrow |a - b| \leq 1$, then $R$ is:

For $x \in R - \{0, 1\}$, let $f_1(x) = \dfrac{1}{x}$, $f_2(x) = 1 - x$ and $f_3(x) = \dfrac{1}{1-x}$ be three given functions. If a function, $J(x)$ satisfies $(f_2 \circ J \circ f_1)(x) = f_3(x)$ then $J(x)$ is equal to:

Let A = {1, 2, 3, 4, 5}. R: xRy ⇔4x ≤5y. Let m = |R| and n = minimum elements to add for symmetry. Find m + n:

The sum of all the elements in the range of $f(x)=\text{Sgn}(\sin x)+\text{Sgn}(\cos x)+\text{Sgn}(\tan x)+\text{Sgn}(\cot x)$, $x\neq\dfrac{n\pi}{2}$, $n\in\mathbb{Z}$, where $\text{Sgn}(t)=\begin{cases}1,&t>0\\-1,&t<0\end{cases}$, is:

Given set S = \{1, 2, 3, \ldots, 50\}. Find the number of non-empty subsets of S such that the product of elements is even.

Given $(p \land {\sim}q) \land (p \land r) \to {\sim}p \lor q$ is false. Now with the given truth table, what is the only possible solution of $(p, q, r)$?

If $p$, $q$ and $r$ are simple propositions with truth value true, false and true respectively, then the truth value of $((\sim p \lor q) \land \sim r) \Rightarrow p$

Contrapositive of the statement 'If a function $f$ is differentiable at $a$, then it is also continuous at $a$', is

If the domain of the function $f(x)=\sin^{-1}\!\left(\dfrac{5-x}{2x+3}\right)+\dfrac{1}{\log_e(10-x)}$ is $(-\infty,\alpha]\cup[\beta,\gamma)-\{\delta\}$, then $6(\alpha+\beta+\gamma+\delta)$ is equal to

In a group of adults, 50 speak all three languages (Hindi, English, Urdu). There are 100 adults who speak Hindi or English, 150 adults who speak English or Urdu, and 80 adults who speak Hindi or Urdu. What is the total number of adults who speak exactly two of the three languages?

The maximum value of z will occur at a corner point of the feasible region, where z = 3x − 4y. The corner points are (0, 0), (12, 6), and (0, 4). Find the maximum value of z.

The negation of the statement, 'if a quadrilateral is a square, then it is a rhombus' is

Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can be formed such that Y ⊆ X, Z ⊆ X and Y ∩ Z is empty, is

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

If $p \rightarrow (q \vee r)$ is false, then the truth values of $p$, $q$ and $r$ are, respectively,

Relations & Functions Questions (810)