Functions Questions - JEE

Let the function f(x) = x² + x + sin x - cos x be defined on the interval [0, 1]. Find the odd and even extensions of f(x) in the interval [-1, 1].

Let $P(x) = kx^3 + 2k^2x^2 + k^3$. If $(x-2)$ is a factor of $P(x)$, find the sum of all real values of $k$.

Let $f(x) = \dfrac{\sin([x]\pi)}{x^2+2x+3} + \sqrt{x(x-1)+\tfrac{1}{4}} + 2x-1$. Determine the nature of $f:\mathbb{R}\to\mathbb{R}$.

Which is an odd function?(A) $|x-2|+(x+2)\text{sgn}(x+2)$(B) $\frac{x}{e^x-1}+\frac{1}{2x}$(C) $\log(\sin x+\sqrt{1+\sin^2 x})$(D) $e^{-4x}(e^{2x}-1)^4$

Find the period of $f(x)=\min\{\sin x,|x|\}+\left\{\dfrac{x}{\pi}\right\}$

Let $P(x) = x^6 + ax^5 + bx^4 + cx^3 + dx^2 + ex + f$ be a polynomial such that $P(1)=1, P(2)=2, P(3)=3, P(4)=4, P(5)=5, P(6)=6$. Find $P(7)$.

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

Functions Questions (992)