Relations & Functions Questions - JEE

Let $'a'$ be the real root of the equation $x^3 - 3x^2 + 5x - 17 = 0$ and $'b'$ be the real root of the equation $x^3 - 3x^2 + 5x + 11 = 0$. Then $a + b = $ ____.

Let $f(x) = 14^{\sin^2 x} + 14^{\cos^2 x}$. The number of integral values that $f(x)$ can take is ____.

The number of real solutions to the equation $3x - 7 = [x^2 - 3x + 2]$ is ____. $[x]$ denotes greatest integer $\leq x$.

Let $f(x) = (x+1)(x+2)(x+3)(x+4) + 5$ ; where $x \in [-6, 6]$. If the range of the function is $[a, b]$; where $a, b \in \mathbb{N}$, then find the value of $(a + b)$.

If $f(2x+1) = 4x^2 + 14x$, then find the sum of the squares of roots of the equation $f(x) = 0$.

Let $g(x) = \frac{e^x - e^{-x}}{2}$ and $g(f(x)) = x$, then evaluate $f\left(\frac{e^{22} - 1}{2e^{11}}\right)$.

Let $f(x) = \sin^3 x - \sin x \cos x + \cos^3 x$, then range of $f(x)$ is:

Let $f(x) = 1 + 2\cos x + 3\sin x$. If real numbers $a, b, c$ are such that $a f(x) + b f(-x) = 1$ holds for any $x \in \mathbb{R}$ then $\frac{b\cos c}{a} =$

The number of positive integers $x$ that satisfy $3^x = x^3 + 3x^2 + 2x + 1$ is:

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \log_{\sqrt{m}}\left\{\sqrt{2}(\sin x - \cos x) + m - 2\right\}$, for some $m$, such that the range of $f$ is $[0, 2]$. Then the value of $m$ 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:

Which of the following is the negation of the statement $p \rightarrow (\sim p \vee \sim q)$?

Let $\mathbb{R}$ be the set of real numbers.Statement-1: $A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : y - x \text{ is an integer}\}$ is an equivalence relation on $\mathbb{R}$.Statement-2: $B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x = \alpha y \text{ for some rational number } \alpha\}$ is an equivalence relation on $\mathbb{R}$.

The contrapositive of the statement "I go to school if it does not rain" is:

Let $ f(x_1) = f(x_2) $, $ x_1, x_2 \in \mathbb{N} $ and $ f(x) = 4x + 3 $. Which of the following is true?

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

Let $ f $ be an invertible function from $ R \to R $ satisfying the equation \[ f^3(x) - (x^3 + 2)f^2(x) + (2x^3 + 1)f(x) - x^3 = 0. \] Then the value of $ f'(8) \times (f^{-1})'(8) $ is:

If $f(y) = \frac{x+1}{x-1}$, then $f(y)$ is equal to

Given f : R → R be defined by $ f(x) = \dfrac{x}{1+x^2},\ x \in R $. Then f(x) ∈

Let $P(A)$ be defined as power set of set $A=\{1,2,3,\ldots,2025\}$. Let $R$ be a relation defined on $P(A)$ as $(B,C)\in R$ if $B$ is superset of $C$, then $R$ is

Domain of definition of the function $f(x) = \dfrac{3}{4 - x^2} + \log_{10}(x^3 - x)$ is

A function $f: R \to R$ satisfies the equation $f(x)f(y) - f(xy) = x + y,\ \forall\ x, y \in R$ and $f(1) > 0$, then:

Let $f(x)$ and $g(x)$ are two functions defined from $R^+ \to R$ such that\[f(x) = \begin{cases} 1 - \sqrt{x}, & \text{if } x \text{ is rational} \\ x^2, & \text{if } x \text{ is irrational} \end{cases}\] and \[g(x) = \begin{cases} x, & \text{if } x \text{ is rational} \\ 1 - x, & \text{if } x \text{ is irrational} \end{cases}\]The composite function $f(g(x))$ is:

Consider the graph of $y = f(x)$ with key points $(-5,-1)$, $(-3,2)$, $(-1,1)$, $(0,3)$, $(2,3)$, $(5,-1)$ and a horizontal asymptote $y=2$. Find the number of solution(s) of $x$ satisfying $f(f(x)) = 2$.

Contrapositive of the statement, "If a number is divisible by 9, then it is divisible by 3" is

If $ A $ and $ B $ are two equivalence relations defined on set $ C $, then which of the following is always true?

The negation of $\sim s \vee (\sim r \wedge s)$ is equivalent to

If $ y = f(x) $ is onto and $ 1 , find the number of integers \([c]$ can take, i.e., find the value of $[c]_{\max}$.

Consider the statement: "P(n) : n2 − n + 41 is prime". Then which one of the following is true?

Given $ f(x) = \dfrac{2x}{x-1} $. Which of the following statements is correct?

If $f(\ln(1+|x|)) = (1 - \ln(1+|x|))^{\frac{1}{7}}$, then $f(f(\cos x))$ is equal to:

Which of the following is true regarding $p \to (q \to p)$?

Let P be the relation defined on the set of all real numbers such that $P = \{(a, b): \sec^2 a - \tan^2 b = 1\}$. Then P is

Given $p \rightarrow (\sim q \vee r) \equiv \sim p \wedge (\sim q \vee r)$. The truth table for $\sim(p \wedge q) \vee r$ is given. Which option correctly represents the equivalent statement?

Consider: Statement I: $(p \wedge \sim q) \wedge (\sim p \wedge q)$ is a fallacy. Statement II: $(p \to q) \leftrightarrow (\sim q \to \sim p)$ is a tautology. (1) Statement I is true; statement II is true; statement II is a correct explanation for statement I. (2) Statement I is true; statement II is true; statement II is not a correct explanation for statement I. (3) Statement I is true; statement II is false. (4) Statement I is false; statement II is true.

Let $S$ be a non-empty subset of $\mathbb{R}$. Consider the following statement: $P$: There is a rational number $x \in S$ such that $x > 0$. Which of the following statements is the negation of the statement $P$?

Consider the following two statements:Statement p: The value of sin 120° can be derived by taking $\theta = 240°$ in the equation $2\sin\dfrac{\theta}{2} = \sqrt{1+\sin\theta} - \sqrt{1-\sin\theta}$.Statement q: The angles A, B, C and D of any quadrilateral ABCD satisfy the equation $\cos\left(\dfrac{1}{2}(A+C)\right) + \cos\left(\dfrac{1}{2}(B+D)\right) = 0$Then the truth values of p and q are, respectively,

Set \(A = \{x : |x| R is

If $A \cup B = A \cup C$ and $A \cap B = A \cap C$, then which of the following is true?

Let $f(x) = (x+2)^2 - 2,\ x \geq -2$. If $g(x)$ is a function whose graph is reflection of the graph of $y = f(x)$ in the line $y = x$, then $g(x)$ is equal to:

The domain of $f(x) = \sqrt{\log\left|\frac{1}{\sin x}\right|}$ is

The range of the function $f(x) = x^2 + \dfrac{1}{x^2 + 1}$ is:

The range of the function $ f(x) = \sin^{-1}\!\left(\log_2 \dfrac{x^2}{2}\right) $ is:

If $ y = \dfrac{x^2 - x + c}{x^2 + x + c} $ and $ y \leq 1 - \dfrac{2}{1 - 2\sqrt{c}} = \dfrac{5}{3} $, find the value of $c$.

Contrapositive of the statement"If two numbers are not equal, then their squares are not equal." is:

Let $ R $ be a relation defined on $ \mathbb{Z} $ by $ (a,b) \in R \Rightarrow a - b = nm $ where $ m \in \mathbb{Z} $ and $ m \neq 0 $. Which of the following is true?

We have $f(x + y) = f(x) + f(y)$ for all $x, y \in \mathbb{R}$, $f(0) = 0$, $f(1) = 7$. Then $\displaystyle\sum_{r=1}^{n} f(r)$ equals:

Find all values of $f(x)$ for which $f(x) = x + \sqrt{x^2}$.

Given $\displaystyle\sum_{k=1}^{10} f(a+k) = 16(2^{10}-1)$, where $f(x+y) = f(x)f(y)$ for all $x, y \in \mathbb{N}$ and $f(1) = 2$. Find the value of $a$.

A function $f: R \to R$ satisfies the equation $f(x)f(y) - f(xy) = x + y$, $\forall\, x, y \in R$ and $f(1) > 0$, then:

Relations & Functions Questions (810)