Relations & Functions Questions - JEE

From the relation $2x + y = 41$, the number of elements in the domain of $R = \{1, 2, 3, \ldots, 20\}$ is:

If $P = \{1,2,3,4,5\}$ and $Q = \{a, b, c\}$, then the number of onto functions from $P$ to $Q$ is

Let a function $f: (4, \infty) \to (0, \infty)$ defined as $f(x) = \frac{x^2}{x^2 - 1}$, then $f$ is

A function $f:\mathbb{I}\to\mathbb{I}$ is: $f(n)=n+3$ (odd $n$), $f(n)=n/2$ (even $n$). $k$ is odd and $f(f(f(k)))=27$. Then sum of digits of $k$ is

If $g(x) = x^2 + 2x + 1$ and $(g \circ f)(x) = 4x^2 - 10x + 5$, then $f'(1)$ is equal to

For the function $f(z) = \sin(\lfloor z \rfloor) < \cos^{-1}(\lfloor z \rfloor)$, choose the correct option. (where $\lfloor \cdot \rfloor$ represents the greatest integer function)

Number of integral values of x, such that \(8 is:

136. Let $f:R\to R$ be given as \(f(x)=\begin{cases}2x+\alpha^2, & x\geq 2\\ \dfrac{\alpha x}{2}+10, & x

Sum of two rational numbers is

Let $f$ be a degree-3 polynomial with $f(k)=-2/k$ for $k=2,3,4,5$. Find $52-10f(10)$.

A relation on the set \(A = \{x : |x| R is

Let f : R → R be such that for all x \in R (2^{1+x} + 2^{1-x}), f(x) and (3x + 3-x) are in A.P., then the minimum value of f(x) is

$(p \wedge \sim q) \wedge (\sim p \wedge q)$ is

Let $f(x) = \sin\!\left(\dfrac{\pi}{6}\sin\!\left(\dfrac{\pi}{2}\sin x\right)\right)$ for all $x \in R$. Then the range of $f(x)$, is:

If $a, b \in \mathbb{R}$ be fixed positive numbers such that $f(a + x) = b + [b^3 + 1 - 3b^2 f(x) + 3b\{f(x)\}^2 - \{f(x)\}^3]^{1/3}$ for all $x \in \mathbb{R}$, then prove that $f(x)$ is a periodic function.

Let $n(X)$ denote the number of elements in $X$. If $A \cap B \cap C = \phi$, then find $n(A \cup B \cup C)$ in terms of $\sum n(A)$ and $\sum n(A \cap B)$. Also, given that $n(A \Delta B) = n(A) + n(B) - 2n(A \cap B)$, find $n(A \cup B \cup C)$ when $n(A \Delta B) = n(B \Delta C) = n(C \Delta A) = n(A) = n(B) = n(C)$. What is the value if the answer is 150?

Let $\mathbb{R}$ be the real line. Consider the following subsets of the plane $\mathbb{R} \times \mathbb{R}$.\(S = \{(x, y) : y = x + 1 \text{ and } 0 Which one of the following is true?

For all $x \in \mathbb{R}$, a function $f$ satisfies $f(2+x) = f(2-x) = f[7-(5+x)] = f[7+(5+x)] = f(12+x)$. It is also given that $f(0) = 0$. Find the number of integers $n$ in $[-2010, 2010]$ for which $f(n) = 0$.

Let $\sum_{k=1}^{10} f(a + k) = 16(2^{10} - 1)$, where the function f satisfies $f(x + y) = f(x)f(y)$ for all natural numbers $x, y$ and $f(1) = 2$. then the natural number 'a' is

Let X = \{1, 2, 3, \ldots, 12\} and N be the number of pairs \{A, B\} such that A ⊆ X, B ⊆ X, A ≠ B and A ∩ B = \{2, 3, 5, 7, 8\}. Then the value of N is

If $f(x) \in (1, 2]$, $[f(x)] = 1, 2$, $\dfrac{2(k+1)}{3} = 3 \Rightarrow k = \dfrac{7}{2}$ and $\dfrac{\mu}{3} = 2 \Rightarrow \mu = 6$, find the value of $2k + \mu$.

Consider the graph of $y = f(x)$ with key points $(-5,-1)$, $(-3,2)$, $(-1,1)$, $(0,3)$, $(2,3)$, $(4,2)$ (approaching $y=2$), $(5,-1)$. Find the number of solution(s) of $x$ satisfying $f(f(x)) = 2$.

Which of the following is a singleton set?(a) $ \{ x : x (b) \( \{ x : x = 5, x \in \mathbb{I} \} $(c) $ \{ x : x = 1, x \in \mathbb{I} \} $(d) $ \{ x : x^2 + x + 1 = 0, x \in \mathbb{R} \} $

Let $x$ and $y$ are real numbers satisfying $x^2 + y^2 = 4$, then find the number of integers in the range of $(x^2 - xy + y^2)$.

Two newspapers A and B are published in a city. It is known that 25% of the city population reads A and 20% reads B while 8% reads both A and B. Further, 30% of those who read A but not B look into advertisements and 40% of those who read B but not A also look into advertisements, while 50% of those who read both A and B look into advertisements. Then, the percentage of the population who look into advertisements is

If $f(x) = 2x - \sin x$, $f: \mathbb{R} \to \mathbb{R}$ and $g(x) = x^2$, $g: \mathbb{R} \to \mathbb{R}$, then

Find range of $f(x)=\log_{\sqrt{5}}\left(3+\cos\left(\frac{\pi}{4}+x\right)+\cos\left(\frac{\pi}{4}-x\right)+\cos\left(\frac{3\pi}{4}+x\right)-\cos\left(\frac{3\pi}{4}-x\right)\right)$.

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

Let $f(x)=x^5+2x^3+3x+1$, $x\in\mathbb{R}$, and $g(x)$ be a function such that $g(f(x))=x$ for all $x\in\mathbb{R}$. Then $\dfrac{g(7)}{g'(7)}$ is equal to:

The number of distinct real roots of the equation $|x||x+2|-5|x+1|-1=0$ is ________.

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

If $f:\mathbb{R}\setminus\{1,-1\}\to A$, $f(x)=\dfrac{x^2}{1-x^2}$ is surjective, find $A$.

Find the domain of $f(x)=\dfrac{1}{4-x^2}+\log_{10}(x^3-x)$.

Let the sum of the maximum and the minimum values of the function $f(x)=\dfrac{2x^2-3x+8}{2x^2+3x+8}$ be $\dfrac{m}{n}$, where $\gcd(m,n)=1$. Then $m+n$ is equal to:

JM Q18 — functional equation.

If a $ \in \mathbb{R} $ and the equation $ -3(x - \lfloor x \rfloor)^2 + 2(x - \lfloor x \rfloor) + a^2 = 0 $ (where $ \lfloor x \rfloor $ denotes the greatest integer $ \leq x $) has non integral real solution, then all possible values of $ a $ lie in the interval :

The complete range of values of a such that (1/2)|x| = x2 − a is satisfied for maximum number of values of x is:

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

\[f(x) = \begin{cases} x, & \text{if } x \text{ is rational} \\ 0, & \text{if } x \text{ is irrational} \end{cases}, \quad g(x) = \begin{cases} 0, & \text{if } x \text{ is rational} \\ x, & \text{if } x \text{ is irrational} \end{cases}\]\nThen, $f \circ g$ is

The following statement $(p \to q) \to [(\sim p \to q) \to q]$ is

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

Given $f(x) = \log_e(\sin x)$ and $g(x) = \sin^{-1}(e^{-x})$. If $a$ and $b$ are real numbers such that $(f \circ g)'(\alpha) = -\alpha$ gives $b = -\alpha$, and $(f \circ g)'(x) = -1 \Rightarrow (f \circ g)'(\alpha) = -1$, then find the values of $a$ and $b$.

If a + a = 1, b + b = 2 and af(x) + af$\left(\frac{1}{x}\right)$ = bx + $\frac{b}{x}$, x ≠ 0, then the value of the expression $\frac{f(x) + f\left(\frac{1}{x}\right)}{x + \frac{1}{x}}$ is

$(D): f^{-1}(x) = \frac{1}{2}\log(x + \sqrt{x^2 + 1})$

Suppose that $ f: \mathbb{R} \to \mathbb{R} $ is a continuous function and satisfies the equation $ f(x)\, f(f(x)) = 1 $ for all $ x \in \mathbb{R} $. Further, if $ f(1000) = 999 $, then which of the following options are necessarily true?$ f(500) = \dfrac{1}{500} $$ f(199) = \dfrac{1}{199} $$ f(2000) = \dfrac{1}{2000} $$ f(235) = \dfrac{1}{235} $$ f(1099) = \dfrac{1}{1099} $$ f(x) = \dfrac{1}{x} \; \forall x \in \mathbb{R} - \{0, 1000\} $No such function existsEnter the product of the number of all correct options. For example, if correct options are 2 and 3, then enter 6.

Let f : (1,3) → R be a function defined by f(x) = \frac{[x]}{1+x^2}, where [x] denotes the greatest integer \leq x. Then the range of f is

Total number of functions = $3^5$. Since each of 1, 2, 3, 4, or 5 can correspond to any of $a$, $b$, or $c$. The number of functions that corresponds to only one element of $B$ is $^3C_1 imes 1^3$ and the number of functions that correspond to almost two elements of $B$ is $^3C_2 imes 2^5$. Total number of onto functions = $3^5 - ^3C_1 imes 1^3 - (^3C_2 imes 2^5)$ (using $^3C_1 imes 1^3$ repeated twice in $^3C_2 imes 2^5$). What is the result?

If $f : \mathbb{R} \to \mathbb{R}$, $f ( x ) = ax + \cos x$ is an invertible function, then complete set of values of $a$ is:

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

Relations & Functions Questions (810)