Relations & Functions Questions (810)

Let A = {1, 2, 3, \ldots , 10} and B = { m n : m, n \in A, m < n and gcd(m, n) = 1}. Then n(B) is equal to :

The number of non-empty equivalence relations on the set $\{1, 2, 3\}$ is:

Let $f : \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = (2+3a)x^2 + \left(\frac{a+2}{a-1}\right)x + b$, $a \neq 1$. If $f(x+y) = f(x) + f(y) + 1 - \frac{2}{7}xy$, then the value of $28\displaystyle\sum_{i=1}^{5}|f(i)|$ is:

If the domain of the function $f(x)=\sin^{-1}\left(\dfrac{x-1}{2x+3}\right)$ is $\mathbb{R}-(\alpha,\beta)$, then $12\alpha\beta$ is equal to:

Consider the function $f:\left[\dfrac{1}{2},1\right]\to\mathbb{R}$ defined by $f(x)=4\sqrt{2}x^3-3\sqrt{2}x-1$. Consider the statements: (I) The curve $y=f(x)$ intersects the $x$-axis exactly at one point (II) The curve $y=f(x)$ intersects the $x$-axis at $x=\cos\dfrac{\pi}{12}$ Then

If the range of $f(\theta)=\dfrac{\sin^4\theta+3\cos^2\theta}{\sin^4\theta+\cos^2\theta}$, $\theta\in\mathbb{R}$, is $[\alpha,\beta]$, then the sum of the infinite G.P., whose first term is 64 and the common ratio is $\dfrac{\alpha}{\beta}$, is equal to ________.

Let $S = \{p_1, p_2, \ldots, p_{10}\}$ be the set of first ten prime numbers. Let $A = S \cup P$, where $P$ is the set of all possible products of distinct elements of $S$. Then the number of all ordered pairs $(x, y)$, $x \in S$, $y \in A$, such that $x$ divides $y$, is ___

For \(f(x)=\max\{x,x^2,x^3,x^4\}\). Which are correct?(A) \(f(x)=x^4\) for \(x\le -1\) (B) \(f(x)=x^2\) for \(-1 (C) \(f(\frac{1}{2})=\frac{1}{2}\) (D) \(f(\frac{1}{2})=\frac{1}{4}\)

Let $[t]$ be the greatest integer $\leq t$. Let $A$ be the set of all prime factors of 2310 and $f:A\to\mathbb{Z}$ be the function $f(x)=\left[\log_2\left(x^2+\left[\dfrac{x^3}{5}\right]\right)\right]$. The number of one-to-one functions from $A$ to the range of $f$ is:

Let $f: \mathbb{R} - \{0,1\} \to \mathbb{R}$ be a function such that $f(x) + f\!\left(\dfrac{1}{1-x}\right) = 1 + x$. Then $f(2)$ is equal to:

Let f : R \to R be a function defined by a+2 2 f (x) = (2 + 3a)x + ( ) x + b, a \ne 1. If a-1 , then the value of 28 \sum is 2 5 f (x + y) = f (x) + f (y) + 1 - xy |f (i)| 7 i=1

The function f : (-\infty, \infty) \to (-\infty, 1), defined by f (x) = 2 -2 x -x is: 2 +2

Let $f(x)=\begin{cases}-a & \text{if } -a\leq x\leq0\\ x+a & \text{if } 0<x\leq a\end{cases}$ where $a>0$ and $g(x)=\dfrac{f(|x|)-|f(x)|}{2}$. Then the function $g:[-a,a]\to[-a,a]$ is:

Let the range of the function $f(x) = 6 + 16\cos x \cdot \cos\!\left(\frac{\pi}{3} - x\right) \cdot \cos\!\left(\frac{\pi}{3} + x\right) \cdot \sin 3x \cdot \cos 6x$, $x \in \mathbb{R}$, be $[\alpha, \beta]$. Then the distance of the point $(\alpha, \beta)$ from the line $3x + 4y + 12 = 0$ is:

Let $f(x) = \log_e x$ and $g(x) = \dfrac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1}$. Then the domain of $f \circ g$ is:

If the domain of the function $f(x)=\log_e\left(\dfrac{2x+3}{4x^2+x-3}\right)+\cos^{-1}\left(\dfrac{2x-1}{x+2}\right)$ is $(\alpha,\beta]$, then the value of $5\beta-4\alpha$ is equal to

Define a relation R on the interval [0, \pi2 ) by xRy if and only if sec 2 x - tan 2 y = 1. Then R is : ​

Let X = R \times R. Define a relation R on X as : (a1 , b1 ) R (a2 , b2 ) \Leftrightarrow b1 = b2 Statement I : R is an equivalence relation. Statement II : For some (a, b) \in X, the set S = {(x, y) \in X : (x, y)R(a, b)} represents a line parallel to y = x. In the light of the above statements, choose the correct answer from the options given below :

Let $f : [0,3] \to A$ be defined by $f(x) = 2x^3 - 15x^2 + 36x + 7$ and $g : [0, \infty) \to B$ be defined by $g(x) = \dfrac{x^{2025}}{x^{2025}+1}$. If both functions are onto and $S = \{x \in \mathbb{Z} : x \in A \text{ or } x \in B\}$, then $n(S)$ is equal to:

The relation R = {(x, y) : x, y \in Z and x + y is even } is:

Let $f(x) = \dfrac{4^{x+2} + 16}{2 \cdot 2^{2x+1} + 2^{x+4} + 32}$. Then the value of $8\!\left(f\!\left(\frac{1}{15}\right) + f\!\left(\frac{2}{15}\right) + \cdots + f\!\left(\frac{59}{15}\right)\right)$ is equal to:

Let $[x]$ denote the greatest integer less than or equal to $x$. Then the domain of $f(x) = \sec^{-1}(2[x] + 1)$ is:

Let $f : \mathbb{R} - \{0\} \to \mathbb{R}$ be a function such that $f(x) - 6f\!\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}$. If $\lim_{x \to 0}\left(\frac{1}{\alpha x} + f(x)\right) = \beta$; $\alpha, \beta \in \mathbb{R}$, then $\alpha + 2\beta$ is equal to:

Let $f(x)=2^x-x^2$, $x\in\mathbb{R}$. If $m$ and $n$ are respectively the number of points at which the curves $y=f(x)$ and $y=f'(x)$ intersect the $x$-axis, then the value of $m+n$ is

Consider \(f(x) = \frac{|x+1|}{x+1} + \frac{x-1}{|x-1|}\). Which are correct?

Let S = {p , p \ldots . , p 1 2 10 } be the set of first ten prime numbers. Let A = S \cup P , where P is the set of all possible products of distinct elements of S . Then the number of all ordered pairs ( x, y ), x \in S , y \in A, such that x divides y , is ______.

Let R = {(1, 2), (2, 3), (3, 3)} be a relation defined on the set {1, 2, 3, 4}. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:

Let N denote the set of all natural numbers. Define two binary relations on N as \(R_1 = \{(x, y) \in N \times N : 2x + y = 10\}\) and \(R_2 = \{(x, y) \in N \times N : x + 2y = 10\}\). Then

Define a relation $R$ on the interval $\left[0, \frac{\pi}{2}\right)$ by $x\,R\,y$ if and only if $\sec^2 x - \tan^2 y = 1$. Then $R$ is:

Let [x] denote the greatest integer less than or equal to x. Then the domain of f (x) = sec -1 (2[x] + 1) is :

Let $A = \left\{x \in (0,\pi) - \left\{\frac{\pi}{2}\right\} : \log_{2/\pi}|\sin x| + \log_{2/\pi}|\cos x| = 2\right\}$ and $B = \{x \geq 0 : \sqrt{x}(\sqrt{x}-4) - 3|\sqrt{x}-2| + 6 = 0\}$. Then $n(A \cup B)$ is equal to:

Let f (x) = 2 2x+1 +16 x+4 . Then the value of 8 (f ( 1 ) + f ( 2 ) + \ldots + f ( 59 )) is equal to 2 +2 +32 15 15 15

If the domain of the function $\log_5(18x - x^2 - 77)$ is $(\alpha, \beta)$ and the domain of the function $\log_{(x-1)}\!\left(\frac{2x^2+3x-2}{x^2-3x-4}\right)$ is $(\gamma, \delta)$, then $\alpha^2 + \beta^2 + \gamma^2$ is equal to:

If \(f:[\frac{7}{2},\infty)\to[-\frac{9}{4},\infty)\), \(f(x)=x^2-7x+10\), find \(f^{-1}(x)\).

Find the period of \(f(x) = \{x\} + \{x+\frac{1}{3}\} + \{x+\frac{2}{3}\}\)

Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined $f(x)=\dfrac{x}{(1+x^4)^{1/4}}$ and $g(x)=f(f(f(f(x))))$, then $18\displaystyle\int_0^{\sqrt{2\sqrt{5}}}x^2g(x)\,dx$

If the function $f:(-\infty,-1]\to(a,b]$ defined by $f(x)=e^{x^3-3x+1}$ is one-one and onto, then the distance of the point $P(2b+4,a+2)$ from the line $x+e^{-3}y=4$ is:

If \(aN = \{ax : x \in \mathbb{N}\}\), then the set \(4N \cap 6N\) is

The feasible region R is unbounded. The minimum value of z = 3x + 2y is to be found, subject to the constraints. The corner points are (12, 0), (4, 2), and (1, 5). Find the minimum value of z.

If \(f : \mathbb{R} \to \mathbb{R}\), \(f ( x ) = \dfrac{x^2 + ax + 1}{x^2 + x + 1}\), then the complete set of values of \(a\) such that \(f ( x )\) is onto is:

(JEE Main / AIEEE 2009) If \(A\), \(B\), and \(C\) are three sets such that \(A \cap B = A \cap C\) and \(A \cup B = A \cup C\), then

Let \(S = \{1, 2, 3, \ldots, 100\}\). The number of non-empty subsets \(A\) of \(S\) such that the product of elements in \(A\) is even is:

Let $f$ and $g$ be functions satisfying $f(x+y)=f(x)f(y)$, $f(1)=7$ and $g(x+y)=g(xy)$, $g(1)=1$, for all $x,y\in\mathbb{N}$. If $\displaystyle\sum_{x=1}^n\left(\dfrac{f(x)}{g(x)}\right)=19607$, then $n$ is equal to:

If $g(x)=3x^2+2x-3$, $f(0)=-3$ and $4g(f(x))=3x^2-32x+72$, then $f(g(2))$ is equal to:

Statement-1: \(\sim (p \leftrightarrow \sim q)\) is equivalent to \(p \leftrightarrow q\).Statement-2: \(\sim (p \leftrightarrow \sim q)\) is a tautology.

Let

Let \(f(x) = \log_e(\sin x)\), \((0

Match the following columns: Column 1: (i) Range of $sgn\{x\}$ is: (where $\{.\}$ represents fractional part function) (ii) Domain of $\sin^{-1} x + \sin^{-1}(1-x)$ is: (iii) Range of $\sqrt{\frac{2\tan^{-1} x}{\pi}}$ is: (iv) Range of $\frac{2}{\pi} \sin^{-1}[x^2 + x + 1]$ is: (where $[.]$ represent greatest integer function) Column II: (a) $\{1\}$ (b) $[0, 1)$ (c) $0, 1]$ (d) $[0, 1]$

If \(f(x) = \cos[\pi^2]x + \cos[-\pi^2]x\), which one is correct?

Given g(x) = f−1(x). Therefore, f(g(x)) = x. If f(x) = x + {x}5, then g′(x) equals: