Sets, Relations and Functions

The sum of the maximum and minimum values of the function $f(x)=|5x-7|+\left[x^2+2x\right]$ in the interval $\left[\dfrac54,2\right]$, where $[t]$ is the greatest integer $\le t$, is ______.

Let $f:\mathbf R\to\mathbf R$ be a continuous function such that $f(3x)-f(x)=x$. If $f(8)=7$, then $f(14)$ is equal to :

If $f(x)=\begin{cases}x+a, & x\le0\\|x-4|, & x>0\end{cases}$ and $g(x)=\begin{cases}x+1, & x<0\\(x-4)^2+b, & x\ge0\end{cases}$ are continuous on $\mathbf R$, then $(g\circ f)(2)+(f\circ g)(-2)$ is equal to :

Let $A=\{1,2,3,4,5,6,7\}$ and $B=\{3,6,7,9\}$. Then the number of elements in the set $\{C\subseteq A:C\cap B\ne\phi\}$ is ______

Let $R_1$ and $R_2$ be two relations defined on $\mathbf R$ by

$a\,R_1\,b\Leftrightarrow ab\ge0$ and $a\,R_2\,b\Leftrightarrow a\ge b$

Then,

Let $f,g:\mathbf N-\{1\}\to\mathbf N$ be functions defined by $f(a)=\alpha$, where $\alpha$ is the maximum of the powers of those primes $p$ such that $p^\alpha$ divides $a$, and $g(a)=a+1$, for all $a\in\mathbf N-\{1\}$. Then, the function $f+g$ is

The domain of the function $f(x)=\sin^{-1}\left[2x^2-3\right]+\log_2\left(\log_{\frac12}\left(x^2-5x+5\right)\right)$, where $[t]$ is the greatest integer function, is :

The number of functions $f$, from the set $A=\left\{x\in\mathbf N:x^2-10x+9\le0\right\}$ to the set $B=\left\{n^2:n\in\mathbf N\right\}$ such that $f(x)\le(x-3)^2+1$, for every $x\in A$, is ______.

Considering only the principal values of the inverse trigonometric functions, the domain of the function $f(x)=\cos^{-1}\left(\dfrac{x^2-4x+2}{x^2+3}\right)$ is :

For $\alpha\in\mathbf N$, consider a relation R on $\mathbf N$ given by $R=\{(x,y):3x+\alpha y\text{ is a multiple of }7\}$. The relation R is an equivalence relation if and only if :

Let $\alpha$, $\beta$ and $\gamma$ be three positive real numbers. Let $f(x)=\alpha x^5+\beta x^3+\gamma x$, $x\in\mathbf R$ and $g:\mathbf R\to\mathbf R$ be such that $g(f(x))=x$ for all $x\in\mathbf R$. If $a_1,a_2,a_3,\ldots,a_n$ be in arithmetic progression with mean zero, then the value of $f\left(g\left(\dfrac1n\displaystyle\sum_{i=1}^nf(a_i)\right)\right)$ is equal to :

Let $S=\left\{x\in[-6,3]-\{-2,2\}:\dfrac{|x+3|-1}{|x|-2}\ge0\right\}$ and $T=\left\{x\in\mathbf Z:x^2-7|x|+9\le0\right\}$. Then the number of elements in $S\cap T$ is :

Let R be a relation from the set {1, 2, 3, . . . , 60} to itself such that R = {(a, b) : b = pq, where p, q ⩾ 3 are prime numbers}. Then, the number of elements in R is :

The domain of the function $f(x)=\sin^{-1}\left(\dfrac{x^2-3x+2}{x^2+2x+7}\right)$ is :

Let $f:\mathbb R-\{5\}\to\mathbb R$ be defined as $f(x)=\dfrac{2x^2+3x-2}{x-5}$. Then $f$ is

Let $f: \mathbb R \to \mathbb R$ be defined by
$$f(x) = \frac{x^2 - 3x - 6}{x^2 + 2x + 4}$$
Then which of the following statements is (are) TRUE?

Let $E$, $F$ and $G$ be three events having probabilities
$$P(E) = \frac18,\quad P(F) = \frac16\quad\text{and}\quad P(G) = \frac14,\quad\text{and let } P(E\cap F\cap G) = \frac1{10}.$$
For any event $H$, if $H^c$ denotes its complement, then which of the following statements is (are) TRUE?

A number is chosen at random from the set $\{1, 2, 3, \ldots, 2000\}$. Let $p$ be the probability that the chosen number is a multiple of 3 or a multiple of 7. Then the value of $500p$ is ____.

Let $f:\mathbf R\to\mathbf R$ be defined as $f(x)=2x-1$ and $g:\mathbf R-\{1\}\to\mathbf R$ be defined as $g(x)=\dfrac{x-\frac12}{x-1}$. Then the composition function f(g(x)) is :

Let A = {n ∈ ℕ : n is a 3-digit number}

B = {9k + 2 : k ∈ ℕ}

and C = {9k + l : k ∈ ℕ} for some l (0 < l < 9)

If the sum of all the elements of the set A ∩ (B ∪ C) is 274 × 400, then l is equal to _______.