Sets, Relations and Functions

If the function $f:\mathbb R\to\mathbb R$ is defined by $f(x) = |x|(x - \sin x)$, then which of the following statements is TRUE?

Let the functions $f:(-1,1)\to\mathbb R$ and $g:(-1,1)\to(-1,1)$ be defined by

$$f(x) = |2x-1| + |2x+1|\quad\text{and}\quad g(x) = x - [x],$$

where $[x]$ denotes the greatest integer less than or equal to $x$. Let $f\circ g:(-1,1)\to\mathbb R$ be the composite function defined by $(f\circ g)(x) = f(g(x))$. Suppose $c$ is the number of points in the interval $(-1,1)$ at which $f\circ g$ is NOT continuous, and suppose $d$ is the number of points in the interval $(-1,1)$ at which $f\circ g$ is NOT differentiable. Then the value of $c + d$ is ______

Let the function $f:[0,1]\to\mathbb R$ be defined by

$$f(x) = \frac{4^x}{4^x + 2}.$$

Then the value of

$$f\left(\frac1{40}\right) + f\left(\frac2{40}\right) + f\left(\frac3{40}\right) + \cdots + f\left(\frac{39}{40}\right) - f\left(\frac12\right)$$

is ______

Let $f:\mathbb R\to\mathbb R$ be given by

$$f(x) = \begin{cases}x^5 + 5x^4 + 10x^3 + 10x^2 + 3x + 1, & x < 0;\\ x^2 - x + 1, & 0 \le x < 1;\\ \frac23x^3 - 4x^2 + 7x - \frac83, & 1 \le x < 3;\\ (x-2)\log_e(x-2) - x + \frac{10}3, & x \ge 3.\end{cases}$$

Then which of the following options is/are correct?

Let $X$ be the set consisting of the first 2018 terms of the arithmetic progression $1, 6, 11, \ldots$, and $Y$ be the set consisting of the first 2018 terms of the arithmetic progression $9, 16, 23, \ldots$. Then, the number of elements in the set $X \cup Y$ is _____.

Let $X$ be a set with exactly 5 elements and $Y$ be a set with exactly 7 elements. If $\alpha$ is the number of one-one functions from $X$ to $Y$ and $\beta$ is the number of onto functions from $Y$ to $X$, then the value of $\frac{1}{5!}(\beta - \alpha)$ is _____ .

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function with $f(0) = 1$ and satisfying the equation $f(x + y) = f(x)f'(y) + f'(x)f(y)$ for all $x, y \in \mathbb{R}$. Then, the value of $\log_e(f(4))$ is _____.

Let $E_1 = \left\{x \in \mathbb{R} : x \ne 1 \text{ and } \frac{x}{x-1} > 0\right\}$ and $E_2 = \left\{x \in E_1 : \sin^{-1}\left(\log_e\left(\frac{x}{x-1}\right)\right) \text{ is a real number}\right\}$.

(Here, the inverse trigonometric function $\sin^{-1} x$ assumes values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.)

Let $f: E_1 \to \mathbb{R}$ be the function defined by $f(x) = \log_e\left(\frac{x}{x-1}\right)$ and $g: E_2 \to \mathbb{R}$ be the function defined by $g(x) = \sin^{-1}\left(\log_e\left(\frac{x}{x-1}\right)\right)$.

The correct option is:

Let $[x]$ be the greatest integer less than or equals to $x$. Then, at which of the following point(s) the function $f(x) = x\cos(\pi(x + [x]))$ is discontinuous?

The function $f:\mathbb R\to\left[-\dfrac12,\dfrac12\right]$ defined as $f(x)=\dfrac x{1+x^2}$, is :

Let a, b, c ∈ ℝ. If $f(x)=ax^2+bx+c$ is such that $a+b+c=3$ and $f(x+y)=f(x)+f(y)+xy$, ∀ x, y ∈ ℝ, then $\displaystyle\sum_{n=1}^{10}f(n)$ is equal to :

For three events A, B and C,

P(Exactly one of A or B occurs) = P(Exactly one of B or C occurs) = P(Exactly one of C or A occurs) = $\dfrac14$ and P(All the three events occur simultaneously) = $\dfrac1{16}$.

Then the probability that at least one of the events occurs, is :

Let $f(x)=2^{10}\cdot x+1$ and $g(x)=3^{10}\cdot x-1$. If $(f\circ g)(x)=x$, then $x$ is equal to :

The function $f:\mathbb N\to\mathbb N$ defined by $f(x)=x-5\left[\dfrac x5\right]$, where $\mathbb N$ is the set of natural numbers and [x] denotes the greatest integer less than or equal to $x$, is :

If $f\left(\dfrac{3x-4}{3x+4}\right)=x+2$, $x\ne-\dfrac43$, and $\displaystyle\int f(x)\,dx=A\log|1-x|+Bx+C$, then the ordered pair (A, B) is equal to :

(where C is a constant of integration)

Let $f:\mathbb R\to\mathbb R$, $g:\mathbb R\to\mathbb R$ and $h:\mathbb R\to\mathbb R$ be differentiable functions such that $f(x)=x^3+3x+2$, $g(f(x))=x$ and $h(g(g(x)))=x$ for all $x\in\mathbb R$. Then

If $f(x)+2f\left(\dfrac1x\right)=3x$, $x\ne0$, and $S=\{x\in\mathbb R:f(x)=f(-x)\}$; then S :

For $x\in\mathbb R$, $x\ne0$, $x\ne1$, let $f_0(x)=\dfrac1{1-x}$ and $f_{n+1}(x)=f_0(f_n(x))$, n = 0, 1, 2, . . . Then the value of $f_{100}(3)+f_1\left(\dfrac23\right)+f_2\left(\dfrac32\right)$ is equal to :

Let $P=\{\theta:\sin\theta-\cos\theta=\sqrt2\cos\theta\}$ and $Q=\{\theta:\sin\theta+\cos\theta=\sqrt2\sin\theta\}$ be two sets. Then :

Let $g:\mathbb R\to\mathbb R$ be a differentiable function with $g(0)=0$, $g'(0)=0$ and $g'(1)\ne0$. Let
$$f(x)=\begin{cases}\dfrac{x}{|x|}g(x),&x\ne0\\0,&x=0\end{cases}$$
and $h(x)=e^{|x|}$ for all $x\in\mathbb R$. Let $(f\circ h)(x)$ denote $f(h(x))$ and $(h\circ f)(x)$ denote $h(f(x))$. Then which of the following is (are) true?