Sets, Relations and Functions

Let $R=\left\{(x,y)\in\mathbf N\times\mathbf N:\log_e(x+y)\le2\right\}$. Then the minimum number of elements, required to be added in R to make it a transitive relation, is ________.

Let $f(x)=\begin{cases}x^3+8; & x<0,\\ x^2-4; & x\ge0,\end{cases}$ and $g(x)=\begin{cases}(x-8)^{1/3}; & x<0,\\ (x+4)^{1/2}; & x\ge0.\end{cases}$ Then the number of points, where the function $gof$ is discontinuous, is ________.

Consider the relation R on the set $\{-2,-1,0,1,2\}$ defined by $(a,b)\in R$ if and only if $1+ab>0$. Then, among the statements :

I. The number of elements in R is 17

II. R is an equivalence relation

Let $f:(1,\infty)\to\mathbf R$ be a function defined as $f(x)=\dfrac{x-1}{x+1}$. Let $f^{i+1}(x)=f\left(f^i(x)\right)$, $i=1,2,\dots,25$, where $f^1(x)=f(x)$. If $g(x)+f^{26}(x)=0$, $x\in(1,\infty)$, then the area of the region bounded by the curves $y=g(x)$, $2y=2x-3$, $y=0$ and $x=4$ is :

Let $f$ be a polynomial function such that
$$\log_2(f(x))=\left(\log_2\left(2+\frac23+\frac29+\dots\infty\right)\right)\cdot\log_3\left(1+\frac{f(x)}{f(1/x)}\right),$$
$x>0$ and $f(6)=37$. Then $\displaystyle\sum_{n=1}^{10}f(n)$ is equal to ________.

Let $\mathbb{N}$ denote the set of all natural numbers, and $\mathbb{Z}$ denote the set of all integers. Consider the functions $f : \mathbb{N} \to \mathbb{Z}$ and $g : \mathbb{Z} \to \mathbb{N}$ defined by

$$f(n) = \begin{cases} (n+1)/2 & \text{if } n \text{ is odd},\\ (4-n)/2 & \text{if } n \text{ is even}, \end{cases}$$

and

$$g(n) = \begin{cases} 3 + 2n & \text{if } n \ge 0,\\ -2n & \text{if } n < 0. \end{cases}$$

Define $(g\circ f)(n) = g(f(n))$ for all $n \in \mathbb{N}$, and $(f\circ g)(n) = f(g(n))$ for all $n \in \mathbb{Z}$.

Then which of the following statements is (are) TRUE?

Let the set of all relations $R$ on the set $\{a, b, c, d, e, f\}$, such that $R$ is reflexive and symmetric, and $R$ contains exactly 10 elements, be denoted by $\mathcal{S}$.

Then the number of elements in $\mathcal{S}$ is _____.

Let $\mathbb{R}$ denote the set of all real numbers. Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(x) > 0$ for all $x \in \mathbb{R}$, and $f(x+y) = f(x)f(y)$ for all $x, y \in \mathbb{R}$.

Let the real numbers $a_1, a_2, \ldots, a_{50}$ be in an arithmetic progression. If $f(a_{31}) = 64f(a_{25})$, and

$$\sum_{i=1}^{50} f(a_i) = 3\left(2^{25} + 1\right),$$

then the value of

$$\sum_{i=6}^{30} f(a_i)$$

is _____.

Let $\mathbb{R}$ denote the set of all real numbers. Let $f : \mathbb{R} \to \mathbb{R}$ and $g : \mathbb{R} \to (0, 4)$ be functions defined by

$$f(x) = \log_e\left(x^2 + 2x + 4\right),\quad\text{and}\quad g(x) = \frac{4}{1 + e^{-2x}}.$$

Define the composite function $f\circ g^{-1}$ by $(f\circ g^{-1})(x) = f\left(g^{-1}(x)\right)$, where $g^{-1}$ is the inverse of the function $g$.

Then the value of the derivative of the composite function $f\circ g^{-1}$ at $x = 2$ is _____.

Let $f : \mathbb{R} \to \mathbb{R}$ and $g : \mathbb{R} \to \mathbb{R}$ be functions defined by

$$f(x) = \begin{cases} x|x|\sin\left(\dfrac{1}{x}\right), & x \neq 0, \\ 0, & x = 0, \end{cases} \quad\text{and}\quad g(x) = \begin{cases} 1 - 2x, & 0 \le x \le \dfrac{1}{2}, \\ 0, & \text{otherwise}. \end{cases}$$

Let $a, b, c, d \in \mathbb{R}$. Define the function $h : \mathbb{R} \to \mathbb{R}$ by

$$h(x) = a\,f(x) + b\left(g(x) + g\left(\frac{1}{2} - x\right)\right) + c\,(x - g(x)) + d\,g(x),\quad x \in \mathbb{R}.$$

Match each entry in List-I to the correct entry in List-II.

The correct option is

Let $f : \mathbb{R} \to \mathbb{R}$ be a function defined by

$$f(x) = \begin{cases} x^2\sin\left(\dfrac{\pi}{x^2}\right), & \text{if } x \neq 0, \\ 0, & \text{if } x = 0. \end{cases}$$

Then which of the following statements is TRUE?

Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(x+y) = f(x) + f(y)$ for all $x, y \in \mathbb{R}$, and $g : \mathbb{R} \to (0, \infty)$ be a function such that $g(x+y) = g(x)g(y)$ for all $x, y \in \mathbb{R}$. If $f\left(\dfrac{-3}{5}\right) = 12$ and $g\left(\dfrac{-1}{3}\right) = 2$, then the value of $\left(f\left(\dfrac14\right) + g(-2) - 8\right)g(0)$ is _____.

Let the function $f : [1, \infty) \to \mathbb{R}$ be defined by

$$f(t) = \begin{cases} (-1)^{n+1}\,2, & \text{if } t = 2n-1,\ n \in \mathbb{N}, \\ \dfrac{(2n+1-t)}{2}f(2n-1) + \dfrac{(t-(2n-1))}{2}f(2n+1), & \text{if } 2n-1 < t < 2n+1,\ n \in \mathbb{N}. \end{cases}$$

Define $g(x) = \displaystyle\int_1^x f(t)\,dt$, $x \in (1, \infty)$. Let $\alpha$ denote the number of solutions of the equation $g(x) = 0$ in the interval $(1, 8]$ and $\beta = \displaystyle\lim_{x\to 1+}\frac{g(x)}{x-1}$. Then the value of $\alpha + \beta$ is equal to _____.

If $n(X) = {}^{m}C_6$, then the value of $m$ is _____.

If the value of $n(Y) + n(Z)$ is $k^2$, then $|k|$ is _____.

Let $S = (0,1)\cup(1,2)\cup(3,4)$ and $T = \{0, 1, 2, 3\}$. Then which of the following statements is(are) true?

Let $f : (0,1) \to \mathbb{R}$ be the function defined as $f(x) = \sqrt n$ if $x \in \left[\dfrac{1}{n+1}, \dfrac1n\right)$ where $n \in \mathbb{N}$. Let $g : (0,1) \to \mathbb{R}$ be a function such that $\displaystyle\int_{x^2}^{x}\sqrt{\frac{1-t}{t}}\,dt < g(x) < 2\sqrt x$ for all $x \in (0,1)$. Then $\displaystyle\lim_{x\to0}f(x)g(x)$

Let $n \ge 2$ be a natural number and $f : [0,1] \to \mathbb{R}$ be the function defined by

$$f(x) = \begin{cases} n(1 - 2nx) & \text{if } 0 \le x \le \dfrac{1}{2n}\\[2mm] 2n(2nx - 1) & \text{if } \dfrac{1}{2n} \le x \le \dfrac{3}{4n}\\[2mm] 4n(1 - nx) & \text{if } \dfrac{3}{4n} \le x \le \dfrac1n\\[2mm] \dfrac{n}{n-1}(nx - 1) & \text{if } \dfrac1n \le x \le 1 \end{cases}$$

If $n$ is such that the area of the region bounded by the curves $x = 0$, $x = 1$, $y = 0$ and $y = f(x)$ is 4, then the maximum value of the function $f$ is _____.

In a study about a pandemic, data of 900 persons was collected. It was found that

• 190 persons had symptom of fever,
• 220 persons had symptom of cough,
• 220 persons had symptom of breathing problem,
• 330 persons had symptom of fever or cough or both,
• 350 persons had symptom of cough or breathing problem or both,
• 340 persons had symptom of fever or breathing problem or both,
• 30 persons had all three symptoms (fever, cough and breathing problem).

If a person is chosen randomly from these 900 persons, then the probability that the person has at most one symptom is _____.

The set of all values of $k$ for which $\left(\tan^{-1}x\right)^3+\left(\cot^{-1}x\right)^3=k\pi^3$, $x\in\mathbb R$, is the interval :