Sets, Relations and Functions

Let $S=\{1,2,3,4,5,6,7,8,9,10\}$. Define $f:S\to S$ as
$$f(n)=\begin{cases}2n, & \text{if } n=1,2,3,4,5\\2n-11, & \text{if } n=6,7,8,9,10\end{cases}$$
Let $g:S\to S$ be a function such that $f\circ g(n)=\begin{cases}n+1, & \text{if } n \text{ is odd}\\n-1, & \text{if } n \text{ is even}\end{cases}$.

Then $g(10)\left(g(1)+g(2)+g(3)+g(4)+g(5)\right)$ is equal to ______.

Let $[t]$ denote the greatest integer $\le t$ and $\{t\}$ denote the fractional part of $t$. The integral value of $\alpha$ for which the left hand limit of the function
$$f(x)=[1+x]+\frac{\alpha^{2[x]+\{x\}}+[x]-1}{2[x]+\{x\}}$$
at $x=0$ is equal to $\alpha-\dfrac43$, is ______.

Let a function $f:\mathbb N\to\mathbb N$ be defined by
$$f(n)=\begin{cases}2n, & n=2,4,6,8,\dots\\ n-1, & n=3,7,11,15,\dots\\ \dfrac{n+1}2, & n=1,5,9,13,\dots\end{cases}$$
then, $f$ is

Let $f:\mathbb R\to\mathbb R$ be defined as
$$f(x)=\begin{cases}\left[e^x\right], & x<0\\ ae^x+[x-1], & 0\le x<1\\ b+[\sin(\pi x)], & 1\le x<2\\ \left[e^{-x}\right]-c, & x\ge2\end{cases}$$
where $a,b,c\in\mathbb R$ and $[t]$ denotes greatest integer less than or equal to $t$. Then, which of the following statements is true ?

Let $R_1$ and $R_2$ be relations on the set $\{1,2,\dots,50\}$ such that

$R_1=\left\{\left(p,p^n\right):p\text{ is a prime and }n\ge0\text{ is an integer}\right\}$ and

$R_2=\left\{\left(p,p^n\right):p\text{ is a prime and }n=0\text{ or }1\right\}$.

Then, the number of elements in $R_1-R_2$ is ______.

Let $A=\{1,a_1,a_2,\dots,a_{18},77\}$ be a set of integers with $1<a_1<a_2<\dots<a_{18}<77$. Let the set $A+A=\{x+y:x,y\in A\}$ contain exactly 39 elements. Then, the value of $a_1+a_2+\dots+a_{18}$ is equal to ______.

Let $R_1=\{(a,b)\in\mathbf N\times\mathbf N:|a-b|\le13\}$ and $R_2=\{(a,b)\in\mathbf N\times\mathbf N:|a-b|\ne13\}$. Then on $\mathbf N$ :

Let $f,g:\mathbf R\to\mathbf R$ be functions defined by
$$f(x)=\begin{cases}[x], & x<0\\ |1-x|, & x\ge0\end{cases}\quad\text{and}\quad g(x)=\begin{cases}e^x-x, & x<0\\ (x-1)^2-1, & x\ge0\end{cases}$$
where $[x]$ denote the greatest integer less than or equal to $x$. Then, the function $f\circ g$ is discontinuous at exactly :

The probability that a randomly chosen one-one function from the set $\{a,b,c,d\}$ to the set $\{1,2,3,4,5\}$ satisfies $f(a)+2f(b)-f(c)=f(d)$ is :

Let $S=\{1,2,3,4\}$. Then the number of elements in the set $\{f:S\times S\to S:f\text{ is onto and }f(a,b)=f(b,a)\ge a\ \forall(a,b)\in S\times S\}$ is ______.

Let a set $A=A_1\cup A_2\cup\dots\cup A_k$, where $A_i\cap A_j=\phi$ for $i\ne j$, $1\le i,j\le k$. Define the relation R from A to A by $R=\{(x,y):y\in A_i\text{ if and only if }x\in A_i,\ 1\le i\le k\}$. Then, R is :

The domain of the function $\cos^{-1}\left(\dfrac{2\sin^{-1}\left(\frac1{4x^2-1}\right)}\pi\right)$ is :

Let $c,k\in\mathbf R$. If $f(x)=(c+1)x^2+\left(1-c^2\right)x+2k$ and $f(x+y)=f(x)+f(y)-xy$, for all $x,y\in\mathbf R$, then the value of $\left|2\left(f(1)+f(2)+f(3)+\dots+f(20)\right)\right|$ is equal to ______.

The probability that a relation R from $\{x,y\}$ to $\{x,y\}$ is both symmetric and transitive, is equal to

Let $f(x)$ and $g(x)$ be two real polynomials of degree 2 and 1 respectively. If $f(g(x))=8x^2-2x$, and $g(f(x))=4x^2+6x+1$, then the value of $f(2)+g(2)$ is ______.

The total number of functions, $f:\{1,2,3,4\}\to\{1,2,3,4,5,6\}$ such that $f(1)+f(2)=f(3)$, is equal to :

Let $f(x)=\begin{cases}\left|4x^2-8x+5\right|, & \text{if }8x^2-6x+1\ge0\\ \left[4x^2-8x+5\right], & \text{if }8x^2-6x+1<0,\end{cases}$ where $[\alpha]$ denotes the greatest integer less than or equal to $\alpha$. Then the number of points in $\mathbf R$ where $f$ is not differentiable is ______.

The number of bijective functions $f:\{1,3,5,7,\dots,99\}\to\{2,4,6,8,\dots,100\}$, such that $f(3)\ge f(9)\ge f(15)\ge f(21)\ge\dots\ge f(99)$, is ______

Let $A=\{1,2,3,4,5,6,7\}$. Define $B=\{T\subseteq A:\text{ either }1\notin T\text{ or }2\in T\}$ and $C=\{T\subseteq A:T\text{ the sum of all the elements of }T\text{ is a prime number}\}$. Then the number of elements in the set $B\cup C$ is ______.

Let $f(x)$ be a quadratic polynomial with leading coefficient 1 such that $f(0)=p$, $p\ne0$, and $f(1)=\dfrac13$. If the equations $f(x)=0$ and $f\circ f\circ f\circ f(x)=0$ have a common real root, then $f(-3)$ is equal to ______.