Sets, Relations and Functions

Let $S=\{1,2,3,\dots,10\}$. Consider the set
$$X=\{R:R\text{ is an equivalence relation on the set }S\text{ such that }R\text{ has exactly 42 elements}\}.$$
Then the number of elements in $X$ is ________.

Let $\mathbb N$ denote the set of all positive integers. Consider the sets
$$A=\{1,2,3,4,5\}\quad\text{and}\quad B=\{1,2,3,4,5,6,7\}.$$
Let $S$ be the set of all functions $f:A\to B$ such that $f(2)\ne2$ and $f(4)\ne4$. Consider the set
$$T=\left\{f\in S:\text{there exists a function }g:B\to\mathbb N\text{ such that }g(f(x))=2^x\text{ for all }x\in A\right\}.$$
Then the number of elements in the set $T$ is ________.

For a real number $\alpha$, let $[\alpha]$ denote the greatest integer less than or equal to $\alpha$. For a finite set $S$, let $|S|$ denote the number of elements in the set $S$.

Consider the functions $f:(-3,3)\to(-\infty,\infty)$ and $g:(-3,3)\to(-\infty,\infty)$ defined by
$$f(x)=\left[x^3\right]\log_e\left(1+\sin^2\left(\pi(x-[x])\right)\right)$$
and
$$g(x)=x^3\sin^2\left(\pi\log_e(1+x-[x])\right).$$
Let
$$A=\{x\in(-3,3):f\text{ is discontinuous at }x\}$$
and
$$B=\{x\in(-3,3):g\text{ is discontinuous at }x\}.$$
Then the value of $|A|+2|B|-|A\cap B|$ is ________.

Let $A$ be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let $B$ be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements in $A\cap B$, which are divisible by 3, is :

Let $[\cdot]$ denote the greatest integer function. Then the value of $\displaystyle\int_0^3\left(\frac{e^x+e^{-x}}{[x]!}\right)dx$ is :

If the domain of the function $f(x)=\sqrt{\log_{(0.6)}\left(\left|\dfrac{2x-5}{x^2-4}\right|\right)}$ is $(-\infty,a]\cup\{b\}\cup[c,d)\cup(e,\infty)$, then the value of $a+b+c+d+e$ is ________.

If $\alpha=\displaystyle\int_0^{2\sqrt3}\log_2\left(x^2+4\right)dx+\int_2^4\sqrt{2^x-4}\,dx$, then $\alpha^2$ is equal to ________.

Let $A=\{2,3,4,5,6\}$. Let R be a relation on the set $A\times A$ given by $(x,y)\,R\,(z,w)$ if and only if $x$ divides $z$ and $y\le w$. Then the number of elements in R is ________.

Let $[\cdot]$ denote the greatest integer function. If the domain of the function $f(x)=\cos^{-1}\left(\dfrac{4x+2[x]}3\right)$ is $[\alpha,\beta]$, then $12(\alpha+\beta)$ is equal to:

The number of functions $f:\{1,2,3,4\}\to\{a,b,c\}$, which are not onto, is:

Let $\displaystyle\int_{-2}^2\left(|\sin x|+[x\sin x]\right)dx=2(3-\cos2)+\beta$, where $[\cdot]$ is the greatest integer function. Then $\beta\sin\left(\dfrac\beta2\right)$ equals:

For the function $f:[1,\infty)\to[1,\infty)$ defined by $f(x)=(x-1)^4+1$, among the two statements:

(I) The set $S=\{x\in[1,\infty):f(x)=f^{-1}(x)\}$ contains exactly two elements, and

(II) The set $S=\{x\in[1,\infty):f(x)=f^{-1}(x+1)\}$ is an empty set,

Let for some $\alpha\in\mathbb R$, $f:\mathbb R\to\mathbb R$ be a function satisfying $f(x+y)=f(x)+2y^2+y+\alpha xy$ for all $x,y\in\mathbb R$. If $f(0)=-1$ and $f(1)=2$, then the value of $\displaystyle\sum_{n=1}^5\left(\alpha+f(n)\right)$ is:

Let $f:\mathbb R\to\mathbb R$ be a differentiable function such that $f\left(\dfrac{x+y}3\right)=\dfrac{f(x)+f(y)}3$ for all $x,y\in\mathbb R$, and $f'(0)=3$. Then the minimum value of the function $g(x)=3+e^xf(x)$, is:

Let $A=\{1,2,3,4,5,6\}$. The number of one-one functions $f:A\to A$ such that $f(1)\ge3$, $f(3)\le4$ and $f(2)+f(3)=5$, is ________.

Let $A=\{1,4,7\}$ and $B=\{2,3,8\}$. Then the number of elements, in the relation $R=\left\{\left((a_1,b_1),(a_2,b_2)\right)\in(A\times B)\times(A\times B):a_1+b_2\text{ divides }a_2+b_1\right\}$ is ________.

Let $f:\mathbf R\to\mathbf R$ be a function such that $f(x)+3f\left(\dfrac\pi2-x\right)=\sin x$, $x\in\mathbf R$. Let the maximum value of $f$ on $\mathbf R$ be $\alpha$. If the area of the region bounded by the curves $g(x)=x^2$ and $h(x)=\beta x^3$, $\beta>0$, is $\alpha^2$, then $30\beta^3$ is equal to ________.

Let $[\cdot]$ denote the greatest integer function. If the domain of the function $f(x)=\sin^{-1}\left(\dfrac{x+[x]}3\right)$ is $[\alpha,\beta)$, then $\alpha^2+\beta^2$ is equal to:

Let $e$ be the base of natural logarithm and let $f:\{1,2,3,4\}\to\{1,e,e^2,e^3\}$ and $g:\{1,e,e^2,e^3\}\to\left\{1,\dfrac12,\dfrac13,\dfrac14\right\}$ be two bijective functions such that $f$ is strictly decreasing and $g$ is strictly increasing. If $\phi(x)=\left[f^{-1}\left\{g^{-1}\left(\dfrac12\right)\right\}\right]^x$, then the area of the region $R=\{(x,y):x^2\le y\le\phi(x),\ 0\le x\le1\}$ is:

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