Sets, Relations and Functions

Let $f(x)=\sin\left(\dfrac\pi6\sin\left(\dfrac\pi2\sin x\right)\right)$ for all $x\in\mathbb R$ and $g(x)=\dfrac\pi2\sin x$ for all $x\in\mathbb R$. Let $(f\circ g)(x)$ denote $f(g(x))$ and $(g\circ f)(x)$ denote $g(f(x))$. Then which of the following is (are) true?

Let A and B be two sets containing four and two elements respectively. Then the number of subsets of the set A × B, each having at least three elements is :

In a certain town, 25% of the families own a phone and 15% own a car ; 65% families own neither a phone nor a car and 2,000 families own both a car and a phone. Consider the following three statements :

(a) 5% families own both a car and a phone.

(b) 35% families own either a car or a phone.

(c) 40,000 families live in the town.

Then,

Let $A=\{x_1,x_2,\dots,x_7\}$ and $B=\{y_1,y_2,y_3\}$ be two sets containing seven and three distinct elements respectively. Then the total number of functions $f:A\to B$ that are onto, if there exist exactly three elements $x$ in A such that $f(x)=y_2$, is equal to :

Let $f:\mathbb R\to\mathbb R$ be a function such that $f(2-x)=f(2+x)$ and $f(4-x)=f(4+x)$, for all $x\in\mathbb R$ and $\displaystyle\int_0^2f(x)\,dx=5$. Then the value of $\displaystyle\int_{10}^{50}f(x)\,dx$ is :

Let $f:(0,\infty)\to\mathbb R$ be given by
$$f(x)=\int_{1/x}^{x}e^{-\left(t+\frac1t\right)}\frac{dt}t.$$
Then

Let $f:\left(-\frac\pi2,\frac\pi2\right)\to\mathbb R$ be given by
$$f(x)=\left(\log(\sec x+\tan x)\right)^3.$$
Then

Let $f:\mathbb R\to\mathbb R$ and $g:\mathbb R\to\mathbb R$ be respectively given by $f(x)=|x|+1$ and $g(x)=x^2+1$. Define $h:\mathbb R\to\mathbb R$ by
$$h(x)=\begin{cases}\max\{f(x),g(x)\}&\text{if }x\le0,\\\min\{f(x),g(x)\}&\text{if }x>0.\end{cases}$$
The number of points at which $h(x)$ is not differentiable is

Let $f:[0,4\pi]\to[0,\pi]$ be defined by $f(x)=\cos^{-1}(\cos x)$. The number of points $x\in[0,4\pi]$ satisfying the equation
$$f(x)=\frac{10-x}{10}$$
is

Let $f_1:\mathbb R\to\mathbb R$, $f_2:[0,\infty)\to\mathbb R$, $f_3:\mathbb R\to\mathbb R$ and $f_4:\mathbb R\to[0,\infty)$ be defined by
$$f_1(x)=\begin{cases}|x| & \text{if } x<0,\\ e^x & \text{if } x\ge0;\end{cases}\qquad f_2(x)=x^2;$$
$$f_3(x)=\begin{cases}\sin x & \text{if } x<0,\\ x & \text{if } x\ge0\end{cases}$$
and
$$f_4(x)=\begin{cases}f_2(f_1(x)) & \text{if } x<0,\\ f_2(f_1(x))-1 & \text{if } x\ge0.\end{cases}$$

If $X=\{4^n-3n-1:n\in N\}$ and $Y=\{9(n-1):n\in N\}$, where N is the set of natural numbers, then $X\cup Y$ is equal to :

If $a\in\mathbf R$ and the equation
$$-3(x-[x])^2+2(x-[x])+a^2=0$$
(where $[x]$ denotes the greatest integer $\le x$) has no integral solution, then all possible values of $a$ lie in the interval :

If $g$ is the inverse of a function $f$ and $f'(x)=\dfrac{1}{1+x^5}$, then $g'(x)$ is equal to :

Let P be the relation defined on the set of all real numbers such that
$$P=\left\{(a,b):\sec^2a-\tan^2b=1\right\}.$$
Then P is :

If $f(x)=\left(\dfrac35\right)^x+\left(\dfrac45\right)^x-1$, $x\in\mathbf R$, then the equation $f(x)=0$ has :

Let $f$ be an odd function defined on the set of real numbers such that for $x\ge0$,
$$f(x)=3\sin x+4\cos x.$$
Then $f(x)$ at $x=-\dfrac{11\pi}6$ is equal to :

A relation on the set $A=\{x:|x|<3,\ x\in Z\}$, where $Z$ is the set of integers is defined by $R=\{(x,y):y=|x|,\ x\ne-1\}$. Then the number of elements in the power set of R is :

If $f(x)=x^2-x+5$, $x>\dfrac12$, and $g(x)$ is its inverse function, then $g'(7)$ equals :

Let $f:\mathbb R\to\mathbb R$ be defined by $f(x)=\dfrac{|x|-1}{|x|+1}$ then $f$ is :

Let $f(n)=\left[\dfrac13+\dfrac{3n}{100}\right]n$, where $[n]$ denotes the greatest integer less than or equal to $n$. Then $\displaystyle\sum_{n=1}^{56}f(n)$ is equal to :