Sets, Relations and Functions

The domain of the function $f(x)=\dfrac{\cos^{-1}\left(\dfrac{x^2-5x+6}{x^2-9}\right)}{\log_e\left(x^2-3x+2\right)}$ is :

The number of one-one functions $f:\{a,b,c,d\}\to\{0,1,2,\dots,10\}$ such that $2f(a)-f(b)+3f(c)+f(d)=0$ is ______.

The number of points where the function
$$f(x)=\begin{cases}|2x^2-3x-7| & \text{if } x\le-1\\ [4x^2-1] & \text{if } -1<x<1\\ |x+1|+|x-2| & \text{if } x\ge1,\end{cases}$$
$[t]$ denotes the greatest integer $\le t$, is discontinuous is ______.

Let $x*y=x^2+y^3$ and $(x*1)*1=x*(1*1)$.

Then a value of $2\sin^{-1}\left(\dfrac{x^4+x^2-2}{x^4+x^2+2}\right)$ is

Let
$$f(x)=\begin{cases}\dfrac{\sin(x-[x])}{x-[x]}, & x\in(-2,-1)\\[2mm] \max\{2x,3[|x|]\}, & |x|<1\\[1mm] 1, & \text{otherwise}\end{cases}$$
where $[t]$ denotes greatest integer $\le t$. If $m$ is the number of points where $f$ is not continuous and $n$ is the number of points where $f$ is not differentiable, then the ordered pair $(m,n)$ is:

The sum of all the elements of the set $\{\alpha\in\{1,2,\dots,100\}:\mathrm{HCF}(\alpha,24)=1\}$ is ______.

Let $f:\mathbb N\to\mathbb R$ be a function such that $f(x+y)=2f(x)f(y)$ for natural numbers $x$ and $y$. If $f(1)=2$, then the value of $\alpha$ for which
$$\sum_{k=1}^{10}f(\alpha+k)=\frac{512}3\left(2^{20}-1\right)$$
holds, is :

Let $f:\mathbb R\to\mathbb R$ be defined as
$$f(x)=x^3+x-5$$
If $g(x)$ is a function such that $f(g(x))=x$, $\forall\ x\in\mathbb R$, then $g'(63)$ is equal to ______.

Let $f:\mathbb R\to\mathbb R$ and $g:\mathbb R\to\mathbb R$ be two functions defined by $f(x)=\log_e\left(x^2+1\right)-e^{-x}+1$ and $g(x)=\dfrac{1-2e^{2x}}{e^x}$. Then, for which of the following range of $\alpha$, the inequality
$$f\left(g\left(\frac{(\alpha-1)^2}3\right)\right)>f\left(g\left(\alpha-\frac53\right)\right)$$
holds ?

Let $f:\mathbb R\to\mathbb R$ be a function defined by $f(x)=\left(2\left(1-\dfrac{x^{25}}2\right)\left(2+x^{25}\right)\right)^{\frac1{50}}$. If the function $g(x)=f(f(f(x)))+f(f(x))$, then the greatest integer less than or equal to $g(1)$ is ______.

Let $A=\{x\in\mathbb R:|x+1|<2\}$ and $B=\{x\in\mathbb R:|x-1|\ge2\}$. Then which one of the following statements is NOT true ?

Let $f(x)=\left[2x^2+1\right]$ and $g(x)=\begin{cases}2x-3, & x<0\\2x+3, & x\ge0\end{cases}$, where $[t]$ is the greatest integer $\le t$. Then, in the open interval $(-1,1)$, the number of points where $f\circ g$ is discontinuous is equal to ______.

Let $f(x)=\dfrac{x-1}{x+1}$, $x\in\mathbb R-\{0,-1,1\}$. If $f^{n+1}(x)=f\left(f^n(x)\right)$ for all $n\in\mathbb N$, then $f^6(6)+f^7(7)$ is equal to :

Let $f,g:\mathbb R\to\mathbb R$ be two real valued functions defined as $f(x)=\begin{cases}-|x+3|, & x<0\\e^x, & x\ge0\end{cases}$ and $g(x)=\begin{cases}x^2+k_1x, & x<0\\4x+k_2, & x\ge0\end{cases}$, where $k_1$ and $k_2$ are real constants. If $(g\circ f)$ is differentiable at $x=0$, then $(g\circ f)(-4)+(g\circ f)(4)$ is equal to :

Let $f(x)=2\cos^{-1}x+4\cot^{-1}x-3x^2-2x+10$, $x\in[-1,1]$. If $[a,b]$ is the range of the function $f$, then $4a-b$ is equal to :

Let $A=\{n\in\mathbb N:\text{H.C.F.}(n,45)=1\}$ and Let $B=\{2k:k\in\{1,2,\dots,100\}\}$. Then the sum of all the elements of $A\cap B$ is ______.

Let $f:\mathbb R\to\mathbb R$ be defined as $f(x)=x-1$ and $g:\mathbb R-\{1,-1\}\to\mathbb R$ be defined as $g(x)=\dfrac{x^2}{x^2-1}$.

Then the function $f\circ g$ is :

Let $f:\mathbb R\to\mathbb R$ satisfy $f(x+y)=2^xf(y)+4^yf(x)$, $\forall x,y\in\mathbb R$. If $f(2)=3$, then $14\cdot\dfrac{f'(4)}{f'(2)}$ is equal to ______.

Let $a$ be an integer such that $\displaystyle\lim_{x\to7}\frac{18-[1-x]}{[x-3a]}$ exists, where $[t]$ is greatest integer $\le t$. Then $a$ is equal to :

Let $f:\mathbb R\to\mathbb R$ be a function defined by $f(x)=\dfrac{2e^{2x}}{e^{2x}+e}$. Then $f\left(\dfrac1{100}\right)+f\left(\dfrac2{100}\right)+f\left(\dfrac3{100}\right)+\dots+f\left(\dfrac{99}{100}\right)$ is equal to ______.