Let $f(x)=\{\begin{array}{l}x-1,x\text{ is even, }\\ 2x,x\text{ is odd, }\end{array}x\in \mathbf{N}$. If for some $\mathrm{a}\in \mathbf{N},f(f(f(\mathrm{a})))=21$, then $\underset{x\to {\mathrm{a}}^{-}}{\lim }\left\{\frac{|x{|}^3}{\mathrm{a}}-\left[\frac{x}{\mathrm{a}}\right]\right\}$, where $[t]$ denotes the greatest integer less than or equal to $t$, is equal to :

Let $f:\mathbf{R}\to \mathbf{R}$ be defined as : $$f(x)=\{\begin{array}{ll}\frac{a-b\cos 2x}{x^2};& x1\end{array}$$ If $f$ is continuous everywhere in $\mathbf{R}$ and $m$ is the number of points where $f$ is NOT differential then $\mathrm{m}+\mathrm{a}+\mathrm{b}+\mathrm{c}$ equals :

Consider the function. $$f(x)=\{\begin{array}{cc}\frac{\mathrm{a}\left(7x-12-x^2\right)}{\mathrm{b}|x^2-7x+12|}& ,x3\\ & \\ \mathrm{b}& ,x=3,\end{array}$$ where $[x]$ denotes the greatest integer less than or equal to $x$. If $\mathrm{S}$ denotes the set of all ordered pairs (a, b) such that $f(x)$ is continuous at $x=3$, then the number of elements in $\mathrm{S}$ is :

If $\mathrm{a}=\underset{x\to 0}{\lim }\frac{\sqrt{1+\sqrt{1+x^4}}-\sqrt{2}}{x^4}$ and $\mathrm{b}=\underset{x\to 0}{\lim }\frac{{\sin }^{2}x}{\sqrt{2}-\sqrt{1+\cos x}}$, then the value of $a{b}^3$ is :

Consider the function $$f:(0,2)\to \mathbf{R}$$ defined by $$f(x)=\frac{x}{2}+\frac{2}{x}$$ and the function $$g(x)$$ defined by $$g(x)=\left\{\begin{array}{ll} \min \lfloor f(t)\}, & 0

Consider the function $$f:(0,\infty )\to \mathbb{R}$$ defined by $$f(x)=e^{-|{\log }_{e}x|}$$. If $$m$$ and $$n$$ be respectively the number of points at which $$f$$ is not continuous and $$f$$ is not differentiable, then $$m+n$$ is

Let $$g(x)$$ be a linear function and $$f(x)=\{\begin{array}{cl}g(x)& ,x\le 0\\ {\left(\frac{1+x}{2+x}\right)}^{\frac{1}{x}}& ,x>0\end{array}$$, is continuous at $$x=0$$. If $$f^{\prime }(1)=f(-1)$$, then the value $$g(3)$$ is

\lim _\limits{x \rightarrow 0} \frac{e-(1+2 x)^{\frac{1}{2 x}}}{x} is equal to

Let $$f:\mathbf{R}\to \mathbf{R}$$ be a function given by $$f(x)=\{\begin{array}{ll}\frac{1-\cos 2x}{x^2},& x0\end{array}$$ where $$\alpha ,\beta \in \mathbf{R}$$. If $$f$$ is continuous at $$x=0$$, then $${\alpha }^2+{\beta }^2$$ is equal to :

If the function $$f(x)=\{\begin{array}{ll}\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}},& x\ne 0\\ a{\log }_{e}2{\log }_{e}3& ,x=0\end{array}$$ is continuous at $$x=0$$, then the value of $$a^2$$ is equal to

For $$\mathrm{a},\mathrm{b}>0$$, let $$f(x)=\{\begin{array}{ll}\frac{\tan ((\mathrm{a}+1)x)+\mathrm{b}\tan x}{x},& x0\end{array}$$ be a continuous function at $$x=0$$. Then $$\frac{\mathrm{b}}{\mathrm{a}}$$ is equal to :

If the function $$f(x)=\frac{\sin 3x+\alpha \sin x-\beta \cos 3x}{x^3},x\in \mathbf{R}$$, is continuous at $$x=0$$, then $$f(0)$$ is equal to :

Let ,$$f:[-1,2]\to \mathbf{R}$$ be given by $$f(x)=2x^2+x+\left[x^2\right]-[x]$$, where $$[t]$$ denotes the greatest integer less than or equal to $$t$$. The number of points, where $$f$$ is not continuous, is :

\lim _\limits{n \rightarrow \infty} \frac{\left(1^2-1\right)(n-1)+\left(2^2-2\right)(n-2)+\cdots+\left((n-1)^2-(n-1)\right) \cdot 1}{\left(1^3+2^3+\cdots \cdots+n^3\right)-\left(1^2+2^2+\cdots \cdots+n^2\right)} is equal to :

Considering only the principal values of inverse trigonometric functions, the number of positive real values of $$x$$ satisfying $${\tan }^{-1}(x)+{\tan }^{-1}(2x)=\frac{\pi }{4}$$ is :

If $$a={\sin }^{-1}(\sin (5))$$ and $$b={\cos }^{-1}(\cos (5))$$, then $$a^2+b^2$$ is equal to

For $$\alpha ,\beta ,\gamma \ne 0$$, if $${\sin }^{-1}\alpha +{\sin }^{-1}\beta +{\sin }^{-1}\gamma =\pi$$ and $$(\alpha +\beta +\gamma )(\alpha -\gamma +\beta )=3\alpha \beta$$, then $$\gamma$$ equals

Let $$x=\frac{m}{n}$$ ($$m,n$$ are co-prime natural numbers) be a solution of the equation $$\cos \left(2{\sin }^{-1}x\right)=\frac{1}{9}$$ and let $$\alpha ,\beta (\alpha >\beta )$$ be the roots of the equation $$m{x}^2-nx-m+n=0$$. Then the point $$(\alpha ,\beta )$$ lies on the line