Let $$I(x)=\int \frac{6}{{\sin }^{2}x(1-\cot x{)}^2}dx$$. If $$I(0)=3$$, then $$I\left(\frac{\pi }{12}\right)$$ is equal to

If $$\int \frac{1}{{\mathrm{a}}^2{\sin }^{2}x+{\mathrm{b}}^2{\cos }^{2}x}\mathrm{d}x=\frac{1}{12}{\tan }^{-1}(3\tan x)+$$ constant, then the maximum value of $$\mathrm{a}\sin x+\mathrm{b}\cos x$$, is :

If the domain of the function $f(x)=\frac{\sqrt{x^2-25}}{\left(4-x^2\right)}+{\log }_{10}\left(x^2+2x-15\right)$ is $(-\infty ,\alpha )\cup [\beta ,\infty )$, then ${\alpha }^2+{\beta }^3$ is equal to :

Let $f:\mathbf{R}\to \mathbf{R}$ and $g:\mathbf{R}\to \mathbf{R}$ be defined as $f(x)=\{\begin{array}{ll}{\log }_{\mathrm{e}}x,& x>0\\ {\mathrm{e}}^{-x},& x\le 0\end{array}$ and $g(x)=\{\begin{array}{ll}x,& x⩾0\\ {\mathrm{e}}^x,& x<0\end{array}$. Then, gof : $\mathbf{R}\to \mathbf{R}$ is :

The function $f:\mathbf{N}-\{1\}\to \mathbf{N}$; defined by $f(\mathrm{n})=$ the highest prime factor of $\mathrm{n}$, is :

Let $$f:\mathbf{R}-\left\{\frac{-1}{2}\right\}\to \mathbf{R}$$ and $$g:\mathbf{R}-\left\{\frac{-5}{2}\right\}\to \mathbf{R}$$ be defined as $$f(x)=\frac{2x+3}{2x+1}$$ and $$g(x)=\frac{|x|+1}{2x+5}$$. Then, the domain of the function fog is :

If $$f(x)=\frac{4x+3}{6x-4},x\ne \frac{2}{3}$$ and $$(f\circ f)(x)=g(x)$$, where $$g:\mathbb{R}-\left\{\frac{2}{3}\right\}\to \mathbb{R}-\left\{\frac{2}{3}\right\}$$, then $$(gogog)(4)$$ is equal to

If $$f(x)=\left\{\begin{array}{cc}2+2 x, & -1 $\le$ x

If the domain of the function $$f(x)={\log }_e\left(\frac{2x+3}{4x^2+x-3}\right)+{\cos }^{-1}\left(\frac{2x-1}{x+2}\right)$$ is $$(\alpha ,\beta ]$$, then the value of $$5\beta -4\alpha$$ is equal to

If the domain of the function $$f(x)={\cos }^{-1}\left(\frac{2-|x|}{4}\right)+{\left\{{\log }_e(3-x)\right\}}^{-1}$$ is $$[-\alpha ,\beta )-\{\gamma \}$$, then $$\alpha +\beta +\gamma$$ is equal to :

Let the range of the function $$f(x)=\frac{1}{2+\sin 3x+\cos 3x},x\in \mathbb{R}$$ be $$[a,b]$$. If $$\alpha$$ and $$\beta$$ ar respectively the A.M. and the G.M. of $$a$$ and $$b$$, then $$\frac{\alpha }{\beta }$$ is equal to

If the domain of the function $$f(x)={\sin }^{-1}\left(\frac{x-1}{2x+3}\right)$$ is $$\mathbf{R}-(\alpha ,\beta )$$, then $$12\alpha \beta$$ is equal to :

Let f(x)=\left\{\begin{array}{ccc}-\mathrm{a} & \text { if } & -\mathrm{a} \leq x \leq 0 \\ x+\mathrm{a} & \text { if } & 0 0 and $$\mathrm{g}(x)=(f(|x|)-|f(x)|)/2$$. Then the function $$g:[-a,a]\to [-a,a]$$ is

Let $$A=\{1,3,7,9,11\}$$ and $$B=\{2,4,5,7,8,10,12\}$$. Then the total number of one-one maps $$f:A\to B$$, such that $$f(1)+f(3)=14$$, is :

Let $$f,g:\mathbf{R}\to \mathbf{R}$$ be defined as : $$f(x)=|x-1|\text{ and }g(x)=\{\begin{array}{ll}{\mathrm{e}}^x,& x\ge 0\\ x+1,& x\le 0.\end{array}$$ Then the function $$f(g(x))$$ is

If the function $$f(x)={\left(\frac{1}{x}\right)}^{2x};x>0$$ attains the maximum value at $$x=\frac{1}{\mathrm{e}}$$ then :

Let $$f(x)=\frac{1}{7-\sin 5x}$$ be a function defined on $$\mathbf{R}$$. Then the range of the function $$f(x)$$ is equal to :

The function $$f(x)=\frac{x^2+2x-15}{x^2-4x+9},x\in \mathbb{R}$$ is

If the shortest distance of the parabola $y^2=4x$ from the centre of the circle $x^2+y^2-4x-16y+64=0$ is $\mathrm{d}$, then ${\mathrm{d}}^2$ is equal to :

Let $$PQ$$ be a chord of the parabola $$y^2=12x$$ and the midpoint of $$PQ$$ be at $$(4,1)$$. Then, which of the following point lies on the line passing through the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ ?