The domain of f(x) = {{{{\log }_{(x + 1)}}(x - 2)} \over {{e^{2{{\log }_e}x}} - (2x + 3)}},x \in \mathbb{R} is

Let $$f:R\to R$$ be a function such that f(x) = {{{x^2} + 2x + 1} \over {{x^2} + 1}}. Then

The number of functions $$f:\{1,2,3,4\}\to \{a\in Z|a|\le 8\}$$ satisfying f(n) + {1 \over n}f(n + 1) = 1,\forall n \in \{ 1,2,3\} is

Let $$f:\mathbb{R}\to \mathbb{R}$$ be a function defined by $$f(x)={\log }_{\sqrt{m}}\{\sqrt{2}(\sin x-\cos x)+m-2\}$$, for some $$m$$, such that the range of $$f$$ is [0, 2]. Then the value of $$m$$ is _________

Let $$f(x)=2x^n+\lambda ,\lambda \in R,n\in N$$, and $$f(4)=133,f(5)=255$$. Then the sum of all the positive integer divisors of $$(f(3)-f(2))$$ is

Let $$f(x)$$ be a function such that $$f(x+y)=f(x).f(y)$$ for all $$x,y\in \mathbb{N}$$. If $$f(1)=3$$ and $$\sum _{k=1}^{n}f(k)=3279$$, then the value of n is

If f(x) = {{{2^{2x}}} \over {{2^{2x}} + 2}},x \in \mathbb{R}, then f\left( {{1 \over {2023}}} \right) + f\left( {{2 \over {2023}}} \right)\, + \,...\, + \,f\left( {{{2022} \over {2023}}} \right) is equal to

The range of $$f(x)=4{\sin }^{-1}\left(\frac{x^2}{x^2+1}\right)$$ is

For $$x\in \mathbb{R}$$, two real valued functions $$f(x)$$ and $$g(x)$$ are such that, $$g(x)=\sqrt{x}+1$$ and $$f\circ g(x)=x+3-\sqrt{x}$$. Then $$f(0)$$ is equal to

Let $$\mathrm{D}$$ be the domain of the function $$f(x)={\sin }^{-1}\left({\log }_{3x}\left(\frac{6+2{\log }_{3}x}{-5x}\right)\right)$$. If the range of the function $$\mathrm{g}:\mathrm{D}\to \mathbb{R}$$ defined by $$\mathrm{g}(x)=x-[x],([x]$$ is the greatest integer function), is $$(\alpha ,\beta )$$, then $${\alpha }^2+\frac{5}{\beta }$$ is equal to

The domain of the function $$f(x)=\frac{1}{\sqrt{[x{]}^2-3[x]-10}}$$ is : ( where $$[\mathrm{x}]$$ denotes the greatest integer less than or equal to $$x$$ )

If f(x) = {{(\tan 1^\circ )x + {{\log }_e}(123)} \over {x{{\log }_e}(1234) - (\tan 1^\circ )}},x > 0, then the least value of f(f(x)) + f\left( {f\left( {{4 \over x}} \right)} \right) is :

Let the sets A and B denote the domain and range respectively of the function $$f(x)=\frac{1}{\sqrt{\lceil x\rceil -x}}$$, where $$\lceil x\rceil$$ denotes the smallest integer greater than or equal to $$x$$. Then among the statements (S1) : $$A\cap B=(1,\infty )-\mathbb{N}$$ and (S2) : $$A\cup B=(1,\infty )$$

Let $$\mathrm{y}=f(x)$$ represent a parabola with focus $$\left(-\frac{1}{2},0\right)$$ and directrix $$y=-\frac{1}{2}$$. Then $$S=\left\{x\in \mathbb{R}:{\tan }^{-1}(\sqrt{f(x)})+{\sin }^{-1}(\sqrt{f(x)+1})=\frac{\pi }{2}\right\}$$ :

Let $A$ be a point on the $x$-axis. Common tangents are drawn from $A$ to the curves $x^2+y^2=8$ and $y^2=16x$. If one of these tangents touches the two curves at $Q$ and $R$, then $(QR{)}^2$ is equal to :

The parabolas : $a{x}^2+2bx+cy=0$ and $d{x}^2+2ex+fy=0$ intersect on the line $y=1$. If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in G.P., then :

If $$\mathrm{P}(\mathrm{h},\mathrm{k})$$ be a point on the parabola $$x=4y^2$$, which is nearest to the point $$\mathrm{Q}(0,33)$$, then the distance of $$\mathrm{P}$$ from the directrix of the parabola $$y^2=4(x+y)$$ is equal to :

If the tangent at a point P on the parabola $$y^2=3x$$ is parallel to the line $$x+2y=1$$ and the tangents at the points Q and R on the ellipse $$\frac{x^2}{4}+\frac{y^2}{1}=1$$ are perpendicular to the line $$x-y=2$$, then the area of the triangle PQR is :

The equations of two sides of a variable triangle are $$x=0$$ and $$y=3$$, and its third side is a tangent to the parabola $$y^2=6x$$. The locus of its circumcentre is :

The distance of the point $$(6,-2\sqrt{2})$$ from the common tangent $$\mathrm{y}=\mathrm{mx}+\mathrm{c},\mathrm{m}>0$$, of the curves $$x=2y^2$$ and $$x=1+y^2$$ is :