The largest interval lying in $\left( { – \frac{\pi }{2},\frac{\pi }{2}} \right)$ for which the function, $f\left( x \right) = {4^{ – {x^2}}} + {\cos ^{ – 1}}\left( {\frac{x}{2} – 1} \right) + \log \left( {\cos x} \right)$  is defined is

The set of values of $'a'$ for which the inequality ${x^2} – (a + 2)x – (a + 3) < 0$ is satisfied by atleast one positive real $x$ , is

If $f(x + ay,\;x – ay) = axy$, then $f(x,\;y)$ is equal to

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by

$f(x)=\frac{2 x}{\sqrt{1+9 x^2}}$. If the composition of $f, \underbrace{(f \circ f \circ f \circ \ldots \circ f)}_{10 \text { times }}(x)=\frac{2^{10} x}{\sqrt{1+9 \alpha x^2}}$, then the value of $\sqrt{3 \alpha+1}$ is equal to………………..

If $y = f(x) = \frac{{ax + b}}{{cx – a}}$, then $x$ is equal to