Let $f(\theta)$ is distance of the line $( \sqrt {\sin \theta } )x + (  \sqrt {\cos  \theta })y +1 = 0$ from origin. Then the range of $f(\theta)$ is –

The minimum value of the function $f(x) = {x^{10}} + {x^2} + \frac{1}{{{x^{12}}}} + \frac{1}{{\left( {1\ +\ {{\sec }^{ – 1}}\ x} \right)}}$ is

Consider a function $f:\left[ { – 1,1} \right] \to R$ where $f(x) = {\alpha _1}{\sin ^{ – 1}}x + {\alpha _3}\left( {{{\sin }^{ – 1}}{x^3}} \right) + ….. + {\alpha _{(2n + 1)}}{({\sin ^{ – 1}}x)^{(2n + 1)}} – {\cot ^{ – 1}}x$ Where $\alpha _i\ 's$ are positive constants and $n \in N < 100$ , then $f(x)$ is

Set $A$ has $3$ elements and set $B$ has $4$ elements. The number of injection that can be defined from $A$ to $B$ is

Let $f$ be a real valued function defined by

$f(x) = sin^{-1} \left( {\frac{{\,\,1 – \,\,\left| x \right|}}{3}} \right) + cos^{-1}\left( {\frac{{\left| x \right|\,\, – \,\,3}}{5}} \right)$ .

Then domain of $f(x)$ is given by :