Let $|M|$ denote the determinant of a square matrix $M$. Let $g:\left[0, \frac{\pi}{2}\right] \rightarrow \mathbb{R}$ be the function defined by

$ g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1} $

where

$ f(\theta)=\frac{1}{2}\left|\begin{array}{ccc} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \end{array}\right|+\left|\begin{array}{ccc} \sin \pi & \cos \left(\theta+\frac{\pi}{4}\right) & \tan \left(\theta-\frac{\pi}{4}\right) \\ \sin \left(\theta-\frac{\pi}{4}\right) & -\cos \frac{\pi}{2} & \log _{e}\left(\frac{4}{\pi}\right) \\ \cot \left(\theta+\frac{\pi}{4}\right) & \log _{e}\left(\frac{\pi}{4}\right) & \tan \pi \end{array}\right| . $

Let $p(x)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g(\theta)$, and $p(2)=2-\sqrt{2}$. Then, which of the following is/are TRUE ?

Let $f(x) = \sin \left( {{\pi \over 6}\sin \left( {{\pi \over 2}\sin x} \right)} \right)$ for all $x \in R$ and g(x) = ${{\pi \over 2}\sin x}$ for all x$\in$R. Let $(f \circ g)(x)$ denote f(g(x)) and $(g \circ f)(x)$ denote g(f(x)). Then which of the following is/are true?

${f_1}\left( x \right) = \left\{ {\matrix{ {\left| x \right|} & {if\,x < 0,} \cr {{e^x}} & {if\,x \ge 0;} \cr } } \right.$

f₂(x) = x² ;

and

${f_4}\left( x \right) = \left\{ {\matrix{ {{f_2}\left( {{f_1}\left( x \right)} \right)} & {if\,x < 0,} \cr {{f_2}\left( {{f_1}\left( x \right)} \right) - 1} & {if\,x \ge 0;} \cr } } \right.$

Let $f:( - 1,1) \to R$ be such that $f(\cos 4\theta ) = {2 \over {2 - {{\sec }^2}\theta }}$ for $\theta \in \left( {0,{\pi \over 4}} \right) \cup \left( {{\pi \over 4},{\pi \over 2}} \right)$. Then the value(s) of $f\left( {{1 \over 3}} \right)$ is(are)

The function $f:[0,3] \to [1,29]$, defined by $f(x) = 2{x^3} - 15{x^2} + 36x + 1$, is

Let $f:(0,1) \to R$ be defined by $f(x) = {{b - x} \over {1 - bx}}$, where b is a constant such that $0 < b < 1$. Then

Let f(x) = x² and g(x) = sin x for all x $\in$ R. Then the set of all x satisfying $(f \circ g \circ g \circ f)(x) = (g \circ g \circ f)(x)$, where $(f \circ g)(x) = f(g(x))$, is

Match the statements given in Column I with the intervals/union of intervals given in Column II :

Consider the polynomial
$f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.$
Let $s$ be the sum of all distinct real roots of $f(x)$ and let $t = \left| s \right|.$

The real numbers lies in the interval

Consider the polynomial
$f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.$
Let $s$ be the sum of all distinct real roots of $f(x)$ and let $t = \left| s \right|.$

The function$f'(x)$ is

Let $f, g$ and $h$ be real valued functions defined on the interval $[0,1]$ by

$f(x)=e^{x^2}+e^{-x^2}$,

$g(x)=x e^{x^2}+e^{-x^2}$

and $h(x)=x^2 e^{x^2}+e^{-x^2}$.

If $a, b$ and $c$ denote, respectively, the absolute maximum of $f, g$ and $h$ on $[0,1]$, then :

If $f''(x)=-f(x)$ and $g(x)=f'(x)$ and $\mathrm{F}(x)=\left(f\left(\frac{x}{2}\right)\right)^{2}+\left(g\left(\frac{x}{2}\right)\right)^{2}$ and given that $\mathrm{F}(5)=5$, then $\mathrm{F}(10)$ is equal to :

Find the range of value of $t$ for which

$2 \sin t=\frac{1-2 x+5 x^{2}}{3 x^{2}-2 x-1}, t \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$