Let \(x=2\) be a root of the equation \(x^2+p x+q=0\) and \(f(x)=\left\{\begin{array}{cc}\frac{1-\cos \left(x^2-4 p x+q^2+8 q+16\right)}{(x-2 p)^4}, & x \neq 2 p \\ 0, & x=2 p\end{array}\right.\) Then \(\lim _{x \rightarrow 22^{+}}[f(x)]\) where [. ] denotes greatest integer function, is \(........\)

Let \(f(x)\) and \(g(x)\) be two functions satisfying \(f\left(x^{2}\right)\) \(+g(4-x)=4 x^{3}\) and \(g(4-x)+g(x)=0\), then the value of \(\int_{-4}^{4} f(x)^{2} d x\) is

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

If \(\log _e y=3 \sin ^{-1} x\), then \((1-x)^2 y^{\prime \prime}-x y^{\prime}\) at \(x=\frac{1}{2}\) is equal to :

In a game, a man wins \(Rs.\,100\)  if he gets \(5\) or \(6\) on a throw of a fair die and loses \(Rs.\,50\) for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/loss (in rupees) is

Let the tangent drawn to the parabola \(y ^{2}=24 x\) at the point \((\alpha, \beta)\) is perpendicular to the line \(2 x\) \(+2 y=5\). Then the normal to the hyperbola \(\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}=1\) at the point \((\alpha+4, \beta+4)\) does \(NOT\) pass through the point.

Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be a function defined by \(f(x)=\|x+2|-2| x\|\). If \(m\) is the number of points of local minima and \(n\) is the number of points of local maxima of \(f\), then \(m+n\) is

Let \(\alpha\) and \(\beta\) be real numbers. Consider a \(3 \times 3\) matrix \(A\) such that \(A ^2=3 A +\alpha I\). If \(A ^4=21 A +\beta I\), then

Let the image of parabola \( x^{2}=4y \) in the line \( x-y=1 \) be \( (y+\alpha)^{2}=b(x-c), \) \( a, b, c \in \mathbb{N} \). Then \( a+b+c \) is equal to

The number of points of discontinuity of the function \(f(\mathrm{x})=\left[\frac{\mathrm{x}^2}{2}\right]-[\sqrt{\mathrm{x}}], \mathrm{x} \in[0,4]\), where \([\cdot]\) denotes the greatest integer function is ________

Let \(k\) be a non-zero real number  If \(f(x) = {\rm{ }}\left\{ {\begin{array}{*{20}{c}}
{\frac{{\left( {{e^x} - 1} \right)^2}}{{\sin {\mkern 1mu} \left( {\frac{x}{k}} \right){\mkern 1mu} \log {\mkern 1mu} \left( {1 + \frac{x}{4}} \right)}}{\mkern 1mu} ,{\mkern 1mu} x \ne 0}\\
{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} 12{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} ,x{\mkern 1mu} {\mkern 1mu}  = 0{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} }
\end{array}} \right.\)  is a continuous function then the value of \(k\) is

The number of ways, \(16\) identical cubes, of which \(11\) are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least \(2\) blue cubes, is

The area of the smaller region enclosed by the curves \(y ^{2}=8 x +4\) and \(x^{2}+y^{2}+4 \sqrt{3} x-4=0\) is equal to.