The sum of all the elements in the range of \(f(x)=\text{Sgn}(\sin x)+\text{Sgn}(\cos x)+\text{Sgn}(\tan x)+\text{Sgn}(\cot x), x\ne\frac{n\pi}{2}, n\in Z\), where \(\operatorname{Sgn}(t)=\left\{\begin{array}{lll}1, & \text { if } & t>0 \\ -1 & \text { if } & t<0\end{array}\right.\), is

If for \(z=\alpha+i \beta,|z+2|=z+4(1+i)\), then \(\alpha+\beta\) and \(\alpha \beta\) are the roots of the equation

Let \(f\) be a differentiable function \(R\) to \(R\) such that \(\left| {f\,(x)\, - \,f(y)} \right|\, \le \,2\,{\left| {x - y} \right|^{\frac{3}{2}}},\) for all \(x,y\,\in R .\) If \(f\,(0)=1\) then \(\int\limits_0^1 {{f^2}\,(x)\,dx} \) is equal to

For the curve \(C :\) \(\left(x^{2}+y^{2}-3\right)+\left(x^{2}-y^{2}-1\right)^{5}=0\), the value of \(3 y^{\prime}-y^{3} y^{\prime \prime}\), at the point \((\alpha, \alpha), \alpha>0\), on \(C\), is equal to.

If \(A\, = \,\left[ {\begin{array}{*{20}{c}}
1&2&x\\
3&{ - 1}&2
\end{array}} \right]\) and \(B\, = \,\left[ {\begin{array}{*{20}{c}}
y\\
x\\
1
\end{array}} \right]\) be such that \(AB\, = \,\left[ {\begin{array}{*{20}{c}}
6\\
8
\end{array}} \right],\) then

The number of real solution(s) of the equation \(x^2+3 x+2=\min \{|x-3|,|x+2|\} \text { is : }\)

Let a straight line \(L\) pass through the point \(P(2,-1,3)\) and be perpendicular to the lines \(\frac{x-1}{2}=\frac{y+1}{1}=\frac{z-3}{-2}\) and \(\frac{x-3}{1}=\frac{y-2}{3}=\frac{z+2}{4}\). If the line \(L\) intersects the \(y z\)-plane at the point \(Q\), then the distance between the points \(P\) and \(Q\) is :

The vertices of a triangle are \(\mathrm{A}(-1,3), \mathrm{B}(-2,2)\) and \(\mathrm{C}(3,-1)\). \(A\) new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is :

If the equation \(\mathrm{a}(\mathrm{b}-\mathrm{c}) \mathrm{x}^2+\mathrm{b}(\mathrm{c}-\mathrm{a}) \mathrm{x}+\mathrm{c}(\mathrm{a}-\mathrm{b})=0\) has equal roots, where \(\mathrm{a}+\mathrm{c}=15\) and \(\mathrm{b}=\frac{36}{5}\), then \(a^2+c^2\) is equal to

Let \(y=y(x)\) be solution of the following differential equation \(e^{y} \frac{d y}{d x}-2 e^{y} \sin x+\sin x \cos ^{2} x=0, y\) \(\left(\frac{\pi}{2}\right)=0\). If \(y(0)=\log _{e}\left(\alpha+\beta e^{-2}\right)\), then \(4(\alpha+\beta)\) is equal to \(....\)

A line with direction ratios \(2,1,2\) meets the lines \(x=y+2=z\) and \(x+2=2 y=2 z\) respectively at the point \(P\) and \(Q\). if the length of the perpendicular from the point \((1,2,12)\) to the line \(\mathrm{PQ}\) is \(l\), then \(l^2\) is

The number of all \(3 \times 3\) matrices \(A\), with enteries from the set \(\{-1,0,1\}\) such that the sum of the diagonal elements of \(\mathrm{AA}^{\mathrm{T}}\) is \(3,\) 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