A vector \(\vec n\) is inclined to \(x-\) axis at \(45^o\), to \(y-\) axis at \(60^o\) and at an acute angle to \(z-\) axis. If \(\vec n\) is a normal to a plane passing through the point \(\left( {\sqrt 2 , - 1,1} \right)\) then the equation of the plane is 

The largest \( n\in N \) for which \(7^n\) divides 101!, is :

Let \(I(x)=\int \frac{6}{\sin ^2 x(1-\cot x)^2} d x\). If \(I(0)=3\), then \(\mathrm{I}\left(\frac{\pi}{12}\right)\) is equal to :

If \(A = \left[ {\begin{array}{*{20}{c}}1&2&2\\2&1&{ - 2}\\a&2&b\end{array}} \right]\) is a matrix satisfying the equation \(AA^T=9I \) where\( I\) is \(3×3\) identity matrix, then the ordered pair \((a, b)\) is equal to:

Let the tangent to the circle \(x^{2}+y^{2}=25\) at the point \(R (3,4)\) meet \(x\) -axis and \(y\) -axis at point \(P\) and \(Q\), respectively. If \(r\) is the radius of the circle passing through the origin \(O\) and having centre at the incentre of the triangle \(OPQ ,\) then \(r ^{2}\) is equal to

The following system of linear equations  \(7 x+6 y-2 z=0\) ; \(3 x+4 y+2 z=0\) ; \({x}-2{y}-6{z}=0,\) has

The value of \(\left(\sin 70^{\circ}\right)\left(\cot 10^{\circ} \cot 70^{\circ}-1\right)\) is

Let \( A=\{x :|x^{2}-10|\le6\} \) and \( B=\{x :|x-2|>1\}. \) Then

The of value the integral \(\frac{48}{\pi^{4}} \int_{0}^{\pi}\left(\frac{3 \pi x ^{2}}{2}- x^{3}\right) \frac{\sin x }{1+\cos ^{2} x } dx\) is equal to

If \(f(x)=\left\{\begin{array}{ll}{\frac{\sin (a+2) x+\sin x}{x}} & {; x<0} \\ {b} & {; x=0} \\ {\frac{\left(x+3 x^{2}\right)^{\frac{1}{3}}-x^{\frac{1}{3}}}{x^{\frac{4}{3}}}} & {; x>0}\end{array}\right.\) is continuous at \(x=0,\) then \(a+2 b\) is equal to

If \(g \) is the inverse of a function \(f \) and \(f'\left( x \right) = \frac{1}{{1 + {x^5}}}\) thne \(g'\left( x \right)\) is equal to :

If \(\cot \alpha=1\) and \(\sec \beta=-\frac{5}{3}\), where \(\pi<\alpha<\frac{3 \pi}{2}\) and \(\frac{\pi}{2}<\beta<\pi\), then the value of \(\tan (\alpha+\beta)\) and the quadrant in which \(\alpha+\beta\) lies, respectively are

Let M denote the set of all real matrices of order \(3 \times 3\) and let \(\mathrm{S}=\{-3,-2,-1,1,2\}\). Let
\(\mathrm{S}_1=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_2=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=-\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_3=\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: a_{11}+a_{22}+a_{33}=0\) and \(a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\}\)
If \(n\left(\mathrm{~S}_1 \cup_2 \mathrm{US}_3\right)=125 \alpha\), then \(\alpha\) equals _______