Let \(f, g: N -\{1\} \rightarrow N\) be functions defined by \(f(a)=\alpha\), where \(\alpha\) is the maximum of the powers of those primes \(p\) such that \(p^{\alpha}\) divides \(a\), and \(g(a)=a+1\), for all \(a \in N -\{1\}\). Then, the function \(f+ g\) is.

The angle of elevation of the summit of a mountain from a point on the ground is \(45^{\circ}\). After climding up one \(km\) towards the summit at an inclination of \(30^{\circ}\) from the ground, the angle of elevation of the summit is found to be \(60^{\circ} .\) Then the height (in \(km\) ) of the summit from the ground is

Let \(S\) be the set of all values of \(\lambda\), for which the shortest distance between the lines \(\frac{x-\lambda}{0}=\frac{y-3}{4}=\frac{z+6}{1}\) and \(\frac{x+\lambda}{3}=\frac{y}{-4}=\frac{z-6}{0}\) is \(13\) Then \(8\left|\sum_{\lambda \in S} \lambda\right|\) is equal to

If \(A = \left[ {\begin{array}{*{20}{c}}2&{ - 3}\\{ - 4}&1\end{array}} \right],\) then \(adj\;\left( {3{A^2} + 12A} \right) = \) . . . .

Let \(\mathrm{f}:(-1, \infty) \rightarrow \mathrm{R}\) be defined by \(\mathrm{f}(0)=1\) and \(f(x)=\frac{1}{x} \log _{e}(1+x), x \neq 0 .\) Then the function \(f\)

For the differentiable function \(f: R -\{0\} \rightarrow R\), let \(3 f(x)+2 f\left(\frac{1}{x}\right)=\frac{1}{x}-10\), then \(\left|f(3)+f^{\prime}\left(\frac{1}{4}\right)\right|\) is equal to

Let \(A=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]\) and \(B=\left[\begin{array}{ccc}9^{2} & -10^{2} & 11^{2} \\ 12^{2} & 13^{2} & -14^{2} \\ -15^{2} & 16^{2} & 17^{2}\end{array}\right]\), then the value of \(A ^{\prime} BA\) is.

Consider three vectors \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}\). Let \(|\overrightarrow{\mathrm{a}}|=2,|\overrightarrow{\mathrm{b}}|=3\) and \(\overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\). If \(\alpha \in\left[0, \frac{\pi}{3}\right]\) is the angle between the vectors \(\vec{b}\) and \(\vec{c}\), then the minimum value of \(27|\overrightarrow{c}-\overrightarrow{a}|^2\) is equal to :

If a random variable \(X\) follows the Binomial distribution \(B (33, p )\) such that \(3 P ( X =0)= P ( X =1)\), then the value of \(\frac{ P ( X =15)}{ P ( X =18)}-\frac{ P ( X =16)}{ P ( X =17)}\) is equal to

Let \(\quad S=109+\frac{108}{5}+\frac{107}{5^2}+\ldots \ldots . .+\frac{2}{5^{107}}+\frac{1}{5^{108}}\). Then the value of \(\left(16 S -(25)^{-34}\right)\) is equal to \(............\).

Let \(f(x)=2 \cos ^{-1} x+4 \cot ^{-1} x-3 x^{2}-2 x+10, x \in[-\) \(1,1]\). If \([ a , b ]\) is the range of the function then \(4 a -\) \(b\) is equal to

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 _______

Let the circle \(C\) touch the line \(x-y+1=0\), have the centre on the positive x -axis, and cut off a chord of length \(\frac{4}{\sqrt{13}}\) along the line \(-3 x+2 y=1\). Let H be the hyperbola \(\frac{x^2}{\alpha^2}-\frac{y^2}{\beta^2}=1\), whose one of the foci is the centre of \(C\) and the length of the transverse axis is the diameter of \(C\). Then \(2 \alpha^2+3 \beta^2\) is equal to ______