Let \(P = \left[ {\begin{array}{*{20}{c}}
  1&0&0 \\ 
  3&1&0 \\ 
  9&3&1 
\end{array}} \right]\) and \(Q = [q_{ij}]\) be two \(3\times3\) matrices such that \(Q -P^5 = I_3\). Then \(\frac{{{q_{21}} + {q_{31}}}}{{{q_{32}}}}\) is equal to

For \(x \in(-1,1]\), the number of solutions of the equation \(\sin ^{-1} x=2 \tan ^{-1} x\) is equal to

Let \({S_n} = 1 + q + {q^2} + ..... + {q^n}\) and \({T_n} = 1 + \left( {\frac{{q + 1}}{2}} \right) + {\left( {\frac{{q + 1}}{2}} \right)^2} + ...... + {\left( {\frac{{q + 1}}{2}} \right)^n}\) where \(q\) is a real number and \(q \ne 1\). If \({}^{101}{C_1} + {}^{101}{C_2}.{S_1} + ...... + {}^{101}{C_{101}}.{S_{100}} = \alpha\, {T_{100}}\) then \(\alpha \) is equal to

Let \(\overrightarrow{ c }\) be a vector perpendicular to the vectors \(\overrightarrow{ a }=\hat{ i }+\hat{ j }-\hat{ k }\) and \(\overrightarrow{ b }=\hat{ i }+2 \hat{ j }+\hat{ k }.\) If \(\overrightarrow{ c } \cdot(\hat{ i }+\hat{ j }+3 \hat{ k })=8\) then the value of \(\overrightarrow{ c } \cdot(\overrightarrow{ a } \times \overrightarrow{ b })\) is equal to ...... .

If the solution of the differential equation \((2 x+3 y-2) d x+(4 x+6 y-7) d y=0, y(0)=3\), is \(\alpha x+\beta y+3 \log _e|2 x+3 y-\gamma|=6\), then \(\alpha+2 \beta+3 \gamma\) is equal to

If \(x = x ( y )\) is the solution of the differential equation \(y \frac{d x}{d y}=2 x+y^{3}(y+1) e^{y}, x(1)=0\); then \(x(e)\) is equal to

Let \(f: \mathbf{R} \rightarrow \mathbf{R}\) be a twice differentiable function such that \((\sin x \cos y)(f(2 x+2 y)-f(2 x-2 y))=(\cos x\) \(\sin \mathrm{y})(f(2 \mathrm{x}+2 \mathrm{y})+f(2 \mathrm{x}-2 \mathrm{y}))\), for all \(\mathrm{x}, \mathrm{y} \in \mathbf{R}\).
If \(f^{\prime}(0)=\frac{1}{2}\), then the value of \(24 f^{\prime \prime}\left(\frac{5 \pi}{3}\right)\) 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

Let \([ x ]\) be the greatest integer \(\leq x\). Then the number of points in the interval \((-2,1)\), where the function \(f(x)=|[x]|+\sqrt{x-[x]}\) is discontinuous is \(........\).

Let \(A=\{0,1,2,3,4,5\}\). Let \(R\) be a relation on A defined by \((x, y) \in R\) if and only if max \(\{x, y\} \in\{3,4\}\). Then among the statements \(\left(\mathrm{S}_1\right)\) : The number of elements in R is 18 , and \(\left(\mathrm{S}_2\right)\) : The relation R is symmetric but neither reflexive nor transitive

\(\frac{2^{3}-1^{3}}{1 \times 7}+\frac{4^{3}-3^{3}+2^{3}-1^{3}}{2 \times 11}+\)\(\frac{6^{3}-5^{3}+4^{3}-3^{3}+2^{3}-1^{3}}{3 \times 15}+\ldots .+\) \(\frac{30^{3}-29^{3}+28^{3}-27^{3}+\ldots+2^{3}-1^{3}}{15 \times 63}\)is equal to.

If \([.]\) represents the greatest integer function, then the value of \(\int_{0}^{\sqrt{\pi / 2}}\left(\left[ x ^{2}\right]+[-\cos x ]\right) d x\) is.............

If the mean deviation about median for the number \(3,5,7,2\,k , 12,16,21,24\) arranged in the ascending order, is \(6\) then the median is