Let the solution curve \(y = y ( x )\) of the differential equation, \(\left[\frac{x}{\sqrt{x^{2}-y^{2}}}+e^{\frac{y}{x}}\right] x \frac{d y}{d x}=x+\left[\frac{x}{\sqrt{x^{2}-y^{2}}}+e^{\frac{y}{x}}\right] y\) pass through the points \((1,0)\) and \((2 \alpha, \alpha), \alpha>0\). Then \(\alpha\) is equal to

Let \(A=\{a, b, c\}\) and \(B=\{1,2,3,4\}\) Then the number of elements in the set \(C =\{ f : A \rightarrow B \mid 2 \in f ( A )\) and \(f\) is not one-one \(\}\) is

Let \(f: R -\{3\} \rightarrow R -\{1\}\) be defined by \(f(x)=\frac{x-2}{x-3} .\) Let \(g: R \rightarrow R\) be given as \(g ( x )=2 x -3\). Then, the sum of all the values of \(x\) for which \(f^{-1}( x )+ g ^{-1}( x )=\frac{13}{2}\) is equal to ...... .

Let \(v\) be the solution of the differential equation \(\left(1-x^{2}\right) d y=\left(xy+\left(x^{3}+2\right) \sqrt{1-x^{2}}\right) d x,-1 < x < 1\) and \(y(0)=0\) if \(\int\limits_{-\frac{1}{2}}^{\frac{1}{2}} \sqrt{1-x^{2}} y(x) d x=k\) then \(k^{-1}\) is equal to:

The value of \(\lim _{n \rightarrow \infty} \frac{[ r ]+[2 r ]+\ldots . .+[ nr ]}{ n ^{2}},\) where
is non-zero real number and \([r]\) denotes the greatest integer less than or equal to \(r\), is equal to ...... .

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 value of \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{1}{[ x ]+4}\right) dx\), where \([\bullet]\) denotes the greatest integer function, is

Let the normal at a point \(P\) on the curve \(\mathrm{y}^{2}-3 \mathrm{x}^{2}+\mathrm{y}+10=0\) intersect the \(\mathrm{y}\) -axis at \(\left(0, \frac{3}{2}\right) .\) If \(\mathrm{m}\) is the slope of the tangent at \(\mathrm{P}\) to the curve, then \(|\mathrm{m}|\) is equal to

Let \(R=\left(\begin{array}{lll}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)\) be a non-zero \(3 \times 3\) matrix, where \(x \sin \theta=y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right)\) \(\neq 0, \theta \in(0,2 \pi)\). For a square matrix \(M\), let trace \((M)\) denote the sum of all the diagonal entries of M. Then, among the statements: \((I)\) \(Trace(\mathrm{R})=0\) \((II)\) If \(trace(\operatorname{adj}(\operatorname{adj}(\mathrm{R}))=0\), then \(R\) has exactly one non-zero entry.

\(\max_{0 \leq x \leq \pi}\left(16\sin\left(\dfrac{x}{2}\right)\cos^3\left(\dfrac{x}{2}\right)\right)\) is equal to:

Let \(\alpha, \alpha + 2, \alpha \in \mathbb{Z}\), be the roots of the quadratic equation \(x(x+2) + (x+1)(x+3) + (x+2)(x+4) + \ldots + (x+n-1)(x+n+1) = 4n\) for some \(n \in \mathbb{N}\). Then \(n + \alpha\) is equal to :

Let \(A=\left[\begin{array}{llc}2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2\end{array}\right]\) and \(P=\left[\begin{array}{lll}1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5\end{array}\right]\). The sum of the prime factors of \(\left|\mathrm{P}^{-1} \mathrm{AP}-2 \mathrm{I}\right|\) is equal to

A square \(ABCD\) has all its vertices on the curve \(x ^{2} y ^{2}=1\). The midpoints of its sides also lie on the same curve. Then, the square of area of \(ABCD\) is