If \(\frac{d x}{d y}=\frac{1+x-y^2}{y}, x(1)=1\), then \(5 x(2)\) is equal to :

Let \(\mathrm{A}\,(\sec \theta, 2 \tan \theta)\) and \(\mathrm{B}\,(\sec \phi, 2 \tan \phi)\), where \(\theta+\phi=\pi / 2\), be two points on the hyperbola \(2 \mathrm{x}^{2}-\mathrm{y}^{2}=2\). If \((\alpha, \beta)\) is the point of the intersection of the normals to the hyperbola at \(\mathrm{A}\) and \(\mathrm{B}\), then \((2 \beta)^{2}\) is equal to ..... .

If for some \(p , q , r \in R\), not all have same sign, one of the roots of the equation \(\left(p^{2}+q^{2}\right) x^{2}-2 q(p+r) x\) \(+q^{2}+r^{2}=0\) is also a root of the equation \(x^{2}+2 x-8=0\), then \(\frac{q^{2}+r^{2}}{p^{2}}\) is equal to-

Let \(\alpha, \beta\) be two roots of the equation \(x^{2}+(20)^{\frac{1}{4}} x+(5)^{\frac{1}{2}}=0\). Then \(\alpha^{8}+\beta^{8}\) is equal to:

Let \(x=x(t)\) and \(y=y(t)\) be solutions of the differential equations \(\frac{\mathrm{dx}}{\mathrm{dt}}+\mathrm{ax}=0\) and \(\frac{\mathrm{dy}}{\mathrm{dt}}+\mathrm{by}=0\) respectively, \(\mathrm{a}, \mathrm{b} \in \mathrm{R}\). Given that \(x(0)=2 ; y(0)=1\) and \(3 y(1)=2 x(1)\), the value of \(t\), for which \(x(t)=y(t)\), is :

Let the mean and the variance of \(5\) observations \(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\) be \(\frac{24}{5}\) and \(\frac{194}{25}\) respectively. If the mean and variance of the first \(4\) observation are \(\frac{7}{2}\) and \(a\) respectively, then \(\left(4 a+x_{5}\right)\) is equal to

Let \(A\left[\begin{array}{cc}1 & 2 \\ -1 & 4\end{array}\right] .\) If \(A^{-1}=\alpha I+\beta A, \alpha, \beta \in R, I\) is a \(2 \times 2\) identity matrix, then \(4(\alpha-\beta)\) is equal to:

Let \(A=\left[a_{i j}\right]\) be a matrix of order \(3 \times 3\), with \(a_{i j}=(\sqrt{2})^{i+j}\). If the sum of all the elements in the third row of \(A^2\) is \(\alpha+\beta \sqrt{2}, \alpha, \beta \in \mathbf{Z}\), then \(\alpha+\beta\) is equal to :

Statement \(-1\) : The system of linear equations \(x + \left( {\sin \,\alpha } \right)y + \left( {\cos \,\alpha } \right)z = 0\) \(x + \left( {\cos \,\alpha } \right)y + \left( {\sin \alpha } \right)z = 0\) \(x - \left( {\sin \,\alpha } \right)y - \left( {\cos \alpha } \right)z = 0\) has a non-trivial solution for only one value of \(\alpha \) lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)  Statement \(-2\) : The equation in \(\alpha \) \(\left| {\begin{array}{*{20}{c}}
  {\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha } \\ 
  {\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha } \\ 
  {\cos {\mkern 1mu} \alpha }&{ - \sin {\mkern 1mu} \alpha }&{ - \cos {\mkern 1mu} \alpha } 
\end{array}} \right| = 0\) has only one solution lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)

Let \(a = lm\left( {\frac{{1 + {z^2}}}{{2iz}}} \right)\), where \(z\) is any non-zero complex number. The set \(A = \{ a:\left| z \right| = 1\,and\,z \ne  \pm 1\} \) is equal to

In an arithmetic progression, if \(S_{40}=1030 \text { and } S_{12}=57 \text {, then } S_{30}-S_{10} \text { is equal to : }\)

If \(\sum_{\mathrm{r}=1}^9\left(\frac{\mathrm{r}+3}{2^{\mathrm{r}}}\right) .{ }^9 \mathrm{C}_{\mathrm{r}}=\alpha\left(\frac{3}{2}\right)^9-\beta, \quad \alpha, \beta \in \mathrm{N}, \quad\) then \((\alpha+\beta)^2\) is equal to

Let \(P ( S )\) denote the power set of \(S =\{1,2,3, \ldots, 10\}\). Define the relations \(R_1\) and \(R_2\) on \(P(S)\) as \(A R_1 B\) if \(\left( A \cap B ^{ c }\right) \cup\left( B \cap A ^{ c }\right)=\varnothing\) and \(AR _2 B\) if \(A \cup B ^{ c }=\) \(B \cup A ^{ c }, \forall A , B \in P ( S )\). Then :