Let \(H\) be the hyperbola, whose foci are \((1 \pm \sqrt{2}, 0)\) and eccentricity is \(\sqrt{2}\). Then the length of its latus rectum is

Let \(A\) and \(B\) be two invertible matrices of order \(3 \times 3\). If det \((ABA^T) = 8\) and \(det\,(AB^{-1}) = 8\), then \(det\, (BA^{-1} B^T)\) is equal to

Let \([t]\) denote the greatest integer less than or equal to \(t.\) Then, the value of the integral \(\int\limits_{0}^{1}\left[-8 x^{2}+6 x-1\right] d x\) is equal to

Let \(\vec a = \hat i - \hat j,\) \(\vec b = \hat i + \hat j + \hat k\) and \(\vec c\) be a vector such that \(\vec a \times \vec c + \vec b = 0\) and \(\vec a.\vec c = 4\), then \({\left| {\vec c} \right|^2}\) is equal to

A perpendicular is drawn from a point on the line \(\frac{{x - 1}}{2} = \frac{{y + 1}}{{ - 1}} = \frac{z}{1}\) to the plane \(x + y + z =  3\) such that the foot of the perpendicular \(Q\) also lies on the plane \(x -y + z = 3.\) Then the co-ordinates of \(Q\) are

Let \(f: R -\{2,6\} \rightarrow R\) be real valued function defined as \(f(x)=\frac{x^2+2 x+1}{x^2-8 x+12}\). Then range of \(f\) is

The system of linear equations  \(3 x-2 y-k z=10\); \(2 x-4 y-2 z=6\) ; \(x+2 y-z=5\, m\) is inconsistent if

Let \(A=\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3\end{array}\right)\). Then the sum of the diagonal elements of the matrix \(( A + I )^{11}\) is equal to:

Let each of the two ellipses \(E _1: \frac{ x ^2}{ a ^2}+\frac{ y ^2}{b^2}=1,( a > b )\) and \(E _2: \frac{ x ^2}{A^2}+\frac{ y ^2}{B^2}=1,(A< B )\) have eccentricity \(\frac{4}{5}\). Let the lengths of the latus recta of \(E_1\) and \(E_2\) be \(\ell_1\) and \(\ell_2\), respectively, such that \(2 \ell_1^2=9 \ell_2\). If the distance between the foci of \(E_1\) is 8 , then the distance between the foci of \(E _2\) is

Consider the following two statements : Statement \(I\) : For any two non-zero complex numbers \(\mathrm{z}_1, \mathrm{z}_2\) \(\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right)\) and Statement \(II\) : If \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) are three distinct complex numbers and a, b, c are three positive real numbers such that \(\frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}\), then \(\frac{\mathrm{a}^2}{\mathrm{y}-\mathrm{z}}+\frac{\mathrm{b}^2}{\mathrm{z}-\mathrm{x}}+\frac{\mathrm{c}^2}{\mathrm{x}-\mathrm{y}}=1\) Between the above two statements,

A bag contains 19 unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was unbiased, is \(\frac{\mathrm{m}}{\mathrm{n}}, \operatorname{gcd}(\mathrm{m}, \mathrm{n})=1\), then \(\mathrm{n}^2-\mathrm{m}^2\) is equal to :

Let for \(n =1,2, \ldots \ldots, 50, S _{ a }\) be the sum of the infinite geometric progression whose first term is \(n ^{2}\) and whose common ratio is \(\frac{1}{(n+1)^{2}}\). Then the value of \(\frac{1}{26}+\sum\limits_{n=1}^{50}\left(S_{n}+\frac{2}{n+1}-n-1\right)\) is equal to

The value of \(\cos \left(\frac{2 \pi}{7}\right)+\cos \left(\frac{4 \pi}{7}\right)+\cos \left(\frac{6 \pi}{7}\right)\) is equal to