If the image of the point \(\mathrm{P}(1,0,3)\) in the line joining the points \(\mathrm{A}(4,7,1)\) and \(\mathrm{B}(3,5,3)\) is \(\mathrm{Q}(\alpha, \beta, \gamma)\), then \(\alpha+\beta+\gamma\) is equal to

Let \(\alpha\) and \(\beta\) be the roots of the equation \(x^{2}+(2 i -\) \(1)=0\). Then, the value of \(\left|\alpha^{8}+\beta^{8}\right|\) is equal to

A symmetrical form of the line of intersection of the planes \(x = ay + b\) and \(z = cy + d\) is

Let \(m_1\) and \(m_2\) be the slopes of the tangents drawn from the point \(P (4,1)\) to the hyperbola \(H: \frac{y^2}{25}-\frac{x^2}{16}=1\). If \(Q\) is the point from which the tangents drawn to \(H\) have slopes \(\left| m _1\right|\) and \(\left| m _2\right|\) and they make positive intercepts \(\alpha\) and \(\beta\) on the \(x\) axis, then \(\frac{(P Q)^2}{\alpha \beta}\) is equal to \(............\).

Let \(A=\left[\begin{array}{cc}i & -i \\ -i & i\end{array}\right], i=\sqrt{-1}\).Then, the system of linear equations \(A^{8}\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{c}8 \\ 64\end{array}\right]\) has :

Let \(A=\{0,3,4,6,7,8,9,10\} \quad\) and \(R\) be the relation defined on A such that \(R =\{( x , y ) \in A \times A : x - y \quad\) is odd positive integer or \(x-y=2\}\). The minimum number of elements that must be added to the relation \(R\), so that it is a symmetric relation, is equal to \(...........\).

For a triangle \(ABC\), the value of \(\cos 2 A +\cos 2 B +\cos 2 C\) is least. If its inradius is \(3\) and incentre is \(M\), then which of the following is NOT correct?

Let \(f:[-1,2] \rightarrow \mathrm{R}\) be given by \(f(x)=2 x^2+x+\left[x^2\right]-[x]\), where \([t]\) denotes the greatest integer less than or equal to \(t\). The number of points, where \(f\) is not continuous, is :

The sum of an infinite geometric series with positive terms is \(3\) and the sum of the cubes of its terms is \(\frac {27}{19}\). Then the common ratio of this series is

If \(a _{1}, a _{2}, a _{3} \ldots\) and \(b _{1}, b _{2}, b _{3} \ldots\) are \(A.P.\) and \(a_{1}=2, a_{10}=3, a_{1} b_{1}=1=a_{10} b_{10}\) then \(a_{4} b_{4}\) is equal to

If \(\alpha = \displaystyle\int_0^{2\sqrt{3}} \log_2(x^2 + 4)\,dx + \displaystyle\int_2^4 \sqrt{2^x - 4}\,dx\), then \(\alpha^2\) is equal to _______.

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,

Let a complex number be \(w =1-\sqrt{3} i\). Let another complex number \(z\) be such that \(|z w|=1\) and \(\arg ( z )-\arg ( w )=\frac{\pi}{2} .\) Then the area of the triangle with vertices origin, \(z\) and \(w\) is equal to ........ .