Let \(\alpha \) and \(\beta \) be two roots of the equation \(x^2 + 2x + 2 = 0\) , then  is equal to \({\alpha ^{15}} + {\beta ^{15}}\) is equal to 

If \(19^{th}\) terms of non -zero \(A.P.\) is zero, then its (\(49^{th}\) term) : (\(29^{th}\) term) is

Let \(x=(8 \sqrt{3}+13)^{13}\) and \(y=(7 \sqrt{2}+9)^9\). If \([t]\) denotes the greatest integer \(\leq t\), then

The position vectors of the vertices \(A, B\) and \(C\) of a triangle are \(2 \hat{i}-3 \hat{j}+3 \hat{k}, \quad 2 \hat{i}+2 \hat{j}+3 \hat{k} \quad\) and \(-\hat{i}+\hat{j}+3 \hat{k}\) respectively. Let \(l\) denotes the length of the angle bisector \(\mathrm{AD}\) of \(\angle \mathrm{BAC}\) where \(\mathrm{D}\) is on the line segment \(\mathrm{BC}\), then \(2 l^2\) equals :

Let \(f\) be a polynomial function such that \(f(x^{2}+1)=x^{4}+5x^{2}+2,\) for all \(x\in\mathbb{R}.\) Then \(\int_{0}^{3}f(x)dx\) is equal to

If the sides \(A B, B C\) and \(C A\) of a triangle \(A B C\) have \(3,5\) and \(6\) interior points respectively, then the total number of triangles that can be constructed using these points as vertices, 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 \(y=y(t)\) be a solution of the differential equation \(\frac{d y}{d t}+\alpha y=\gamma e^{-\beta t}\) Where, \(\alpha > 0, \beta > 0\) and \(\gamma > 0\). Then \(\operatorname{Lim}_{t \rightarrow \infty} y(t)\)

lf a line \(L\) is perpendicular to the line \(5x - y\,= 1\) , and the area of the triangle formed by the line \(L\) and the coordinate axes is \(5\), then the distance of line \(L\) from the line \(x + 5y\, = 0\) is

Let \(\alpha, \beta\) be roots of \(x^2+\sqrt{2} x-8=0\). If \(\mathrm{U}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}\), then \(\frac{\mathrm{U}_{10}+\sqrt{2} \mathrm{U}_9}{2 \mathrm{U}_8}\) is equal to ............

If \(f: R \rightarrow R\) is a function defined by \(f(x)=[x-1] \cos \left(\frac{2 x-1}{2}\right) \pi,\) where \([.]\) denotes the greatest integer function, then \(f\) is

Let \(a_1 , a_2, a_3, .... , a_n\), be in \(A.P\). If \(a_3 + a_7 + a_{11} + a_{15} = 72\) , then the sum of its first \(17\) terms is equal to

Let \([ x ]\) denote greatest integer less than or equal to \(x .\) If for \(n \in N ,\left(1-x+x^{3}\right)^{n}=\sum_{j=0}^{3 n} a_{j} x^{j}\), then \(\sum_{j=0}^{\left[\frac{3 n}{2}\right]} a_{2 j}+4 \sum_{j=0}^{\left[\frac{3 n-1}{2}\right]} a_{2 j+1}\) is equal to