Let \(z = 1 + ai\) be a complex number, \(a > 0\), such that \(z^3\) is areal number. Then the sum \(1 + z + z^2 + .... + z^{11}\) is equal to

Three numbers are in an increasing geometric progression with common ratio \(\mathrm{r}\). If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference \(\mathrm{d}\). If the fourth term of GP is \(3 \mathrm{r}^{2}\), then \(\mathrm{r}^{2}-\mathrm{d}\) is equal to:

If \(A=\left(\begin{array}{cc}0 & \sin \alpha \\ \sin \alpha & 0\end{array}\right)\) and \(\operatorname{det}\left(A^{2}-\frac{1}{2} I\right)=0,\) then a possible value of \(\alpha\) is

Let \(\alpha\) and \(\beta\) be two real roots of the equation \((\mathrm{k}+1) \tan ^{2} \mathrm{x}-\sqrt{2} \cdot \lambda \tan \mathrm{x}=(1-\mathrm{k})\) where \(\mathrm{k}(\neq-1)\) and \(\lambda\) are real numbers. If \(\tan ^{2}(\alpha+\beta)=50,\) then a value of \(\lambda\) is :

The mean and variance of a data of 10 observations are 10 and 2, respectively. If an observation \( \alpha \) in this data is replaced by \( \beta \), then the mean and variance become 10.1 and 1.99, respectively. Then \( \alpha+\beta \) equals.

Consider two vectors \(\overrightarrow{\mathrm{u}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}\) and \(\overrightarrow{\mathrm{v}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\lambda \hat{\mathrm{k}}, \lambda \gt 0\). The angle between them is given by \(\cos ^{-1}\left(\frac{\sqrt{5}}{2 \sqrt{7}}\right)\). Let \(\overrightarrow{\mathrm{v}}=\overrightarrow{\mathrm{v}}_1+\overrightarrow{\mathrm{v}}_2\), where \(\overrightarrow{\mathrm{v}}_1\) is parallel to \(\overrightarrow{\mathrm{u}}\) and \(\overrightarrow{\mathrm{v}}_2\) is perpendicular to \(\overrightarrow{\mathrm{u}}\). Then the value \(\left|\overrightarrow{\mathrm{v}}_1\right|^2+\left|\overrightarrow{\mathrm{v}}_2\right|^2\) is equal to

Let \(y=y(x)\) be the solution curve of the differential equation \(\sin \left(2 x^{2}\right) \log _{c}\left(\tan x^{2}\right) d y+\left(4 x y-4 \sqrt{2} x \sin \left(x^{2}-\frac{\pi}{4}\right)\right) d x=0\) \(0 < x < \sqrt{\frac{\pi}{2}}\), which passes through the point \(\left(\sqrt{\frac{\pi}{6}}, 1\right)\). Then \(\left|y\left(\sqrt{\frac{\pi}{3}}\right)\right|\) is equal to \(.....\)

Let \(y=f(x)\) be the solution of the differential equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{x y}{x^2-1}=\frac{x^6+4 x}{\sqrt{1-x^2}},-1 \lt x \lt 1\) such that \(f(0)=0\). If \(6 \int_{-1 / 2}^{1 / 2} f(x) \mathrm{d} x=2 \pi-\alpha\) then \(\alpha^2\) is equal to _______ .

Let \( y=y(x) \) be a differentiable function in the interval \( (0, \infty) \) such that \( y(1)=2 \) and \( \lim_{t\rightarrow x}(\frac{t^{2}y(x)-x^{2}y(t)}{x-t})=3 \) for each \( x>0. \) Then \( 2y(2) \) is equal to

If the system of linear equations, \(x+y+z =6\) ; \(x+2 y+3 z =10\) ; \(3 x+2 y+\lambda z =\mu\) has more two solutions, then \(\mu-\lambda^{2}\) is equal to

A value of \(x\) satisfying the equation \(\sin \left[ {{{\cot }^{ - 1}}\left( {1 + x} \right)} \right] = \cos \left[ {{{\tan }^{ - 1}}\,x} \right]\) , is

The system of equations
\(\begin{aligned}
& x+y+z=6 \\
& x+2 y+5 z=9, \\
& x+5 y+\lambda z=\mu,
\end{aligned}\) has no solution if

Let \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) be three points on the parabola \(y^2=6 x\) and let the line segment \(A B\) meet the line \(L\) through \(\mathrm{C}\) parallel to the \(\mathrm{x}\)-axis at the point \(\mathrm{D}\). Let \(\mathrm{M}\) and \(\mathrm{N}\) respectively be the feet of the perpendiculars from \(\mathrm{A}\) and \(\mathrm{B}\) on \(\mathrm{L}\). Then \(\left(\frac{\mathrm{AM} \cdot \mathrm{BN}}{\mathrm{CD}}\right)^2\) is equal to ...........