Let \(\mathrm{n}\) denote the number of solutions of the equation \(z^{2}+3 \bar{z}=0\), where \(\mathrm{z}\) is a complex number. Then the value of \(\sum_{k=0}^{\infty} \frac{1}{n^{k}}\) is equal to:

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 \(............\).

Consider \(10\) observation \(\mathrm{x}_1, \mathrm{x}_2, \ldots, \mathrm{x}_{10}\). such that \(\sum_{i=1}^{10}\left(x_i-\alpha\right)=2\) and \(\sum_{i=1}^{10}\left(x_i-\beta\right)^2=40\), where \(\alpha, \beta\) are positive integers. Let the mean and the variance of the observations be \(\frac{6}{5}\) and \(\frac{84}{25}\) respectively. The \(\frac{\beta}{\alpha}\) is equal to :

Let \(y=f(x)\) be a thrice differentiable function in \((-5,5)\). Let the tangents to the curve \(y=f(x)\) at \((1, \mathrm{f}(1))\) and \((3, \mathrm{f}(3))\) make angles \(\frac{\pi}{6}\) and \(\frac{\pi}{4}\), respectively with positive \(x\)-axis. If \(27 \int_1^3\left(\left(f^{\prime}(t)\right)^2+1\right) f^{\prime \prime}(t) d t=\alpha+\beta \sqrt{3}\) where \(\alpha, \beta\) are integers, then the value of \(\alpha+\beta\) equals

Let \(A=\begin{bmatrix} 1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7 \end{bmatrix}\) and \(\det(A-\alpha I)=0\), where \(\alpha\) is a real number. If the largest possible value of \(\alpha\) is \(p\), then the circle \((x-p)^2+(y-2p)^2=320\), intersects the co-ordinate axes at

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three units vectors such that \(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0} .\) If \(\lambda=\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}} \) and \(\overrightarrow{\mathrm{d}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}},\) then the ordered pair \((\lambda, {\mathrm{\vec d}})\) is equal to 

Let three vectors \(\overrightarrow{\mathrm{a}}=\alpha \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}\), \(\vec{b}=5 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{c}=x \hat{i}+y \hat{j}+z \hat{k}\) from a triangle such that \(\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}\) and the area of the triangle is \(5 \sqrt{6}\). if \(\alpha\) is a positive real number, then \(|\overrightarrow{\mathrm{c}}|^2\) is :

The number of integral values of \(k\), for which one root of the equation \(2 x^2-8 x+k=0\) lies in the interval \((1,2)\) and its other root lies in the interval \((2,3)\), is :

The area (in sq, units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse \(\frac{{{x^2}}}{9} + \frac{{{y^2}}}{5} = 1\) is :

If \(\int {\frac{{dx}}{{{x^3}{{\left( {1 + {x^6}} \right)}^{2/3}}}} = xf\left( x \right){{\left( {1 + {x^6}} \right)}^{\frac{1}{3}}} + C} \) where \(C\) is a constant of integration, then the function \(f(x)\) is equal to

If \(\frac{1^3+2^3+3^3+\ldots \ldots \text {.upto } n \text { terms }}{1 \cdot 3+2 \cdot 5+3 \cdot 7+\ldots \ldots \text { upto } n \text { terms }}=\frac{9}{5}\), then the value of \(n\) is

If \(\alpha\) is a root of the equation \(x^2+x+1=0\) and \(\sum_{\mathrm{k}=1}^{\mathrm{n}}\left(\alpha^{\mathrm{k}}+\frac{1}{\alpha^{\mathrm{k}}}\right)^2=20\), then n is equal to

The area (in \(sq. units\)) of the region \(A = \left\{ {\left( {x,y} \right):\frac{{{y^2}}}{2} \le x \le y + 4} \right\}\) is