Let \(\alpha \) and \(\beta \) be the roots of the equation \(x^2 + x + 1 = 0.\) Then for \(y \ne 0\) in \(R,\) \(\left| {\begin{array}{*{20}{c}}
{y\, + \,1}&\alpha &\beta \\
\alpha &{y\, + \,\beta }&1\\
\beta &1&{y\, + \,\alpha }
\end{array}} \right|\) is equal to

\(\lim\limits_{x \rightarrow 0}\left(\frac{3 x^{2}+2}{7 x^{2}+2}\right)^{\frac{1}{x^{2}}}\) is equal to

Let \(\vec{v}=\alpha \hat{i}+2 \hat{j}-3 \hat{k}, \vec{w}=2 \alpha \hat{i}+\hat{j}-\hat{k}\), and \(\overrightarrow{ u }\) be a vector such that \(|\vec{u}|=\alpha > 0\). If the minimum value of the scalar triple product \([\vec{u} \vec{v} \vec{w}]\) is \(-\alpha \sqrt{3401}\), and \(|\vec{u} . \hat{i}|^2=\frac{m}{n}\) where \(m\) and \(n\) are coprime natural numbers, then \(m + n\) is equal to \(.........\).

Let \(A = \begin{bmatrix} 1 & 1 & 2 \\ -2 & 0 & 1 \\ 1 & 3 & 5 \end{bmatrix}\). Then the sum of all elements of the matrix \(\text{adj}(\text{adj}(2(\text{adj}A)^{-1}))\) is equal to:

Let the area of the triangle formed by a straight Line L: \(\mathrm{x}+\mathrm{by}+\mathrm{c}=0\) with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line L makes an angle of \(45^{\circ}\) with the positive x -axis, then the value of \(\mathrm{b}^2+\mathrm{c}^2\) is:

Let the length of the latus rectum of an ellipse with its major axis long \(x -\) axis and center at the origin, be \(8\). If the distance between the foci of this ellipse is equal to the length of the length of its minor axis, then which one of the following points lies on it?

If the function \(f(x)=\frac{1}{x} \log _{e}(\frac{1+\frac{x}{a}}{1-\frac{x}{b}}) , \quad x<0\) \(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad k \quad, \quad x=0\) \(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\frac{\cos ^{2} x-\sin ^{2} x-1}{\sqrt{x^{2}+1}-1} ,\,\,\, x>0\) is continuous at \(x=0\), then \(\frac{1}{a}+\frac{1}{b}+\frac{4}{k}\) is equal to :

Let \(H _{ n }=\frac{ x ^2}{1+ n }-\frac{ y ^2}{3+ n }=1, n \in N\). Let \(k\) be the smallest even value of \(n\) such that the eccentricity of \(H _{ k }\) is a rational number. If \(l\) is length of the latus return of \(H _{ k }\), then \(21 l\) is equal to \(.......\).

If the distance of the point \((a, 2, 5)\) from the image of the point \((1, 2, 7)\) in the line \(\dfrac{x}{1} = \dfrac{y-1}{1} = \dfrac{z-2}{2}\) is \(4\), then the sum of all possible values of \(a\) is equal to :

Let \(\quad \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=-\hat{\mathrm{i}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{c}}=\beta \hat{\mathrm{j}}-\hat{\mathrm{k}}\), where \(\alpha\) and \(\beta\) are integers and \(\alpha \beta=-6\). Let the values of the ordered pair \((\alpha, \beta)\) for which the area of the parallelogram of diagonals \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) is \(\frac{\sqrt{21}}{2}\), be \(\left(\alpha_1, \beta_1\right)\) and \(\left(\alpha_2, \beta_2\right)\). Then \(\alpha_1^2+\beta_1^2-\alpha_2 \beta_2\) is equal to

Let for some real numbers \(\alpha\) and \(\beta\), a \(=\alpha-i \beta\). If the system of equations \(4 ix +(1+ i ) y =0\) and \(8\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right) x+\bar{a} y=0\) has more than one solution then \(\frac{\alpha}{\beta}\) is equal to

The mean deviation about the mean for the data

\(x_i\)	\(5\)	\(7\)	\(9\)	\(10\)	\(12\)	\(15\)
\(f_i\)	\(8\)	\(6\)	\(2\)	\(2\)	\(2\)	\(6\)

is equal to:

If the function \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
  {\frac{{\sqrt {2  + \cos \,x} - 1}}{{\left( {\pi  - {x^2}} \right)}},}&{x \ne \pi } \\ 
  {k\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,}&{x = \pi } 
\end{array}} \right.\) is continuous at \(x\, =\pi \) , then \(k\) equals