\(2.{}^{20}{C_0} + 5.{}^{20}{C_1} + 8.{}^{20}{C_2} + 11.{}^{20}{C_3} + ......62.{}^{20}{C_{20}}\) is equal to

Let \(a_1=8, a_2, a_3, \ldots a_n\) be an \(A.P.\) If the sum of its first four terms is \(50\) and the sum of its last four terms is \(170\) , then the product of its middle two terms is

\( \text { If } S(x)=(1+x)+2(1+x)^2+3(1+x)^3+\ldots . \) \( +60(1+x)^{60}, x \neq 0 \text {, and }(60)^2 S(60)=a(b)^b+b\) where \(a, b N\), then \((a+b)\) equal to ...............

If the eccentricity \(e\) of the hyperbola \(\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1\), passing through \((6, 4\sqrt{3})\), satisfies \(15(e^2 + 1) = 34e\), then the length of the latus rectum of the hyperbola \(\dfrac{x^2}{b^2} - \dfrac{y^2}{2(a^2+1)} = 1\) is:

For some \(\theta \in\left(0, \frac{\pi}{2}\right),\) if the eccentricity of the hyperbola, \(x^{2}-y^{2} \sec ^{2} \theta=10\) is \(\sqrt{5}\) times the eccentricity of the ellipse, \(x^{2} \sec ^{2} \theta+y^{2}=5,\) then the length of the latus rectum of the ellipse is

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 \(A =\left(\begin{array}{ll}2 & -2 \\ 1 & -1\end{array}\right)\) and \(B =\left(\begin{array}{ll}-1 & 2 \\ -1 & 2\end{array}\right)\). Then the number of elements in the set \(\left\{( n , m ): n , m \in\{1,2, \ldots . .10\}\right.\) and \(\left.nA ^{ n }+ mB ^{ m }= I \right\}\) is

If \([ t ]\) denotes the greatest integer \(\leq t\), then the value of \(\int_{0}^{1}\left[2 x-\left|3 x^{2}-5 x+2\right|+1\right] d x\) is.

Let \(A\) be a \(2 \times 2\) real matrix and \(I\) be the identity matrix of order \(2\) . If the roots of the equation \(|A-x I|=0\) be \(-1\) and \(3\) , then the sum of the diagonal elements of the matrix \(A^2\) is ...........

Let the coefficients of third, fourth and fifth terms in the expansion of \(\left(x+\frac{a}{x^{2}}\right)^{n}, x \neq 0,\) be in the ratio \(12: 8: 3 .\) Then the term independent of \(x\) in the expansion, is equal to ...... .

Let the system of linear equations \(x+y+\alpha z=2\) \(3 x+y+z=4\) \(x+2 z=1\) have a unique solution \(\left(x^{*}, y^{*}, z^{*}\right)\). If \(\left(\alpha, x^{*}\right),\left(y^{*}, \alpha\right)\) and \(\left(x^{*},-y^{*}\right)\) are collinear points, then the sum of absolute values of all possible values of \(\alpha\) is

If for the solution curve \(y=f(x)\) of the differential equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}+(\tan x) y=\frac{2+\sec x}{(1+2 \sec x)^2}\), \(x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right), f\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{10}\), then \(f\left(\frac{\pi}{4}\right)\) is equal to :

Let the image of the point \(P(0, -5, 0)\) in the line \(\dfrac{x-1}{2} = \dfrac{y}{1} = \dfrac{z+1}{-2}\) be the point \(R\) and the image of the point \(Q\left(0, \dfrac{-1}{2}, 0\right)\) in the line \(\dfrac{x-1}{-1} = \dfrac{y+9}{4} = \dfrac{z+1}{1}\) be the point \(S\). Then the square of the area of the parallelogram \(PQRS\) is __________.