If \(\cos \,\left( {\alpha  + \beta } \right) = \frac{3}{5},\,\sin \,\left( {\alpha  - \beta } \right) = \frac{5}{{13}}\) and \(0 < \alpha ,\beta  < \frac{\pi }{4}\) then \(\tan \,\left( {2\alpha } \right)\) is equal to

The numbers of pairs \((a, b)\) of real numbers, such that whenever \(\alpha\) is a root of the equation \(x^{2}+a x+b=0, \alpha^{2}-2\) is also a root of this equation, is :

The value of \({1^2} + {3^2} + {5^2} + ....... + {25^2}\) is

If the sum of the first \(20\) terms of the series \(\log _{\left(7^{\frac{1}{2}}\right)} x+\log _{\left(7^{\frac{1}{3}}\right)} x+\log _{\left(7^{\frac{1}{4}}\right)} x+\ldots\) is \(460,\) then \(x\) is equal to

A vector \(\vec{a}\) is parallel to the line of intersection of the plane determined by the vectors \(\hat{i}, \hat{i}+\hat{j}\) and the plane determined by the vectors \(\hat{i}-\hat{j}, \hat{i}+\hat{k}\). The obtuse angle between \(\vec{a}\) and the vector \(\vec{b}=\hat{i}-2 \hat{j}+2 \hat{k}\) is

A circle touching the \(x-\) axis at \((3, 0)\) and making an intercept of length \(8\) on the \(y-\) axis passes through the point 

Let M denote the set of all real matrices of order \(3 \times 3\) and let \(\mathrm{S}=\{-3,-2,-1,1,2\}\). Let
\(\mathrm{S}_1=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_2=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=-\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_3=\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: a_{11}+a_{22}+a_{33}=0\) and \(a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\}\)
If \(n\left(\mathrm{~S}_1 \cup_2 \mathrm{US}_3\right)=125 \alpha\), then \(\alpha\) equals _______

Let [t] denote the greatest integer less than or equal to t. If the function
\(f(x)=\left\{\begin{array}{cl}b^2 \sin \left(\frac{\pi}{2}\left[\frac{\pi}{2}(\cos x+\sin x) \cos x\right]\right), & x<0 \\ \frac{\sin x-\frac{1}{2} \sin 2 x}{x^3} & , x>0 \\ a & , x=0\end{array}\right.\)
is continuous at \(x =0\), then \(a ^2+ b ^2\) is equal to

The number of real roots of the equation \(5 + |2^x - 1| = 2^x(2^x - 2)\) is

Let \(s _1, s _2, s _3, \ldots \ldots, s _{10}\) respectively be the sum to 12 terms of 10 A.P.s whose first terms are \(1,2,3, \ldots, 10\) and the common differences are \(1,3,5, \ldots, 19\) respectively. Then \(\sum \limits_{i=1}^{10} s _{ i }\) is equal to

\(\mathop {\lim }\limits_{y \to 0} \frac{{\sqrt {1 + \sqrt {1 + {y^4}} }  - \sqrt 2 }}{{{y^4}}} = \)

Let \(f(x)=\int \frac{2 x}{\left(x^2+1\right)\left(x^2+3\right)} d x\) . If \(f(3)=\frac{1}{2}\left(\log _e 5-\log _e 6\right)\), then \(f(4)\) is equal to

For some \( \alpha, \beta\in R \) let \( A=\begin{bmatrix}\alpha&2\\ 1&2\end{bmatrix} \) and \( B=\begin{bmatrix}1&1\\ 1&\beta\end{bmatrix} \) be such that \( A^{2}-4A+2I=B^{2}-3B+I=O \). Then \( (\text{det}(\text{adj}(A^{3}-B^{3})))^{2} \) is equal to ....