If \(tan\, A\) and \(tan\, B\) are the roots of the quadratic equation, \(3x^2 - 10x - 25 = 0\) then the value of \(3\, sin^2\, (A +B)- 10\, sin\,(A +B). cos\,(A+ B)- 25\, cos^2\, (A+B)\) is

If \({I_1} = \int\limits_0^1 {{e^{ - x}}} {\cos ^2}\,x\,dx\,;\,{I_2} = \int\limits_0^1 {{e^{ - {x^2}}}} {\cos ^2}\,x\,dx\) and \(\,{I_3} = \int\limits_0^1 {{e^{ - {x^3}}}} dx\) ; then

The \(20^{\text {th}}\) term from the end of the progression \(20,19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \ldots,-129 \frac{1}{4}\) is :-

Let \(f: \left(-\frac{\pi}{4}, \frac{\pi}{4}\right) \rightarrow \mathrm{R}\) be defined as \(f(x)=(1+|\sin x|)^{\frac{3 a}{\sin x \mid}} ,\quad -\frac{\pi}{4}\,<\,x\,<\,0\) \(\quad\quad\quad\quad\quad\quad b ,\quad\quad\quad\quad\quad x=0\) \(\quad\quad\quad\quad e^{\cot 4 x / \cot 2 x} ,\quad\quad\quad 0\,<\,x\,<\,\frac{\pi}{4}\) If \(\mathrm{f}\) is continuous at \(\mathrm{x}=0\), then the value of \(6 \mathrm{a}+\mathrm{b}^{2}\) is equal to:

For the function \(f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right) \text {, where } x \in\left[0, \frac{\pi}{2}\right] \text {, }\) consider the following two statements : (\(I\)) \(\mathrm{f}\) is increasing in \(\left(0, \frac{\pi}{2}\right)\). (\(II\)) \(\mathrm{f}^{\prime}\) is decreasing in \(\left(0, \frac{\pi}{2}\right)\). Between the above two statements,

If \(2x = {y^{\frac{1}{5}}} + {y^{ - \frac{1}{5}}}\) and \((x^2 -1) \frac{{{d^2}y}}{{d{x^2}}} + \lambda x\frac{{dy}}{{dx}} + ky = 0\) , then \( \lambda + k\) is equal to

The number of ways to distribute \(30\) identical candies among four children \(C _{1}, C _{2}, C _{3}\) and \(C _{4}\) so that \(C _{2}\) receives atleast \(4\) and atmost \(7\) candies, \(C _{3}\) receives atleast \(2\)and atmost \(6\) candies, is equal to

For \(x\, \in \,R\,,\,x\, \ne \, - 1,\) if \({(1 + x)^{2016}} + x{(1 + x)^{2015}} + {x^2}{(1 + x)^{2014}} + ....{x^{2016}} = \sum\limits_{i = 0}^{2016} {{a_i\,}{\,x^i}} ,\) then \(a_{17}\) is equal to

The integral \(\int \frac{(2 x-1) \cos \sqrt{(2 x-1)^{2}+5}}{\sqrt{4 x^{2}-4 x+6}} d x\) is equal to (where \(c\) is a constant of integration)

Statement \(-1\) : The system of linear equations \(x + \left( {\sin \,\alpha } \right)y + \left( {\cos \,\alpha } \right)z = 0\) \(x + \left( {\cos \,\alpha } \right)y + \left( {\sin \alpha } \right)z = 0\) \(x - \left( {\sin \,\alpha } \right)y - \left( {\cos \alpha } \right)z = 0\) has a non-trivial solution for only one value of \(\alpha \) lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)  Statement \(-2\) : The equation in \(\alpha \) \(\left| {\begin{array}{*{20}{c}}
  {\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha } \\ 
  {\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha } \\ 
  {\cos {\mkern 1mu} \alpha }&{ - \sin {\mkern 1mu} \alpha }&{ - \cos {\mkern 1mu} \alpha } 
\end{array}} \right| = 0\) has only one solution lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)

Let \(R\) be a relation on \(N \times N\) defined by \((a, b) R\) (c, d) if and only if \(a d(b-c)=b c(a-d)\). Then \(R\) is

\(\mathop \smallint \limits_{\frac{\pi }{4}}^{\frac{{3\pi }}{4}} \frac{{dx}}{{1 + \cos x}} = \) . . . .

If \({ }^{2n } C _3:{ }^{n } C _3=10: 1\), then the ratio \(\left(n^2+3 n\right):\left(n^2-3 n+4\right)\) is