If \(m\) is the \(A.M\) of two distinct real numbers \( l\)  and \(n (l,n>1) \) and  \(G_1, G_2\) and \(G_3\) are three geometric means between  \(l\) and \(n\) then \(G_1^4 + 2G_2^4 + G_3^4\) equals :

Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be defined as \(f(x+y)+f(x-y)=2 f(x) f(y), f\left(\frac{1}{2}\right)=-1 .\) Then, the value of \(\sum_{\mathrm{k}=1}^{20} \frac{1}{\sin (\mathrm{k}) \sin (\mathrm{k}+\mathrm{f}(\mathrm{k}))}\) is equal to:

The sum of all possible values of \(\theta \in[-\pi, 2 \pi]\), for which \(\frac{1+i \cos \theta}{1-2 i \cos \theta}\) is purely imaginary, is equal to

Let \(p(x)\) be a quadratic polynomial such that \(p(0)= 1\) . If \(p(x)\) leaves remainder \(4\) when divided by \(x-1\) and it leaves remainder \(6\) when divided by \(x+ 1\) ; then

A player \(X\) has a biased coin whose probability of showing heads is \(p\) and a player \(Y\) has a fair coin . They start playing a game with their own coins and play alternately . The player who throws a head first is a winner. If \(X\) starts the game, and the probability of winning the game by both the players is equal, then the value of \('p'\) is

Consider the following two statements : Statement \(I\) : For any two non-zero complex numbers \(\mathrm{z}_1, \mathrm{z}_2\) \(\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right)\) and Statement \(II\) : If \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) are three distinct complex numbers and a, b, c are three positive real numbers such that \(\frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}\), then \(\frac{\mathrm{a}^2}{\mathrm{y}-\mathrm{z}}+\frac{\mathrm{b}^2}{\mathrm{z}-\mathrm{x}}+\frac{\mathrm{c}^2}{\mathrm{x}-\mathrm{y}}=1\) Between the above two statements,

If the function \(f(x)=\left\{\begin{array}{cc}\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}} & , x \neq 0 \\ a \log _e 2 \log _e 3 & , x=0\end{array}\right.\) is continuous at \(x=0\), then the value of \(a^2\) is equal to

Let \(\alpha \) and \(\beta \) be two roots of the equation \(x^2 + 2x + 2 = 0\) , then  is equal to \({\alpha ^{15}} + {\beta ^{15}}\) is equal to 

Let \(f: R \rightarrow R\) be defined as \(f(x)=e^{-x} \sin x\). If \(F :[0,1] \rightarrow R\) is a differentiable function such that \(F ( x )=\int_{0}^{ x } f ( t ) dt ,\) then the value of \(\int_{0}^{1}\left( F ^{\prime}( x )+ f ( x )\right) e ^{ x } dx\) lies in the interval

Let \(y=y(x)\) be the solution of the differential equation \(\left(x^2-3 y^2\right) d x+3 x y d y=0, y(1)=1\). Then \(6 y^2( e )\) is equal to \(......\)

\(\left( {\left( {\begin{array}{*{20}{c}}
{21}\\
1
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
1
\end{array}} \right)} \right) + \left( {\left( {\begin{array}{*{20}{c}}
{21}\\
2
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
2
\end{array}} \right)} \right)\)\( + \left( {\left( {\begin{array}{*{20}{c}}
{21}\\
3
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
3
\end{array}} \right)} \right) + \;.\;.\;.\)\( + \left( {\left( {\begin{array}{*{20}{c}}
{21}\\
{10}
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
{10}
\end{array}} \right)} \right) = \)

A plane \(P\) contains the line \(x+2 y+3 z+1=0=x-y-z-6\)  and is perpendicular to the plane \(-2 x+y+z+8=0\). Then which of the following points lies on \(\mathrm{P}\) ?

The value of the limit \(\lim _{\theta \rightarrow 0} \frac{\tan \left(\pi \cos ^{2} \theta\right)}{\sin \left(2 \pi \sin ^{2} \theta\right)}\) is equal to :