\(\cos ^2\left(\frac{7 \pi}{8}\right)+\cos ^2\left(\frac{5 \pi}{8}\right)+\cos ^2\left(\frac{3 \pi}{8}\right)+\cos ^2\left(\frac{\pi}{8}\right)=\)

If \(f(x)=3 e^{x^2}\) then \(f^{\prime}(x)-2 x f(x)+\frac{1}{3} f(0)-f^{\prime}(0)=\)

Solve the following equation \(\sin x+\sqrt{3} \cos x=\sqrt{2}\)

The derivative of \(y=\sin ^{-1}\left(\frac{\sqrt{1+x}-\sqrt{1-x}}{2}\right)\) is

The interval in which \(y=\ln (\ln (x)), x>1\) is decreasing is

\(\log _4 2-\log _8 2+\log _{16} 2-\ldots\) is equal to

If \(A+2 B=\left(\begin{array}{ccc}1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1\end{array}\right)\) and \(2 A-B=\left(\begin{array}{ccc}2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2\end{array}\right)\) then \(\operatorname{Tr}[\mathrm{A}]-\operatorname{Tr}[\mathrm{B}]=\)

One of the latus recta of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) subtends an angle \(2 \operatorname{Tan}^{-1}\left(\frac{3}{2}\right)\) at the centre of the hyperbola. If \(\mathrm{b}^2=36\) and e is the eccentricity of the given hyperbola, then \(\sqrt{\mathrm{a}^2+\mathrm{e}^2}=\)

Noble metals, like gold and platinum are soluble in which of the following mixtures?

If \(x_1, x_2, x_3 \ldots x_n\) are \(n\) observations such that \(\Sigma\left(x_i+2\right)^2=28 n\) and \(\Sigma\left(x_i-2\right)^2=12 n\), then the variance is:

If a circle \(S\) passing through the point \((3,4)\) cuts the circle \(x^2+y^2=36\) orthogonally, then the locus of the centre of \(S\) is

Match the item given in Column-I (method used for determining colligative property) and Column-II (for the corresponding colligative property) and find the correct order.
\(\begin{array}{|c|c|c|}
\hline & \text { Column I } & \text { Column II } \\
\hline \text { A. } & \text { Beckmann method } & \text { 1. Osmotic pressure } \\
\hline \text { B. } & \begin{array}{l}
\text { Ostwald-Walker method }
\end{array} & \text { 2. Elevation in B.P } \\
\hline \text { C. } & \begin{array}{l}
\text { Berkeley-Hartley method }
\end{array} & \text { 3. Depression in F.P. } \\
\hline \text { D. } & \text { Landsberger method } & \begin{array}{l}
\text { 4. Relative lowering of vapour pressure }
\end{array} \\
\hline
\end{array}\)

Two persons \(A\) and \(B\) are throwing an unbiased six faced dice alternatively, with the condition that the person who throws 3 first wins the game. If \(A\) starts the game, then probabilities of \(A\) and \(B\) to win the same are, respectively