The set of all values of \(\theta\) such that \(\frac{1-i \cos \theta}{1+2 i \sin \theta}\) is purely imaginary is

If \(\cot A=\frac{11}{60}, \cos B=\frac{7}{25}\) and neither \(A\) nor \(B\) is in the first quadrant, then \(\left(A+\frac{B}{2}\right)\) lies in the quadrant

The height (in cm) of a cylinder of the greatest volume that can be inscribed in a sphere of radius $3 \mathrm{cm}$, is

If the distance between the points \((a \cos \theta, a \sin \theta)\) and \((a \cos \phi, a \sin \phi)\) is \(2 a\), then \(\theta\) is equal to

A pair of dice is thrown twice in succession. The probability of getting prime numbers on both the dice in first throw and composite numbers on both the dice in second throw is

If \(\mathrm{I}_n=\int \frac{1}{\left(x^2+1\right)^n} d x\), then \(2 n \mathrm{I}_{n+1}-(2 n-1) \mathrm{I}_n=\)

If \(\alpha, \beta\) and \(\gamma\) are the roots of the equation \(x^3+p x^2+q x+r=0\), then the coefficient of \(x\) in the cubic equation whose roots are \(\alpha(\beta+\gamma), \beta(\gamma+\alpha)\) and \(\gamma(\alpha+\beta)\) is

\(9 \mathrm{~kg}\) solution is poured into a glass U-tube as shown in the figure below. The tube's inner diameter is \(2 \sqrt{\frac{\pi}{5}} \mathrm{~m}\) and the solution oscillates freely up and down about its position of equilibriwm \((x=0)\). The period of oscillation in seconds is \(\left(1 \mathrm{~m}^3\right.\) of solution has a mass \(\mu=900 \mathrm{~kg}, g=10 \mathrm{~m} / \mathrm{s}^2\), ignore frictional and swrface tension effects)

Match the items given in List-I to the items given in List-II.
The correct match is Codes \(\begin{array}{llll}A & B & C & D\end{array}\)

With reference to the redox properties of hydrogen peroxide \(\left(\mathrm{H}_2 \mathrm{O}_2\right)\), which of these reactions are feasible? (i) \(2 \mathrm{Fe}^{2+} 2 \mathrm{H}^{+}+\mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{Fe}^{3+}+2 \mathrm{H}_2 \mathrm{O}\) (ii) \(2 \mathrm{MnO}_4^{-}+6 \mathrm{H}^{+}+5 \mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{Mn}^{2+}\) \(+8 \mathrm{H}_2 \mathrm{O}+5 \mathrm{O}_2\) (iii) \(2 \mathrm{Fe}^{2+}+\mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{Fe}^{3+}+2 \mathrm{OH}^{-}\) (iv)\(2 \mathrm{MnO}_4^{-}+3 \mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{MnO}_2+2 \mathrm{H}_2 \mathrm{O}\) \(+3 \mathrm{O}_2+2 \mathrm{OH}^{-}\)

Match the following:

The correct answer is

The alloy formed by beryllium with ' X ' is used in the preparation of high strength springs. ' X ' is