The refractive index of a transparent liquid filled in an equilateral hollow prism is \(\sqrt{2}\). The angle of minimum deviation for the liquid will be

Three students \(S_{1}, S_{2}\) and \(S_{3}\) perform an experiment for determining the acceleration due to gravity \((g)\) using a simple pendulum. They use different lengths of pendulum and record time for different number of oscillations. The observations are as shown in the table.

Student No.	Length of pendulum \((cm)\)	No. of osci \((n)\)	Total time for osci	Time period \((s)\)
\(1.\)	\(64.0\)	\(8\)	\(128.0\)	\(16.0\)
\(2.\)	\(64.0\)	\(4\)	\(64.0\)	\(16.0\)
\(3.\)	\(20.0\)	\(4\)	\(36.0\)	\(9.0\)

(Least count of length \(=0.1 \,{m}\), least count for time \(=0.1\, {s}\) ) If \(E_{1}, E_{2}\) and \(E_{3}\) are the percentage errors in \('g'\) for students \(1,2\) and \(3\) respectively, then the minimum percentage error is obtained by student no. ....... .

The surface tension of a soap solution is \(3.5 \times 10^{-2}\) N/m. The work required to increase the radius of a soap bubble from \(1\) cm to \(2\) cm is \(\alpha \times 10^{-6}\) J. The value of \(\alpha\) is _____. (\(\pi = 22/7\))

If one mole of the polyatomic gas is having two vibrational modes and \(\beta\) is the ratio of molar specific heats for polyatomic gas \(\left(\beta=\frac{ C _{ P }}{ C _{ v }}\right)\) then the value of \(\beta\) is:

The radius of a sphere is measured to be \((7.50 \pm 0.85) \,cm .\) Suppose the percentage error in its volume is \(x\). The value of \(x\), to the nearest integer is .....\(\%\)

A radioactive nucleus \(n_2\) has 3 times the decay constant as compared to the decay constant of another radioactive nucleus \(n_1\). If initial number of both nuclei are the same, what is the ratio of number of nuclei of \(n_2\) to the number of nuclei of \(n_1\), after one half-life of \(n_1\) ?

Figure shows a circuit that contains four identical resistors with resistance \(R =2.0\, \Omega,\) two identical inductors with inductance \(L =2.0 \,mH\) and an ideal battery with \(e m f E =9 \,V .\) The current \('i'\) just after the switch \('S'\) is closed will be .... \(A\)

Spherical insulating ball and a spherical metallic ball of same size and mass are dropped from the same height. Choose the correct statement out of the following (Assume negligible air friction)

Two particles are located at equal distance from origin. The position vectors of those are represented by \(\bar{A}=2 \hat{i}+3 n \hat{j}+2 \hat{k}\) and \(\bar{B}=2 \hat{i}-2 \hat{j}+4 p \hat{k}\), respectively. If both the vectors are at right angle to each other, the value of \(\mathrm{n}^{-1}\) is _____ .

A \(LCR\) circuit behaves like a damped harmonic oscillator. Comparing it with a physical springmass damped oscillator having damping constant \(\mathrm{b}\), the correct equivalence would be:

For equal point charges \(Q\) each are placed in the \(xy\) plane at \((0, 2), (4, 2), (4, -2)\) and \((0, -2)\). The work required to put a fifth change \(Q\) at the origin of the coordinate system will be

A gun mounted on the ground fires bullets in all directions with same speed. The farthest distance the bullets could reach is \(6.4\) m. The speed of the bullets from the gun is ______ m/s.
(take \(g=10\) m/s\(^2\))