The critical angle for a denser-rarer interface is \(45^{\circ}\). The speed of light in rarer medium is \(3 \times 10^8\) ms. The speed of light in the denser medium is:

If \(R , X _{ L }\). and \(X _{ C }\) represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless:

What will be the effect on the root mean square velocity of oxygen molecules if the temperature is doubled and oxygen molecule dissociates into atomic oxygen?

Suppose a long solenoid of 100 cm length, radius 2 cm having 500 turns per unit length, carries a current \(I=10 \sin (\omega t ) A\), where \(\omega=1000 rad . / s\). A circular conducting loop (B) of radius 1 cm coaxially slided through the solenoid at a speed \(v =1 cm / s\). The r.m.s. current through the loop when the coil B is inserted 10 cm inside the solenoid is \(\alpha / \sqrt{2} \quad \mu A\). The value of \(\alpha\) is ___________.
[Resistance of the loop \(=10 \Omega\) ]

A sample of radioactive material \(A\), that has an activity of \(10\, mCi\, (1\, Ci = 3.7 \times 10^{10}\, decays/s)\), has twice the number of nuclei as another sample of different radioactive material \(B\) which has an activity of \(20\, mCi\). The correct choices for half-lives of \(A\) and \(B\) would then be respectively

The critical angle of a medium for a specific wavelength, if the medium has relative permittivity \(3\) and relative permeability \(\frac{4}{3}\) for this wavelength, will be.....\(^o\)

If \(1000\) droplets of water of surface tension \(0.07\,N / m\). having same radius \(1\,mm\) each, combine to from a single drop. In the process the released surface energy is \(\left(\text { Take } \pi=\frac{22}{7}\right)\)

The explosive in a Hydrogen bomb is a mixture of \({ }_1 \mathrm{H}^2,{ }_1 \mathrm{H}^3\) and \({ }_3 \mathrm{Li}^6\) in some condensed form. The chain reaction is given by \( { }_3 \mathrm{Li}^6+{ }_0 \mathrm{n}^1 \rightarrow{ }_2 \mathrm{He}^4+{ }_1 \mathrm{H}^3 \) \( { }_1 \mathrm{H}^2+{ }_1 \mathrm{H}^3 \rightarrow{ }_2 \mathrm{He}^4+{ }_0 \mathrm{n}^1\) During the explosion the energy released is approximately [Given : \(\mathrm{M}(\mathrm{Li})=6.01690\  \mathrm{amu} . \mathrm{M}\left({ }_1 \mathrm{H}^2\right)=2.01471 \  amu.\) \(\mathrm{M}\left({ }_2 \mathrm{He}^4\right)=4.00388\  \mathrm{amu}\),\( and\ \)\(1 \ \mathrm{amu}=931.5\) \(\mathrm{MeV}]\)

A Carnot engine operates between two reservoirs of temperatures \(900\; \mathrm{K}\) and \(300 \;\mathrm{K}\) The engine performs \(1200\; \mathrm{J}\) of work per cycle. The heat energy (in \(\mathrm{J}\) ) delivered by the engine to the low temperature reservoir, in a cycle. is

Let \(\left\{a_{n}\right\}_{n=0}^{\infty}\) be a sequence such that \(a _{0}= a _{1}=0\) and \(a _{ n +2}=2 a _{ n +1}- a _{ n }+1\) for all \(n \geq 0\). Then, \(\sum\limits_{ n =2}^{\infty} \frac{ a _{ n }}{7^{ n }}\) is equal to

If \(\lim \limits_{x \rightarrow 1} \frac{x+x^{2}+x^{3}+\ldots+x^{n}-n}{x-1}=820,(n \in N)\) then the value of \(n\) is equal to

Let \(\vec{a}=a_i \hat{i}+a_2 \hat{j}+a_3 \hat{k}\) and \(\vec{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}\) be two vectors such that \(|\vec{a}|=1 ; \quad \vec{a} \cdot \vec{b}=2\) and \(|\vec{b}|=4\). If \(\vec{c}=2(\vec{a} \times \vec{b})-3 \vec{b}\), then the angle between \(\vec{b}\) and \(\vec{c}\) is equal to :

In the given figure switches \(S_{1}\) and \(S_{2}\) are in open condition. The resistance across \(a b\) when the switches \(S_{1}\) and \(S_{2}\) are closed is \(...\,\Omega\)