Let the refractive index of a denser medium with respect to a rarer medium be \(n_{12}\) and its critical angle be \(\theta_C\).  At an angle of incidence \(A\) when light is travelling from denser medium to rarer medium, a part of the light is reflected and the rest is refracted and the angle between reflected and refracted rays is \(90^o\).  Angle \(A\) is given by

The magnetic potential due to a magnetic dipole at a point on its axis situated at a distance of \(20 \mathrm{~cm}\) from its center is \(1.5 \times 10^{-5} \  \mathrm{Tm}\). The magnetic moment of the dipole is _______ \(\mathrm{Am}^2\). (Given : \(\frac{\mu_0}{4 \pi}=10^{-7} \  \mathrm{TmA}^{-1}\) )

A current of \(5\; {A}\) is passing through a non-linear magnesium wire of cross-section \(0.04\; {m}^{2}\). At every point the direction of current density is at an angle of \(60^{\circ}\) with the unit vector of area of cross-section. The magnitude of electric field at every point of the conductor is ....\({V} / {m}\) (Resistivity of magnesium is \(\rho=44 \times 10^{-8}\, \Omega m\))

The de Broglie wavelength of an electron having kinetic energy \(E\) is \(\lambda\). If the kinetic energy of electron becomes \(\frac{E}{4}\), then its de-Broglie wavelength will be :

Two identical thin biconvex lenses of focal length \(15\, cm\) and refractive index \(1.5\) are in contact with each other. The space between the lenses is filled with a liquid of refractive index \(1.25\). The focal length of the combination is \(cm\).

Charge is distributed within a sphere of radius \(R\) with a volume charge density \(\rho (r) = \frac{A}{{{r^2}}}{e^{ - 2r/a}}\) where \(A\) and \(a\) are constants. If \(Q\) is the total charge of this charge distribution, the radius \(R\) is.

A force \(\mathrm{f}=\mathrm{x}^2 \mathrm{y} \hat{\mathrm{i}}+\mathrm{y}^2 \hat{\mathrm{j}}\) acts on a particle in a plane \(\mathrm{x}+\mathrm{y}=10\). The work done by this force during a displacement from \((0,0)\) to \((4 \mathrm{~m}, 2 \mathrm{~m})\) is Joule (round off to the nearest integer)

In a screw gauge, \(5\) complete rotations of the screw cause it to move a linear distance of \(0.25\, cm\). There are \(100\) circular scale divisions. The thickness of a wire measured by this screw gauge gives a reading of \(4\) main scale divisions and \(30\) circular scale divisions . Assuming negligible zero error, the thickness of the wire is

A hanging mass \(M\) is connected to a four times bigger mass by using a string-pulley arrangement. as shown in the figure. The bigger mass is placed on a horizontal ice-slab and being pulled by \(2\,Mg\) force. In this situation. tension in the string is \(\frac{x}{5}\) \(Mg\) for \(x =\) Neglect mass of the string and friction of the block (bigger mass) with ice slab. (Given \(g=\) acceleration due to gravity)

Let \( \alpha, \beta \in \mathbb{R} \) be such that the function
\( f(x)=\begin{cases}2\alpha(x^{2}-2)+2\beta x&,x<1\\ (\alpha+3)x+(\alpha-\beta)&,x\ge1\end{cases} \)
be differentiable at all \( x \in \mathbb{R} \). Then \( 34(\alpha+\beta) \) is equal to

A line passing through the point \(\mathrm{P}(\mathrm{a}, 0)\) makes an acute angle \(\alpha\) with the positive x -axis. Let this line be rotated about the point \(P\) through an angle \(\frac{\alpha}{2}\) in the clock-wise direction. If in the new position, the slope of the line is \(2-\sqrt{3}\) and its distance from the origin is \(\frac{1}{\sqrt{2}}\), then the value of \(3 a^2 \tan ^2 \alpha-2 \sqrt{3}\) is

If in a \(G.P.\) of \(64\) terms, the sum of all the terms is \(7\) times the sum of the odd terms of the \(G.P,\) then the common ratio of the \(G.P\). is equal to

Let \(k\) be a non-zero real number  If \(f(x) = {\rm{ }}\left\{ {\begin{array}{*{20}{c}}
{\frac{{\left( {{e^x} - 1} \right)^2}}{{\sin {\mkern 1mu} \left( {\frac{x}{k}} \right){\mkern 1mu} \log {\mkern 1mu} \left( {1 + \frac{x}{4}} \right)}}{\mkern 1mu} ,{\mkern 1mu} x \ne 0}\\
{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} 12{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} ,x{\mkern 1mu} {\mkern 1mu}  = 0{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} }
\end{array}} \right.\)  is a continuous function then the value of \(k\) is