Consider a light ray travelling in air is incident into a medium of refractive index \(\sqrt{2 n}\). The incident angle is twice that of refracting angle. Then, the angle of incidence will be

In a Young's double slit experiment, three polarizers are kept as shown in the figure. The transmission axes of \(P_1\) and \(P_2\) are orthogonal to each other. The polarizer \(P_3\) covers both the slits with its transmission axis at \(45^{\circ}\) to those of \(P_1\) and \(P_2\). An unpolarized light of wavelength \(\lambda\) and intensity \(I_0\) is incident on \(P_1\) and \(P_2\). The intensity at a point after \(P_3\) where the path difference between the light waves from \(s_1\) and \(s_2\) is \(\frac{\lambda}{3}\), is _______.

An emf of \(20\; \mathrm{V}\) is applied at time \(\mathrm{t}=0\) to a circuit containing in series \(10\; \mathrm{mH}\) inductor and \(5 \;\Omega\) resistor. The ratio of the currents at time \(\mathrm{t}=\infty\) and at \(t=40\; s\) is close to : (Take \(e^{2}=7.389\) )

Heat energy of \(184\,kJ\) is given to ice of mass \(600\,g\) at \(-12^{\circ}\,C\), Specific heat of ice is \(2222.3\,J\,kg ^{-1^{\circ}}\,C ^{-1}\) and latent heat of ice in \(336\,kJ / kg ^{-1}\) \((A)\) Final temperature of system will be \(0^{\circ} C\). \((B)\) Final temperature of the system will be greater than \(0^{\circ} C\). \((C)\) The final system will have a mixture of ice and water in the ratio of \(5: 1\). \((D)\) The final system will have a mixture of ice and water in the ratio of \(1: 5\). \((E)\) The final system will have water only. Choose the correct answer from the options given below:

The height of liquid column raised in a capillary tube of certain radius when dipped in liquid A vertically is, \(5 cm\). If the tube is dipped in a similar manner in another liquid \(B\) of surface tension and density double the values of liquid \(A\), the height of liquid column raised in liquid \(B\) would be \(........m\).

Primary coil of a transformer is connected to \(220 \mathrm{~V}\) ac. Primary and secondary turns of the transforms are \(100\) and \(10\) respectively. Secondary coil of transformer is connected to two series resistance shown in shown in figure. The output voltage \(\left(\mathrm{V}_0\right)\) is _______.

As shown in the figure, a potentiometer wire of resistance \(20\,\Omega\) and length \(300\,cm\) is connected with resistance box (R.B.) and a standard cell of emf \(4\,V\). For a resistance ' \(R\) ' of resistance box introduced into the circuit, the null point for a cell of \(20\,mV\) is found to be \(60\,cm\). The value of ' \(R\) ' is \(.....\Omega\)

Given below are two statements: One is labelled as Assertion \((A)\) and the other is labelled as Reason \((R).\) Assertion \((A)\) : In an uniform magnetic field, speed and energy remains the same for a moving charged particle. Reason \((R)\) : Moving charged particle experiences magnetic force perpendicular to its direction of motion.

The current sensitivity of a galvanometer can be increased by: \((A)\) decreasing the number of turns \((B)\) increasing the magnetic field \((C)\) decreasing the area of the coil \((D)\) decreasing the torsional constant of the spring Choose the most appropriate answer from the options given below.

If the system of linear equations :
\(\begin{aligned}
& x+y+2 z=6 \\
& 2 x+3 y+\mathrm{a} z=\mathrm{a}+1 \\
& -x-3 y+\mathrm{b} z=2 \mathrm{~b}
\end{aligned}\)
where \(a, b \in \mathbf{R}\), has infinitely many solutions, then \(7 a+3 b\) is equal to :

For \(p, q \in R\), consider the real valued function \(f ( x )=( x - p )^{2}- q , x \in R\) and \(q >0\). Let \(a _{1}, a _{2}, a _{3}\) and \(a _{4}\) be in an arithmetic progression with mean \(P\) and positive common difference. If \(\left| f \left( a _{ i }\right)\right|=500\) for all \(i=1,2,3,4\), then the absolute difference between the roots of \(f ( x )=0\) is.

Let \(y=y(x)\) be the solution of the differential equation \(\left(x^2+1\right) y^{\prime}-2 x y=\left(x^4+2 x^2+1\right) \cos x\), \(y(0)=1\). Then \(\int_{-3}^3 y(x) d x\) is :

Let \(S=\left\{E, E_{2} \ldots . E_{8}\right\}\) be a sample space of random experiment such that \(P\left(E_{n}\right)=\frac{n}{36}\) for every \(n =1,2 \ldots .\). Then the number of elements in the set \(\left\{ A \subset S : P ( A ) \geq \frac{4}{5}\right\}\) is