For a rolling spherical shell, the ratio of rotational kinetic energy and total kinetic energy is \(\frac{x}{5}\). The value of \(x\) is ................

In Young's double slit experiment, monochromatic light of wavelength \(5000\) \(A\) is used. The slits are \(1.0 \mathrm{~mm}\) apart and screen is placed at \(1.0 \mathrm{~m}\) away from slits. The distance from the centre of the screen where intensity becomes half of the maximum intensity for the first time is _______ \(10^{-6} \mathrm{~m}\).

A sample contains \(10^{-2}\, kg\) each of two substances A and \(B\) with half lives \(4 \,s\) and \(8 \,s\) respectively. The ratio of then atomic weights is \(1: 2\) The ratio of the amounts of \(A\) and \(B\) after \(16 \,s\) is \(\frac{x}{100}\). the value of \(x\) is........

Three conductors of same length having thermal conductivity \(\mathrm{k}_1, \mathrm{k}_2\) and \(\mathrm{k}_3\) are connected as shown in figure.

Area of cross sections of \(1^{\text {st }}\) and \(2^{\text {nd }}\) conductor are same and for \(3^{\text {rd }}\) conductor it is double of the \(1^{\text {st }}\) conductor. The temperatures are given in the figure. In steady state condition, the value of \(\theta\) is _______ \({ }^{\circ} \mathrm{C}\).
(Given : \(\mathrm{k}_1=60 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}, \mathrm{k}_2=120 \mathrm{Js}^{-1}\mathrm{~m}^{-1}\) \( \mathrm{~K}^{-1}, \mathrm{k}_3=135 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}\) )

In the given circuit the internal resistance of the \(18\,V\) cell is negligible. If \(R_1 = 400 \,\Omega ,\,R_3 = 100\,\Omega \) and \(R_4 = 500\,\Omega \) and the reading of an ideal voltmeter across \(R_4\) is \(5\,V,\) then the value of \(R_2\) will be ........... \(\Omega\)

Two bodies of masses \(m_{1}=5\,kg\) and \(m _{2}=3\,kg\) are connected by a light string going over a smooth light pulley on a smooth inclined plane as shown in the figure. The system is at rest. The force exerted by the inclined plane on the body of mass \(m _{1}\) will be\(....N\) [Take \(g=10\,ms ^{-2}\) ]

A plane electromagnetic wave of frequency 20 MHz travels in free space along the \(+x\) direction. At a particular point in space and time, the electric field vector of the wave is \(\mathrm{E}_y=9.3 \mathrm{Vm}^{-1}\). Then, the magnetic field vector of the wave at that point is:

Identify the logic gate given in the circuit.

The speed of a longitudinal wave in a metallic bar is \(400~m/s.\) If the density and Young's modulus of the bar material are increased by 0.5% and 1% respectively then the speed of the wave is changed approximately to ___________ \(m / s\).

Let \(e\) be the base of natural logarithm and let \(f: \{1, 2, 3, 4\} \rightarrow \{1, e, e^2, e^3\}\) and \(g: \{1, e, e^2, e^3\} \rightarrow \left\{1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}\right\}\) be two bijective functions such that \(f\) is strictly decreasing and \(g\) is strictly increasing. If \(\phi(x) = \left[f^{-1}\left\{g^{-1}\left(\dfrac{1}{2}\right)\right\}\right]^x\), then the area of the region \(R = \{(x, y): x^2 \leq y \leq \phi(x), 0 \leq x \leq 1\}\) is:

The sum of the first \(20\)  terms of the series  \(1 + \frac{3}{2} + \frac{7}{4} + \frac{{15}}{8} + \frac{{31}}{{16}} + ...\) is?

A plane \(P\) meets the coordinate axes at \(A, B\) and \(C\) respectively. The centroid of \(\Delta ABC\) is given to be \((1,1,2)\) . Then the equation of the line through this centroid and perpendicular to the plane \(P\) is

Let \(\vec{u}=\hat{i}-\hat{j}-2 \hat{k}, \vec{v}=2 \hat{i}+\hat{j}-\hat{k}, \vec{v} \cdot \vec{w}=2\) and \(\vec{v} \times \vec{w}=\vec{u}+\lambda \vec{v}\), then \(\vec{u} \cdot \vec{w}\) is equal to