A body starts falling freely from height \(\mathrm{H}\) hits an inclined plane in its path at height \(\mathrm{h}\). As a result of this perfectly elastic impact, the direction of the velocity of the body becomes horizontal. The value of \(\frac{\mathrm{H}}{\mathrm{h}}\) for which the body will take the maximum time to reach the ground is _______.

The electric field component of a monochromatic radiation is given by \(\vec E = 2{E_0}\,\hat i\,\cos\, kz\,\cos\, \omega t\) Its magnetic field \(\vec B\) is then given by

The translational degrees of freedom \((f)\) and rotational degrees of freedom \((f)\) of \(\mathrm{CH}_4\) molecule are _______.

The diameter and height of a cylinder are measured by a meter scale to be \(12.6 \pm 0.1\, cm\) and \(34.2  \pm  0.1\, cm\), respectively. What will be the value of its volume in appropriate significant figures?

A satellite of mass \(\frac{M}{2}\) is revolving around earth in a circular orbit at a height of \(\frac{R}{3}\) from earth surface. The angular momentum of the satellite is \(M \sqrt{\frac{G M R}{x}}\). The value of \(x\) is ______ , where \(M\) and \(R\) are the mass and radius of earth, respectively. ( G is the gravitational constant)

A \(750\, Hz,\) \(20\) \(V\) (rms ) source is connected to a resistance of \(100\, \Omega\) an inductance of \(0.1803\, H\) and a capacitance of \(10\mu ,\) \(F\) all in series. The time in which the resistance (heat capacity \(2 J /{ }^{\circ} C\) ) will get heated by \(10^{\circ} C\). (assume no \(105 s\) of heat to the surroundings) is close to\(.....s\)

A wire of resistance \(20 \Omega\) is divided into \(10\) equal parts. A combination of two parts are connected in parallel and so on. Now resulting pairs of parallel combination are connected in series. The equivalent resistance of final combination is _______\(\Omega\).

The position vectors of two 1 kg particles, (A) and (B), are given by
\(\overrightarrow{\mathrm{r}}_{\mathrm{A}}=\left(\alpha_1 \mathrm{t}^2 \hat{i}+\alpha_2 \mathrm{t} \hat{j}+\alpha_3 \mathrm{t} \hat{k}\right) \mathrm{m}\) and \(\overrightarrow{\mathrm{r}}_{\mathrm{B}}=(\beta_1 \mathrm{t} \hat{i}+\beta_2 \mathrm{t}^2 \hat{j}\) \(+\beta_3 \mathrm{t} \hat{k}) \mathrm{m}\), respectively; \((\alpha_1=1 \mathrm{~m} / \mathrm{s}^2, \alpha_2=3 \mathrm{n} \mathrm{m} / \mathrm{s},\) \(\alpha_3=2 \mathrm{~m} / \mathrm{s},\) \(\beta_1=2 \mathrm{~m} / \mathrm{s}, \beta_2=-1 \mathrm{~m} / \mathrm{s}^2, \beta_3=4 \mathrm{pm} / \mathrm{s})\), where t is time, n and p are constants. At \(t=1 \mathrm{~s},\left|\overrightarrow{V_A}\right|=\left|\vec{V}_B\right|\) and velocities \(\vec{V}_A\) and \(\vec{V}_B\) of the particles are orthogonal to each other. At \(t=1 \mathrm{~s}\), the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is \(\sqrt{\mathrm{L}} \mathrm{kgm}^2 \mathrm{~s}^{-1}\). The value of L is _______ .

A particle is rotating in a circular path and at any instant its motion can be described as \(\theta = \dfrac{5t^4}{40} - \dfrac{t^3}{3}\). The angular acceleration of the particle after \(10\) seconds is _______ rad/s\(^2\).

A monochromatic beam of light has a frequency \(v = \frac{3}{{2\pi }} \times {10^{12}}\,Hz\) and is propagating along the direction \(\frac{{\hat i + \hat j}}{{\sqrt 2 }}\). It is polarized along the \(\hat k\) direction. The acceptable form for the magnetic field is

10 kg of ice at \(-10^{\circ}C\) is added to 100 kg of water to lower its temperature from \(25^{\circ}C.\) Consider no heat exchange to surroundings. The decrement to the temperature of water is ___________ \(^{\circ}C.\)
(specific heat of ice \(= 2100\ J/Kg.^{\circ}C\), specific heat of water \(= 4200\ J/Kg.^{\circ}C\), latent heat of fusion of ice \(= 3.36 \times 10^{5}\ J/Kg\))

Two waves of intensity ratio \(1: 9\) cross each other at a point. The resultant intensities at the point, when (a) Waves are incoherent is \(I_1\) (b) Waves are coherent is \(I_2\) and differ in phase by \(60^{\circ}\). If \(\frac{I_1}{I_2}=\frac{10}{x}\), then \(x=\) _______.