Two bodies are having kinetic energies in the ratio \(16: 9\). If they have same linear momentum, the ratio of their masses respectively is

For an object placed at a distance \(2.4\,m\) from a lens, a sharp focused image is observed on a screen placed at a distance \(12\,cm\) from the lens. A glass plate of refractive index \(1.5\) and thickness \(1\,cm\) is introduced between lens and screen such that the glass plate plane faces parallel to the screen. By what distance should the object be shifted so that a sharp focused image is observed again on the screen \(\dots\,m\)

Velocity-time graph for a body of mass \(10\, kg\) is shown in figure. Work-done on the body in first two seconds of the motion is  ................ \(\mathrm{J}\)

The coordinates of centre of mass of a uniform flag shaped lamina (thin flat plate) of mass \(4\;kg\). (The coordinates of the same are shown in figure) are

There are ' \(n\) ' number of identical electric bulbs, each is designed to draw a power \(p\) independently from the mains supply. They are now joined in series across the main supply. The total power drawn by the combination is :

In an adiabatic process, which of the following statements is true?

A proton and an electron have the same de Broglie wavelength. If \(K_p\) and \(K_e\) be the kinetic energies of proton and electron respectively. Then choose the correct relation :

An unstable heavy nucleus at rest breaks into two nuclei which move away with velocities in the ratio of \(8:27\). The ratio of the radii of the nuclei (assumed to be spherical ) is

In photoelectric effect, the stopping potential \(\left(\mathrm{V}_0\right) \mathrm{v} / \mathrm{s}\) frequency \((\nu)\) curve is plotted.
( h is the Planck's constant and \(\phi_0\) is work function of metal )
(A) \(\mathrm{V}_0 \mathrm{v} / \mathrm{s} \nu\) is linear.
(B) The slope of \(\mathrm{V}_0 \mathrm{v} / \mathrm{s} \nu\) curve \(=\frac{\phi_0}{\mathrm{~h}}\)
(C) h constant is related to the slope of \(\mathrm{V}_0 \mathrm{v} / \mathrm{s} \nu\) line.
(D) The value of electric charge of electron is not required to determine \(h\) using the \(V_0 \mathrm{v} / \mathrm{s} \nu\) curve.
(E) The work function can be estimated without knowing the value of \(h\).
Choose the correct answer from the options given below :

\(\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \frac{{\cot x - \cos x}}{{{{\left( {\pi - 2x} \right)}^3}}} = \) . . . .

Two thin metallic spherical shells of radii \({r}_{1}\) and \({r}_{2}\) \(\left({r}_{1}<{r}_{2}\right)\) are placed with their centres coinciding. A material of thermal conductivity \({K}\) is filled in the space between the shells. The inner shell is maintained at temperature \(\theta_{1}\) and the outer shell at temperature \(\theta_{2}\left(\theta_{1}<\theta_{2}\right)\). The rate at which heat flows radially through the material is :-

If the wavelength of the first member of Lyman series of hydrogen is \(\lambda\). The wavelength of the second member will be _______.

If the shortest distance between the lines \(\vec{r}_{1}=\alpha \hat{i}+2 \hat{j}+2 \hat{k}+\lambda(\hat{i}-2 \hat{j}+2 \hat{k}), \lambda \in R, \alpha>0\) and \(\vec{r}_{2}=-4 \hat{i}-\hat{k}+\mu(3 \hat{i}-2 \hat{j}-2 \hat{k}), \mu \in R\) is \(9\), then \(\alpha\) is equal to \(.....\)