$A$ train leaves Pune at $7:30\,am$ and reaches Mumbai at $11:30\,am$. Another train leaves Mumbai at $9:30\,am$ and reaches Pune at $1:00\,pm$. Assuming that the two trains travel at constant speeds,at what time do the two trains cross each other?

The amplitude and the time period in a $S.H.M.$ are $0.5 \, cm$ and $0.4 \, s$ respectively. If the initial phase is $\pi / 2 \, rad$,then the equation of $S.H.M.$ will be

The condition for the lines $lx + my + n = 0$ and $l_1x + m_1y + n_1 = 0$ to be conjugate with respect to the circle $x^2 + y^2 = r^2$ is

$A$ bimetallic strip is formed out of two identical strips,one of copper and the other of brass. The coefficients of linear expansion of the two metals are $\alpha_C$ and $\alpha_B$. On heating,the temperature of the strip increases by $\Delta T$ and the strip bends to form an arc of radius $R$. Then $R$ is proportional to

The half-life of a radioactive isotope $X$ is $50$ years. It decays to another element $Y$,which is stable. The two elements $X$ and $Y$ were found to be in the ratio of $1:15$ in a sample of a given rock. The age of the rock was estimated to be ......... years.

When a metallic surface is illuminated with radiation of wavelength $\lambda$,the stopping potential is $V$. If the same surface is illuminated with radiation of wavelength $2\lambda$,the stopping potential is $\frac{V}{4}$. The threshold wavelength for the metallic surface is: