If $\tan A$ and $\tan B$ are the roots of the quadratic equation $x^2-px+q=0$,then $\sin^2(A+B)$ is equal to

The adjoining diagram shows the spectral energy density distribution $E_{\lambda}$ of a black body at two different temperatures. If the areas under the curves are in the ratio $16 : 1$,the value of temperature $T$ is ........ $K$. (in $,000$)

If $x=e^{y+e^{y+e^{y+\ldots}}}$,then $\frac{d y}{d x}=$

$A$ particle is executing a simple harmonic motion. Its maximum acceleration is $\alpha$ and maximum velocity is $\beta$. Then,its time period of vibration will be

Water rises in a capillary tube of radius $r$ to a height $h$. The mass of water in the capillary is $m$. What will be the mass of water that rises in a capillary tube of radius $\frac{r}{3}$?

An object $2.4 \; m$ in front of a lens forms a sharp image on a film $12 \; cm$ behind the lens. $A$ glass plate $1 \; cm$ thick,of refractive index $1.50$ is interposed between the lens and the film with its plane faces parallel to the film. At what distance (from the lens) should the object be shifted to be in sharp focus on the film (in $; cm$)?