The expression $\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}$ is equal to

$A$ plano-convex lens fits exactly into a plano-concave lens. Their plane surfaces are parallel to each other. If the lenses are made of different materials with refractive indices $\mu_1$ and $\mu_2$ and $R$ is the radius of curvature of the curved surface of the lenses,then the focal length of the combination is:

$A$ body projected vertically upwards crosses a point twice in its journey at a height $h$ just after $t_1$ and $t_2$ seconds. Maximum height reached by the body is

In a compound microscope,the focal lengths of the objective and the eye lens are $2.5 \, cm$ and $5 \, cm$ respectively. An object is placed at $3.75 \, cm$ before the objective and the image is formed at the least distance of distinct vision. Then,the distance between the two lenses (i.e.,the length of the microscope tube) will be......$cm$.

$A$ double slit experiment is performed by using light of wavelength $6000\,\mathring{A}$. If the distance of the screen is $1\,m$ and the slits are $0.1\,cm$ apart,then calculate the angular position of the $10^{th}$ bright fringe.

Two bodies of mass $4 ~kg$ and $5 ~kg$ are moving along east and north directions with velocities $5 ~m/s$ and $3 ~m/s$ respectively. The magnitude of the velocity of the centre of mass of the system is