How many litres of oxygen (at $STP$) are required for complete combustion of $39 \text{ g}$ of liquid benzene? (Atomic weights: $C=12, O=16, H=1$)

If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ in four points $(x_i, y_i)$,for $i=1, 2, 3, 4$,then $y_1+y_2+y_3+y_4$ equals

$A$ parallel plate capacitor with air between the plates has a capacitance of $9 \ pF$. The separation between its plates is $d$. The space between the plates is now filled with two dielectrics. One of the dielectrics has a dielectric constant $K_1 = 6$ and thickness $\frac{d}{3}$,while the other has a dielectric constant $K_2 = 12$ and thickness $\frac{2d}{3}$. The new capacitance of the capacitor is ........ $pF$.

The equations of the normals at the ends of the latus rectum of the parabola $y^2 = 4ax$ are given by

Two particles of masses $m_1$ and $m_2$ are separated by a distance $d$. The shift in the centre of mass when the two particles are interchanged is:

$A$ convex lens has its radii of curvature equal. The focal length of the lens is $f$. If it is divided vertically into two identical plano-convex lenses by cutting it,then the focal length of the plano-convex lens is ($\mu$ = the refractive index of the material of the lens).