The distance between the pair of lines $x^2+2 \sqrt{2} x y+2 y^2+4 x+4 \sqrt{2} y+1=0$ is:

Particles of masses $m, 2m, 3m, \ldots, nm$ grams are placed on the same line at distances $l, 2l, 3l, \ldots, nl$ $cm$ from a fixed point. The distance of the centre of mass of the particles from the fixed point in centimetres is

Sunlight of intensity $1.3 ~kW ~m^{-2}$ is incident normally on a thin convex lens of focal length $20 ~cm$. Ignore the energy loss of light due to the lens and assume that the lens aperture size is much smaller than its focal length. The average intensity of light,in $kW m^{-2}$,at a distance $22 ~cm$ from the lens on the other side is.

An observer moves towards a stationary source of sound with a speed $1/5^{th}$ of the speed of sound. The wavelength and frequency of the source emitted are $\lambda$ and $f$ respectively. The apparent frequency and wavelength recorded by the observer are respectively:

Which of the following oxides will produce hydrogen peroxide upon reaction with dilute acid?

The plates of a parallel plate capacitor are charged up to $200 \ V$. $A$ dielectric slab of thickness $4 \ mm$ is inserted between its plates. Then,to maintain the same potential difference between the plates of the capacitor,the distance between the plates is increased by $3.2 \ mm$. The dielectric constant of the dielectric slab is: