The two lenses of an achromatic doublet should have:

If $\sin (y+z-x), \sin (z+x-y)$ and $\sin (x+y-z)$ are in $AP$,then

Explain the reason for the fusion of an organic compound with metallic sodium for testing nitrogen,sulphur and halogens.

The area (in sq units) of the quadrilateral formed by two pairs of lines $\lambda^2 x^2 - m^2 y^2 - n(\lambda x + m y) = 0$ and $\lambda^2 x^2 - m^2 y^2 + n(\lambda x + m y) = 0$ is:

An electron revolves in a circle of radius $0.4 \text{ \AA}$ with a speed of $10^6 \text{ m/s}$ in a hydrogen atom. The magnetic field produced at the centre of the orbit due to the motion of the electron (in Tesla) is: $\left[\mu_0 = 4\pi \times 10^{-7} \text{ H/m}, e = 1.6 \times 10^{-19} \text{ C}\right]$

Given $\lambda \in [0, 20]$,find the number of integral values of $\lambda$ for which the function $f(x) = x^3 - 12x + \lambda$ has a point of local maxima.