If y = $${\left[ {x + \sqrt {{x^2} - 1} } \right]^{15}} + {\left[ {x - \sqrt {{x^2} - 1} } \right]^{15}},$$

then (x² $$-$$ 1) $${{{d^2}y} \over {d{x^2}}} + x{{dy} \over {dx}}$$ is equal to :

The integral

$$\int {\sqrt {1 + 2\cot x(\cos ecx + \cot x)\,} \,\,dx} $$

$$\left( {0 < x < {\pi \over 2}} \right)$$ is equal to :

(where C is a constant of integration)

There is a uniform electrostatic field in a region. The potential at various points on a small sphere centred at $$P,$$ in the region, is found to vary between the limits 589.0 V to 589.8 V. What is the potential at a point on the sphere whose radius vector makes an angle of 60^o with the direction of the field ?

A single slit of width b is illuminated by a coherent monochromatic light of wavelength $$\lambda $$. If the second and fourthminima in the diffraction pattern at a distance 1 m from the slit are at 3 cm and 6 cm respectively from the central maximum, what is the width of the central maximum ? (i.e. distance between first minimum on either side of the central maximum)