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)

$$\mathop {\lim }\limits_{x \to 3} $$ $${{\sqrt {3x} - 3} \over {\sqrt {2x - 4} - \sqrt 2 }}$$ is equal to :

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 ?

Let the refractive index of a denser medium with respect to a rarer medium be n₁₂ and its critical angle be θ_C . At an angle of incidence A when light is travelling from denser medium to rarer medium, a part of the light is reflected and the rest is refracted and the angle between reflected and refracted rays is 90^o. Angle A is given by :