Consider the following series:

(i)

(ii)

(iii)

Choose the correct option.

Consider the function , defined as

Which of the following statements is/are correct?

(Here,  is the set of real numbers.)

Consider a non-negative function  which is continuous and bounded over the interval . Let  and  denote, respectively, the maximum and the minimum values of  over the interval.

Among the combinations of  and  given below, choose the one(s) for which the inequality   is guaranteed to hold.

Consider the Earth to be a perfect sphere of radius R. Then the surface area of the region, enclosed by the N latitude circle, that contains the north pole in its interior is__________.

Let  and  represent density and velocity, respectively, at a point  and time t. Assume  is continuous. Let  be an arbitrary volume in space enclosed by the closed surface  and  be the outward unit normal of  Which of the following equations is/are equivalent to  ?

Let  be functions of (x, y, z). Suppose that for every given pair of points A and B in space, the line integral evaluates to the same value along any path C that starts at A and ends at B. Then which of the following is/are true? 

The rate of increase, of a scalar field  in the direction  at a point  is

The value of the line integral  along the straight line joining the points  and  is

The value of the integral  over the region , given in the figure, is __________(rounded off to the nearest integer).