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(Subjective Type) A spherical ball of mass m is kept at the highest point in the space between two fixed, concentric spheres A and B as shown in figure. The smaller sphere A has a radius R and the space between the two spheres has a width d. The ball has a diameter very slightly less than d. All surfaces are frictionless. The ball is given a gentle push (towards the right in the figure). The angle made by the radius vector of the ball with the upward vertical is denoted by

.
(a) Express the total normal reaction force exerted by the sphere on the ball as a function of angle

.
(b) Let N_A and N_B denote the magnitudes of the normal reaction forces on the ball exerted by the sphere A and B, respectively. Sketch the variations of N_A and N_B as functions of

in the range

by drawing two separate graphs in your answer book, taking

on the horizontal axes.

Chemistry

(Objective Type) The IUPAC name of C₆H₅COCl is

(a) Benzene chloro ketone
(b) Benzoyl chloride
(c) Chloro phenyl ketone
(d) Benzene carbonyl chloride

Mathematics

If arg (z) <0, then arg (-z) – arg (z) =