Physics: Electrostatics Problem

Part 1: Physics

Section I

• A point charge (+q) is released from a distance from the center of a solid sphere of radius R and charge Q. The speed of the point charge when it reaches a distance R from the surface of the sphere is to be found.
• A uniformly charged ring of charge Q and radius R is folded from its diameter, and the net electric field at the center of the ring (O) is to be found.
• The thickness of a flat disc (H) is needed to experience the same gravitational acceleration as on the surface of a spherical Earth.
• Six stars of equal mass m move about the center of mass of the system, and their common time period of revolution around the center is to be found.
• Statement 1: If three forces are represented by three sides of a triangle, they are concurrent forces and satisfy the condition for equilibrium. Statement 2: A triangle made up of three forces as its sides taken in the same order satisfies the condition for translatory equilibrium.

Section II

• A particle of mass m and charge q is projected towards a non-conducting fixed sphere having the same charge uniformly distributed throughout the volume of the sphere.
• A small dipole is placed at a distance R from the surface of a charged solid conducting sphere, and the options are:
  • Charge distribution on the surface of the sphere will be non-uniform.
  • Electric field due to charges on the sphere at point C (center of the sphere) will be zero.
  • Potential at point C will be zero.
  • Potential at point D due to the charges on the sphere will be zero.
• A narrow smooth tunnel of L-shape is made into the Earth up to the center, and a small ball is released from 'A'. The collision of the ball is perfectly inelastic.
• A planet of mass m revolves around the Sun in an elliptical orbit of eccentricity e, and its speed at apogee and perigee is to be found.

Section III

• The distance of a point from the Earth's center where the resultant gravitational field due to the Earth and the Moon is zero is to be found.
• The vector is given, and its magnitude and derivative at t = 1s are to be found.

Part 2: Chemistry

Section I

• NaN3 is decomposed into Na and N2, and the number of moles of N2 produced is to be found.
• CaC2 reacts with H2O to produce CaO and C2H2, and the number of moles of C2H4 produced is to be found.

Section II

• The reaction between K2Cr2O7 and FeSO4 is to be balanced, and the number of moles of Fe2+(aq) produced is to be found.
• The oxidation of MnO2 by alkaline oxidative fusion is to be balanced.
• The molarity of a solution of H2O2 is to be found.

Part 3: Mathematics

Section I

• The function f(x) = x^100 is to be found, and its value at x = 1 is to be found.
• The polynomial P(x) = (x-1)(x-2)(x-3)(x-4)(x-5) + x^2 is to be evaluated at x = 6.
• The equation (x^2-1)^3 - (2x-1)^3 = x^3(x-2)^3 is to be solved.

Section II

• The equation x^2-2x+2 = x is to be solved, and the number of solutions is to be found.
• The equation x^2-1 = x-1 is to be solved, and the number of solutions is to be found.

Section III

• The equation |x^2-1| + |x-1| = 3 is to be solved, and the number of solutions is to be found.
• The equation (2)^(2/x) - 3(2)^(1/x) + 8 = 0 is to be solved, and the number of solutions is to be found.
• The equation (5x+7)(13y+1) = 156x is to be solved, and the value of x and y is to be found.