4. Of the three plane angles which form a trihedral angle, any two are together greater than the third. II. 1. State and prove the rule for finding the square root of any rational integral algebraic expression which is a perfect square. Find the square root of Qx2 (y + z)2 + 2y2 (z + x)2 + Qz2 (x+y)2 + 4xyz(x + y + z). 2. State the meanings given to a" when m is fractional or negative, and explain why the meanings are given. shew that each of these ratios is also equal to la + mb + no la' + mb' + nc'' Solve the equations ax + by + cz = 0, x2 + y2 + z2 = 1. Z Ꮓ 4. Find the sum of any number of terms of an arithmetical progression. If a, b, c are respectively the ph, qth, and 7th terms of an arithmetical progression, find the relation between a, b, c, p, q, r. III. 1. Find an expression for all the angles having the same sine as a given angle a. Find the general solution of the equation 3. Discuss the ambiguous case in the solution of triangles. If a, b, A are given, and c1, c2 are the two values of c, prove that c12 + c22 = 2a2 + 262 cos 2A. 4. Find an expression for the radius of an escribed circle of a triangle. Prove that 4rrers r(a + b + c)2. PURE MATHEMATICS.-PART II. The Board of Examiners. Write short Essays on― (1) Polar coordinates. (2) Properties of the ellipse. (3) Successive differentiation. (4) Evaluation of indeterminate forms. (5) Integration of rational algebraic expressions. (6) Volumes of solids of revolution. PURE MATHEMATICS.-PART III. The Board of Examiners. Write short Essays on— (1) Asymptotes. (2) Differentiation of a definite integral. (3) Generating lines of a conicoid. (5) Simultaneous ordinary differential equations. (6) Linear partial differential equations with constant coefficients. MIXED MATHEMATICS.-PART I. The Board of Examiners. 1. Define Acceleration, Force, Momentum, and state the fundamental relations between them. Two elastic strings of natural lengths 2 and 3 feet, and moduli of elasticity 50 and 100 lbs. weight respectively, are attached to a mass of 1 lb. lying on a smooth horizontal plane. The strings are pulled horizontally at an angle of 60° with each other till their extensions are 2 and 3 inches respectively, and the mass is then set free. Assuming that the directions and tensions of the strings are kept constant, find the position of the mass at the end of 3 secs. 2. State Galileo's laws for the motion of a projectile in a vacuum at the earth's surface, and find the horizontal range of a projectile whose initial velocity is V at an elevation a. Two particles are projected at times 0, T from the same point, in the same direction, and with horizontal and vertical velocities u, v. Shew that they are at a minimum distance uT apart at time g+T 3. Two planes of inclinations a, ß in opposite directions have a horizontal line of intersection. Two particles of masses m1, m2 on the planes are connected by an inextensible string which passes over the ridge of the planes and lies along lines of slope on them. Find the acceleration of the particles (a) when the planes are smooth, (b) when their coefficients of friction with the particles are μ, μ, respectively, and m1 is moving downwards. An inextensible string which passes over two pulleys close together hangs between them in a loop on which is slung a ring of mass M. On one end of the string is hung a mass m, and the other end is hauled down with a given acceleration f Find the accelerations of the masses, neglecting friction. Find 4. Define the dimensions of a physical quantity, and find the dimensions of Force and Power. the ratio of a Horse-power to a Kilowatt. 5. Shew that it is necessary and sufficient for the equilibrium of a rigid body acted on by any number of forces in one plane, that the sums of the resolved parts of the forces in two directions and the sum of the moments about one point, should separately vanish. A uniform heavy rectangle ABCD is supported in a vertical piane, with the side AB at an inclination a to the horizontal and below the side CD, between two pegs, one, rough, at the middle point of AB, the other, smooth, at a point P of the opposite side CD. Find the reactions at the pegs, and shew that, if the angle of friction with the rough peg is <a, the distance of P from the middle point of CD cannot be greater |