6. Prove the formula sut + ft2 for a particle moving with initial velocity u and acceleration f. If a particle starts from rest, and the acceleration =kt, where t is the time from rest, show that the space described in time t is kt3. 7. Enunciate Newton's laws of motion. of the third law. Discuss and give illustrations A shot of m lb. is fired from a gun of M lb. placed on a smooth horizontal plane and elevated at an angle a. Find the angle of projection. 8. If a mass M is connected with a mass m by the first system of pulleys, find the accelerations of m and M, the number of movable pulleys being n. 9. Show that the force which must act on a particle of mass m in order to make the particle describe a circle of radius r with uniform velocity v is directed towards the centre of the circle and is of magnimv2 tude A railway engine of 7 tons moves on a carve of 800 ft. radius with a velocity of 40 miles an hour; find the horizontal force exerted on the rails in pounds. 10. What is meant by the dimensions of a unit? Find the dimensions of the following units: (a) acceleration, (b) force, (c) impulse, (d) power. If the acceleration due to gravity be taken as the unit of acceleration, and the velocity generated in 15 seconds be the unit of velocity, find the unit of length. MATHEMATICS. SECOND HONOUR PAPER. Paper set by-C. W. PEAKE, ESQ., M.A. The figures in the margin indicate full marks. 1. Prove that the whole pressure of a liquid on a surface is equal to the weight of a column of liquid, of which the base is equal to the area of the surface and the height is equal to the depth of its centre of gravity below the surface of the liquid. 5 A cylindrical vessel of height 2h and radius of base h is filled with two 20 liquids, which do not mix. The volume occupied by each liquid is the same, but the density of one is twice that of the other. Find the whole pressure on each of the two portions into which the curved surface of the cylinder is divided by the surface of separation of the liquids. 2. A cylinder 4 feet high and 5 square feet in section, with its upper 20 end closed and lower end open, is lowered into water until its top is 14 feet below the level of the surface of the water. Air is then pumped into it till it is three-fourths full of air. Given the height of the waterbarometer to be 34 feet, find the volume the air would occupy at atmospheric pressure. 3. Find the centre of pressure of a triangle whose base is in the sur- 10 face of a homogeneous liquid. A square lamina ABCD is immersed with one side AB in the surface of a homogeneous liquid Find the distance between the centres of pressure of the two triangles ABC, BCD. 4. Describe Nicholson's Hydrometer, and show how it can be used to compare the specific gravities of a solid and a liquid. The weight required to sink a Nicholson's Hydrometer to the fixed mark is 372 grammes. A substance is placed in the upper cup and it requires 3.34 grammes to sink the instrument to the fixed level. When the substance is placed in the lower cup a weight of 3:53 grammes is required. Find the specific gravity of the substance. 5. The spout of a common pump is 20 feet above the surface of the water in the reservoir, the diameter of the piston is one foot and the piston range is 2 feet. Given also that the height of the water-barometer is 33 feet and that the piston makes 10 strokes a minute, find the weight of water discharged per minute and the tension of the piston rod when the pump is full. 6. Show that at a place whose latitude is λ the time of apparent revolution of the plane of oscillation of a pendulum is Tcosec λ, where T is the time of revolution of the earth about its axis. 20 15 15 25 15 7. Show that the refraction of a heavenly body, the temperature and 15pressure being constant, varies as the tangent of the apparent zenith distance. Describe some method of determining the coefficient of refraction. 8. What are precession and nutation? To what physical causes are 15 they due? By what observations can their existence be detected? 9. Show that each star aberrates towards a point on the ecliptic 90° 15behind the sun, and that the aberration varies as the sine of the earth's way. 10. If Q be Jupiter's synodic period and T the time between two suc- 15cessive quadratures, then the greatest angle which the distance of the earth from the sun subtends at Jupiter is 90° (1–277). MATHEMATICS. THIRD HONOUR PAPER. Paper set by THE HON'BLE MR. JUSTICE ASUTOSH MOOKERJEE, SARASWATI, M.A., D.L., F.R.A.S., F.R.S.E. Examiner-C. LITTLE, ESQ., M.A. The figures in the margin indicate full marks. 1. Prove that the orthocentre of the triangle formed by the lines ∞➡ay+ka2=0, x-By+kB2=0, x−yy+k2=0 ➡k, k(a + B +y+aßy). 20 3. Form the equation of a circle through three fixed points, and interpret the result geometrically. 4. Find the condition that the circle conjugate to that throngh (§, n). Form the equations of the axes. 7. Prove that from any point three normals can, in general, be drawn to a parabola, the feet of which lie on a circle throngh the vertex. 8. Prove that the circle circumscribing a triangle self-conjugate with regard to an equilateral hyperbola passes through the centre. 9. If a variable conic pass through three fixed points and have an asymptote parallel to a given line, the locus of its centre is a parabola. 10. If x2+ y2+2gx+2fy+c=0 intersects in points whose eccentric angles are a, 8, y, d, prove that MATHEMATICS. FOURTH HONOUR PAPER. Paper set by-THE HON'BLE MR. JUSTICE ASUTOSH MOOKERJER, SARASWATI, M.A., D.L., F.R.A.S. F.R.S.E. Examiner-C. LITTLE, ESQ., M.A. The figures in the margin indicate full marks. 1. Calculate from first principles the differential coefficient of sec ☛ with regard to ≈. 231 6. Define an integral as the limit of a sum, and prove that the limit, 20 when n = ∞, of and deduce that the length of the arc from the vertex to the point (b, c) is approximately consists of a single oval; trace it and calculate its area. 20 20 :0 20 PHYSIOLOGY. FIRST PASS PAPER. Examiner-S. B. MITRA, Esq., B.Sc., M.B. The figures in the margin indicate full marks. 1. Describe the structure of an artery, a vein, and a capillary 25 showing how the structure of each is adapted to its functions. 2. Give an account of the digestion and the absorption of ghee. 20 |