2. Find the maximum and minimum value of (x2 −x+1)/(x2 + x − 1) 3. Show that the equations for cycloid are x=a(0―sin 0), y=a(1 − cos 0). 4. Write down the values of sin30 do, dx/x/x-a, and dy de/cos e 5. Show that Y-y= (X-x) is tangent to the dx curve y=f(x); and deduce the normal equation. Illustrate by finding the tangent and normal to the curve y=x3-3x2+2x at the origin. 6. Explain the radius of curvature, and prove 3/2 7. Obtain the general solutions for the following problems, (a) To find the area of a polar curve, (b) To find the volume of a solid of translation, (c) To find the C. of G. of a thin uniform plate, (d) To find the moment of inertia of a uniform plate about a perpendicular axis, (e) To find the centre of pressure on a plane side of a vessel filled with water. 8. Expand cos 0 in powers of 0. FACULTY OF PRACTICAL SCIENCE. MATHEMATICS II. SECOND PAPER. I. A and B are places 12 miles apart on a straigh shore, and a man is in a row boat at sea opposite A 4 miles out. Where must he land in order to reach B in the shortest possible time, if he can walk 4 miles while he rows 3? 2. Find the length of the equiangular spiral r=12° traced out as the radius vector sweeps over the first quadrant. 3. A chord of a parabola, perpendicular to the axis, is 12 in length and a distance 9 from the vertex. Find the volume of the solid generated by the segment of the parabola revolving about the chord. 4. Find the coordinates of the C. of G. of the uniform plate bounded by the coordinate axes and the part of the curve y=64-x2 in the first quadrant. 5. A segment of a circular plate has height 18 and chord 48. Find the moment of inertia about a di ameter of the circle parallel to the chord. the radius of gyration. Find also FACULTY OF PRACTICAL SCIENCE. PHYSICS I. A. Dynamics and Properties of Matter. 1. Describe the application of the Pitot tube to the recording of the speed of steamboats; and derive the formula on which the measurement depends. 2. An empty car (mass 20 tons) is at rest at the bottom of a grade (2%). A second car (mass 60 tons) is shunted against the first and strikes with a velocity of 4 feet per second. The coupling is made with the impact and they move on together. If friction be 15 lbs. per ton, how far up the grade do they move? 3. A train (150 tons) approaches a 1% grade 300 feet long with a speed of 50 miles per hour. It reaches the upper level with a speed of 49 miles per hour. If the resistance be 14 lbs. per ton, find the work done by the engine during the ascent. 4. A 2 horse-power motor is attached to a pump of 60% efficiency to empty a pit of water. The pit is 10 ft. in diameter and 20 ft. deep and is 34 full of water. (a) How long will it take to empty the pit, if there be no inflow during the pumping? (b) How many gallons per minute are discharged. by the pump when the water is five feet deep on the bottom of the pit? 5. A window (4 ft. by 6 ft.) is swung on hinges from the top of the sash. It is held open by two 12-inch hooks (one on each side) that slip into eyes 5 ft. from the hinges. If the hooks meet the plane of the window normally, and if the mass of the window be 50 lbs., find the compressive strain on the hooks and the reactions on the hinges. Take 26 5.1. |