GROUP C. I. Elements of Geometry. I. 1. Define-parallel straight lines, a rectangle, a segment of a circle, a point of contact.' When is a circle said to be described about a rectilineal figure? 2. If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them of the other, the base of that which has the greater angle shall be greater than the base of the other triangle. 3. If a straight line be bisected, and produced to any point, the rectangle contained by the whole line thus produced and the part of it produced, together with the square on half the line bisected, is equal to the square on the straight line which is made up of the half and the part produced. 4. The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same part of the circumference. 5. About a given circle describe a triangle equiangular to a given triangle. 6. If two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals, the triangles shall be equiangular to one another, and shall have those angles equal which are opposite to the homologous sides. 7. Define-ratio, duplicate ratio, similitude of ratios, similar rectilineal figures, reciprocal figures. Explain the terms-invertendo, convertendo. 8. If the square described on one of the sides of a triangle be equal to the squares described on the other two sides, the angle contained by these two sides is a right angle. 9. In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing, the obtuse angle, by twice the rectangle contained by the side on which, when produced, the perpendicular falls and the straight line intercepted without the triangle between the perpendicular and the obtuse angle. 10. A segment of a circle being given, describe the circle of which it is a segment. 11. Describe an equilateral and equiangular pentagon about a given circle. 12. On a given straight line describe a rectilineal figure similar and similarly situated to a given rectilineal figure. II. Elements of Geometry. II. 1. Prove that the circumference of a circle has a constant ratio to the diameter: give an approximate value of this ratio; and employ it to find the number of degrees, minutes, and seconds in the angle subtended at the centre by an arc equal to the radius. 2. Define the sine, cosine, and tangent of an angle. Draw the angles whose tangents are 1,, and respectively; and give the values of their sines and cosines. 3. Find the values of sin (90°+4), sin (180° +A),. and sin (270°+4), when sin A, cos A = = If sin (90+34) = sin (270+ 4), find A. 4. Prove the following identities: (1) sin 24+ cos 2A = 1, 3 (2) sin A tan A+ cos A cot A = (1+cot 4) (sec 4-sin 4).. 5. Prove that (1) sin (4+B) = sin A cos B+ cos ♫ sin B, (2) cos B—cos A = 2 sin A+B A-B sin 2 2 Find sin (4+B+C) in terms of the sines and cosines of A, B, C. 6. Prove geometrically that cos 60°=}. In a right angled triangle, if the sides which contain the right angle be to one another as the angles opposite to them will be spectively. 3+1 to √3-1, 75° and 15 re 7. In any triangle ABC, of which the sides are a, b, c, and the perpendiculars from the angles on the sides opposite to them are p, q, r, (1) a2 = b2+c2 — 2bc cos A, (2) p sin A = q sin B = r sin C. 8. Find the logarithms of the numbers (1) 1287.65, (2) 3'4 x 14'1 x '00017; and the numbers corresponding to the logarithms (3) 54682017, (4) 40107600. 9. Find the tabular logarithm of the sine of the vertical angle of an isosceles triangle, in which each of the angles at the base is three times the vertical angle. If tan 4 = 343, find A. 10. In the triangle ABC, if a = 17, 6 = 127, B = find c, 4, and C. : 15°; * 11. A, B, C are three stations, and D is a point in BC; and it is found by observation and measurement that the angles BAD and DAC are each 30°, the length of AD is 120 yards, and the angle ADC is 60°. Find the distance from B to C. III. Elements of Mechanics. I. 1. Define the terms composition and resolution, as applied to forces. What is the effect of compounding two forces which are represented in direction and magnitude by the diagonals AC, BD of the parallelogram ABCD? 2. Prove that if two commensurable forces are represented in direction and magnitude by the straight lines AB, AC, the resultant acts in the direction of the diameter of the parallelogram of which AB, AC are adjacent sides. If the resultant is double of one of the forces and inclined to it at an angle of 60°, shew that the forces are at right angles to each other. 3. Three forces, P, Q, R, applied at a point are in equilibrium; P and R are each 55 lbs., and the angle between P and Q is 120°. Find Q. 4. Prove that any number of parallel forces, acting in the same direction, have always a resultant which is parallel to each of them. 5. Explain the principle of the lever. A power of 5 lbs., applied at one end of a straight lever, 3 feet from the fulcrum, will lift a weight of 15 lbs. Find (1) the length of the lever, and (2) the position of the fulcrum when the same power will lift a weight of 75 lbs. 6. Find the centre of gravity of a triangular lamina. If the sides of the triangle are 15, 20, and 25 inches, find the perpendicular distance of the centre of gravity from each of them. 7. A beam, 9 feet long, rests horizontally on two props at its extremities, and produces a pressure of 8 lbs. on each prop. Where must a weight of 72 lbs. be placed, that the whole pressure on one of the props may be 40 lbs. ? 8. Apply the principle of virtual velocities to find the relation between the power and weight in a system of pulleys consisting of two blocks, with the same string passing round all the sheaves. If each block contains four sheaves, and a power of 30 lbs. supports a weight of 230 lbs., what is the weight of the lower block? 9. A weight of 120 lbs. is supported on a smooth plane, inclined at 30° to the horizon. Find (1) the sustaining force and the pressure on the plane when the force acts along the plane, and (2) the force, and the angle at which it must act, to reduce the pressure to one half of its former value. 10. How is the velocity of a moving body estimated, when it describes equal spaces in equal times, and when it is variable? If the acceleration of gravity be taken to be 32-2 feet per second, compare the velocities of a falling body at the end of the third and fifth seconds of motion. 11. Prove that the spaces described from rest by a body falling in vacuo under the action of an uniform acceleration varies as the square of the time. 12. If a body, after falling 15 feet, strikes a horizontal plane, and rebounds with half the velocity of impact, to what height will it ascend? IV. Elements of Mechanics. II. 1. Express the range of a projectile on a horizontal plane in terms of the velocity and angle of projection. A shot, fired with an elevation of 15°, strikes the ground on a level with the mouth of the gun at a distance of 900 feet. feet. Find the velocity of projection. 2. Find the time of flight of a projectile, the range being on an inclined plane. A shot is fired at the base of a plane, inclined at an angle of 45°, with an initial velocity of 322 ft. per second. It strikes the plane in 14'14 seconds. Find the angle of projection. 3. A cistern 3 ft. by 6 ft. has 2 feet of water in it; find the pressure on the bottom of the cistern, and the whole pressure on the bottom and sides. How is either of these quantities related to the whole weight of water to be supported? 4. Show that the pressure of a fluid upon a surface is independent of the quantity of the fluid. 5. The specific gravity of ice is about 1. If a cubical block of ice float in water with two faces horizontal, how much of it will stand above the water? And if the density of the water be increased by one tenth, how much more of the cube will stand above it? |