8. Given a cos B+ b cos A = c. a sin B-b sin A = 0. a2+b2 - 2ab cos C = 2. A = 42°, a = 141, 6 = 172°5, find all solutions of the triangle ABC. 9. From the point O the three straight lines OA, OB, OC are drawn in the same plane, of lengths 1, 2, 3 respectively, and with the angles AOB and BOC each equal to 60°. Find the angle ABC correct to one minute. IO. Find the area of the greatest circle which can be cut out of a triangular piece of paper whose sides are 3, 4, 5 feet respectively. 11. A conical extinguisher, whose section through the vertex is an isosceles triangle with vertical angle 30°, is placed over a cylindrical candle whose diameter is one inch, and rests so that the point of contact of the top of the candle with each generating line of the cone bisects that line. Find the whole inside surface of the extinguisher. V. STATICS AND DYNAMICS. [It may be assumed that π = 2, and that g= 32, when a foot and a second are the units of length and time.] 1. Explain why it is that forces can be completely represented by straight lines. ABCD is a square; find the resultant of the forces represented by the straight lines AB, AC, and AD. 2. Enunciate and prove the theorems of the triangle of forces and the polygon of forces, and state whether the converses of these theorems are true. 3. A heavy pole, weighing 140 lbs., is carried on the shoulders of two men, one at each end; the centre of gravity of the pole being two feet from one end and five feet from the other, find the weight supported by each man. Also find what would be the effect of placing each man one foot nearer to the centre of gravity of the pole. 4. Find the ratio of the power to the weight when there is equilibrium in a system of three moveable pulleys, each of which is supported by a separate string, and in which the free portions of the strings are vertical. Also, if the weights of the pulleys, supposed to be equal, are taken into account, find the relation between the power, the weight, and the weight of a pulley. 5. A heavy uniform rod is supported by a string fastened to its ends, of double its own length, which passes over a smooth horizontal rail. Find the tension of the string first, when the rod is hanging at rest in a vertical position, and secondly, when the rod is at rest in a horizontal position. 6. Explain what is meant by saying that a point is moving in a straight line with uniform acceleration, and show how this acceleration is measured. What is the measure of the acceleration of a body falling freely when eight feet and half a second are the units of length and time? 7. A body is projected vertically upwards with the velocity of 256 feet per second; find the greatest height to which it rises and the time in which it will return to the point of projection. Also find the times, during the ascent and descent, at which it passes the level of 768 feet above the point of projection. 8. Prove that the path of a projectile is a parabola, and that, if u is the horizontal component of the velocity of projection, the latus rectum of the parabola is equal to u2 16' 9. The top of the spire of a church, standing on a level plane is 200 feet above the plane. From a position on the plane, at the distance of 400 feet from the vertical line through the top of the spire, a bullet is fired off so as to pass horizontally just over the top of the spire. Find the initial direction and the initial velocity of the bullet. IO. Find the direction and magnitude of the acceleration of a point moving uniformly in a circle. A mass of 7 pounds, on a smooth horizontal plane, is fastened to one end of a string, 7 feet in length, and the other end is fastened to a fixed peg on the plane. The string is then straightened, and the particle is projected horizontally, at right angles to the string, with such a velocity as to describe its circular path in 5 seconds. Find the tension of the string in poundals, and also in pounds' weight. FURTHER EXAMINATION. I. VI. PURE MATHEMATICS. [Full marks may be obtained for about two-thirds of this paper.] Show that the two perpendiculars, erected at the extremities of any chord of a circle, meet any diameter of the circle at two points equidistant from the centre; and contain a rectangle equal to the difference of the squares on the radius and on half the interval they intercept on the diameter. 2. Describe a circle passing through two given points and intercepting, on a given line, a segment of given length. 3. Define a homogeneous integral function of any number of variables. Write down the most general function, of degree 4, in 3 variables. 4. If the increase in population be 8 per cent. every decade, the rate of increase being constant, find the population of a town of 100,000 inhabitants 5 years hence. Find also, employing a table of logarithms, the percentage of increase per annum. 5. Find the coefficient of xyz in the expansion of as a rational integral function of x, y, and z. 6. If the circle escribed to the side BC of a triangle ABC touch AB and AC produced in D and E respectively, prove that AD = AE = half the sum of the sides of the triangle. The longest side of a triangular plot of ground is 100 yards, the perimeter is 250 yards and one angle is 40°. Determine the remaining angles. in a form adapted to logarithmic computation and evaluate it where 8. = 17° 4′, ' = 22° 27′, p′′ = 38° 19′, p= 21. There is a rectangular plot of ground. Show how, by means of a cord, an ellipse may be inscribed so as to touch the sides at the middle points. Prove the propositions on which the construction depends. 9. Give a geometrical construction for drawing tangents to an hyperbola from an external point. IO. Find the equation to a straight line passing through a fixed point (h, k) and making an angle of π/3 with the axis of x. If the straight line rotate, in a counter clock-wise direction, about the fixed point through an angle of 1', show that the intercept on the axis of y is diminished by a length equal to π 2700 h approximately. II. If the point (h, k) do not lie on the perimeter of the circle interpret the expression geometrically. (x − a)2 + ( y − ẞ)2 – p2 = 0 (h-a)2+(k-ẞ)2-p2 Transpose the above equation to polar coordinates and find the angle between the two tangents from the pole. 12. Find an equation to a parabola. Prove, analytically, that, at every point of the curve, the diameter and focal radius make equal angles with the tangent. |