5. Find two factors of 23 - y), and four of x6y6. If two numbers differ by 4, show that the difference of their cubes is equal to three times the square of their sum increased by sixteen. 6. Show that bc (c-a) (a−b)+ca (a-6)(b-c) + ab (b-c) (c-a) = abc (a +b+c) — 62,2ca— a. 7. Solve the equations : (1) 15 (a — 3) +4 (0 - 1)-6 = (0— 4) + 5; x + 2y = 16. 8. Divide 1000 into two parts so that one part shall be at of the other. 9. If I buy apples at x for a shilling and sell them at y pence per dozen, how much per cent. do I gain ? 10. Three men A, B, C each have a field of the same area, all three fields being rectangular in shape. A's field is 5 yards broader than B's and 10 yards broader than C's, B’s field is 8 yards longer than A's and C's field is 20 yards longer than A's. What is the area of the fields ? XIX. TRINITY TERM, 1906. 1. If a = -3, b = 7, c= -2, find the value of 2 a? (6-2c)– 3 V72 — 4ac +(bc + 2 a) (1 - a)(b + 3c) 2. Find the G. C. M. of + + 4x3 + 6x2 + 5x + 2 and 24 + 2 x3 + 3x2 + 2x+1; and the L. C. M. of Q2 + x – 12, 22 – X – 20, x? — 8x + 15, 22 6x +9. 1 + a? + c2-62 6 c 3. Simplify: 1 1 1 1 2 ac ū+b+c7-atc . 22 - 2x - 99, -a(26-c)+ a (62-2 bc) + b2c. 5. Find the square root of p2__ 2P _ 29 + 3. qapaq P 6. Solve the equations: (1) (2 — 1)(x-2)+ (0-3) ' = *(-3) (— 4)+ (OC — 4) (@— 5); al - 62 ab + a ax – 62 bæ+azi 323 + 26 y = 18 (2C — y) +19 (+ 2y)+1. 7. Show that the difference between the fourth powers of two consecutive numbers is equal to the product of their sum into the number next greater than twice their product. 8. A man spends £50 of his money and then a half of what he has left. He receives £15 and then has a third of what he had at first. How much was that ? 9. A debt of £1. 118. 10d. is paid in shillings and francs. The number of coins is 36, and a franc is taken as worth 9fd. Find the number of francs which were used. 10. A, B, C working together can do a piece of work in æ hours. A by himself can do the work in 2x + 6 hours, B in 3x hours, and C in 3x +9 hours. Find 2. XX. SEPTEMBER, 1906. 1. If X = 3a + 2b +c, z = 2a + b + 30, show that 2x+2-5 (+ y +2) – = 3a. 6 2. In the previous question, if a =-1, b = 1, c = 2, find the value of (1) 202 + y2 +22; (2) (x – Y->) (2 + y-2). 3. Find the G.C. M. of 2c4 + 5x3 + 10x2 + 10x + 4 and 2c4 + 3x3 + 5x2 + 4x +2, and the L. C. M. of X, – 7X+6, 22 — 12x + 36, 2 – 2x + 1. 4. Find the square root of 2® + 2 206 + 3x4 + 4 x3 + 3x2 + 2x + 1. la + b a? _627 1 - 9x 1 +9x81x2–1 5x + 32 7 X - 81 2x + 3 8x + 9 6 (3) ã + 7 = a +6+ 7. If the sum of two numbers is twice their difference, show that the difference of their squares is twice the square of their difference. 8. How much sugar at 4 d. per lb. must be mixed with 20 lb. at 5 d., so that the mixture may be worth 4 d. per lb. ? 9. A man invests £5500, partly in a 4 per cent. and partly in a 5 per cent. investment. If he were to change the places of the two sums invested, his yearly income would be increased by £5. What is his present income? 10. A man's age is at present equal to the sum of the ages of his two sons. Ten years ago, the sum of the ages of all three was 62 years. What is the father's present age ? GEOMETRY A. Stated Subject. I. MICHAELMAS TERM, 1904. 1. (6) Prove that the diagonals of a rhombus bisect each other at right angles. . 2. Given in a plane a fixed straight line and two fixed points, find the point on the straight line which is equally distant from the two points. Is there any case in wbich your construction fails ? 3. On the hypotenuse BC of a right-angled triangle ABC a point D is taken so that DA = DB. Prove that D must be the middle point of BC. 4. (b) Describe a rectangle equal in area to a given triangle. 5. (6) Prove that the area of a right-angled triangle is one-half of the rectangle contained by the two sides. 6. A finite straight line AB is bisected at C, and P is any external point. Prove that (a) PA? + PB2 = 2 PC2 + 2 AC2; (6) PA + PB is not less than 2 PC. 7. Show how to draw the two tangents to a circle from an external point, and prove that they are equal in 2\\2\/?/????? Show also how to draw the two tangents to a circle which are parallel to a given straight line. 8. (6) A, B, C are three fixed points on a circle, and P is any point on the tangent at Ĉ. Prove that the angle APB is always less than the angle ACB. 9. Through a fixed point 0 on a given circle, chords OP, OQ, OR ... are drawn to meet the circle in the points P, Q, R .... Find the locus of the middle points of all these chords. M.-B. |