a (2) (a—b) (a–c) + b (a−b) (a–c) * (b−c)(b−a) 4. Find the G. C. M. and the L. C. M. of 5. Multiply x2 + y2+22 + y2+zx-xy by x+y-z. 6. If a+b=2, prove that a3+b3 +6ab = 8. 7. Solve the equations: 9. A butcher bought 8 sheep for a certain sum of money. If he had bought 10 for the same sum, he would have paid 108. per head less. What was the price of a sheep? 10. A number consists of two digits of which the first is double the second. Prove that the difference between the number and that formed by reversing the digits is three-quarters of the latter. 11. A train moving at the rate of 30 miles per hour reaches a station which is three times as long as the train. The time that elapses between the moment at which the engine enters the station and the moment at which the last carriage leaves it is 15 seconds. Find the length of the train. 2. Resolve into factors the expressions: (1) 12x2+2x-70; (2) x4+3x2+4; 3. Find the G. C. M. of 3 x − x2 + 23x2-6x-4 4x59x+11x3+7x2-22x-12. and (1) 4. Simplify: x2 + y2 x¤—y¤ ̧ x2+x2y2+y* . x2+2xy + y2 x2-y2 x + y ; (2) (x+1)(x+2) (x + 5) = x (x + 4)2 ; abc 7. Find two consecutive numbers such that the differ ence between their squares is 159. 8. Show that (a−b)3 + (b −c)3 + (e− a)3 = 3 (a−b) (b −c) (c — a). 9. A man has two sons whose ages differ by 2 years. Ten years ago their combined ages were one-third of their father's age; and 8 years hence the father will be twice as old as the elder son. Find the present ages of the three. 10. A sum of money is made up of half-crowns and florins, there being 56 coins in all. If half the halfcrowns were exchanged for florins, and a quarter of the florins for half-crowns, the total value would be reduced in the ratio of 19 to 20. How many of each coin are there? IX. MICHAELMAS TERM, 1903. 1. Simplify the expressions: (1) (b+c-a) (c+ a−b) + (c + a−b) (a+b−c) x-b +(a+b−c)(b+c− a); (b−c) (b-a) * (c—a) (c—b) * 2. Find the cube of + х а a X -1; and the remainder when x+2x-x+1 is divided by x2+x+1. and the G. C. M. of 2(x3-1)-3(x2-1) and 3-3x+2. 4. Extract the square root of (a+b+c)3=a3+ b3 + c3 + 3 (b + c) (c + a) (a + b) ; and deduce that if a+b+c= 0, a3 + b3 + c3 = 3abc. 8. In a football match the winning side scores as many goals and tries respectively as the losing side scores tries and goals, and wins by 2 points. If one side had converted all the tries into goals and the other none, the difference would have been 6 points. Find the scores, it being given that a goal counts as 5 points and a try as 3. 9. Find a fraction such that if 5 be added to the numerator and to the denominator, it becomes equal to ; and if 4 be subtracted from each, it equals 10. The average rainfall per rainy day in a certain place is of an inch. If it had rained on 76 fewer days this year, the rainfall would have been of what it is. Find the rainfall and the number of rainy days in the whole year. 2. Multiply x2+ y2+3xy by x2+ y2-3xy; and divide a − 3 x1 + 3 x2 −1 by x3-x2-x+ 1. M.-B. D 5. Find the G. C. M. of 2x2-5x+3 and 2x2+-6; and the L. C. M. of 2x2+x-3, 2x2-x-6, and x2+x-2. 6. Solve the equations: (2) = 2 1 5x 1-2x 10x2-7x+1 (3) 7x-6y= 23 6x-7y= 3 S 7. Find the square root of 8. A had a number of halfpence. He bargained with B to double them, and then A would give him a shilling. With the balance he made a similar bargain with Č, after which he had sixpence left. How much had he lost? 9. A and B have legacies left them of such amounts that if half of B's were added to A's, A would receive one-third more than was bequeathed to him; and if £40 |