were taken from this larger sum and given to B, he would now receive one-third of what was left for A. What were the two legacies ? 10. A bag contains a certain number of black and white balls. If the black balls had been 4 less in number there would have been 2 black to every 3 white, but if there had been 4 white balls more there would have been only 3 black to 4 white. How many were there of each ? 2. Find the H. C. F. and L. C. M. of x3+x2-5x+3 and 3+4x2-11x+6. 3. Prove that (b−c)3 + (c− a)3 + (a−b)3 = 3 (b−c) (c − a) (a−b). 4. Resolve into factors: (1) x2-10x-119; 6. Find the remainder, if x2-2x + 3 be multiplied by +2x-3 and the product divided by 2-2-3. 7. Find the first three terms of the square root of: 36 x1 + 48 x3 – 20μ2 — 39. What integral value of x would make the expression a perfect square? 8. There is a fraction, whose numerator is one less than the denominator, and, if you add to its numerator the sum of the numerator and denominator, and subtract from its denominator twice the difference between the numerator and denominator, it will become 4. Find the fraction. 9. A boy receives a fixed sum as pocket-money at the beginning of every week, and in each week he spends half of all he had at its beginning. He had no money before the first pocket-money was given him, and at the end of the third week he has 18. 2d. What is his weekly allowance? XII. SEPTEMBER, 1904. 1. If a = x2 + 3x2 + 3x + 1, b = x2+2x+1, c=x+1, find the value of a-3b+3c-1 when x = - ૩. 2. Multiply 3+2x2 −x+4 by x3 −2x2+x−4. 3. Divide 23-2x2+3x-4 by x2-5x+6, and find the value of x in the case where the remainder is zero. 4. Find the H. C. F. of 7x2-8x2+50x-7 and x3-2x2+8x-7. 5. If a 2x (2x2+3) and b = 4x+11x2+9, prove that the square root of a2+b2 is 4x2+13x2+9. (2) 3x-4y=3, 5x+6y=— 70; (3) ax + by = a + b, x+cy=1+c. Verify the results in the case of equations (1) and (2). 8. Prove that the sum of two odd numbers is always even, and that the product of the product and sum of any two numbers is always even. 9. A is twice as old as B, and C is five times as old as A, B together. Ten years hence the united ages of all three will be 84. Find their present ages. 10. A sum of £1000 is invested partly at 3 per cent. and partly at 5 per cent. The total interest obtained is £45. 128. How much is invested at each rate? root of 25x-30x3 +49 x2-24x+16. and the square 4. Find the highest common factor of 25x30x+49x-24x+16 and 50a3-45x2+49x-12. 5. Simplify 7. Write out a description of any method of solving two simultaneous equations of the first degree in two variables. Illustrate the method by solving the equations y-1=-(x-4) 8. Prove that the product of any two consecutive numbers is always even, and that the difference of their squares is always odd. 9. If a tons of coal are bought for a total sum of £y, and are sold at z shillings per ton so as to gain r per cent., prove that 5xzy (100+ r). 10. A certain sum of money is divided equally between a number of persons. If there had been one more to share the money, each would have received £20 less, and if there had been three more, each would have received £50 less. Find the share of each person, and the number of persons. 11. Two numbers have the same two digits in inverted order. The difference of the numbers is 63, and the sum of the digits is 11. Find the numbers. XIV. HILARY TERM, 1905. 1. If a, b, c = 3, find the value of 2. Divide the product of (x3-3x2+3x-1), (x-1), and (x2-2x+1) 3. Find the H. C. F. and L. C. M. of (x3-4x2+5x-2) and (3-5x2+8x-4). 4. Simplify (1) 1 + (a —b) (a — c) * (b−c) (b− a) (c—a) (c —b)° Show that (2) + b2 (a−b) (a−c) * (b−c) (b−a) + (c—a) (c—b) 5. Extract the square root of 6x2-7x-20 x2+2x-3 (c—a)(c—b) 9x4-24x3-26x2+56x+49. 6. Simplify 3x2-2x-8 ÷ 2x2+x-15 x2+6x-7 2 +5–14 7. Solve the equations : (1) 15x-5(60—x) = 240; = 1. a2c b (2) (x+a) (x−b) + a (b + c) = ~~~ + x2; (3) (x+3y= 21 2x - y = 0. 8. The length of a field is twice its breadth; another field which is 50 yards longer and 10 yards broader contains 6800 square yards more than the former. Find the length of each field. 9. A club manager buys some cigars at a cost of two pounds, and means to retail them so as to make a profit of 25 per cent. After sixty are sold at this rate, he finds that half the remainder are spoiled. He sells the unspoiled cigars at the same rate as the former sixty and then finds that he has made no profit or loss. Find the original number of cigars, and the sale price of each. 10. A and B together possess £570. If A's money were three times as much as it is and B's five times as much, they would together possess £2350. How much has each? |