III. TRINITY TERM, 1902.

1. Find the value of

(1) 2[1-{1-(1−1−p)+p}]2;

(2) a + ab + abc + abcd+b+be+ bed+c+cd when a = 1, b = −2, c = 3, d=0.

2. Multiply 3x2-4xy+5y2 by 5x2+4xy+3y2; and divide

45 (a3 + b3) — 24 a2b2 (a* — b1) + 54a2b1

3. Resolve into factors:

(1) 21x2+5x-44;

and show that

by 15a+12 a2b2 +9b1.

(2) 8 p3-27 q3;

4

(p + 2 + 2) (p − 2 + 2) = p2 + ·

4. Find the G. C. M. and L. C. M. of
2x2-5x-3, 3x2-11x+6, 6x3-19x2+x+6.

(a + 1)* *

6. Find the square root of:

(1) 9x+36x3y + 48 x2y2 + 24xy3 + 4 y1 ;

y

8. A number is divided into two parts such that their difference is half their sum, and the smaller part is 25; find the number.

9. A and B each go into business, A starting with thrice B's capital. Each makes £50, and then A has 2 times B's capital. With what capital did each start?

10. The price of a wine being raised 10 shillings a dozen, 4 dozen now cost the same as 5 dozen cost before the change of price. Find the price of the wine.

IV. SEPTEMBER, 1902.

1. Find the value of p-{q-[-(q-p)]} p = 1, q=-3, r = −4.

when

2. Simplify (p+q+r)2 - (p−q− r)2, and divide

60p + 193 p5q +382 p*q2 + 456 p3q3 + 386 p2q1 +191pq5+60q6 by 4p2+3pq+5q2. 3. Find the coefficient of a3 in the product of a3 (b−c)+b3 (c− a)+c3 (a−b) by a+b+c.

4. Obtain the factors of

(p-q)2-36, 1+125p3, 8p2 — 6 pq — 9 q2. 5. Find the G. C. M. of

and

6.

p1+p3q+2p2 q2+3pq3 +q*,
p1+p3q+2 p2q2 + 3pq3 + pqr2+qa + q2p2.

Find the L. C. M. of

6 (p+q)2, 3p2 - 3 q2, 15 (p3+q3).

7. Simplify:

(1) √p1+2 p3 (q + r') + p2 (q2 + r22 + 4 qr) +2 pqr (q+r)+q2p2 ;

9. Divide 810 into two parts such that one part may be two-thirds of the other.

10. A bill of £6. 38. was paid in half-crowns and florins, and the whole number of coins was 56: how many coins were there of each kind?

11. A bicycle is travelling at the rate of a feet in b seconds. Express this speed in miles per hour.

V. MICHAELMAS TERM, 1902.

1. Simplify: 5-[4-(3-(2-1-x)}].

What value of x will make the expression equal to zero ?

2. Multiply 3p+5q+7r by 7p+5q+3r, and divide

a3+b3—3ab +1 by [(a−b)2+(b− 1)2 + (a −1)2]. 3. Resolve into factors

p3-pq2, 2p2+pq-3 q2, 3 p2 -pq — 2 q2.

4. Find the G. C. M. of

and

3x2+2xy +7xz-5y2-7yz,

7x2-2xy +3xz-5y2-3yz.

5. Find the L. C. M. of

and

4 (x + y)3, 6 (x2-xy+y2), 8(x2 — y2),
12 (x3 — 2 x3y + 2 xy2 — y3).

7. Find the square root of

9. Divide 54 apples among A, B, and C, so that A gets 1 more than B, and B 1 more than C.

10. I borrow equal sums from A, B, and C. A asks no interest, B asks 5 per cent., C asks 3 per cent. per annum. At the year's end I have to pay £77 in all. What did I borrow of each?

11. A cistern is filled by 3 pipes A, B, and C. A alone fills it in x minutes, B in y minutes, C in 60 minutes; B and C together in 36 minutes; A, B, and C all together in 24 minutes. Find x and y.

(2) Write down an expression for the number consisting of three digits, of which the first is a, the second zero, and the third b.

2. Multiply

p2 −2 pq + q2 + r2 by p2+2pq+q2 — p2.

3. Prove that, if p2 (q+r) + q2 (r+p) + r2 (p + q) +3pqr is divided by qr+rp+pq, the quotient is p+q+r. 4. Find the G. C. M. of

and

(p + q)3 + 3 (p + q)3 r + 3 (p + q) r2 + pů p2 (q+r) + q2 (r+p) + r2 (p + q) + 3 pqr.

5. Resolve into factors:

(1) -p2 (a+b)2 + 2 pq (a2 —b2) — q2 (u — b)2 ;
(2) (4p+3q)3 — (3p+4q)3.

8. Divide 100 into two parts, so that one half of their difference may be 15.

9. If 2 lb. of tea and 18 lb. of sugar cost 9s. 6d., and 3 lb. of tea and 44 lb. of sugar cost 188. 6d., find the price of 4 lb. of sugar.

10. Seven years ago A was three times as old as B then was, and seven years hence A will be twice as old as B will be: find their present ages.

2. Express in as many factors as possible:

(1) 6-1; (2) x3-2x2y + 2xy2 —— y3.