E.—la.

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4. Eeduce to their simplest forms — 3a -2{a - 2(6 + 4c) + 3(2a - 6)} + 4(a - 26"c); 2a-36_lr 3 , _3a-c_ t„ 36-2 c _ 26-(a- 2c)) n "~5~" 8L 2 1 5 3 JJ' 5. Find the highest common factor of 4a 4 - 9a 2 6 2 + 30a6 3 - 256 4 and 6a 3 + 5a 2 6 - 6a6 2 + 356 s 6. Simplify— x y 2x y x y 1 and 1 — 6 _ 1 — a 1+ b f+q + (l + a)(l + 6) 7. Solve the equations,— 3a; - \ 2a; +8 - (3a; - 10)} = 6a; +4 - {3a; -4 - (16 - 2a;)} ; 4a; — s_sa; — B_a; — 5. 71 Hi QjT ' b-\-c — a _ c + a — 6 aa; + 6c 6a; + c ' 8. A man possessed £2,500; he invested part of it in the 4-per-cent. stock at 96, and the remainder in the 6-per-cents at £117. If his income was £116 13s. 4d. per annum, find how much he invested in each kind of stock. 9. In a school the number of boys bears to the number of girls the ratio of ato 6; if the number of boys be increased by 6, and the number of girls by a, the ratio is that of cto d. Find how many boys and how many girls there are in the school.

Algebra. — For Senior Civil Service. Time alloived: 3 hours. 1. If a = 1, 6 = 2, c = 3, find the values of — (1.) 4'a" +6 3 + 3a6c; 1 ; ; 'V 2a 2 + 2c 2 + 6 2 + V 26 2 + 2c 2 - 2a 2 2. State the rule of signs in multiplication. Prove it by the example of the multiplication of a — b into c — d. Multiply 2x 4 - 3ar ! + x 2 - 1 by 2ai 4 + 3x s - x 2 - 1. 3. Prove the rule for finding the highest common factor of two algebraical expressions. Find the highest common factor of 2x 3 + 5a; 2 - x - 6 and 2a; 4 + 3a; 3 - 2a; - 3. 4. Eeduce to their simplest forms— ,-. v 2a; 8 + 5a; 2 — x — 6 . ( '' 2a; 4 + 3a; 3 - 2a; - "3' (2 'i a 3 . _ &" , c 3 . y'' (a- b) (a -c)*(b-a) (b - c) (c -a)(c - b)' mi fa + b a— b\ fa +b_a — b\ {'' \a-b a+ b) ' ~ a~+~b) ' 5. Prove that, if a rational integral algebraical expression of the form aa: 4 + bx s + cx 2 + dx + c have the value zero, when x has the value p the expression must be exactly divisible by x — p. Find the factors of— (1.) a\b -c) + b 2 (c -a) + c\a -b) ; (2.) a 3 (b -c) + b\c -a) + c\a - b). 6. Solve the equations,— (1) 2a; +5j- 5 + 3a; = 6a; + 8 13. ['' 7 5 - 3a; 21 +3 ' /on a; — 3 . a; — 6 __ a; — 4 ai — 5 1 '' + ~ + aT=^6' 7. Solve the equations,— (ax -j- by = c, (ax + b'y= c, and from your result, or otherwise, find the values of x and y which satisfy the equations 7a; + iy = 37, 15a; -ly = 17. 8. Two men, A and B, run a race over a given course. If A have ten yards start, they arrive at the end of the course together. If they start together from the beginning of the course, B will arrive at the end y seconds before A does. Find how many yards A can ran in a minute. If it be also given that B can run the whole course in five minutes, find the length of the course, and B's rate of running. 9. A certain fraction becomes -| if its denominator be increased by 8, while if its numerator be doubled, and its denominator increased by 3, it becomes unity.