E.—la.

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Algebra. — For Class D, and for Junior Civil Service. Time allowed: 3 hours. 1. Write down the algebraical expression for the following: — The number x is mutiplied by itself and is added to the result obtained by multiplying the number y by itself, and the sum is divided by the sum of the numbers x and y ; the quotient is subtracted from that number which when multiplied by itself gives the product of the numbers x and y. If a. =4 and y=9, find the numerical value of the final result obtained. 2. Multiply 3 — 2a. + 4;r 2 — 5a; 8 by 2x 2 —x+ 4, and find the value of x which makes the product exactly divisible by x 2 — _c +3. 3. Find the factors of— 6a 2 -lab-3b 2 ; x 3 -\-a 3 — a 2 x — ax 2 ; 2a?-2V 2 -2c 2 +3ac-sbc. 4. Simplify [1- (1 +(1 -x)) + x 2] x [-1 + (1- (1-x + x 2))} ; and find the quotient when (a. +yy + (y + z) 3 +(z+x) 3 -3(x+y)(jj + z)(z+x) is divided by x+y +z. „ _~ ... 2a; 3 -5k 2 -4a.+3 5. Simplify te ._ Ba o_ 6a . +6 a — la-1 a+3, a+'A 3 ~7~'~o+a+2 a+'i ' a-2 a— 'i ~_T + ~3 +a-l 6. Show that if 2s = a+b +c, then (s-a) 2 +(s-by+(s-cy+s 2 -d 2 +b 2 +c 2 , and 1 1 _1 I__ abc s — a's-b'~s-c s sls — a)(s — b)(s —c) 1111 7. Solve the equations r=_ -r-, - —r-cc-a x— b x+ b x+a 8. A rectangular field is j. feet long and g feet wide, and from one corner there is cut off a portion in the shape of a right-angled triangle, of which the right angle coincides with one of the corners of the field, and whose sides, which contain the right angle and which are each a feet long, are in the direction of the sides of the field. Find the cost of fencing in the remainder at &x a chain. 9. How much tea at Is. Bd. a pound must be mixed with 1501b. at 2*s. a pound so that when the mixture is retailed at 2s. a pound a gain of 8 per cent, is made ? 10. A cyclist sets out from P and travels at a uniform speed towards Q, which is 180 miles distant. At the same instant as he starts another cyclist starts from Q and travels uniformly towards P. They reach their destinations respectively 12| hours and 8 hours after they meet: find the rate at which each travels.

Algebra. — For Senior Civil Service. Time allowed: 3 hours. 1. Define the terms power, index, cube root, reciprocal. Show that a 0 =1, that a$ = Va, and that a~ u is the reciprocal of a". 2. Divide x* - 2a.V +V s by x% + 2a%* + y%, and 25a: 6 " - 16a; 4 " - 8a. 2 " -1 by 5a. 3n + 6a; 2 " + 2x n + 1. 3. Explain what is meant by the highest common factor and the lowest common multiple of two quantities, and show that the lowest common multiple is the product of the two quantities divided by their highest common factor. 4. Find the H.C.F. and the L.C.M. of— (a.) 4 (x 2 y + xy 2 ), 6 (a. 3 — xy 2 ), and 9 (xy 2 + y 3 ) (b.) X s +'3x 5 +x + 3 and X s -8x + 3 5. Simplify the expressions— <*> G+i)-(*-S). i i ... x — (x 2 - y' 2 )' 2 x + (x' 2 - y' 2 )' 2 (x + y)i -(x - y)i (x + y)i +(x - y)l 6. Extract the square roots of— a m+2 + 6a sn + i + 9a 2,_ _ i e n-2 _ 30 a " c -- 2 + 25c 2 "- 1 , and of 38 -12 \/W. 7. Solve the equations— (<i \ -4. _____ = b + 1 \ a, > x+ a x+ b (b.) Via~+le = 2 V¥+x - \/x (c.) x -±l+2 Z = 21,y±1-Bx=-65,^±y^z r 4: (d.) a -+ b - = c,l+ C - = a, C -+?=b >•'xyy z ' z x