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 Examination Manuals 
 
 No. II. 
 
 4 ^GEBB^ 
 
 -^-O-H^ 
 
IN MEMOMAM 
 Irving Stringham 
 
 n 
 
WENTWORTH & HILL'S 
 
 EXAMINATION MANUALS. 
 
 No. II. 
 
 ALGEBRA. 
 
 BOSTON: 
 
 GINN, HEATH, & COMPANY. 
 
 1884. 
 

 Entered, according to Act of Congress, in the year 1883, 
 
 By GEORGE A. WENTWORTH and GEORGE A. HILL, 
 
 In the Office of the Librarian of Congress at Washington. 
 
 J. S. Cushing & Co., Printers, Boston. 
 
L-% 
 
 PREFACE. 
 
 THIS Manual consists of two parts : The first part contains one 
 hundred and fifty examination papers, the questions for 
 which have been selected mainly from the best English, French, 
 and German collections of problems. These papers may be divided 
 into three groups. The first fifty papers embrace the subjects 
 of Elementary Algebra as far as Quadratic Equations; the next 
 fifty papers also include Quadratic Equations and Radical Expres- 
 sions; the last fifty papers extend over still more ground, includ- 
 ing several topics usually regarded as belonging to Higher 
 Algebra. 
 
 In each of these groups the earlier papers will be found 
 somewhat easier than the later ones. The papers are intended 
 to be hour papers, but if any of them are thought to be too 
 long for one hour, the time may be increased or the length of 
 the paper reduced by omitting one or more of the questions. 
 
 The second part of the Manual is a collection of recent 
 papers actually set in various American and English institutions 
 of learning. 
 
 There are two ways in which the Manual may be used. 
 
 First : To test the learner's knowledge in the usual way by 
 means of an examination. For this purpose the class will come 
 to the recitation-room provided with the Manual and blank 
 books, and the teacher will simply designate by number the 
 paper to be worked. 
 
 Secondly : To exercise the learner from day to day in the 
 various rules and processes, to detect his weak points, and ascer- 
 
 800575 
 
IV PREFACE. 
 
 tain where he most needs assistance. This may be done by- 
 assigning exercises to be worked in the class-room, or by assign- 
 ing to each member of the class a paper with directions to 
 hand in the solutions, neatly worked out, at a subsequent reci- 
 tation. 
 
 The Manual will be found especially useful in reviewing the 
 subject of Algebra, and in preparing for examinations. 
 
 Answers to the problems in the first one hundred and fifty 
 papers, bound separately in paper covers, can be had by teachers 
 only, on application to the publishers. 
 
 G. A. WENTWORTH. 
 G. A. HILL 
 
SPECIMEN PAPER WORKED OUT. 
 
 1. Simplify 
 
 a _ [26 + {3 c - 3a - (a + b)} + {2a - (b + c)}] 
 
 = a-[26+{3c-3a-a-&}+{2a-6-c}] 
 
 = a — [26 + 3c -3a — a -6 + 2a — 6 -c] 
 
 = a-26 — 3c + 3a + a + 6 — 2a + 6 + c 
 
 = 3a-2c. 
 
 Ans. 3 a — 2 c. 
 
 2. Resolve into factors 
 
 x i _ ^2 + 2 2 _ a 2 _ 2 X 2 + 2ay 
 
 = (a; 2 - 2xz + z 2 ) - (a 2 - 2 ay + y 2 ) 
 - (s - z) 2 - (a - 3/) 2 
 = {(ir-2)-(a-2/)}{(a;-2) + (a-y)} 
 = (a; — 2 - a + y) (x — z + a - 3/). 
 
 3. Find the H.O.F. of 
 
 5a; 2 (12a 8 + 4^ + 17a -3) 
 12a; 3 + 4a; 2 + 17s -3 
 12a 3 + 4a; 2 - a; 
 
 5x 2 (12ar» + 4a; 2 +17a;-3) 
 
 and 10a; (24ar» - 52a; 2 + 14 ar - 1). 
 
 10a:(24ar J -52a; 2 + 14a;-l) 
 24a* - 52a; 2 + 14a; -1 
 24x 3 + Sx 2 + 34z-6 
 
 3 )18 a; -3 
 6*-l 
 
 -5)- 
 
 -60 a; 2 - 
 
 20 a; + 
 
 5 
 
 
 12a; 2 + 
 12 a; 2 - 
 
 4a;- 
 2x 
 
 1 
 
 
 
 6a:- 
 6x- 
 
 1 
 
 1 
 
 Reserve 5 a? 
 
 2 
 
 x 
 
 2z + l 
 
 Ans. 5x(Qx — l). 
 
VI EXAMINATION MANUAL. 
 
 4. 
 
 Simplify 
 
 r J 3 7 4 -20a; 
 
 
 
 
 
 l-2x l + 2a; 4a; 2 -1 
 
 
 
 3 7 4 -20a; 
 
 
 
 
 
 l-2a? l + 2a; 4a; 2 -1 
 
 
 
 3 7 4-20a; 
 
 
 
 
 
 l-2ar 1+2sb l-4a; 2 ' 
 
 
 
 L.C.D. = l-4a; 2 . 
 
 
 
 
 
 3 + 6 x = first numerator, 
 
 
 
 
 
 — 7 + 14 a; = second numerator, 
 
 
 
 
 
 4 — 20 a; = third numerator. 
 
 
 
 
 
 = sum of numerators. 
 
 
 
 Arts. 0. 
 
 5, 
 
 Solve 10a; + 152/ -242= 41 
 15s-12y + 16«- 10 
 18a; -14y- 7s — —13. 
 
 
 
 (1) 
 
 (2) 
 
 ■ (3) 
 
 
 Multiply (1) by 2, 20 x + 30 y - 48 z = 
 Multiply (2) by 3, 45 x - 36 y + 48 z = 
 
 82 
 30 
 
 
 
 Add, 65a;- 62/ 
 
 
 112 
 
 (4) 
 
 
 Multiply (2) by 7, 105 a; - 84 y + 112 2 = 
 Multiply (3) by 16, 288 x - 224y - 1122 = 
 
 70 
 
 -208 
 
 
 
 Add, 393 x - 
 
 308 y = 
 
 -138 
 
 (5) 
 
 
 Multiply (4) by 154, 10010 a; - 
 Multiply (5) by 3, 1179 a;- 
 
 924y = 
 924y = 
 
 17248 
 -414 
 
 
 
 Subtract, 8831 x 
 
 = 
 
 17662 
 2 < 
 
 
 
 Substitute value of x in (4), 130 
 
 . *. x — 
 
 -6y = 
 
 112. 
 
 
 .-.y = 3. 
 Substitute values of x and y in (1), 20 + 45 — 24 2 = 41. 
 
 .-. 2=1. 
 Ans. a; = 2, y = 3, 2 = 1. 
 
 6, A passenger train, after travelling an hour, is detained 
 15 minutes; after which it proceeds at three-fourths 
 of its former rate, and arrives 24 minutes late. If 
 the detention had taken place 5 miles farther on, 
 the train would have been only 21 minutes late. 
 Determine the usual rate of the train. 
 
SPECIMEN PAPER WORKED OUT. Vll 
 
 Let x = usual rate of train per hour, 
 
 and y = number of miles train has to run. 
 
 Then y — x = number of miles train has to run after detention, 
 
 y ~ x — number of hours usually required to run y — x miles. 
 
 and ^-r — = number of hours actually required to run y — x miles, 
 
 Since the detention was 15 minutes, and the train was 24 minutes 
 late, the loss in running-time is 9 minutes, or ^ of an hour. 
 
 • V~ x .. V~ x - 3 m 
 
 'fa? a; 20 K ' 
 
 If the detention had occurred 5 miles farther on, the loss in run- 
 ning-time would have been 6 minutes, or ^ of an hour. 
 
 . y-x-5 y- x -5_l f ~ 
 
 Simplify (1), 
 Simplify (2), 
 
 Subtract (1) from (2), 
 
 . Solve 
 
 1 + 
 
 2a; 2 + a; + l 2ar 
 
 1 
 (2a;-l)(a; + l) (2a;- 
 
 Simplify, abx — ab H 
 
 U 2 x 2 -a 2 
 
 \x x 10 K J 
 
 202/- 29a; = 
 20y-26a;=100 
 
 3a; = 100 
 .-. a; = 33J. 
 
 Ans. 33| miles, 
 
 1 a 2 bx + b 
 
 ! — 3 a; + 1 2 bx — b ax 2 — a 
 1 a 2bx + b 
 
 l)(a;-l) b(2x-l) a(sc-l)(aj + l) 
 
 L.C.D. = ab(x- l)(x + l)(2a; - 1). 
 - abx + ab = a 2 x 2 -a 2 -4:b 2 x 2 + b 2 , 
 2abx = a 2 x 2 -a 2 -4cb 2 x 2 + b 2 , 
 x 2 + 2abx = b 2 -a 2 , 
 
 (U 2 - a 2 )x 2 + 2abx = b 2 - a 2 . 
 Complete the square, multiplying by 4 times the coefficient of 
 and adding the square of the coefficient of x, 
 
 4(46 2 - a 2 ) 2 x 2 + ( ) + (2ab) 2 = 16¥-I6b 2 a 2 + 4a 4 , • 
 (4Z> 2 - a 2 ) 2x + 2ab = ± (4 b 2 - 2a 2 ), 
 (4& 2 
 
 2b + a 
 
EXAMINATION MANUAL. 
 
 8. Solve 
 
 
 
 
 
 a; + y = 4 
 
 
 (1) 
 
 
 x* + 2/* = 82 
 
 
 (2) 
 
 Put u + v for x, and u — v 
 
 for y. 
 
 
 
 (1) becomes 
 
 2u = 4. 
 .-. u = 2. 
 
 
 
 (2) becomes 
 
 w 4 +16w 2 v 2 +'y* = 41 
 
 
 (3) 
 
 Substitute 2 for u in (3), 
 
 16 + 24v 2 + ^=41, 
 t^ + 24v 2 = 25, 
 ^ + () + 144 = 169, 
 a 2 + 12 = ± 13, 
 
 
 
 
 v 2 = 1 or 
 
 -25, 
 
 
 
 a = ± 1 or ± V^ 
 
 -25. 
 
 
 ( a; = 3, 1, 
 
 or 2± 
 
 V^25, 
 
 
 ^ iy-1,3, 
 
 or 2 + 
 
 V-25. 
 
 9. Show that 2a/« 3 6 2 , -^o^ 2-vl^ are similar surds. 
 
 2^5 2 = 2a^6 2 ", 
 y/a9>=2bW, 
 
 Since they all have the game surd factor they are similar surds. 
 
 10. Simplify 
 
 = (2ab)$x(3ab 2 )* + (5ab*)l 
 = V{2^bfx^(Mb 2 J 2 - y/5ab 3 
 
 V2 3 X 3 2 X a 5 X V -r- 
 
 
 2 3 x3 2 a 5 6 7 
 5a& 3 
 «/2 3 x3 2 x5 5 a 4 6* 
 
 r>« 
 
 = |V225000a*6 4 
 
 iV225000a 4 6 4 . 
 
SPECIMEN PAPER WORKED OUT. ix 
 
 11. Expand f— - </f 
 
 ( 2x 2 y- 1 -y$f=(2x 2 y- 1 f-6(2x 2 y- 1 Y(yt) + 15 (2 as* y" 1 )*^)* 
 - 20(2x 2 y- 1 ) 3 (y$f + 15(2x 2 3/" 1 ) 2 (y^ 
 -6(2x*y-i)(yi? + (yty 
 - 64z 12 y- 6 - 192a; 10 y-* + 240cc 8 y- § - lBO^y' 1 
 + 60rr*3/^-12x 2 2/ 5 + t/*. 
 
 Am. 64:X 12 y- 6 -192x 10 y-^ } etc. 
 
 12. Find the value of 
 
 j/ 3.1416 x 4771.21 x 2.7183* 
 30.103* X 0.4343* X 69.897*' 
 
 log 3.1416= 0.4971 =0.4971 
 
 log 4771.21 - 3.6786 =3.6786 
 
 £log 2.7183= 0.4343^2 =0.2172 
 
 4colog 30.103 = (8.5214 - 10) X 4 = 4.0856 - 10 
 
 £colog 0.4343= 0.3622-^2 =0.1811 
 
 4colog 69.897 = (8.1555 - 10) x 4 = 2.6220- 10 
 
 11.2816-20 
 30 -30 
 
 5 ) 41.2816 -50 
 
 i|41 
 
 8.2563 - 10 
 
 Am. 0.01804. 
 
EXAMINATION MANUAL. 
 
 ALGEBRA. 
 
 1. 
 
 1. Ifa=l, 5 = 3, c = 5,d=0 t 
 
 find the value of a 2 + 2 b 2 + 3 c 2 + 4 rf 2 . 
 
 2. Add a + 6-2c, 8a + 4e + 26, 3c-2a-65, 
 
 and 25 - 2c + 3a. 
 
 3. Multiply l + x + x 4 + x 5 by 1 — a? -f a* — s*. 
 
 4. Resolve into the simplest factors a 2 x 2 y — b 2 xy 2 . 
 
 5. Find the L.C.M. of xy, x — y, and y 3 — x 2 y. 
 
 o «■ i7 7# — 10 Sx—7 27x — 30 . 
 
 6. Simplify — 
 
 r J 5 6 30 
 
 7. Solve 2(a?-3) + 6(a?-2) = 54. 
 
 8. Divide a line 12 inches long into two parts, such that 
 
 the greater shall be five times the less. 
 
 2. 
 
 1. Ifa = l, b = -l, c = 0, d=2, 
 
 find the value of (ac — bd) (ad— be) (ab — cd). 
 
 2. Divide a 2 + ab + 2a<? - 2b 2 + 7fo - 3c 2 by a- 5 + 3c. 
 
 3. Resolve into the simplest factors, a 2 x 2 — a 2 y 2 . 
 
 4. Find the H.O.F. of 9* 2 - 25 and 9* 2 + 3a; - 20. 
 
 5. Add^, *=£, and 5 "* 
 
 ab ac be 
 
 6. Solve 5 - 3(4 -x) + 4(3- 2x) = 0. 
 
 7. A and B together have $4900. A invests £ of his 
 
 money in business, and B \ of his, and each has the 
 same sum remaining. How much money had each ? 
 
 8. Solve 2o;-f 3y = 221 
 
 3s + 4y = 31J 
 
EXAMINATION MANUAL. 
 
 3. 
 
 1. If a = l, & = 0, <? = -!, 
 
 evaluate (1 — ab) (1 — ac) (1 — be). 
 
 2. "Write down the product of x + -> a? — ^> x 2 -}-—- 
 
 3. Simplify 3a- J[5+{2o-(5-a;)}]. 
 
 4. Find the H.C.F. of 6 a 6 - 6 a 3 a; 3 and 8a 6 -8a 6 #. 
 
 5. Subtract — — — from 
 
 ax ax — x 2 
 
 6. Solve \x+\(x — l) = a?-4. 
 
 7. Nine years ago A was three times as old as B, but now 
 
 he is only twice as old. Find the present ages of 
 A and B. 
 
 8. Solve * + 1 = l 
 
 3 2 6 
 
 7 3 21 
 
 4. 
 
 1. Multiply a — b-\-chy a + b — c, and find the value of 
 
 the product when a = 9, b = 4, c = 3. 
 
 2. From 5a + 3c -45 - 7a 7 - e 
 
 take 4a + 7c 7 +5e — 55— 6c. 
 
 3. Divide x G — y 6 by a; — y. 
 
 4. Solve ^-U* = l£-(^ + 5). 
 
 8 12 ? \ •' 
 
 5. Solve 7x+ 4y = 17yl 
 
 6a;-10y = 8 J 
 
 6. Besolve into factors y 2 + 25 y — 150. 
 
 7. Simplify a '~ 4 X ft2 ~ 25 . 
 
 ^ J a 2 +5a a 2 + 2a 
 
 8. Find a number such that the sum of its fifth and its 
 
 seventh parts shall exceed the difference of its 
 fourth and its seventh parts by 99. 
 
ALGEBRA. 
 
 5. 
 
 1. Simplify 3 + a) (y - b) - { - aty - (c -d)]\. 
 
 2. Find the H.O.F. of 12* 4 - 108a; 2 and 16# 4 - 48* 3 . 
 
 3. Find the L.C.M. of 4 ab (a 2 - b 2 ) 3 and 6 a 3 b 2 - 6 a 2 b 3 . 
 
 4. Multiply-—^- by 
 
 x 2 -2x-S J x 2 + 2x-Z 
 
 5. Solve 1+JL_JL = L 
 
 a; 2x ox 3 
 
 6. What sum is that which is as much greater than $20 as 
 
 its half is less than $20? 
 
 7. Two trains travelling, one at 26 and the other at 30 
 
 miles an hour, start at the same time from two places 
 120 miles apart, and move towards each other. How 
 long will it be before the trains meet ? 
 
 6. 
 
 1. Add 2a — Bb + c, 2b-Sc + a, 2c-Ba + b, 
 
 and a-\-b-\-c. 
 
 2. From 5a 3 -7a 2 b + 6ab 2 - b 3 + 5 
 
 take 3a 3 + 4a 2 b-3 + 8ab 2 -3b*. 
 
 3. Multiply x 3 — 2 x 2 y + 2 xy 2 — y 3 by x 2 -\-2xy + y*. 
 
 4. Divide a 3 — b 3 — c 3 — 3 abc by a—b — c. 
 
 5. Solve i(;r-4Hi(:r + 4) = T VO*; + 20). 
 
 6. Simplify ^--1---^. 
 
 7. Solve #-l-2y + 3z = 10 
 
 2x+3y + 4:z = 16 
 4# + 4?/ + 5z = 25 
 
 8. A man bought 3 horses and 5 cows, and gave the same 
 
 sum for the 3 horses as for the 5 cows. If he had 
 bought 4 horses and 10 cows, his outlay would have 
 been $ 600 more. Find what his outlav was. 
 
EXAMINATION MANUAL. 
 
 7. 
 
 1. If a = 1, b = 3, c == 5, c? = 0, find the value of 
 
 12a 3 -5 2 . 2c 2 g + 6 2 + c 3 g 2 -^ 2 
 
 3a 2 a + 6 2 6 2 -2fo + c 2 6a-c' 
 
 2. Simplify 2-3a?-(4-6a?)-{7-(9-2a?)j. 
 
 3. Multiply 1 -f 2x + 3a; 2 by 1 + 4a; + 5a: 2 + 6a; 3 . 
 
 4. Find the continued product of 
 
 x-\-y, x — y, and a; 2 + 2 xy -f- y 2 . 
 
 5. Divide a; 3 -f- (a -|- ^ + c ) & + («& + b° + ^) # + °bc by 
 
 a; + a. 
 
 6. Eeduce to the simplest form — ^ — , 7q x ■ a ,. • 
 
 r a 2 — ab-\-b 2 a 3 — b* 
 
 7. Solve the equation — = — r— - 
 
 4 o o 
 
 8. A is twice as old as B and four years older than 0. The 
 
 sum of the ages of A, B ? and is 96 years. Find 
 B's age. 
 
 8. 
 
 1. Find the value of a ! + ?!~ C ! + ^ if a = 4 t S = },c=l. 
 
 a 2 — b 2 — c 2 -f 2 be 
 
 2. What number diminished by 42 becomes 18 ? 
 
 3. Simplify \ y *-(l-y)]x-\x 2 + (x-y)y\. 
 
 4. Multiply px 2 + qx — r by mx — n, and bracket coeffi- 
 
 cients of the powers of x in the product. 
 
 5. Divide 1 -far 5 - 8?/* + 6a;y by l + x-2y. 
 
 6. Solve 3a? ~ 2 + 90a? "- 9 = 5s + 4*. 
 
 7. Find the length and breadth of a rectangular field if 
 
 the length is 140 yards more than the breadth, and 
 if it requires 1000 yards of fence to enclose it. 
 
 8. Point out first the factors, secondly the terms, and thirdly 
 
 the coefficients, in the expression 5 ax — 2 ay + abz. 
 
ALGEBKA. 
 
 9. 
 
 1. If a = 1, b = 2, find the value of 
 
 a e _ 3a 5 6 + 6a*b 2 - 7a 3 b 3 -f 6a 2 6 4 - 3a& 5 + 5 s . 
 
 2. Divide £ 4a +4y 45 by x 2a +2^<y + 2y 2b . 
 
 3. Resolve into the simplest factors 
 
 ax 2 ~a; x'-Sla 4 ; 8a 3 + 27; ^ + 7^+10. 
 
 4. Find the L.O.M. of a + b, b-c, c + a, a 2 -b 2 , b 2 + c\ 
 
 c 2 -a 2 . 
 
 5. Find the H.C.F. of 
 
 x 2 — y 2 -{■ xz -\- yz and x 2 + y 2 + xz ~ V z — ^ X V- 
 
 6. Simplify 2(a?-y)-2J3( a ;-y)-^-3y-^U. 
 
 7. Solve 2a(3a-a7)-3^(26-a) = a5. 
 
 8. The sum of .two numbers is 70, and their difference is 
 
 23 ; find the numbers. 
 
 10 
 
 1. If a = 25, 5 = 9, c = 4, d = 1, find the value of 
 
 3 Va + 2 VU - V9 c +V16 d. 
 
 2. From 7a; 3 - 6x 2 + 2a? - 1 take ±x 3 -3x 2 - x + 2. 
 
 3. Multiply a 2n + a n a; w + a; 2n by 2a n - 2x n . 
 
 4. Divide x K — 9 a; 2 — 6xy — y 2 by x 2 + y + 3 a?. 
 
 5. Resolve into the simplest factors 
 
 x't-y 16 ; 8x« + - 3 ) lQx n -y i ; x 2 + 6x-7. 
 
 6. Find the H.C.F. of 
 
 6a 2 + 7aa?-3a? and 6 a 2 + 11 as; + 3 a; 2 . 
 
 7. solve 2x + a ( * ~ h) ~ SaX + (a ~ 5y 
 
 b a ab 
 
 8. A is now twice as old as B, but 10 years ago he was three 
 
 times as old. Find the present ages of A and B. 
 
EXAMINATION MANUAL. 
 
 11. 
 
 1. If a = 0, b = 2, c = 4, rf= 6, find the value of 
 
 3</2b 2 -a + 2$V+j + 1 - ^/2(b + c) 2 ~(b + d) 2 . 
 
 2. From-fa: 4 -f-£a; 2 + 3a:-lf take %x*-%x-2\. 
 
 3. Divide l-6x 5 +5Vby \-2x-\-x 2 . 
 
 4. Simplify (a; + y) 2 - 2* (3a; + 2y) - (y - x) (- x + y). 
 
 5. Resolve into the simplest factors 
 
 2^-3#-5 and a 2 - b 2 + c* - 2ac. 
 
 6. Find the H.O.F. of 
 
 y K -\-m 2 y l -\- ra 4 and y 4 + my 3 — m z y — m 4 . 
 
 7. Solve 5_.2£zil!_£l±3 = o. 
 
 2 3 4 
 
 8. A person bought 16 yards of cloth, and if he had bought 
 
 one yard less for the same money, each yard would 
 have cost 25 cents more. What did the cloth cost 
 per yard ? 
 
 12. 
 
 1. Find the H.C.F. of 
 
 12a 2 +lSab + 3b 2 and 6 a 2 + 23 ab -f W. 
 
 2. Simplify 2(x-y)-2{s(x-y)-(2x-Sy-^\ 1 . 
 
 3. Find the square root of 
 
 9 a 4 - 12 a*b + 34 a 2 b 2 - 20 ab 3 -f 25 6 4 . 
 
 4. A is twice as old as B ; 22 years ago he was four times 
 
 as old ; what is A's age ? 
 
 5. Multiply x 2 + ^ - y 2 by x 2 - ^ + ^ . 
 
 6. Solve ar + ii^ = ^^- 
 
 7. What fraction is that to the numerator of which if 7 be 
 
 added, its value is I ; but if 7 be taken from the 
 denominator, its value is f ? 
 
 8. Solve # + 2?/ = 23, 3# + 4z = 57, 5y + 6z = 94. 
 
ALGEBRA. 
 
 13. 
 
 1. Add .5a; 3 - 3a; + 2y, -x* + 2x -y, and 7a; 3 -4a; -j- 3y. 
 
 2. From2a; + lla + 10&-5e-23 take2c-10+5a-36. 
 
 3. Multiply 9x 2 + Sx7/ + f-6x + 27/ + 4:hy Zx — y — 2. 
 
 4. Divide a; 3 — 5a; 2 — x + 14 by a; 2 — 3a;— 7. 
 
 5. Solve K^+l) + K^+2)-16 + i(a; + 3) = 0. 
 
 6. Resolve into factors 2xy — x 2 —y 2 -\- z 2 . 
 
 7. Simplify ^ + —£- + — ^ — 
 
 y x-\-y x 2 -\-xy 
 
 8. Solve3a;-22/=:5 
 
 4a;-3y+2z=ll 
 x— 2y — 5z = — 7 
 
 14. 
 
 1. Find the square root of a; 4 + 2x % — x + -J-. 
 
 2. Find the H.C.F.ofa; 4 +3a; 3 -6a;-4anda; 4 -3a; 3 + 6a;-4. 
 
 3. Resolve into factors x 2 -f- 20 a; + 91. 
 
 4. Divide 86 into two parts such that their difference shall 
 
 be 18. 
 
 5. Simplify «=* + _^L. _ ^!+^. 
 
 o a — b a 2 b — b 3 
 
 6. Divide 1 — by 1 + — S- — 
 
 a; 2 + a 2 J a; 2 -a 2 
 
 7. Solve ar + y + z = 90 
 
 2a; + 40 = 3y + 20 
 
 2* + 40 .— 4* + 10 
 
8 EXAMINATION MANUAL. 
 
 15. 
 
 1. Multiply a 3 - 2a 2 b + 3ab 2 + U 3 by a 2 - 2ab -3b\ 
 
 2. Divide x 4 + (2b 2 - a 2 ) x 2 -f b* by x 2 + ax + 6 2 . 
 
 3. Find the square root of — + 8 + — 
 
 6 2 a 2 
 
 4. Solve (3a;-5)(6^-7)-2(3^-5) 2 = 7. 
 
 5. Solve 3x — y + z = l7 
 
 5x + 3y~2z = l0 
 7^ + 4y — 5z = 3 
 
 6. Kesolve into factors x 2 — y 2 -f z 2 — a 2 — 2xz -f 2 ay. 
 
 7. Simplify 
 
 1 2,1 
 
 + 
 
 (2-ra)(3-m) (m-l)(m-3) (m-l)(m-2) 
 
 16* 
 
 1. A body moves a feet toward the north and then b feet 
 
 towards the south. How many feet is the body, at 
 the end of the motion, from the starting point ? On 
 which side of the starting point is the body (i.) if 
 a > 6, (ii.) if a < b ? 
 
 2. Reduce to the simplest form 
 
 l-{l-(l-4:x)\+{2x-(S-5x)\-\2-(-4:+5x)\. 
 
 3. Divide a 2 + ab + 2ac-2b 2 + 7bc-3c 2 by a- b + Sc. 
 
 4. Resolve into elementary factors a 6 — 9 a 4 b 6 . 
 
 5. Find by inspection the H.C.F. of 
 
 4 O 3 + a 3 ) and 6 (x 2 - 2ax - 3a 2 ). 
 
 6. Simplify £±y___2g_ + s*y-< 
 
 y %-\-y x 2 y — y 3 
 
 7. Solve T \(2x~S)-^(3x-2) = }(4:x-S)-S^. 
 
 8. A and B have together $8, A and C have $10, B and C 
 
 have $12. What have they each ? 
 
ALGEBRA. 9 
 
 17. 
 
 1. Resolve into four factors (a 2 — b 2 ~ c 2 ) 2 — 4 b 2 &. 
 
 2. Find the H.C.F. of 4:X*-xy 2 and 4^ 3 - 10x 2 y + 4:xy 2 . 
 
 3. Simplify Sa-(a~b-c)-2\a + c-2(b-c)\. 
 
 4. Divide x 8 + xUf + y* hy x* + x 2 y 2 + y\ 
 
 5. Solve 2^-^=7 + ^2* - 
 
 4 5 
 
 4y-f 
 
 a? -2 = 26J _2 2 M 1 l 
 
 3 2 
 
 6. Divide 91 into two such parts that the quotient of the 
 
 greater part divided by the difference between the 
 parts may be 7. 
 
 7. Extract the square root of 16 x* — 32 # 3 -f- 24 # 2 — 8x -f- 1. 
 
 18. 
 
 1. From ax 2 -f bxy -f cy 2 
 
 take (b — c)x 2 -\-(c — a) xy-\-(a — b) y*. 
 
 2. Multiply x 2 — (a — b) x — ab by x 2 + (a — 5) x — a&. 
 
 3. Resolve into the simplest factors 
 
 32# 5 + 2/ 5 ; hx 2 + %xy-2ly 2 ; 6a 2 x 2 -7ax-S. 
 
 4. Simplify 3a-52a-(3a-Z0 2 S+3aJ25-3a--^-l. 
 
 I 3a J 
 
 5. Find the L.C.M. of 3(1 -a; 2 ), 6(1 -a;) 2 , 5(a:+a; 2 ). 
 
 6. Find the H.C.F. of 7a; 2 - 12*; + 5 and 2x 3 +x 2 ^8x-\-5. 
 
 7. Solve 2>x-^{x-n) = 9~\(px-l). 
 
 8. If A can do a piece of work in 8 days, and A and B 
 
 together can do it in 6 days, how long would B 
 take to do it alone? 
 
10 EXAMINATION MANUAL. 
 
 19. 
 
 1. From ax m+n + (a -f b) x m + 2 bx m ~ n 
 
 take bx m + (a + b) x m ~ n — ax™- 271 . 
 
 2. Find the coefficient of x* in the product of 
 
 a* - bx 2 - 3 abx + a 2 and 2s; 2 - 3 ax - b 2 . 
 
 3. Reduce to lowest terms a ' 2+ h *~ c * + 2ab . 
 
 a 2 -b 2 -c 2 +2bc 
 
 4. Simplify ^+ * <*±£ 
 
 5. Find the H.O.F. of 
 
 7a 2 -23a£ + 65 2 and5a 3 -18a 2 £ + llaZ> 2 -66 3 . 
 
 6. Reduce to a single fraction a — b -) 
 
 7. Solve _?£---- 3 = —^~ 
 
 4a 
 
 26 2a-6 
 
 8. A square floor would contain 17 square yards more if 
 each side were 1 yard longer. Find its area. 
 
 20. 
 
 1. Divide 2a Sn - 6a 2n b n + 6a n b 2n ~ 2b 3n by a n -b n . 
 
 2. Find the H.O.F. of 20x 5 +x s -x and 25^+5.^- x 2 -x. 
 
 3. Simplify 2o^+8a« x 4tts»-3a'a» x a 2 
 
 4:x 2 -Sax a 2 x 2 -ct 2* 2 +3a# 
 4. Resolve into factors 256 x 8 — 1 and x 2 — 7 x — 18. 
 
 5. Simplify 
 
 f * | y V ( x y \ 
 
 \x + y x-yj \x-y x + y) 
 
 6. Solve (a + x)(b + x)~a(b + 6) = ?^ + x 2 . 
 
 b 
 
 7. A laborer was engaged for 36 days on condition that for 
 
 every day he worked he was to receive $3, and 
 for every day he was idle to forfeit $2. At the end 
 of the time he received $68. How many days did 
 he work ? 
 
ALGEBRA. 11 
 
 21. 
 
 Resolve into their simplest factors 
 
 a 3 - a 2 x - ax 2 + x z ; x* + 3 x 2 y - 4 :n/ 2 - 12 y\ 
 a;-f-l a — 1 1 — 3a; 
 
 2. Simplify 
 
 2a?-l 2a; +1 a;(l — 2ar) 
 
 Simplify ../„ , n + „/., , 2 x + 
 
 «0 + 6) a(a 2 -b 2 ) b(a-b) 
 4 Solve 8a- + 5 7*-3 _ 16*;+ 15 2£ 
 14 6* + 2 28 7 
 
 5. Find the L.C.M. of 
 
 Sx^-Ux+6, 4:x 2 + ±x-3, and ±x 2 + 2x-6. 
 
 6. Add l-{2-(3-a?)j, 3#— (4-5*), 4-r-(-5 + 6ar). 
 
 7. Twenty-four persons subscribed the cost of a new boat, 
 
 but four of the subscribers proving defaulters, each 
 of the others had to pay $ 2 more than his due share. 
 Find the cost of the boat. 
 
 22. 
 
 1. Resolve into their simplest factors 
 
 a V-fiy 4 ; 27bx i -b i xy s ; 20ax 2 +l8a 2 x-18a\ 
 
 2. Simplify V XJ , ^ XJ - 
 
 3. Find the L.C.M. of x 2 -±y 2 and x 2 + xy-6y 2 . 
 
 4. Solve 14(7ar+5)-[84a;-(9a?-21)] = 210. 
 
 5. Solve **±±- 4^ + 2 = 2 £lL 5 
 
 3 Sx-9 3 
 
 6. A and B can do a piece of work in 8 days, working to- 
 
 gether ; A working alone can do it in 12 days. In 
 how many days can B do it, working alone ? 
 
 7. State the distinction between an algebraic identity and 
 
 an algebraic equation, and give an example of each. 
 
12 EXAMINATION MANUAL. 
 
 23. 
 
 1. Ifa = 0, b = 2, c = -S, d = 4, find the value of 
 
 4:(ad-bc)-{(a-b)-(c-d)\\ 
 
 2. Add $a + $b — \e + Jrf-andf « — f & + c — fi 
 
 3. Divide a; 6 - 2 a 3 ^ 3 + a 6 by x 1 - 2 a^ + a\ 
 
 X* ~\~ Q? 
 
 4. Eeduce to the lowest terms 
 
 x 2j r2ax-\- a 2 
 
 5. Solve 4a? + 9y = 121 # 
 
 6^-32/= 7j" 
 
 6. Extract the square root of x 4, — 2x z -| — — + — • 
 
 7. A person walking at the rate of 4f miles an hour starts 
 
 li hours after another person who walks only 4 
 miles an hour. When and where will he overtake 
 him? 
 
 24. 
 
 1. If a = 1, b = 3, c = 5, d = 7, find the value of 
 
 a- 25- {3c-^~[3a-(56-c-8c?)]-25}. 
 
 2. Simplify (*-8) — (8-*)(l.-a?)-(a?-8)(5-2a?). 
 
 3. From (&+6) 2 -2&r take \(a-\-b)(a-x)-(a—b){b—x)\. 
 
 4. Find the H.C.F. of 
 
 I2xy(x , y-2x i y* + xtf) 1 20(x*-2x h y i + x 1 y«), and 
 &y@-xf). 
 
 5 . Solve E±^_ g -y- 2 =i 
 
 12 5 
 
 9 y _5^ = 10. • 
 
 6. Find a number which is as much greater than 63 as its 
 
 half is less than 93. 
 
ALGEBRA. 
 
 13 
 
 25. 
 
 1. Resolve into the simplest factors x 12 — y 12 . 
 
 2. Simplify a 2 - (b 2 - c 2 ) -\b 2 - (c 2 - a 2 ) \ + \c 2 - (b 2 -a 2 )}. 
 
 3. Find the H.C.F. of 8aV + 12aV and 12a; 5 + 18atf*. 
 
 4. Divide a 2 + ah + ac + bc , a_+i 
 
 a 2 — ae + ac? — cd a-\-d 
 
 5. Solve 2ax + b = 3cx-}-4:a. 
 
 Solve - + - = 7 
 a; y 
 
 The sum of the two digits of a certain number is six 
 times their difference, and the number itself exceeds 
 six times their sum by 3. Find the number. 
 
 26. 
 
 1. Solve 
 
 2^3 
 
 2 + *=4 
 2^8 
 
 2 X 1 
 
 16 l2~6 
 
 2. A person swimming in a stream which runs \\ miles an 
 
 hour finds that it takes him four times as long to 
 swim a mile up the stream as it does to swim the 
 same distance down. Find his rate of swimming in 
 still water. 
 
 3. Find the H.C.F. of 3z 4 -4a; 3 +l and 4^-5^-.^+^+ 1. 
 
 4. Solve 
 
 x — 7 . x x-\- 7 
 
 2 ^9" 3 
 
 5. Divide a 2 b + (a — b) 2 x 
 
 6. Simplify £±£— 
 
 iCbsr- 
 
 x 
 
 x 3 by b -J- x. 
 
 y z + y 
 
14 EXAMINATION MANUAL. 
 
 27. 
 
 1. Subtract 
 
 (a+y> 3 -( 2 a 2 -f)x 2 ~2 arf from aty-(2 x 2 -a)y 2 + ax\ 
 and arrange your answer in descending powers of y. 
 
 2. Divide a 2 - ±ab + U 2 by a2 ~ 2ab . 
 
 a 2 b 2 a + 26 
 
 3. Solve = a — b. 
 
 x x 
 
 4. Simplify X 1±3L + ±ZlL 
 
 x — y (x-y) 1 
 
 5. Find two consecutive numbers, the difference of the 
 
 squares of which is equal to 51. 
 
 6. Two trains pass a station at an interval of 4 hours, trav- 
 
 elling respectively at the rates of 11 J and 17 J miles 
 an hour. How far will the slower train have run 
 before it is overtaken by the other ? 
 
 7. Solve |* + $y = 16; \x-\y = 2. 
 
 28. 
 
 1. Divide x 3 by 1 — x 2 to three terms. What is the differ- 
 
 ence between the answer thus obtained and the true 
 answer ? 
 
 2. Arrange according to powers of x 
 
 3(a 2 -2x)(a-x*) + 2x(a-6x 2 ). 
 
 3. Find the H.C.F. of 
 
 6a 3 - 6a 2 y + 2ay 2 - 2tf and 12a 2 - Way + 3y 2 . 
 
 4. How do you find the L.C.M. of two expressions when 
 
 their factors cannot be determined by inspection ? 
 
 r- a . ,-.- a? + a 3 sx x z 4:a 2 x 2 -\~2ax 3 
 
 5. Simplify - — ! X ■ — - -= — - J — - — 
 
 Aax — x l x z — ax-\-a l 4a -r 
 
 6. A, B, and can reap a field in 30 hours. A can do 
 
 half as much again as B, and B two-thirds as much 
 again as 0. How many hours would each require 
 to do the work alone ? 
 
 7. Solve x-}(y-2) = 5; iy - $(s + 10) = 3. 
 
ALGEBRA. 15 
 
 29. 
 
 1. Divide a L±I + a -^ by 2±1 _£=£. 
 <z — a; a-\-x a — x a-\-x 
 
 1 , x 
 
 2. Simplify l ~\ X l X * 
 
 1-x 2 l + x 
 
 3. Solve 5^ = 43-7y; ll# + 9y = 69. 
 
 4. Find the H.O.F. of %o*-\0a*+\ba?-&c and 2x 6 -Ax 4 +2x 2 . 
 
 5. Find all the expressions whose L.C.M. is x 3 — 4xy 2 
 
 6. A number consists of two digits. The sum of the digits 
 
 is 7, and if 27 be subtracted from the number the or- 
 der of the digits will be reversed. Find the number. 
 
 3 3 
 
 7. Find the value of j ■ when x — £. 
 
 l-\-x 1—x 
 
 30. 
 
 1. A is x years old ; B is 20 years younger. Express in 
 
 algebraic language the age of B ; the sum of their 
 ages ; the age of A five years ago ; the age of B 
 five years hence. 
 
 2. Simplify (2a-b) 2 +2b(a+b)--Za 2 -(a-b) 2 +(a+b)(a-b); 
 
 3. Multiply a-- by -+-. 
 
 4l Simplify ct(x 2 + c 2 )-2acx ^ 
 F y b(x 2 + c 2 )-2bcx 
 
 5. Find the H.C.F. of 
 
 3a 3 -7a 2 6-33a&-7a6 2 +ll£ 2 -{-36 3 and 3a 2 -l0ab+3b 2 .. 
 
 6. Solve 5x-\-2y— z = 2z — x — y — y + z — 2x — 6. 
 
 7. If a father was four times as old as his son seven years 
 
 ago, and will be twice as old as his son in seven 
 years more, what is the present age of each ? 
 
16 EXAMINATION MANUAL. 
 
 31. 
 
 1. What is meant by a term, an index, a factor? Give 
 
 examples. 
 
 2. Solve ™ + ^ = «; 
 x y 
 
 3. Solve 2x- 4y + 
 7x+ 3y- 
 9o; + 10y-llz = 
 
 4. Find the H.C.F. of 
 
 2a; 2 + 3xy + 6a; + 9y and 3^ - 2a;y -f 9a: — 6y. 
 
 5. Simplify 2 **-/ a ^E±£ 
 
 a; 2 — 2a;y-f-y 2 x 6 — ^ 
 
 6. Find a fraction such that if 5 be added to the numerator 
 
 the fraction equals 1, but if 3 be taken from the de- 
 nominator the fraction equals i. 
 
 7. Arrange according to descending powers of x 
 
 x 2 — ax — c 2 x 2 — bx-\- bx 3 — ex 2 -f- a 2 x z — x 2 — ex. 
 
 32. 
 
 1. What is a common multiple of two or more quantities? 
 
 What is meant by their least common multiple ? 
 
 2. Find the H.C.F. of 
 
 4a; 4 +22a; 3 -26a; 2 -198a;-90 and 4o; 3 -14o; 2 -92a;-42. 
 
 rnn 
 
 (m+nf 
 
 a 2 c 
 
 3. Bwn%-H_ + /_»_y. 
 
 m + n \m + nj 
 
 4, Solve (a + x)(b + x)-a(b + c)=~+x r . 
 
 b 
 
 5l Solve ^ + | = 5; §-4y==i 
 
 6. Solve a- + y = 16; 2z-2a; = 8; 2y + 3z = 51. 
 
 7. At an election there were two candidates, and 1296 per- 
 
 sons voted ; the successful candidate had a majority 
 of 120. How many votes were cast for each ? 
 
 8. What is the rule for transposing terms in equations? 
 
 What is the reason for the rule ? 
 
ALGEBRA. 17 
 
 33. 
 
 a-\-x . a—x 
 
 1. Solve ^ + ^ = a; __^_. 
 
 a — b a-\-b a 1 — b 2 
 
 2. Solve x + y = a ; bx — cy = d. 
 
 3. Solve 5*-%-f2s=*19; 4<*f5y-3*=81; 3a+7y-4z=31. 
 
 4. Simplify a + h a ~ b * ab 
 
 a — b a-j-b b 2 —a 2 
 
 5. Eeduce to lowest terms 21 s*+14sy t 
 
 18a^+12ay 
 
 6. Divide x«- 2a 3 x 3 + a 6 by x 2 - 2 ax + a 2 . 
 
 7. Simplify 2\a{a~b)-{a-{-b) 2 \{a 2 -2b 2 \. 
 
 8. The difference between two numbers is one-fourth of 
 
 their sum, and the half of the greater diminished by 
 7 is equal to one-fifth of the less increased by 2Jr . 
 Find the numbers. 
 
 34. 
 
 1. Define factor, common factor, highest common factor. 
 
 2. Find the H.C.F. and the L.C.M. of 
 
 (2 a 2 bx + 2 aba*)* and (6 a 2 b 2 - 6 J 2 x 2 ) 3 . 
 
 3. Multiply a 6 + a 5 - a - 1 by 1 - a + a 2 - a 3 + a\ 
 
 4. Simplify - 
 
 1-M4 
 
 1 + * + 
 
 1+37 
 
 5. Solve "« +JL- S *+* 
 
 era; or# 
 
 6. Solve x + y = 5 (a? — y) = 10. 
 
 7. A, B, C, and D have $290 between them. A has twice 
 
 as much as C, and B has three times as much as D. 
 Also, C and D together have $50 less than A. Find 
 how much each has. 
 
 8. Expand (2x 2 yzJ-, (-7a 8 6V) s ; (-3a5V)*. 
 
18 EXAMINATION MANUAL. 
 
 35. 
 
 1, Simplify tx a + b f h + e + 
 
 (b-c){a-c) (a-b)(c-a) (b-a)(b-c) 
 
 2. Simplify q'+3q'6*3qy + y a 2 -2ab + b 2 , 
 
 P * a 3 -3a 2 6 + 3a& 2 -6 3 a 2 + 2a6 + 6 2 
 
 3. Solve i*+iy=12-iz; £y+t* = 8+£z; ^+^ = 10. 
 
 4. Find the L.C.M. of 9a 2 -81, 12a 2 -36a, and 16a 2 +48a. 
 
 5. Extract the square root of x i -8x*-2e>x 2 +W8x+4Al. 
 
 6. Two persons start at noon from towns 60 miles apart. 
 
 One walks at the rate of 4 miles an hour, but stops 
 2 £ hours on the way. The other walks at the rate 
 of 3 miles' an hour, without stopping. When and 
 where will they meet? 
 
 7. How many different equations must be given when the 
 
 values of two or more unknown quantities are re- 
 quired ? In what sense must the equations differ ? 
 
 36. 
 
 3/— 
 
 3 I- 
 
 1. Find the value of x-(Vx±l+2)- x ^ x when x = S. 
 
 Vx— 4 
 
 2. Divide a 2 x*— c 2 y*— bx 2 y(2ax— by) by (ax — by) x — cy 1 . 
 
 3. Find the H.C.F.'of 
 
 Sax 2 +2mj 2 -Sbx 2 -2by 2 and 3az s -2ay 2 -3bx 2 +2by 2 . 
 
 4. Find the L.C.M. of a 2 +x 2 , a 2 -x 2 , a?+x 3 , and a z -x\ 
 
 5. g^phfy — x a2 __ 2x _ a{a _ 2) X (g , + ^ (g+g) - 
 
 6. A is 5 years more than twice as old as B, and is also 25 
 
 years older than B. Find their present ages. 
 
 7. "What are simultaneous equations ? Give an example. 
 
 8. Expand (« + |J. 
 
ALGEBRA. 19 
 
 37. 
 
 ., q. yn be | _ae ab 
 
 ' impiy (a-b)(a-cy (b-a)(b-c) (b-e)(c-a) 
 
 'a b\( a + b a + b' 
 
 J) aj\ a 
 2. Simplify 7^ b , n ' ± 
 
 b 2 a 2 J \a b 
 
 3. solve i+M-g; M+5-& hh s =^- 
 
 x^y^z 20 y^z^x 60 z^ x^ y 30 
 "4. If a rectangular court had its length diminished by 7 
 feet, and its width increased by 5 feet, it would be- 
 come an exact square, enclosing the same area. Find 
 the dimensions of the court. 
 
 5. Extract the square root of x^ — x 3 -j- — + -• 
 
 6. Expand g-fj 
 
 2 ' 4 
 
 38. 
 
 1. Find the product of — — — -- and — ■— -\ f- -f- -£r« 
 
 a o a 1 ab b l 
 
 2. Find the H.C.F. of x*+e>x 2 +llx+6 and ar , -|-9rc 2 +27a?+27. 
 
 q «• V o 3:z 2 + l hx-2 , 2x + b 
 
 3. Sl mphf y _____ + _^. 
 
 4. Extract the square root of x* + — - -j — f- - -f- -• 
 
 3 9 3 4 
 
 £2 
 
 5. Solve ax x ~ a -f- 5. 
 
 « 
 
 6. A and B have each an annual income of $4000. A 
 
 spends 'every year $400 more than B. At the end 
 of 4 years their joint savings amount to $4000. 
 What does each spend annually ? 
 / 3V 
 
 7. Expand [2a — 
 
 8. Explain how to clear an equation of fractions, and the 
 reason for the method. 
 
20 EXAMINATION MANUAL. 
 
 1. Simplify 
 
 39. 
 
 1 , 1 
 
 a(a—b) (a—c) b(a—b) (c—b) c(c—a) (b—c) 
 SI # — 3 x— 6 _ * + 12 x — 6 . x 
 0VG 7 6- ~ 12 3 ^8 
 
 9h 
 
 3. Solve 7 + | = 8; - + ^ = 6. 
 
 4 2 2 6 
 
 4. Extract the square root of — 2 + 
 
 5. Simplify 1- {l-(l-4x)\+{2z-@-5x) }-{2-(-4+5s){/ 
 
 6. Find the L.O.M. of a'-l, (aJ - 5) <?, and (ac + c) b. 
 
 7. A and B can mow 225 square rods in a day ; B and C 
 
 can mow 223 square rods in a day ; A and C can 
 mow 230 square rods in a day. How many square 
 rods can each mow by himself? 
 
 40. 
 
 1. Simplify M\a(a-b)(Za-b)-(a-by\. 
 
 2. Find the H.C.F. of 
 
 12 a 2 + 13 ab + 3 b 2 and 6a 2 + 23 ab + lb 2 . 
 
 3. Find the L.C.M. of 
 
 4(1 -x)\ 8(1-*), 8(1 + *), 4(1 + * 2 ). 
 
 4. Find the cube root of 
 
 l-6* + 21* 2 -44* 3 +63* 4 -54* 5 +27* 6 . 
 
 6. Solve |-1W| + 1 = |; x + y + z^-ll. 
 
 7. At what time between two and three o'clock are the 
 
 hour and minute hands of a clock in a straight line 
 and pointing in opposite directions ? 
 
ALGEBRA. 21 
 
 41. 
 
 1. Find the square root of 1 — xy — *£■ x 2 y 2 -f 2x*y* -f 4or 4 y 4 . 
 
 2. Simplify K 2 - i°0 - a _ fr( 6a + 4 ) + i a + 2 
 
 r * 2-i(2 + 2o) l + 2a 
 
 3. Solve ^ + y + 2 = 6; # -f- y — z = 4 ; a; — y -}- z = 2. 
 
 4. Multiply f -| byf -| + 1 
 
 5. A boat is rowed 1 mile in 5-^ minutes ; another is rowed 
 
 ' 1 mile in 5} minutes. If they start at the same time 
 from opposite ends of a 4-mile course, at what dis- 
 tance from each starting-point will they me*et ? 
 
 6. 
 
 Expand (f + f J. 
 
 7. Solve o 2 / + ® + ^ x ~ a N = 24. 
 
 1. Simplify 
 
 42. 
 3 , x x—3 
 
 2. Find the H.C.F. of Ss?-3a?y+xy i -y i and itf-xy-Sy 2 . 
 
 , a& 2 & 3 
 
 3. Find the cube root of a 3 — a 2 b A 
 
 3 27 
 
 4. Solve (2 + x)(8 + x)-22 = % + x\ 
 
 5. Solve the simultaneous equations 
 
 3* = 2<j/ + 2)--|; 2y=3(*~*)-Hh 2=^+3/ 
 
 -t 
 
 6. Find the value of 2 (1 + Va) - ^""^ 
 
 a 
 
 3 Va + vV 
 
 when a? = lo a. 
 
 7. 36 octavo volumes exactly fill a box whose length and. 
 breadth are 1-J- feet and 11-J- inches, respectively. 
 The dimensions of each volume are 9 inches by 5| 
 inches by 1| inches. Find the depth of the box. 
 
22 EXAMINATION MANUAL. 
 
 43. 
 
 1. Find the L.C.M. of 9a*+5Bx*-9x-rl8 and fcM-llar+3(X 
 
 2. Find the square root of x* — 4 x z -f re 8 + 6 # + f . 
 
 3. Solve Sx+4y-6z==Z2] 4^-5y+32=18; 5;r-3y-42=2. 
 
 4. Divide 5^ + ^-^ 7 «V + ^+^by2^-^-^ 
 
 5. Simplify 
 
 6 —x 
 
 x — 1,3/ — 1,2- 1 
 
 6. Simplify §32 ~ "T" ^~ 
 
 yz-\-xz — xy 1 1 1 
 
 # y z 
 
 7. Eeduce to lowest terms — — 
 
 6;r — x—1 
 
 44. 
 
 1. Simplify — JL - + 
 
 (2a; -l) 2 (2a7+l)(l-2s) 
 
 o tv • i 4a 2 — 4a# , 
 2. Divide -— — by 
 
 (a + xf 4aa:4-4a; 2 
 
 o \iji * a 2 b 2 . c 2 b 2 c 2 , a 2 c 2 a 2 , b 2 
 
 3. Add together \-—, — . 
 
 8 2 3 4 2 3 4 2 3 4 
 
 4. Find the L.C.M. of 
 
 x z -f x 2 y + a;?/ 2 + y 3 and # 3 — tf 2 ?/ + a:?/ 2 — y 3 . 
 
 5. Extract the square root of a 2 +4 5 2 +c 2 +26zc+4 a6-f-4 be. 
 
 6. Solve - = 
 
 x — b x—a 
 
 7. Divide the number 90 into four parts so that the first 
 
 part increased by 2, the second diminished by 2, the 
 third multiplied by 2, and the fourth divided by 2, 
 may be all equal. 
 
 8. What powers of a quantity are the same whether the 
 
 quantity be positive or negative ? 
 
ALGEBRA. 23 
 
 45. 
 
 1. Simplify I 4- 1 
 
 r - J l + 3a + 2a 2 ' l-a-2a 2 
 
 2. Multiply 1 — x — x 2 by x 2 — a? — 1. 
 
 3. Find the H.C.F. of 
 
 x H + bx 2 -{-10x-{-8 and x 5 + 2x i -x-2. 
 
 4. Find the L.C.M. of 
 
 l-x 2 , \-2x + x 2 , l- x + x 2 -x\ l-x-tf + x 3 . 
 
 5. Reduce to lowest terms " x ~ X V~V . 
 
 y 2 -\-xy-2x 2 
 
 6. Expand (<*-£)' 
 
 7. Solve i + l_i = 6; 1 + 1-1 = 4- 1+1-1 = 2 
 
 y z x z x y ' x y z 
 
 46. 
 
 1. Reduce to a simple fraction x 
 
 2. Reduce to its lowest terms ^ % x * + 7x — §1 
 
 2a* + 19 a* + 35 
 
 3. Find the square root of J~\ , , 2£x_lL 
 
 16# 2 y 2 
 
 4. Solve 3*-a = ^±^-^-^. 
 
 3 a 
 
 5. The united ages of a father and son make 75 years. 
 
 The father was 25 years old when the son was born. 
 "What is the age of each ? 
 
 6. Solve bx -Zy -28 = 0) 
 
 3a;- 2y- 13 = /' 
 
 7. Subtract ax from 2«* + «'*» + 2* , 
 
 2-\-ax-2x 2 
 
24 EXAMINATION MANUAL. 
 
 47. 
 
 1. Multiply a*+a 5 -a-l by 1- a + a 2 - a 3 +a*. 
 
 2. Find the H.O.F. of 
 
 4^ 4 +22.t 3 -26^ 2 -198^-90 and 4z 3 -14:r 2 -92tf-42. 
 
 3. Find the L.O.M. of 
 
 x 2 -l, x 3 ~l, x*+l, (x+1) 2 , (x-1) 2 . 
 
 4. Solve x + 3y + 2z = 11 
 
 2x+ y+3z = 14 
 Bx + 2y + z = ll 
 
 5. What sum is that which is as much greater than $2 as 
 
 its half is less than $ 3 ? 
 
 6. A number consisting of two digits when divided by the 
 
 sum of the digits gives 7 for the quotient ; and the 
 number formed by interchanging the digits, when 
 divided by 6, gives 4 for the quotient. Find the 
 number. 
 
 7. What powers of a quantity are always positive ? What 
 
 powers of a negative quantity are always negative ? 
 
 1. Simplify 
 
 48. 
 
 x+1 x—1 1 
 
 Simplify 
 
 / x 2 -y 2 . ff 2 +3aA y ,( x 5 -9x s . x^-Sx 2 ^ 
 \x 2 —2xy-\-y i ' x-3 J \ x+y x—y J ' 
 
 S0lVG 2^i-l + 3^+L+l" 
 
 2x 2 —3x- 
 Reduce to lowest terms 
 
 Simplify 
 
 5x*-7x-e> 
 ^ + 5a7 + 4 . a?+2x+\ 
 # 2 + 7z + 12 ' # 2 + 3# + 2' 
 
 6 . simplify 7 2 + f + m ^+ m ; What wo » ld the re " 
 bm 2 + on 2 + m 2 n 2 + nr 
 
 suit be if a = — m 2 ? 
 
ALGEBRA. 25 
 
 49. 
 
 t • « -, a + h i fa — b\ be - a 
 
 L Solve —^--|c —— ] = -f— — . 
 
 26 \ ox J {a-\-o)x a + b 
 
 2. Solve «# + £?/ = <?; a'# + £'y = c'. 
 
 3. Solve #+2/+z = 5; 3#-5y+7z=75; 9#-llz+10 = 0. 
 
 * a i 1,1 1,1 1,1 
 
 a; y y z x z 
 
 5. Solve 
 
 3 + 5 + 7 ' 4 + 6 + 3 ''2 5 + 40 *' 
 
 50. 
 
 1. A person walking at the rate of 4f miles per hour starts 
 
 H hours after another person who walks only 4 
 miles an hour. When and where will he overtake 
 him? 
 
 2. A number consists of three digits, the right-hand digit 
 
 being 0. If the left-hand and the middle digits be 
 interchanged, the number will be diminished by 180. 
 If the left-hand digit be halved, and the middle and 
 right-hand digits be interchanged, the number will 
 be diminished by 454. Find the number. 
 
 3. A had twice as much money as B ; but when B had re- 
 
 ceived $5 from A, he had three times as much as A. 
 How much had they each at first ? 
 
 4. A tank of water may be filled by three pipes. By the 
 
 first pipe it can be filled in 4 hours, by the second 
 in 10 hours, by the third in 15 hours. In what time 
 will it be filled if all three pipes are opened ? 
 
 What will be the solution of this question if the 
 general symbols m, n, p are used in place of the 
 numbers 4, 10, 15, respectively? 
 
 5. What are simultaneous equations ? Give an example. 
 
26 EXAMINATION MANUAL. 
 
 51. 
 
 1. Bracket like powers of x in the expression 
 
 ax 3 — bx 2 — ex — bx 3 + ex 2 — dx + ex 3 — dx 2 — ex. 
 
 2. Divide a 3 - 6 3 + c 3 + 3a£c by a- b + <?. 
 
 3. Simplify g- (g=Q* + ff -y?f 
 
 4. Divide 150 into two parts such that if one is divided by 
 
 23, and the other by 27, the sum of the quotients 
 is 6. 
 
 5l Solve gg- 15 ** + 8 = 2a»-3. 
 4 6 
 
 6. Solve 2x = 4-f-. 
 
 52. 
 
 1. Subtract (a—b)x—(b — c)y from (a +5) a? +(b-\-c)y. 
 
 2. Find the difference in value between the arithmetical 
 
 expression 57 and the algebraical expression ab, 
 when a = 5 and b = l. 
 
 3. Find by inspection the H.C.F. of 
 
 9(aV-4) and 120V + 4 a* + 4). 
 
 4. Simplify (4 «-§)-*. 
 
 5. Solve £±|-^lf=|. 
 
 x-1 2x 3 
 
 6. There is a rectangular field whose length exceeds its 
 
 breadth by 16 yards. The field contains 960 square 
 yards. Find its dimensions. 
 
ALGEBRA. 27 
 
 53. 
 
 1. Find the L.C.M. of 
 
 4 (a 3 - ab 2 ), 12 (ab 2 + b 5 ), and 8 (a" - a 2 b). 
 
 % Simplify 2^ + 2^ + ^- 
 
 3. Simplify (64<? 16 )-i 
 
 4. Divide 16 a; — y 2 by 2#* — yi 
 
 5. Solve 1--*±A = £zl6 
 
 2z+l a; — 2 
 
 6. Find two numbers in the ratio of 2\ to 2, such that 
 
 when diminished each by 5 they shall be in the 
 ratio of 1 J to 1. 
 
 54. 
 
 1. Solve x + 2y=l 
 
 y + 2z=2 
 
 3x + 2y = z-l 
 
 2. The sum of $330 is laid out in two investments, on one 
 
 of which 15 per cent is gained, and on the other 8 
 per cent is lost. The total amount returned from 
 the investments is $345. Find each investment. 
 
 3. Simplify [(<*-»&*)*{-* 
 
 4. Divide x~ l — y~ l by x~\ - y~\. 
 
 5. Reduce to an entire surd the expression ■£-(£)-*. 
 2 1 3 
 
 6. Solve 
 
 1 x-{-3 8" 
 
28 EXAMINATION MANUAL. 
 
 55. 
 
 1. Find the L.C.M. of 
 
 6(^y + a^), 9(V-:n/ 2 ), and 4^ + ay*). 
 
 x 2 , x 3 
 
 2. Simplify -^ ^— - 
 
 1-s (I-*) 2 ' (l-*) 3 
 
 3. Arrange in order of magnitude 2-^22, 3^7, 4V2. 
 
 4. Find the cube root of 
 
 x«- Sax" + 5aV - 3a 5 a; - a 6 . 
 
 5. Simplify 3^432. 
 
 6. Solve _^L_3*-2 
 
 a; + 4 2^-3 
 
 56. 
 
 1. Show that V20, V45, VJ are similar surds. 
 
 2. Simplify 2y/a~x xVSaFb X^JWx. 
 
 3. Simplify -^128+^/686 +^16. 
 
 4. Simplify ^ 2 -8* + 6 x **-0* + 20 x **- 7 *. 
 
 5. Solve i^ 2 + H^ = 3i|. 
 
 6. If each side of a square plot were enlarged by 10 yards, 
 
 it would contain 1£ acres more. Find the dimen- 
 sions of the plot. (1 acre = 4840 sq. yds.) 
 
ALGEBKA. 29 
 
 57. 
 
 1. Simplify \ar%y{xff^(ar 1 y)'^\\ 
 
 2. Simplify VT28 - 2 V50 + V72 - Vl8. 
 
 3. By selling a cow for $ 24 I lose as much per cent as the 
 
 cow cost me. What was the cost of the cow ? 
 
 1 Solve m + it=-\ 
 
 5. Solve x + y + z = 5~\ 
 
 x+y=z-7 >• 
 x — 3 =y-\-z) 
 
 6. A person has travelled 3036 miles. He has gone 7 miles 
 
 by water to 4 on foot, and 5 by water to 2 on horse- 
 back. How many miles has he travelled each way ? 
 
 58. 
 
 , c -, 27 — x , Sx — 4 5x — 2 
 
 1. Solve - T - + — b — = 2. 
 
 2. Find by inspection the L.C.M. of 
 
 a 2 -l, a 2 + 2a-3, and a?-7a 2 + 6a. 
 
 3. A steamer takes two hours and forty minutes less time 
 
 to travel from A to B than from B to A. The 
 steamer travels at the rate of 14 miles per hour in 
 still water, and the stream flows at the rate of If 
 miles per hour. Find the distance from A to B. 
 
 4. Which is the greater W% or £a/27 ? 
 
 5. Solve !*_«-*«-- 
 
 2x 2 
 
 6. A and B enter into partnership with a joint capital of 
 $ 3400. A put in his money for 12 months ; B put 
 in his money for 16 months. In closing the business, 
 A's share was $2070 including his profit, and B's 
 share $ 1920. Find the sum put in by each. 
 
30 EXAMINATION MANUAL. 
 
 59. 
 
 1. Solve x -f- ay = b ; ax — by = c. 
 
 2. If a = 4, & = 3, c=l,. i=_7, find the value of 
 
 a 2 + ac + 5 2 V4a5 + 6 2 + ^ c 
 
 a?-ac + b 2 -y/f^b^P^tid a + b + c+d 
 
 3. Find the H.O.F. of 
 
 x' + Sx'-Qx-A and x* -Sx 3 + 6x-4:. 
 
 4. Simplify 1 
 
 x — y x-\-y x 
 
 5. Find the square root of a 2 — 4 ax -f<6 a + 4 # 2 — 12 x -f 9. 
 
 6. When are the hour and minute hands of a clock to- 
 
 gether between 10 and 11 o'clock ? 
 
 7. Solve K*-l) = «* + l)--K*-l) 2 . 
 
 60. 
 
 1. If x = 8, y= 1, find the value of 
 
 ^{ /+ ^S±Zj. 
 
 j£ . <y+y 3 
 
 2. Simplify -_^ + 2/2 . ^ 
 
 3. Find the H.O.F. of 
 
 x s -Sx 2 +lx-21 and 2# 4 + 19 a; 2 -f 35. 
 
 4. Find the square root of {x 2 -\ 2a; ][! + -] + 1. 
 
 5. Solve ^x-i(8-x)-i(5 + x) + ¥- = 0. 
 
 6. Find a fraction such that if 5 be added to the numer- 
 
 ator, it will be equal to 1, but if 3 be taken from the 
 denominator, it will be equal to £. 
 
 7. A boat's crew can row 5 miles an hour in still water. 
 
 If they row 3£ miles down a river and then back, 
 again to the starting-point in 1 hour 40 minutes, 
 what is the velocity of the current? . 
 
ALGEBRA. 31 
 
 61. 
 
 1. Solve 7 a; 2 -11a; = 6. 
 
 2. Solve _f^_^ = - 1 A_.. 
 
 a+b a—b a 2 —b 2 
 
 3. Solve - + ^=2; bx-ay = 0. 
 
 a b 
 
 4. Find the H.C.F. of 
 
 48a; 2 -f 16a; -15 and 24a; 3 - 22a; 2 + 11 x- 5. 
 
 5. Divide a? + b* + & -Sabc by a + b + c. 
 
 6. Extract the square root of x 4, — 2a; 3 + — — - + — 
 
 H 2 2 16 
 
 7. A vessel containing 120 gallons was emptied in 10 min- 
 
 utes by two pipes running successively; the. first 
 pipe discharged 14 gallons in a minute, and the 
 second pipe discharged 9 gallons in a minute. How 
 many minutes did each pipe run ? 
 
 62. 
 
 1. Subtract a 2 -13 ab +10 ac-c 2 from la 2 +12ab-3ac-b 2 , 
 
 and find the value of the remainder, when a = 1, 
 5=2, <? = -3. 
 
 2. Divide x 5 -. by x 
 
 x 5 J x 
 
 3. Simplify ^^^[ ; m2 _j )( 3^ + J) ] 
 
 4. Find the H.C.F. of 
 
 96a; 4 +8a; 3 -2a; and 32a; 3 - 24a; 2 - 8a; + 3. 
 
 5. Extract the square root of 
 
 a 2 -4a& 2 -6ac 3 + 4£ 2 (6 2 + 3c 2 ) + 9c 6 . 
 
 6. Solve 5a; + fy = 52; f a; + 4y - 27. 
 
 7. Solve 5, + ^+g = 10 ^- 7 + 9. 
 
 3a;-2 2s -1 
 
 8. When are the hour and minute hands of a watch at 
 
 right angles between 3 and 4 o'clock ? 
 
32 EXAMINATION MANUAL. 
 
 63. 
 
 1. Solve ^Lii§ + 2^ z 3 = 9 
 
 x — 1 2a; — 2 
 
 2. Solve a; + 2y + 3z=101 
 
 2a; + 3y + 4z = 16 [• 
 4a; + 4y + 5z = 25 J 
 
 3. Solve *f+l -Sx=^=± + 3. 
 
 3a;-2 4a;-3^ 
 
 4. Find the L.C.M. of 
 
 x 3 - 4a; 2 + 9a? - 10 and x 3 + 2a; 2 - 3a; + 20. 
 , b — a 
 
 5. Simplify y. -■ 
 
 1 a(b — a) 
 
 l + ba 
 
 6. A and B have the same income. A saves -J- of his in- 
 
 come, and B spends $50 a year more than A. At 
 end of the 4 years B finds himself $100 in debt. 
 Find their income. 
 
 64. 
 
 1. Add together 
 
 1 x - (1 - x) and 10a; 2 -x -3 
 
 1 , 1 a; + 3-(l-3a;y 4a; 2 +4a; + l' 
 
 1 + x 
 
 2. Divide a; 4 — ±£x 3 + a; 2 + fa; — 2 by fa; — 2. 
 
 3. Find the L.C.M. of a 3 - a 2 b - ab + 6 2 and a 4 - 6 2 . 
 
 4. Extract the square root of a 2 -f- b to three terms. 
 
 5. Solve aa; 2 + 6a;-f c = a(l — #) 2 + &(l — x) — c. 
 
 6. Solve -JL-Jfi-i 
 
 x-\- i a; 4 
 
 7. Two numbers consist of the same two digits, but in an 
 
 inverse order. The sum of the two numbers is 55, 
 and their difference is 27. Find the numbers. 
 
ALGEBRA. 33 
 
 65. 
 
 1. Simplify - 
 
 x-l + - 
 
 4 — a; 
 
 2. Find the L.C.M. of a; 2 - a;, x 2 -l, x*-2x 2 , and x*-4:X. 
 
 o a -\ ' 10a; — 7 9a: — 1 
 
 3. Solve - = 2. 
 
 2a;— 1 3 a; + 1 
 
 4. Solve 2a; + 3y + 4z = 34; 3a;— 2y + 2z=13; y— 2=1. 
 
 5. Simplify 10a?- {3a;- [- 4a;- (- 2a; + 3a;)]J. 
 
 6. Solve (a; + «)(> 2 -aa; + 6 2 ) = a3 +£a; 2 - 
 
 66. 
 
 1. The sum of two numbers is 1000, and their difference is 
 ■^ of the less ; find them. 
 
 „ a . ,., 4a; 4 — 1 K .Sx 3 — y 
 
 3. Find the H.C.F. of 20a;*-a; 2 -l and 25a;*-5a; 3 -a;-l. 
 
 2 1 
 
 4. Extract the square root of m 2 -f-2ra— 1 — — H =• 
 
 5. Solve ***2LzJL=** 
 
 6 . Solve . + |„£. 
 
 7. When a term is transposed from one side of an equation 
 
 to the other side, the sign of the term must be 
 changed. Why? 
 
34 
 
 EXAMINATION MANUAL. 
 
 67. 
 
 1. Ify 
 
 1 
 
 and 
 
 1 
 
 1 + z 2 1 + x 
 
 of x, in its simplest form. 
 
 Add together 
 
 i-ft-(i-*)l, ±-(*-5*), 
 
 Simplify {a p - q ) p+q X (a q ) q+r -f- (a?) p - q . 
 
 x 2 + a 2 + b 2 + c 2 
 
 find the value of y in terms 
 
 i-(i + 4*)- 
 
 Solve x -\- a -\- b -\- c 
 
 a-\- b — c-\-x 
 
 Solve *+2-2y 
 3 J 
 
 2X— 4:7/ , 
 
 = 9. 
 
 y 
 
 23 
 5 J 
 
 6. A person out walking has 18 miles to go, and finds that 
 at the rate at which he is going he will be half an 
 hour late, but if he quickens his pace by half a mile 
 an hour, he will arrive just at the proper time. At 
 what rate is he going ? 
 
 68. 
 
 1. Solve ax 2 +b(2a + b)x-(a i -V) = 0. 
 
 2. Multiply x* — £as>f f by $x + 2. 
 
 3. Divide x* + lOrr 3 + 35tf 2 + 24 + 50a: by (x +!)(# + 4). 
 
 4, Simplify 
 
 x 2 -f- xy x 4, — y 4 
 x — y ' (x — y) 2 
 
 5. Solve £+2-5=0 
 3 5 
 
 2* + |-17 = 
 6.E X pand(|-MJ. 
 
ALGEBRA. 35 
 
 69. 
 
 1. What values of x will make the value of the expression 
 
 (x — a) (x — b) (x — c)(x — d) equal to ? 
 
 2. Resolve into the simplest factors 
 
 ra 2 + n 2 -\-p 2 + q 2 + 2(mq —pn). 
 
 ooi o # + 3 , -,,- 12# + 26 
 
 3. Solve 2x j- 1- 15 = -7 
 
 3 5 
 
 4. Divide $20,000 between A, B, and C so that A's share 
 
 shall be i of C's, and A's and B's together shall be 
 equal to C's. 
 
 5. Find the H.C.F. of 
 
 6a 3 - 6a 2 y + 2af- 2f and 12a 2 - 15ay + Zy 2 . 
 
 6. Solve ax 2 — a 2 (x-\- b 2 ) = ab(x — ab). 
 
 70. 
 
 1. If a =3, 6 = 1, c = -2, d=0, find the value of 
 
 a?-b 3 . g?+b 2 +c 2 +d 2 
 
 2ab+2b?+2ed+2ad ' ^/( a -c}(b+c)+6(-c-bXa-2b') 
 
 2. Divide (a 2 + 2 be) 3 - 4 be (3 a' + 4 b 2 c 2 ) by a 2 - 2 be. 
 
 3. Find the H.C.F. of 
 
 6a 3 — s* + 16 and 6x*- 3a; 3 + 9x 2 + 2>x+ 12. 
 1 1 
 
 4. Simplify ** + x +} x 
 
 + 1 
 
 x- f- ^ + l + i 
 
 5. Simplify 
 
 1 a? + l x 2 + x+l 
 
 6. Solve - + i=7; -+i=6; -+i=5. 
 
 y^ a: z a; y 
 
 7. Solve J^ = 15_l 
 
 07+ \ a; 4 
 
36 EXAMINATION MANUAL. 
 
 71. 
 
 1. Subtract Bx-~-(2x + ^x 2 ){rom^-^—(^-x\ 
 8 2 6 \ 4 / 
 
 * Sim P lif y4(IT^ + 4(I^) + 2(IT^)' 
 
 3. Find the square root of 
 
 (a + b) 2 - c 2 + (a + c) 2 - Z> 2 + (6 + c) 2 - a 2 . 
 
 4. There are 1200 books in my library. There are twice 
 
 as many English books as French books, and five 
 times as many French books as books in other for- 
 eign languages. How many English books and 
 how many French books do I have ? 
 
 5. Solve (4 °'7r f £ +1) -2«- 
 
 4a 2 + b 2 
 
 6. A man walking -^ of a mile above his ordinary rate gains 
 
 f of an hour in 39 miles. At what rate does he 
 ordinarily walk? 
 
 72. 
 
 1. D ivide a " x + ax * + X * by .-*-■ 
 
 a 3 — x 3 a — x 
 
 2. Extract the square root of x 2 + £ to three terms. 
 
 3. Divide ^— ^ by — r 
 
 Vc 2 Vfl' 
 
 4. Solve ^+y + 2; = 0; a# -f- 6?/ + ez = ; 2#+3y + z = l. 
 
 5i Solve ifo+ir-i- 
 
 6. Find the H.C.F. of 
 
 x" + 5a: 3 — 7^ 2 — 9x — 10 and 2a; 4 — 4a; 3 + 8* — 4. 
 
ALGEBRA. 37 
 
 73. 
 
 1. Simplify </a 2 bcX</ab^~a^b^cl 
 
 2. Find the L.C.M. of x 2 - 9x + 20 and x 2 + 6 a? — 55. 
 
 3. Solve 2 ^ 2 + l = _g 1. 
 
 9^ 2 -16 4 + 3^ 9 
 
 4. The sum of two numbers exceeds three times their dif- 
 
 ference by 8, and their difference is equal to -J- of the 
 smaller number. Find the numbers. 
 
 5. Solve x + m ~ 2n ^ n + 2m — 2x 
 
 x-\-m-\-2n n — 2m-{-2x 
 
 6. Extract the square root of 
 
 8^ 2 (2x 2 -f 6ax + a) + c 2 (SQx 2 + 12ar+ 1). 
 
 7. Solve (^+10J) 2 = 4(^ + 9^). 
 
 74. 
 
 1. Two men run a mile. The winner finishes in 4 minutes 
 
 44 seconds, and wins by 2 seconds. How many 
 yards start must the loser of the race have for a tie, 
 if each runs at a uniform rate ? 
 
 2. Find two consecutive numbers, the difference of whose 
 
 squares is 51. 
 
 3. Solve U12-x) + Sx -i = 4x -i+H- x . 
 
 5\ j 3 2 7 
 
 4. Find the H.C.F. of 4^-f 4^ 2 — %x—9 and x 3 +2x 2 —x—2. 
 
 Simplify — — +%' 
 
 a — b a-\- b 
 Solve 6O-l)O + i) = 0. 
 
38 EXAMINATION MANUAL. 
 
 rye 
 
 Find the value of 
 
 a +VV + 6 2 v , -, 7 
 
 when a = — 4 and o 
 
 a 3 -2Z>(> 2 -6 2 ) 
 
 2. Bracket together the equivalent expressions in the fol- 
 
 lowing list: 
 
 — , a 2 , at, a~ z . Va, Va 4 , a*, vV, — , a 3 X a -1 . 
 a 3 a 7 
 
 3. Divide x* ; bv x 
 
 # 4 J x ■ 
 
 4. Solve (s + 5)(y + 7) = (*+l)(y + 9) + 112) 
 
 2tf+10 = 3y+l j 
 
 5. Simplify 5 3 A/'8^F + 4a-v / ^F 2 --^125a 6 6 12 . 
 
 6. A traveller walks a certain distance at a certain rate. 
 
 If he had gone -J a mile an hour faster, he would 
 have walked the distance in % of the time. If he 
 had gone ^ a mile slower, he would have been 2-J- 
 hours longer on the road. Find the distance, and 
 the rate at which he travelled. 
 
 7. A man walking £ of a mile above his ordinary rate gains 
 
 f of an hour in 39 miles. Find his ordinary rate of 
 walking. 
 
 76. 
 
 1. Multiply x% — 4#* + 2 by x — xi. 
 
 2. Solve _^i_=4+^^I 
 
 ■Vax + 1 2 
 
 3. Show that if - = - = -, then - = ' y — - 
 
 J d / o xb + ya + zf 
 
 4. Divide xiy-% by x~iyt. 
 
 5. Simplify (3V8)(2V6). 
 
 6. Simplify [\a\atff\*J. 
 
 7. Solve a? —15f + — ^— = 6. 
 
 x — 151 
 
ALGEBRA. 39 
 
 77. 
 
 1. Two ships sail at the same time from the same port ; one 
 
 due north at the rate of 9 miles an hour, the other 
 due east at the rate of 12 miles an hour. After how 
 many hours will they be 60 miles apart? 
 
 a , a(a—b). . , (b — a)b 
 
 2. Solve — * 4- -\-a-\-b — x= v , , ' — 
 
 x — a — o a-f- o — x 
 
 3. Solve -=-^-. 
 
 x a — x 
 
 4. Two bodies move toward each other, starting from points 
 
 1800 feet apart. The first body starts 5 seconds 
 after the second body, and the two bodies meet half- 
 way between the points. If the rate of the first 
 body is 6 feet a second more than that of the second 
 body, find the rate of each body. 
 
 5. Solve 9x 2 — 9x + 2 = 0. 
 
 78. 
 
 1. Multiply a 2 — a + 1 — a~ x by a + 1 + a~ l + a~\ 
 
 2. If ?= £ show that ^ = £Z± 
 
 o a c + a c — d 
 
 3. What is the value of [{^)t{-»? 
 
 4. Simplify (a* &"*)*+ (a~* 5 s )*. 
 
 5. Divide $ 560 between A and B, so that for every dollar 
 
 A receives B shall receive $ 2.50. 
 
 6. Solve x 2 — ay = 101 
 
 xy — y 2 = 6 J 
 
 7. For what positive values of m and n does the expression 
 
 «» 
 (— a) « denote an imaginary quantity ? 
 
40 EXAMINATION MANUAL. 
 
 79. 
 
 1. Find the value of x * ~ xy + ^ when 
 
 x 2 + xy + y 2 
 
 V3 + 1 -, V3 - 1 
 x — ■ — and ?/ = — 
 
 2 J 2 
 
 2. Find the L.C.M. of a-1, 1 - a 2 , a -2, 4 
 
 3. Solve a + x+-V2ax + x 2 = b. 
 
 4. Find two numbers, the sum of which is 15, and the sum 
 
 of their squares 113. 
 
 5. Two freight trains pass a station at an interval of 4 
 
 hours, proceeding at the rates of 11 ? and 17£ miles 
 an hour, respectively. At what distance from the 
 station will the faster train overtake the other ? 
 
 6. Solve ax 2 -\-bx-\-c = 0. What is the relation between 
 
 a and c when the product of the roots of this equa- 
 tion is equal to unity? 
 
 80. 
 
 1. Two passengers have together 400 pounds of baggage. 
 
 One pays $1.20, the other $1.80, for excess above the 
 weight allowed. If all the baggage had belonged to 
 one person he would have had to pay $4.50. How 
 much baggage is allowed free ? 
 
 2. Find the fourteenth term of (2 a - b) l \ 
 
 3. Find the square root of 
 
 16 m 4 + Jjf m 2 n + 8m 2 + |?z 2 + \n + 1. 
 
 4. Find the H.C.F. of x" - ax 3 + a*x - a* and x 3 - a 3 . 
 
 5. Find the simplest value of v64# 3 yV + -v# 4 2/V 2 . 
 
 6. Solve 2bx + b_ 1 1 
 
 2bx-b ax 2 -a 2* 2 + *-l 2* 2 -3*+l 
 
 7. Find two numbers whose sum is 7, and the sum of their 
 squares exceeds their product by 19. 
 
ALGEBRA. 
 
 41 
 
 81. 
 
 1. Find the value of ax + by when 
 
 91. 
 
 br 
 
 and y 
 
 ar — cp 
 
 aq — bp~ " aq — 6^? 
 
 2. Add * 2 -3^-fy 2 , 2^— §y 3 +z 2 , 37y-i 2 / 2 + 2 / 3 , ^y-\if. 
 
 , 637+13 337 + 5 2^ 
 
 3. Solve _^t —X^ = _ 
 
 4. Solve V4a; + 9-2Va; = l. 
 
 5. Solve ^+1_^±4 £=7 _ 9^+33 
 
 7 14 
 
 o 5a: — 4?/ 11 y— 19 
 y _3_. . — ^=37 z- 
 
 u 2 4 
 
 6. Find two numbers whose difference is 1, and the sum of 
 
 their squares 313. 
 
 7. Find the L.C.M. of 
 
 37 2 + 537 + 4, ^ + 2*7-8, and 37 2 +737 + 12. 
 
 82. 
 
 37 (a + 37) 
 
 (f+ 12 ) 
 
 1. Solve ^--337 
 
 2. The difference between a number and its square is 18 
 
 more than the same difference would be if the num- 
 ber were 1 less. Find the number. 
 
 3. Find the L.C.M. of a -b, a + b, a 2 -b\ a?-b 3 , a z + b\ 
 
 37 2 + y 2 
 
 4. Simplify 
 
 ~ x 2 2 
 
 v x y ■ 
 
 l_l 
 
 y x 
 
 37 3 + y 3 
 
 5. Solve aa7 2 + £37 + c = 0, and state the conditions under 
 
 which the values of 37 will be (i.) real, (ii.) equal, 
 (iii.) imaginary. 
 
 6. Find the product of 2 2/3V23; 3 , VQxy, and • 
 
 7 ql 1 13 V3 ^ 
 
 7. bolve 37 = y = -• 
 
 y y x 2 
 
42 EXAMINATION MANUAL. 
 
 83. 
 
 1. Find the H.C.F. of 
 
 x*-2a? + x 2 -8x + 8 and 4:z 3 - 12^ 2 + 9^+ 1. 
 
 2. Divide -i^- by 2Q ^. 
 
 25 ^" 3 21^^a" 2 
 
 3. A can do a piece of work in 10 days which B can do in 
 
 8 days ; after A has been at work upon it 3 days, B 
 comes to help him ; in what time will they finish it? 
 
 4. Solve ^±^- x -ZlA = a. 
 
 x-2> x + 3 
 
 5. What number is that to which if 1, 5, and 13, severally, 
 
 be added, the first sum shall be to the second as the 
 second to the third ? 
 
 6. Expand (a -3 x)\ 
 
 84. 
 
 1. If s = | (a -f- b -j- c), show that 
 
 s 2 + (s - a) 2 + - b) 2 + (s - c) 2 = a 2 + b 2 + c 2 . 
 
 2. Subtract L=L^ f rom 
 
 l + 3a; 1 — 3a; 
 
 3. A man has 6 hours for an excursion. How far can he 
 
 ride out in a carriage which travels 8 miles an hour 
 so as to return home in time, walking back at the 
 rate of four miles an hour ? 
 
 4. Kesolve into factors x 8 — -^q. 
 
 5. Show that, if \ = % then 7a + 5ft = 7 g + 5rf ■ 
 
 b d 8a-Sb' 8c-Zd 
 
 so- 
 
 6. Solve x — y = \\ x 2 — y 2 = -£ ( 
 
 7. Solve 2*-«-2" 4a-7a 
 
 bx ax — bx ab — b 2 
 
ALGEBRA. 
 
 85. 
 
 43 
 
 1. Show that the sum of the squares of any two different 
 
 numbers is always greater than twice their product. 
 
 « ' , ,, , , , 4 + 12a?+93* 
 
 2. Reduce to its lowest terms ' — » 
 
 2 -f- 13 a; + 15 a; 2 
 
 3. Find the value of a: in 
 
 96~ 3 
 
 V5' 
 
 \3**/ "'\4^tfP 
 
 4. Solve V*(3 - a?) = Va; + 1 + V2(ar- 1). 
 
 5. Extract the square root of 33 — 20 V2. 
 
 6. Expand (l-^Y- 
 
 86. 
 
 , ^, a — 2b — x bb — x . 2a— -a;— 195 _ q 
 a 2 -4:b 2 ax + 2bx 2bx-ax 
 
 2. Simplify 1 + x 
 
 I. i ZiX 
 
 + X + 
 
 3. Solve 
 
 1-x 
 3a; -1 4a;-2 1 
 
 2a;- 1 3.T-2 6 
 
 4. Expand (2a;-32/) 4 . 
 
 5. A vessel which has two pipes can be filled in 2 hours 
 
 less time by one than by the other, and by both to- 
 gether in 2 hours 55 minutes. How long will it take 
 each pipe alone to fill it ? 
 
44 EXAMINATION MANUAL. 
 
 87. 
 
 1. If a = 2, b = 3, x = 6, y — 5, what is the value of 
 
 a-f 2a;- j6 + y-[a-a;-(6-2y)]|. 
 
 2. Simplify | V3 + 4 V=l +^3 - 4 V^ i *• 
 
 3. Reduce to the lowest terms 
 
 2a; 2 - 8^ + 6 a: 2 -9a; + 20 , 2^ 2 -10a: 
 x 2 -bx + 4: x*-\0x + 2l ' x 2 -7x ' 
 
 4. Solve 1 -f \ =1. 
 
 x-{- o — a x + b — c 
 
 5. When are the hands of a clock together between the 
 
 hours of 6 and 7 ? 
 
 6. The plate of a looking-glass is 18 inches by 12 inches, 
 
 and it is to be framed with a frame of uniform width, 
 whose area is to be equal to that of the glass. Find 
 the width of the frame. 
 
 7. Show that a°=l. 
 
 88. 
 
 1. Find the fourth term of (2 a: - 5y) 12 . 
 
 2. Divide ? M-L- by ^M_^Zf. 
 
 x a-\-x a — x a — x a-\-x 
 
 3. Solve (a — b) x = (a -\- b)y; x-\-y = c. 
 
 4. A carriage, horse, and harness are worth, together, $ 720. 
 
 The carriage is worth £ of the value of the horse, 
 and the harness f of the difference between the 
 values of the horse and carriage. Find the value of 
 each. 
 k qi x-\-13a-\-3b -, _ a — 2b 
 5a — 35 — x x-\-2b 
 
ALGEBRA. 45 
 
 89. 
 
 1. Solve x i -5x i + 4: = 0. 
 
 o, , ^2d 1 -x i + b^/2a-x ^/a + b 
 
 2. Solve — J == —- 
 
 V2a 2 -o; 2 -6V2a-a; V a - b 
 
 1 
 
 3. S^phfy —X^x^ 
 Mc-i 
 
 4 Solve 2-1-1. ?'+*-?• 
 
 a 6 b a b 
 
 5. A hare is 50 of her leaps before a greyhound, and takes 
 four leaps to his three. Two of the greyhound's 
 leaps are equal to three of the hare's. How many 
 leaps must the greyhound take to catch the hare ? 
 
 90. 
 
 1. Extract the square root of 81 a~%b$c. 
 
 2. Express in the simplest form i ( \/3 + \ \/l92 + \/81). 
 
 3. Simplify {(a-^}~i. 
 
 4. If a:b = c:d = e:f, show that • 
 
 a\b = a J r c-{-e:b-\-d +/. 
 
 5. For building 108 rods^of stone-wall, 6 days less would 
 
 have been required if 3 rods more a day had been 
 built. How many rods a day were built ? 
 
 6. Solve x -\-y = h\ 
 
 x 2 -Sxy + 4:y 2 = 7)' 
 
46 EXAMINATION MANUAL. 
 
 91. 
 
 1. 
 
 *"*■* cw 
 
 2. A railway train, after travelling an hour, meets with an 
 
 accident, which delays it 1 hour ; after this it pro- 
 ceeds at f of its former rate, and arrives 5 hours 
 late. If the accident had occurred 50 miles farther 
 on, the train would have been 3 'hours 20 'minutes 
 late. Find the length of the line. 
 
 3. Prove that if a : b = c : d, then 
 
 a-\-b : a — b = c-{-d: c — d. 
 
 4. Solve x~ x -f ax~% = 2 a 2 . 
 
 5. Write the 5th term of (3 a* - 4y*) 9 . 
 
 6. What is a surd ? Give an example. What are similar 
 
 surds? Give an example. 
 
 92. 
 
 g& qfl 
 
 1. Find the value of when x = 2? cfi and y= 2~*at- 
 
 2, Simplify 
 
 x-y 
 
 4a?-{-5y 4# — 5y 
 
 2x 2 -2xy 4(a; 2 -3/ 2 ) ±{x-y) 2 
 
 a :** (*)'*(^V(^. 
 
 4. Solve VS + 2a =i V£ + 4a 
 
 Vaf+2 6 V*+3& 
 
 5. Find a? and 3/ from the proportions 
 
 2a7 + 2/:y:: 32/ : 2?/ — a?j 
 
 2x + l:y::2x + e>:y + 2. 
 
 6. A boat's crew row 3 \ miles down the river and back 
 
 again in 1 hour and 40 minutes. If the current of 
 the river is 2 miles per hour, determine their rate 
 of rowing in still water. 
 
ALGEBRA. 47 
 
 93. 
 
 1. Add ■_ 
 
 V20a 2 m— 20a<?ra+5(? 2 ra and V20e 2 ra— Q0acm-\-4:ba 2/ m. 
 
 2. Find the H.O.F. of 
 
 4x 2 yXx 2 -\-xy+y 2 )\ §xy(x«-y«), and \&x z (x 3 ~y i )' i . 
 
 m m ■• , _ _ , 3a 2 a? 6a 2 -a6-26 2 6 2 a; 
 
 3. Solve ate 2 H — : 
 
 1 $ c 2 c 
 
 4. The circumference of the fore-wheel of a carriage is 9 
 
 feet, and that of the hind-wheel 12 feet. What is 
 the distance gone over when the fore-wheel has 
 made two more revolutions than the hind- wheel ? 
 
 5. Which is the greater, Vli + V7 or Vl9 + V2 ? 
 
 Prove your answer. 
 
 6. Solve x+y = 6 1 
 
 ^ + y 3 = 72J " 
 
 94. 
 
 1. Solve x 2 + xy-j-2y* = U) 
 
 2:z 2 -:ry + 3/ 2 = 16 J 
 
 2. Find the two middle terms of ( - — ^ ) . 
 
 3. Solve 2^ 2 -21a; + 55 = 0. 
 
 4. If x' and #" are the roots of the equation x 2 — px + q = 0, 
 
 show that #' + x" =p, and #'#" = q. 
 
 5. Form and solve the quadratic equation, the product of 
 
 whose roots is 42 and their sum 13. 
 
 6. A number having two digits is to the number formed by- 
 
 interchanging the digits as 7 is to 4, and the sum of 
 the two numbers is 132. Find them. 
 
48 EXAMINATION MANUAL. 
 
 95. 
 
 1. For what value of m will the equation 2# 2 + 8;r + m = 
 
 have equal roots? 
 
 2. Find the value of a?+xy +y 2 if z = 1+ V2, 3/ = 1- V2. 
 
 3. A and B distribute % 100 in charity. A relieves 5 per- 
 
 sons more than B, and B gives to each % 1 more than 
 A. How many did they each relieve ? 
 
 4. Solve x 6 - 20* 3 = 189. 
 
 5. Mul ti pl y (fly. (^. and (£)* 
 
 6. Find the H.C.F. of . 
 
 x i + x z -2x 2 +3x-S and at- ±x"-2x 2 + 3x + 2. 
 
 96. 
 
 t w vj # 3 — 8V 3 — z 3 — 6#?/2: 
 L Simi>llfy * 3 -^-2^ ' 
 
 2. Find the H.C.F. and the L.C.M. of 
 
 34a 2 &ry, 27#y& 3 , SlyW*. 
 
 3. Extract the square root of a; 4 -) h 2aH — 
 
 1 9 T s 2 3 Sx 
 
 4. Solve a: 2 — xy = f 1 
 
 #y + y 2 = i J 
 
 5. Solve ax 2 -\-2bx-\- c = 0, and show that the two roots 
 
 - are equal if b is a mean proportional between a and c. 
 
 6. Solve V# + V4-f-# = ~ 
 
ALGEBRA. 49 
 
 97. 
 
 1. Expand (V^-^J- 
 
 2. Rationalize the denominator, and then find the approxi- 
 
 . , , 7 + 2 VlO 
 mate value ot — ! 
 
 7-2VT0 
 
 3. Solve 15x-3x 2 +4:(x 2 -5x+&)$=-l. 
 
 4. A man starts from the foot of a mountain to walk to its 
 
 summit. His rate of walking during the second 
 half of the distance is £ mile per hour less than his 
 rate during the first half, and he reaches the summit 
 in 5 2 hours. He descends in 3f hours, by walking 
 1 mile more per hour than during the first half of 
 the ascent. Find the distance to the top, and the 
 rates of walking. 
 
 98. 
 
 1. Divide 2V3 + 3V2+ V30 by 3 V6. 
 
 1.1,1 
 
 2. Simplify 
 
 4(1 + VV) 4(1 -Vx) 2(1 + *) 
 
 3. Solve V* + 3 + V* + 8 = 5V*. 
 
 4. Find the mean proportional between (a+5) 2 and (a— by 
 
 5. Solve x — y = 2 ) 
 
 15(x 2 -y 2 ) = lQxy) ' 
 
 6. Expand (2^?-^ 2 ) 4 . 
 
50 EXAMINATION MANUAL. 
 
 99. 
 
 1. Simplify 2 ^40 + 3^/108 + ^5^-^320-2^1372. 
 
 2. Extract the square root of 
 
 1 + 4y~* - 2y-l - 4y~ x + 25 y"*- 24y"* + 16y~ 2 . 
 
 3. Expand [ — — £yV# J . 
 
 4. Solve V2o;+l-V^ = f4 = iV^ :r 3. 
 
 5. Solve ^±l_iA_2\ = 3^+l t 
 
 6 x\b a) a 
 
 6. The volume of a sphere varies as the cube of its diame- 
 
 ter. Compare the volume of a sphere 6 inches in 
 diameter with the sum of the volumes of three 
 spheres whose diameters are 3, 4, 5 inches, respec- 
 tively. 
 
 100. 
 
 i i 
 
 1. Simplify 
 
 • Va 2 — x 2 a + Va 2 — x 2 
 
 2. Eationalize the denominator of -. and find the 
 
 3-2V2 
 approximate value of the fraction. 
 
 3. Form the quadratic equation whose roots are a ±V— 56. 
 
 4. Solve (or* + 2)(>-3 + 5) = or 1 + 8. 
 
 5. A vessel which has two pipes can be filled in 2 hours 
 
 less time by one than by the other, and by both to- 
 gether in 1 hour 52 minutes 30 seconds. How long 
 will it take each pipe alone to fill the vessel ? 
 
ALGEBRA. 51 
 
 101. 
 
 1. Multiply and free from negative exponents 
 by - 4 ar % b~ V. 
 
 2, Solve £±XEz£=| 
 
 x — V9 — x 3 
 
 3. Find an equivalent expression with a rational denomi- 
 V2l+ Vl2 
 
 nator for 
 
 V7-V3 
 
 4. Solve 2x 2 + 2>y 2 -xy = 2>l')^ 
 
 x~y = 3J 
 
 5. Find the limit of the series 1 + $ + £• 
 
 6. Reduce to partial fractions by means of indeterminate 
 
 rn • , 6 a; 2 4 # — 6 
 
 coefficients — — • 
 
 (x -l)(x- 2) (x - 3) 
 
 7. Expand (l-^Y. 
 
 102. 
 
 1. Simplify and free from negative exponents 
 (-f a- , 5 8 c-*rf- 1 ) X (f a 2 Z>- 3 <rW). 
 
 2. Multiply -\/5-2^/6 by 3^4-^36. 
 
 3. Solve x*-y 3 = e>3\ 
 
 x — y = 3 J 
 
 4. How many terms of the series 54, 51, 48, etc., amount to 
 
 513 ? Explain the two answers. 
 
 5. Solve 3a*-a;-i + 2 = 0. 
 
 6. Find the first five terms, and the (r + l)th term, in the 
 
 expansion of (1 — x)~%. 
 
52 EXAMINATION MANUAL. 
 
 103. 
 
 2. Divide x~ 6 + y~' by x~* + y~ 3 . 
 
 3. Solve * + V2 
 
 x — V2 — a: 2 o 
 
 4. A debt can be discharged in a year by paying $1 the 
 
 first week, $3 the second week, $5 the third week, 
 etc. Eequired the amount of the debt, and the last 
 payment. 
 
 5. Find the coefficient of x 12 in the expansion of (a b — &V)i 
 
 6. Solve x 2 + xy + y 2 = 52) 
 
 xy — x 2 = 8 ) 
 
 104. 
 
 1. Simplify V5 X ^2 X </l. 
 
 2. Find an equivalent expression with a rational denomi- 
 
 4 + V2 
 
 nator for 
 
 4 + V3 
 
 3. Solve x 2 — 2x+6^/x 2 — 2a; + 5 = ll. 
 
 4. The height of a certain triangle is 4 inches less than the 
 
 base. If the base be increased by 6 inches, and the 
 height diminished by 6 inches, the area is dimin- 
 ished by i. Required the base of the triangle. 
 
 5. Find the fifth term of (3ar - 2y)" 10 . 
 
 x 2 
 
 6. Resolve into partial fractions 
 
 (*»_!)(* -2) 
 
ALGEBRA. 53 
 
 105. 
 
 1. Simplify the expression "\-^_|_ £ x \ Va — 6. 
 
 2. Solve — — ^ r = ^±^- 
 
 rrvp (x + «) w px 
 
 3. Solve a; 2 = 21 + V(.* 2 - 9). 
 
 4. Solve bxy = 84 — a; 2 ?/ 2 1 
 
 x — 3/= 6 J 
 
 5. The sum of 11 terms of an arithmetical progression is 
 
 22, and the common difference is -f. Find the first 
 term. 
 
 6. Find the coefficient of x 9 in (5 a 3 — 4a; 3 ) 7 . 
 
 106. 
 
 1. Simplify and free from negative exponents 
 
 ■(- 7 a-'b'c- 5 ) X (Sa 2 b~ s c). 
 
 2. Find the square root of 28 + 5 Vl2. 
 
 3. Solve x 2 — X + 5-V2X 2 — 5x + 6 = J (3a; + 33). 
 
 4. Prove the formula for finding the sum of n terms of an 
 
 ' arithmetical progression. 
 
 5. There are three numbers in arithmetical progression. If 
 
 1, 3, 9 are added to them, respectively, they are 
 then in geometrical progression. Find the numbers. 
 
 6. Expand to five terms Vl + 2x. 
 
 7. Give an example of a series of terms in harmonical pro- 
 
 gression. 
 
54 EXAMINATION MANUAL. 
 
 107. 
 
 1. Multiply 5V14 + 3V5 by 7VI4-2V5. 
 
 2. Simplify ^(a?b</tfxbxc)\ 
 
 3. Solve ar\ + 2 = x ~\ + 8 . 
 
 ari + 5 
 
 4. Solve 2^-3a:y + 3/ 2 = 24\ 
 
 3^-5^ + 2^ = 33]' 
 
 5. How many terms of the series 19 + 17 + 15 amount 
 
 to 91? 
 
 6. Prove the formula for finding the sum of n terms in 
 
 geometrical progression. 
 
 7. Find the fifth term in the expansion of Vl + x. 
 
 108. 
 
 1. Divide 5-7V3 by 1+V3. 
 
 2. Find an arithmetical progression such that the second, 
 
 third, fourth, and sixth terms may form a proportion. 
 
 3. Solve f-JU2* 
 
 y x 4 
 
 3 
 
 x — ?/ = - 
 y 2 
 
 4. A person bought a number of shares in a railroad, that 
 
 cost him % 3000. He reserved 10 of the shares, and 
 sold the remainder for $2700, gaining $4 a share. 
 How many shares did he buy ? 
 
 5. Find the limit of the sum of 9 — 6 + 4 — 
 
 1 
 
 6. Resolve into partial fractions 
 
 x* — cC 
 
 7. Find the fourth term in the expansion of [ 5 — - j • 
 
ALGEBRA. 55 
 
 109. 
 
 1. Three numbers are as 6 : 11 : 20. If 1 is added to each 
 
 they are in geometrical progression. Find the 
 numbers. 
 
 2. Solve g + J + « = ** + <*&. 
 
 a -f- b b bx 
 
 3. Divide 28 into two parts such that their squares shall be 
 
 in the ratio 3 : 10. 
 
 4. The first term of a series in arithmetical progression is 
 
 17, the last term — llf , and the sum 25^-. Find 
 the common difference. 
 
 5. Find the limit of the sum of 1 — ■§■ -f- Y V ~~ 
 
 6. Find the first five terms, and the (r -f 1) th term, of the 
 
 expansion of (1 — x 2 )~%. 
 
 _ 110. 
 
 2. Find an equivalent expression with a rational denomi- 
 
 , , V20-V8 
 nator lor *-. 
 
 V5+V2 
 
 3. Solve ff-f-5 = V# + 5 + 6. 
 
 4. A certain capital is invested at 4 per cent. If the num- 
 
 ber of dollars in the capital is multiplied by the 
 number of dollars in the interest for 5 months, the 
 result is 117,0411. Required the capital. 
 
 5. The first and the ninth terms of a series in arithmetical 
 
 progression are 5 and 22. Find the sum of twenty- 
 one terms. 
 
 6. Find the middle term of (1 + x) 2n . 
 
 7. Resolve into partial fractions 
 
 * 2 — 9#-f-14 
 
56 EXAMINATION MANUAL. 
 
 111. 
 
 2. Find the H.O.F. of 
 
 ^+7^+14^+5^-3 tmd2x i +9x*+8x 2 -x + e>. 
 
 3. A merchant buys some goods for $40, and sells them to 
 
 another merchant, who, in his turn, sells them for 
 $48£. If each merchant makes a profit of the same 
 rate per cent, determine what that rate is. 
 
 4. If a : b — c : d = e : /, show that 
 
 a + 2c + 3e _ 2a + 3c + 4e 
 b + 2d+3f 26 + 3^+4/ 
 
 5. The sum of five terms of an arithmetical series is 30, and 
 
 the product of the first and last terms is 20. Form 
 the series. 
 
 6. If a, /S are the roots of the equation x 1 — x + 1 = 0, show 
 
 thata + £ = l, anda 2 + /? 2 = -l. 
 
 - 112. 
 
 1. Solve x* + 3y 2 = 28; xy + y 2 = \2. 
 
 2. In each of 3 battles 36 officers and 10 per cent of the 
 
 men engaged are killed. At the end of the second 
 battle the percentage of officers to men is two-thirds 
 as great as at its commencement. The number of 
 men at the end of the third battle is equal to the 
 square of the number of officers at its commence- 
 ment. How many officers and men were engaged 
 in the first battle ? 
 
 3. Find an arithmetical progression such that the sum of n 
 
 terms shall be equal to n 2 . 
 
 4. Solve z 2 + ^-2V2a; 2 + 5* + 3 = ?. 
 
 A 2 
 
 5. The terms of a ratio are 7 and 3 ; what number must be 
 
 added to each in order that the ratio may be halved? 
 
ALGEBRA. 57 
 
 113. 
 
 1. Solve ;r 2 + 2y 2 = 22; 2a;y-fy 2 :=21. 
 
 2. Simplify 2 V| + V60-Vl5 -Vf. 
 
 3. Solve V9 x + 40 + 2 Va; + 7 = Vx + 44. 
 
 4. What is a ratio ? Is it a quantity ? 
 
 If w -f w : m — n = x -f- y : a; — y, show that 
 # 2 -f- ra 2 : a; 2 — m 2 = y 2 + n* : y 2 — ?z 2 . 
 
 5. The thickness of a hollow cylinder varies directly as the 
 
 amount of material, and inversely as the length of 
 the cylinder and the sum of the radii of its internal 
 and external surfaces. If the amount of material in 
 a cylinder 50 feet long, whose radii of external and 
 internal surfaces are 4 feet and 3 feet, respectively, 
 he 1100 cubic feet, find the thickness of a cylinder 
 84 feet long, having the sum of its radii 5 feet, and 
 containing 1650 cubic feet of iron. 
 
 114. 
 
 1. In an arithmetical progression, if s lf s 2 , s 3 denote the sums 
 
 to the nth, 2 nth, 3 nth terms, respectively, prove that 
 
 53 = 3(52 — 8!). 
 
 2. In a geometrical progression, if l lf l 2 , l 3 , denote the nth, 
 
 2 nth, 3 nth terms, respectively, prove that 4 s — hb 
 
 a r> a 4. -4. i 4.4. 2x 3 —llx 2 y + l9xy 2 —10y 3 
 
 3. Keduce to its lowest terms ^ ' * -f— 
 
 x 3 -7x 2 y +13^y 2 -6y* 
 
 4. Find the limit of 1 + 4 + i + 
 
 5. A committee of 7 members is to be chosen out of a body 
 
 of 20 protestants and 15 catholics in such a way 
 that there shall be 3 protestants and 4 catholics. In 
 how many different ways can such a committee be 
 chosen. 
 
58 EXAMINATION MANUAL. 
 
 115. 
 
 1. If A varies as- B\ B* as C\ C 5 as D r \ and D 7 as E\ 
 
 show that — X — - X — X — , does not vary at all. 
 E E E E J 
 
 2. In how many ways can 2 white balls and 3 red ones be 
 
 selected out of an urn containing 7 white balls and 
 10 red balls? 
 
 3. Solve — *■■ 1 = 1. 
 
 y/x — V2 — x Va? + V2 — x 
 
 4. Insert 10 arithmetical means between 6 and 61, and find 
 
 the sum of the whole series. 
 
 5. An express-train, travelling at uniform speed, after be- 
 
 ing an hour in motion, was delayed half an hour by 
 an accident ; after which it proceeded at three-fourths 
 of its original rate of speed, and arrived at the end 
 of its journey 1 hour and 50 minutes late. Had the 
 accident occurred after the train had travelled a 
 distance of 60 miles, it would have been 1 hour and 
 40 minutes late. Find the length of the line. 
 
 116. 
 
 1. Find an arithmetical progression such that the second, 
 
 fourth, and eighth terms are in geometrical progres- 
 sion. 
 
 2. Solve x 2 — ax-\-by\ if — ay -\- bx. 
 
 3. If a be the greatest of the four proportionals a, b, c, d, 
 
 show that a—by-c — d. 
 
 4. If four numbers be in proportion, and the first three be 
 
 in arithmetical progression, show that the reciprocals 
 of the last three are in arithmetical progression also. 
 
 5. In what scale of notation will 540 be the square of 23 ? 
 
ALGEBRA. 59 
 
 117. 
 
 2. Prove that a ratio of greater inequality is diminished if 
 
 the same quantity be added to both its terms. 
 
 3. Solve 4:X 2 + xy = Q; Sxy + y 2 = 10. 
 
 4. The sum of 10 terms of a certain geometrical series is 33 
 
 times the sum of 5 terms of the same series. What 
 is the common ratio ? 
 
 5. Sum the series 3, 2 T 7 7 , 2 T 4 7 , , to 21 terms. 
 
 6. The volume of a sphere varies as the cube of its radius, 
 
 and that of a circular plate of given thickness as the 
 square of its radius. If the volume of a sphere of 
 radius 1 inch be equal to that of a plate of radius 2 
 inches, find the radius of a plate which is equal in 
 volume to a sphere of radius 4 inches. 
 
 118. 
 
 1. In an arithmetical series the common difference is 2, and 
 
 the square roots of the first, third, and sixth terms 
 also form an arithmetical series. Find the series. 
 
 2. Find a geometrical progression such that the sum of 
 
 an infinite number of terms shall be 4, and the 
 second term shall be f . 
 
 3. What is the equation whose roots are double those of 
 
 the equation x A + x 2 — 6 = ? 
 
 4. A cattle dealer buys sheep, and sells them at a profit of 
 
 20 per cent. With the proceeds he again buys sheep, 
 and sells them so as to gain 25 per cent. Once more 
 he invests the proceeds in sheep, and this time he 
 gains 16 per cent. If his last profit amounted to 
 $300, how much money did he invest at first ? 
 
 5. Solve x* — 9f = a s ; x — y — b. 
 
60 EXAMINATION MANUAL. 
 
 119. 
 
 1. Solve + a) 5 -{x -of = 242 a 5 . 
 
 2. Solve x 2 - x + 3 VV -3# + 36 = 2(> + 26). 
 
 3. If the roots of tlie equation 3 # 2 — 8# -f- 5 = De a anc ] ^3 
 
 show that the equation whose roots are £ and — is 
 15a; 2 -Z4:x + 15 = 0. ^ 
 
 4. The sum of three quantities is y. The first varies in- 
 
 versely as x 2 , the second inversely as x, and the third 
 is constant. When # = 4, 2, 1, then 2/ = 3, 7, 21, 
 respectively. Find the equation between x and y. 
 
 5. Three numbers in geometrical progression, if multiplied 
 
 by 3, 2, and 1, respectively, are in arithmetical pro- 
 gression. If the middle number is 18, what are the 
 others ? 
 
 120. 
 
 n tc -l z. -1/u b — ca-\-b a — bb-\-c 
 
 1. lia\o = o\c, then — = — : — ! — = : — ; — 
 
 b a a o 
 
 2. If x and y are two numbers, A their arithmetical mean, 
 
 and G their geometrical mean, then 
 x 2 + y 2 = 2(A 2 -G 2 ). 
 
 3. If z vary inversely as 3x -f- £/, and y vary inversely as #, 
 
 and if, when x = 1 and y = 2, 3 = 3, find the' value 
 of z when # = 2. 
 
 4. Find two numbers such that their product, their sum, 
 
 and three times their difference are in the projDortion 
 5:2:4. 
 
 5. The cost of an entertainment was $120, which was to 
 
 have been divided equally among the party ; but 
 four of them leave without paying, and the rest have 
 each to pay $2.50 extra, in consequence. Of how 
 many did the party consist? 
 
ALGEBRA. 61 
 
 121. 
 
 1. If the number of visitors to a fair varies as the square of 
 
 the number of degrees above 42° F., and if there are 
 1152 visitors when the temperature is 68°, how 
 many visitors will there be if the temperature is 55° ? 
 
 2. Insert between 6 and 16 two numbers such that the first 
 
 three numbers may be in arithmetical progression, 
 and the last three in geometrical progression. 
 
 3. If £ of the sum of the squares of the roots of the equa- 
 
 tion ax 2 -f- bx + c = is equal to their product, find 
 the relation between a, b, and c. 
 
 4. Solve x-\-y=a\ x 2 -{-mxy-\-y 2 — b. 
 
 5. What is the price of eggs per dozen when two more in a 
 
 dollar's worth lowers the price one cent per dozen ? 
 
 122. 
 
 1. Solve x — y = a\ bx 2 — cx 2 = d. 
 
 What do the values of x and y become when b — c? 
 
 2. The side of a square is a. By joining the middle points 
 
 of its sides another square is formed ; by joining the 
 middle points of the sides of this square a third 
 square is formed. If the operation is continued in- 
 definitely, find the limit of the sum of the areas of 
 the squares. 
 
 3. Separate into partial fractions ^— t ^— 
 
 r ^ (x-l)(x-3)(x-5) 
 
 4. Find the fourth term of 
 
 (d 2 -bx)$ 
 
 5. How many different signals can be made with ten flags, 
 of which three are white, two red, and the rest blue, 
 if all are hoisted together, one above another ? 
 
62 EXAMINATION MANUAL. 
 
 123. 
 
 1. What number must be added to 20, 50, and 100, respec- 
 
 tively, that the results may be in geometrical pro- 
 gression ? 
 
 2. Find an arithmetical progression such that the second, 
 
 fourth, and eighth terms are in geometrical progres- 
 sion. 
 
 3. The volume of a sphere varies as the cube of its radius. 
 
 If three spheres, whose radii are 9 inches, 12 inches, 
 15 inches, respectively, are melted into one, what 
 will be its radius ? 
 
 4. Solve x 2 — y 2 ~x 2 -}-y i — xy = 3. 
 
 5. In how many ways can a base-ball nine be arranged if 
 
 three men can play in any position, and the others 
 ' in any position except those of pitcher and catcher ? 
 
 124. 
 
 1. What number must be added to 20, 50, and 100, respec- 
 
 tively, that the results may be in harmonical pro- 
 gression ? 
 
 2. The sum of four numbers in geometrical progression is 
 
 170, and the third exceeds the first by 30. Find 
 them. 
 
 3. Solve a? — 11 ar 8 + 37a- — 35 = 0, one root being 5. 
 
 4. Solve 4a? + 4V33*-7ar+3 = 3ar(a?-l) + 6. 
 
 5. Transform 3256 from the septenary to the duodenary 
 
 scale. 
 
 6. Two steamers ply between the same two ports, a distance 
 
 of 420 miles. One travels half a mile an hour faster 
 than the other, and is two hours less on the journey. 
 Find their rates. 
 
ALGEBRA. 63 
 
 125. 
 
 1. In an arithmetical series the second term is 21, the sev- 
 
 enth term 41, the sum 1625. Find the number of 
 terms. 
 
 2. The first and the seventh terms of a geometrical series 
 
 are 2 and i. Find the intermediate terms. 
 
 3. Prove that the difference of the roots of the equation 
 
 x^—px -f- q = is equal to the difference of the roots 
 of the equation x 2 — Spx -f 2p 2 -f- q — 0. 
 
 4. A vessel is half full of a mixture of wine and water. If 
 
 filled with water, the ratio of the quantity of water 
 to the quantity of wine will be 9 times as great as if 
 filled with wine. Determine the original quantities 
 of water and wine. 
 
 1 
 
 5. Find the tenth term of 
 
 126. 
 
 1. Jf a : b = c : d prove that 
 
 a (a + b + c + d) = (a + b) (a + c). 
 
 2. Solve g + V a'-a^ m 
 
 x — Vcf-a? n 
 
 3. Solve r J + y 3 = 225yj 
 
 3*-y»== 75 3 
 
 4. Insert 9 arithmetical means between 1 and — 1. 
 
 5. There are four numbers, of which the first three are in 
 
 geometrical progression and the last three in arith- 
 metical progression. The sum of the first and last 
 numbers is 16, and the sum of the second and third 
 numbers is 12. Find the numbers. 
 
 6. Find the fortieth term of (1 — x)~~°. 
 
64 EXAMINATION MANUAL. 
 
 127. 
 
 1. Find an equivalent expression with a rational denomi- 
 
 nator for t— £ 
 
 3V2-2V3 
 
 2. Solve f-i-tL^ia. a? + y = 12. 
 
 3. Two vessels, one of which sails faster than the other by 
 
 2 miles an hour, start at the same time on voyages 
 of 1152 and 720 miles, respectively. The slower 
 vessel reaches its destination one day before the 
 other. What is the rate per hour of the faster vessel ? 
 
 4. There are five numbers in arithmetical progression. 
 
 Their sum is to the sum of their squares as 9 : 89. 
 The sum of the first four numbers is 32. Find the 
 numbers. 
 
 5. Find the limit of the sum of 4 + 3 + f + 
 
 6. Find the value of x in an infinite series in terms of y, if 
 
 y=l — 2x + 2>x\ 
 
 128. 
 
 1. Eeduce to the simplest form 
 
 2 \/3 (v^9 - 2 </M + 4 -\/J - 3 </2). 
 
 2. Solve 3xt-x~i+2 = 0. 
 
 3. In t seconds a body falling freely describes 16 f feet. 
 
 The velocity of sound is 1100 feet a second, very 
 nearly. If a stone dropped from the top of a tower 
 is heard to strike the ground after 4 seconds, find 
 the height of the tower. 
 
 4. The sum of three numbers in arithmetical progression is 
 
 24, and their product is 480. Find them. 
 
 5. The third and the seventh terms of a geometrical series 
 
 are 12 and 192. Find the tenth term. 
 
 6. Find the tenth term of — — — • 
 
 (1 — bxy 
 
 7. In what scale is 4954 expressed by 20305 ? 
 
ALGEBRA. 65 
 
 129. 
 
 1. Find the arithmetical mean, the geometric mean, and 
 
 the harmonic mean between a + b and a — b. 
 
 2. A man divided $216 equally among a certain number 
 
 of persons. If there had been three more persons, 
 each would have received $1 less. Required the 
 number of persons. 
 
 3. Solve #* + #* = 706; x — y=2. 
 
 4. Find the value of the last term of an arithmetical 
 
 series, having given the first term, the common dif- 
 ference, and the sum of the series. 
 
 5. Develop (3 or 1 — 2 a:) -4 to five terms. 
 
 6. Separate into partial fractions — — ■ ■ -^— 
 
 a? — bx l — x-\- 30 
 
 7. How many different arrangements can be made with 
 
 the letters of the word freshman, if the letters mna 
 are never separated ? 
 
 130. 
 
 1. Divide iVF.by V2 + 3vT 
 
 2. Solve * + 4+^+iY=-12 r 
 
 \x—y x—A 
 
 3. A sculptor purchased two cubical blocks of marble for 
 
 $2960, at $5 per cubic foot. The length of the two 
 together was 12 feet. Find the length of each. 
 
 4. An army on 'the march is advancing at the rate of 12 
 
 miles a day, when a detachment 55 miles in the 
 rear is ordered to join it. How long will it take to 
 do so, if it can advance 25 miles the first day, 24 
 the next, 23 the next, and so on? 
 
 5. The third term of a geometric series is 20, the eighth 
 
 term is 640, and the sum of the series 20475. How 
 many terms are there ? 
 
 6. Insert five harmonic means between — 1 and h 
 
 7. What is the sixth term of (a 2 - x 2 )~ 2 ? 
 
EXAMINATION MANUAL. 
 
 131. 
 
 1. Solve 9rr-3.? 2 -f 4(s» — 3s + 5)* = 11. 
 
 2. Solve x 2 + a?y + y 2 = 52 ; ^y — s 8 = 8. 
 
 3. From two towns, 168 miles apart, A and B set out to 
 
 meet each other. A went 3 miles the first day, 5 
 the second, 7 the third, and so on. B went 4 miles 
 the first day, 6 the second, 8 the third, and so on. 
 In how many days did they meet? 
 
 4. The sum of three numbers in harmonical progression is 
 
 37, and the sum of their squares is 469. Find the 
 numbers. 
 
 5. Expand to six terms, in a series of ascending powers of x, 
 
 the fraction - — ^— 
 
 1 — x — x 2 
 
 6. How many numbers between 10,000 and 15,000 can be 
 
 formed with the digits 0, 1, 2, 3, 4, 5, 6, no repeti- 
 tions being allowed, and each number to be divisi- 
 ble by 5? 
 
 132. 
 
 1. Bationalize the denominator of the fraction -^— — 
 
 2. Solve a* + 5a* -22 = 0. V5+V2 
 
 3. Solve (^ + . ? /) 2 + (^-: ? /) 2 = 8; a; 2 + ?/ 2 = 2(a 2 + & 2 ). 
 
 4. There are two arithmetical series which have the same 
 
 ■ # common difference. The first terms are 3 and 5, 
 respectively, and the sum of seven terms of one is 
 to the sum of seven terms of the other as 2:3. 
 Determine the series. 
 
 5. What is the least integer which added to the ratio 9 : 23 
 
 will make it greater than the ratio 7 : 11 ? 
 
 6. Ten papers are to be set at an examination, four of 
 
 them being in mathematics. In how many differ- 
 ent orders can they be set so that the mathematical 
 papers shall always come together ? 
 
ALGEBRA. 67 
 
 133. 
 
 1. Solve 2x-y = 2; 8x 3 -y 3 = 98. 
 
 2. A and B start at the same time from the places C and 
 
 D, A to travel from C to D, B to travel from D to 
 C. When they meet, A had travelled thirty miles 
 more than B. If they should continue at the same 
 rates, A would finish his journey in 4 days and B 
 in 9 days. Find the distance from Cto D. 
 
 3. The first term of an arithmetical progression is f , and 
 
 the difference between the third and the seventh 
 terms is 3. Find the sum of n terms. 
 
 4. Find the limit of the sum of f — \ -f f — 
 
 5* Resolve into partial fractions 
 
 ^-7^+36 
 
 6. From a society consisting of 12 men and 8 women, in 
 how many ways can a committee of 4 men and 3 
 women be selected? In how many of these ways 
 will a particular woman always be included ? 
 
 134. 
 
 1. If a varies directly as V&, and inversely as c 3 , and if 
 
 a = 3 when b = 256 and e == 2, find b when a = 24 
 and c = J. 
 
 2. Solve ■v'a? + 22 -:-&e+1j = 1. 
 
 3. Solve o; 2 + ?/ 2 + ^ + y = 78; a; + y + ay = 39. 
 
 4. Find four numbers in arithmetical progression, such 
 
 that the sum of the squares of the first and the 
 second shall be 29, and the sum of the squares of 
 the third and the fourth shall be 185. 
 
 5. Insert three geometrical means between \ and 128. 
 
 6. Fin d the sixth term of- (4 a 2 ex — 3 0$yi)i 
 
68 EXAMINATION MANUAL. 
 
 135. 
 
 1. Simplify V2a/2V1-a/ V^l. 
 
 2. Solve x n — ax ln = b. 
 
 3. A stone is dropped from the top of a tower, and when it 
 
 has fallen exactly half-way to a window a feet below 
 the top, another stone is dropped from the window. 
 After how many seconds, reckoned from the time 
 when the first stone is dropped, will the first stone 
 overtake the second? The space described by a 
 falling body in t seconds is hgi 1 . 
 
 4. The difference between two numbers is 48, and the 
 
 arithmetical mean exceeds the geometrical mean by 
 18. What are the numbers ? 
 
 5. Find the value of x in an infinite series in terms of y, if 
 
 y = 1 -f x — 2x 2 -f x\ 
 
 6. A boat's crew consists of 8 men ; 2 men can row only 
 
 on the port side, and 2 others only on the starboard 
 side. In how many ways can the crew be arranged? 
 
 136. 
 
 1. Solve ay 2 -f hxy — 5 = 0; bx 2 -f- axy — a = 0. 
 
 2. Find in the simplest form the value of (1 + V — 3) 6 . 
 
 3. If the second term of a geometric series is 24, and the 
 
 fifth term 81, find the series. 
 
 4. If a : b = c : d, prove that Va + b : V3 = Vc + d : V5. 
 
 .5. On how many nights may a different guard of 5 men 
 be selected from a company of 36 men, and on how 
 many of these would the oldest man have to serve ? 
 
 6. Ify = # — — -f— , find the value of a;, in terms of y, 
 
ALGEBRA. 69 
 
 137. 
 
 1. Find the three cube roots of 1. 
 
 , Solve ^ + ^ = 2; ^_ y = 75 . 
 
 3. How many signals can be made with 3 blue and 2 
 
 white flags which can be displayed singly, or any 
 number at a time, one above another ? 
 
 4. Find the least number which, if divided by 15, leaves 
 
 a remainder of 14, and if divided by 13, leaves a 
 remainder of 12. 
 
 5. Find by the method of finite differences the sum of 
 
 3 + 11 + 31 + 69 + 131 + , to 20 terms. 
 
 6. How many balls are there in an incomplete square pile 
 
 if the upper course consists of 529 balls and the 
 base of 5184? 
 
 138. 
 
 1. Solve x 2 -f y 2 = axy ; x -f y = bxy. 
 
 2. From a horse-car station, a closed car leaves every 9 
 
 minutes, beginning at 7 o'clock a.m., and an open 
 car every 16 minutes. At what times will an open 
 car leave exactly 3 minutes after a closed car ? 
 
 3. If on an average A solves 3 problems out of 5 which 
 
 he tries, and B solves 2 out of 5, what is the proba- 
 bility that a problem which both try will be solved ? 
 
 4. Find by the method of indeterminate coefficients the 
 
 sum of 1 + 2 3 -f 3 3 -f 4 3 + , to twenty terms. 
 
 5. If y = x -f x 2 -f or 3 , find the value of x in terms of y, de- 
 
 veloping the series to six terms. 
 
70 EXAMINATION MANUAL. 
 
 139. 
 
 1. Solve _ + _=— ^ ; _+ _= _. 
 
 x y x + y x 2 f a? 
 
 2. A man wishes to make up as many different parties as 
 
 he can out of 20 friends, each party consisting of 
 the same number. How many should he invite at 
 a time, and how many parties will there be ? 
 
 3. If on an average 2 ships out of 5 return safe to port, 
 
 what is the chance that out of 5 ships expected at 
 least two will return ? 
 
 4. A man wishes to build a wall 31 feet long, and has 
 
 stones of 4 different lengths : viz., 2.2 feet, 2.5 feet, 
 3 feet, 4.1 feet. How many of each kind may he 
 use in laying a course ? 
 
 5. Solve by resolution into factors x* — 3x 2 -\- Sx — 1 = 0. 
 
 6. Find by continued fractions the fourth convergent 
 
 value for VTO. . 
 
 140. 
 
 1. What relation exists between the two roots of the equa- 
 
 tion ax 2 -f- bx -f- a = 0? 
 
 2. Solve a*— f=S\ O + y) (** + f) = 7. 
 
 3. Find by means of the binomial theorem the value of 
 
 V16.16 correct to five decimal places. 
 
 4. Find by the method of finite differences the sum of the 
 
 infinite series 1 + bx + 9x 2 + 13* 3 + llx 4 + 
 
 5. Expand to four terms by the method of indeterminate 
 
 coefficients ■ — -• 
 
 1 + x + x 2 
 
 6. Find the least integral values of x and y which will 
 
 satisfy the equation x 2 — 52?/ = 1. 
 
ALGEBRA. 71 
 
 141. 
 
 1. Define a logarithm, its characteristic, its mantissa. Prove 
 
 that logmn = logm + logn. 
 
 2. If 2 is the base of a system of logarithms, find the loga- 
 
 rithms of 8, -^j, and Vl6. 
 
 3. Given loga, log b, logo, and logo?; how is the value of 
 
 log — found ? 
 
 4. Find by logarithms the value of 24.13 X 6.052. 
 
 5. Find by logarithms the cube root of 3852. 
 
 6. Find the value of 8352x3.69 
 
 (30.57) 3 
 
 142. 
 
 1. State and prove the rule for finding the logarithm of a 
 
 quotient. 
 
 2. If the base of a system of logarithms is 3, find log 9, 
 
 logA.,log27Mog^S. 
 
 3. What is colog 10 in the common system ? 
 
 4. If log 6492 = 3.81238, find log 0.00006492. 
 
 5. Find the mean proportional between (0.01)* and (0.2) 4 . 
 
 6. Find the value of (0-005234)* x (0.017)*. 
 
 24* 
 
72 EXAMINATION MANUAL. 
 
 143. 
 
 1. "What is the logarithm of 1 in any system ? of the base 
 
 of the system ? of the reciprocal of the base ? If the 
 base is greater than 1, for what numbers are the loga- 
 rithms positive, and for what negative? Give rea- 
 sons for your answers. 
 
 2. If the base of a system of logarithms is J- , find log 1 ^- } . 
 
 logW^ndlog-^S- 
 
 3. If log a = 0.78241, and log b = 0.63575, find logVa 2 +Z> 2 . 
 
 4. Find the value of , 
 
 -\/047 2 
 
 5. Solve 20* =100. 
 
 144. 
 
 1. If the base of a system of logarithms is 64, find log 4, 
 
 logl6, log32, log^. 
 
 2. Of what number is — h the logarithm in a system whose 
 
 base is 16 ? 
 
 35* X 12" 
 
 3. Find the value of 
 
 13 
 
 a xv A ., . f VO0125 X V31.15 
 
 4. Imd the square root ol ttt^t^j 
 
 0.00081 
 
 5. Solve 10* = 2.45. 
 
 6. A man owes $14,720.20. At the end of each year he 
 
 pays §2000. In how many years will the debt be 
 paid, the rate of interest being 6 per cent ? 
 
ALGEBRA. 73 
 
 145 
 
 1. If the base of a system of logarithms is — f, find log— ^-, 
 
 log |, log ft. 
 
 2. Of what number is — 5 the logarithm in the common 
 
 system ? in the system whose base is 3 ? in the sys- 
 tem whose base is $ ? 
 
 3. For what base is log 1 ^ equal to 4 ? 
 
 4. Find the sixth root of 0.000000004096. 
 
 5. Find the value of *2 X (O0016)* . 
 
 a/108 
 
 6. Solve (!•)*= 17.4. 
 
 146. 
 
 1. If the base of a system of logarithms is — 6, find log 36, 
 
 log 1296, log ~*fr. 
 
 2. Find the value of (8.31)-°- 27 . 
 
 3. If </M7 : (0.84) 2 = -^0.054321 : x, find the value of x. 
 
 4. Find the value of (15-6)» X (0.0015)*. 
 
 (0.00065)* 
 
 5. Solve (i)» = i, 
 
 6. What sum will amount to $1000 in 10 years at 5 per 
 
 cent per annum compound interest ? 
 
74 EXAMINATION MANUAL. 
 
 147. 
 
 1. What is the logarithm of 512 to the base 2 V2? 
 
 2. What is the base of the system in which log 81 = — 4 ? 
 
 . „. , ., , , 1.265 X 0.01628 
 
 3. Fmd the value of —^-^j— 
 
 4. Find the mean proportional between 
 
 V4?756and ^(O0078)" 2 . 
 
 5. Find the value of \381 + V58. 
 
 6. In how many years will a given sum of money treble 
 
 itself, at 3 per cent per annum compound interest, 
 payable semi-annually? 
 
 148. 
 
 1. What is the logarithm of T ^ to the base 2 V3? 
 
 2. Find the value of V(24.4)"*. 
 
 3. Given the Napierian base 2.71828; find the Napierian 
 
 logarithms of 19, 23, 29, and 31, to five decimal 
 places. 
 
 4. Find the value of ( . ) • 
 
 W0.01/ 
 
 5. A man places $25,000 at 5 per cent compound interest, 
 
 and draws out § 1000 at the end of each year. What 
 will be due at the end of 12 years? 
 
 6. Solve 2* = 769. 
 
ALGEBRA. 75 
 
 149. 
 
 1. What is log 1 in any system ? Why ? 
 
 2. If the base of a system of logarithms is a, what is log a, 
 
 log-, log a 3 , log^a 7 , log—, logf- 
 
 3. Find the value of A/^4 + \/5. 
 
 9 347x-n/0.0073 
 
 4. Find the value of 
 
 5. Given the Napierian base e = 2.71828, log e 5 = 1.60943, 
 
 log e 11 = 2.39789; find to five decimal places the 
 common logarithms of 5 and 11. 
 
 150. 
 
 1. If the base of the system is a m , what is log a m , log a 5 ™, 
 
 1 Cb m 
 
 log-, log 1, log a™, log—? 
 
 2. Find the value of 4^5. 
 
 ^0.01 
 
 A^69 + 3 -yTl9 
 
 3. Find the value of 
 
 18V0.95 
 
 4. Find the modulus for changing common logarithms to 
 
 those whose base is 7. 
 
 5. Find to five decimal places log 54 in a system whose 
 
 base is 12. 
 
 6. A spendthrift, at the age of 30, comes into possession of 
 
 a fortune of $200,000, which pays 5 per cent. He 
 spends each year all his income, and also borrows 
 $2000 at 5 per cent compound interest. How old 
 will he be when his fortune is all gone ? 
 
EXAMINATION PAPEKS IN AIGEBEA. 
 
 I. 
 
 Bowdoin College, Brunswick, Me. 
 
 Examination for Admission, June, 1883. 
 
 1. Divide a^ar 5 + 5^-4^ -3 by x 2 -2x-2>. 
 Divide a 6 - b« by a 3 + 2a 2 b + 2ab 2 + b\ 
 
 2. Resolve x 16 — 3/ 16 into five factors. 
 
 3. Find the L.C.M. of (x + 2a) 3 , (x-2af, and (x 2 - 4a 2 ). 
 
 4. From subtract 
 
 a — x a? — x 2 
 
 5. Multiply together i^^ 2 , — ~-£, a ndl+— £-• 
 
 1 -j~ y x -j- x J. — x 
 
 6. (* + f)(*-f) + t = (*7 + 5)(*-3). Finda;. 
 
 7 437 + 81 R 12a? + 97 i' if. % -, 
 
 '• vk TTr ~ «• tt — ^t; — 4. Find a; and y. 
 
 lOy — 17 15y-17 y 
 
 8. Find the square root of x* — x 3 + — + 4 ar - 2 + —^ 
 
 9, —^ Reduce this to a fraction with ra- 
 
 1 + a-Vl-a 2 
 
 tional denominator. 
 
 10. Ja?-ia? + 7i=8. Find a:. 
 .r 6 - 8ar» = 513. Find ar. 
 
78 
 
 EXAMINATION PAPERS. 
 
 II. 
 
 Dartmouth College, Hanover, KH. 
 
 Entrance Examination, June, 1883. 
 
 1. Divide a 2 + — 2 — 2 by a. 
 
 a 2 J a 
 
 2. Resolve a 12 — x 12 into six factors. 
 
 3. Find the L.C.M. of a? - x, x 2 - x - 2, and r> + 1. 
 
 4. Reduce f „ - 2 — 4 _ 4 ) (y 2 + ar 2 ) to its sinplest form. 
 
 * q — ) = ~4T' 
 
 3 
 
 (iii.) • 
 
 (ii.) 
 
 x y 
 
 2 + 4 
 
 1 1_1 
 
 x^~ y~~ 4 
 
 2* y_ 9 
 
 a? y~12 
 
 6. Write the values of 8"!, 8°, 16*, and (8a*£ 2 )i 
 It Multiply together Vabc 2 , aib^cb, and a'SbSc 1 . 
 8. Divide a 3 - b 2 by a* - V&. 
 
 1. Multiply together 
 
 III. 
 
 Boston University, Boston, Mass. 
 
 Examination for Admission, June, 1882. 
 I-* 2 l-v 2 * 1 
 
 -£-, and 
 
 id sim- 
 
 a; 
 
 1+2/ a; + x 2 1 
 
 plify the result as much as possible. 
 
 n . y — 'z x + z 1 x — y x — 
 
 2. Given ^— -+- — -» — r-^ r 
 
 2 4 2 5 6 
 
 2HL? = a? + .V _ 4. Find the values of #, y, and z. 
 
 = 0, and 
 
ALGEBRA. 79 
 
 3. Simplify _JL. -_£- + ;£_. 
 
 4. Extract the square root of 40 x + 25 — 14 ar* + 9 x* -r- 24 s*. 
 
 5. Given 3 a? 2 — 2sy =15; 2# + 3y = 12. Find values of 
 
 a; and y. 
 
 v^X a/3 
 
 6. Express = with a single radical sign. 
 
 V2 
 
 7. Expand (1 — 2x 2 f by the binomial theorem. 
 
 IV. 
 
 Brown University, Providence, R.I. 
 
 Examination for Admission, June, 1883. 
 
 1. Factor x* — y* ; also, factor 4 a 4 — 8 a 3 # + 4 a 2 a 8 . 
 
 6#+7 2#-2 2# + l « ■, , £ 
 
 2. — — ; -r = — -i — Find value of x. 
 
 15 7x — 6 5 
 
 3. A sum of money is divided equally among a certain 
 
 number of persons; if there had been four more, 
 each would have received a dollar less than he did ; 
 if there had been five fewer, each would have re- 
 ceived two dollars more than he did ; find the num- 
 ber of persons and what each received. 
 
 4. Multiply a§ - c& + 1 - or I + cr\ by oh + 1 - or*. 
 
 5. Va-frz + Va — x = V&- Find value of x. 
 
 6. ;r + y = 4; --f.-s-l. Find values of x and y. 
 
 x y . . 
 
 7. A boat's crew row 3 } miles down a river and back again 
 
 in 1 hour and 40 minutes ; supposing the river to 
 have a current of 2 miles per hour, find the rate at 
 which the crew would row in still water. 
 
 8. Find sum of six terms of the geometrical progression of 
 
 which -§ is the first term and •§ the second term. 
 
80 EXAMINATION PAPERS. 
 
 V. 
 
 Mass. Institute of Technology, Boston, Mass. 
 
 Entrance Examination, June, 1882. 
 
 1. Factor the following expressions : 
 
 4a; 2 -12a;y + 9y 2 ; x 2 + 5cc + 6; x 3 -8f. 
 
 2. Find the G.C.D. of 
 
 2a; 3 - 4a; 2 - 13a; - 7 and 6a? - 11 x 2 - 37 x - 20. • 
 
 3. Find the L.C. M. of 4(1 + x), 4(1 -x), and 2(1 -x 2 ). 
 
 4. Simplify ^ + 5_2a-5 6a&_. 
 
 a — b a-\-b a 2 — b 2 
 
 m 
 
 5. Multiply a n + a 2 and Va together. 
 
 6. Solve the equation -^- + a + — = 0. 
 
 b — ex c 
 
 7. Solve the simultaneous equations 
 
 5 " 10 ~ U anct 6 + ~T~~ 3 ' 
 
 8. Extract the square root of x* — 2 x?y + 3 ^y 2 — 2 rzy 3 + y 4 . 
 
 9. Solve the quadratic equation x x ~ = — Hr — 
 
 ^ ^ 8a? -3 a; + l 
 
 10. Solve the simultaneous quadratic equations 
 
 -+- = 5 and i+~==13. 
 
 x y xr y 2 
 
 VI. 
 
 Entrance Examination, Sept., 1882. 
 
 1. Factor the following : 
 
 9m*-24m+16; x 2 ~2xy + y 2 - z 2 . 
 
 2. Find the G.C.D. of 
 
 12a; 3 - 9 a; 2 + 5x + 2 and 24a; 2 + 10a? + 1. 
 
 3. Find the L.C.M. of x 2 - 1 ; a; 2 + 2a;-3; 6a? 2 -a;-2. 
 
ALGEBRA. 81 
 
 4a ba 2 
 
 4. Simplify , 
 
 r J x — a \x — a) {x — a) 
 
 5. Show that 
 
 (a+bV^l)(a-bV^)=(a^b+V2aJ)(a+b--y/2a~b). 
 
 6. Solve the equation — ■ — - — = — 
 
 ox box 
 
 7. Solve the simultaneous equations 
 
 y — x 8 ' 7 
 
 8. Extract the square root of 
 
 ^ + 3^ + 6^+ 7a; 3 + 6^ + 3^+1. 
 
 9. Solve the quadratic equations 
 
 % + ~-f- = 0a,ndl9x i +216x 1 = x. 
 6 4 3a 
 
 10. Solve the simultaneous quadratic equations 
 
 xv , ., a b 
 
 - + 1 = 1, and - + - = 4. 
 
 a o x y 
 
 VII. 
 
 Harvard College, Cambridge, Mass. 
 
 Examination for Admission, June, 1883. 
 
 1. Solve the equation 1 = 2- 4aa'-3ft(*-2X 
 
 * x 2a(x 2 + l) + 3b 
 
 2. A man walks, at a regular rate of speed, on a road 
 
 which passes over a certain bridge, distant 21 miles 
 from the point which the man has reached at noon. 
 If his rate of speed were half a mile per hour 
 greater than it is, the time at which he crosses the 
 bridge would be an hour earlier than it is. Find 
 his actual rate of speed, and the time at which he 
 crosses the bridge. Explain the negative answer. 
 
82 EXAMINATION PAPERS. 
 
 3. Find the prime factors of the coefficient of the 6th term 
 
 of the 19th power of (a — b). What are the expo- 
 nents in the same term, and what is the sign ? 
 
 4. Eeduce the following fraction to its lowest terms : 
 
 X*-4:3? + 10^-12^+9' 
 
 5. Prove that, if a : b = c : d, ' = . = - = -• 
 
 c -fa c — d c d 
 
 6. Solve the equations, xy = 4 — y 2 ; 2x 2 — y 2 =17. Find 
 
 all the answers, and show what values of x and y 
 belong together. 
 
 VIII. 
 
 Yale College, New Haven, Ct. 
 
 Examination for Admission, Jy,ne, 1883. 
 1. Reduce the following expression to its simplest form: 
 
 x(x — a)(x — b) a(a—x)(a—b) b(b—x)(b — a) 
 
 2. Resolve y*—b 9 into three factors. 
 
 3. Change xy~ 2 — 2#*y~ 1 2T 1 -f z -1 to an expression which 
 
 will contain no negative exponents. 
 
 ^ lf a + h + c + d = a-b + c-d b the • ci les 
 
 a+b — c — d a — b — c + d 
 
 of proportion that -= — 
 b d 
 
 5. Find the value of 2 a Vl + # 2 when x = -^(■\t ~\~ )' 
 
 6. Given (7-4V3)* 2 + (2 - V3)rr = 2, to find x. 
 
 7. The sum of two numbers is 16, and the sum of their 
 
 reciprocals is \. What are the numbers ? 
 
ALGEBRA. 83 
 
 8. Compute the value of the continued fraction 
 1 • 
 
 2 + -U 
 
 * + l 
 
 9. Convert — into an infinite series by the method 
 
 of indeterminate coefficients, or by the binomial 
 theorem. 
 
 10. Insert three geometrical means between £ and 128. 
 
 IX. 
 
 Sheffield Scientific School, New Haven, Ct. 
 
 Entrance Examination, June, 1883. 
 
 Candidates for examination in this subject as a whole should 
 take the whole of this paper ; those for the first year's partial exam- 
 ination, the first part of it ; those for the second year's partial exam- 
 ination, the second part. 
 
 State what text-book you have studied, and to what extent. 
 
 I. 
 
 1. Reduce to their simplest forms the fractions 
 
 ,. x ac-\- bd+ ad-\-bc , ,•• n ax" 1 — bx m+1 
 ^ af+2bx + 2ax + bf W a'bx-b'x 3 ' 
 
 2. Given — ^ — ^—l bx = ae — 3 bx, to find x. 
 
 a a 
 
 3. A sum of money, at simple interest, amounted in m 
 
 years to a dollars, and in n years to b dollars. Find 
 the sum and the rate of interest. 
 
 4. Prove that if < 1 — -, and m is positive, then x<y> 
 
 7Tb X 
 
84 EXAMINATION PAPERS. 
 
 5. (i.) Simplify (a 2 b s )l + (aV)i 
 
 (ii.) Extract the square, root of 6 hm 2n -f h 2 -f 9 m 4n - 
 
 (iii.) Reduce . to an equivalent fraction 
 
 V# + a — V# — a 
 
 with a rational denominator. 
 
 II. 
 
 6. Given 15^ 3 — 20^ = 35, to find x. 
 
 7. Given a7 +^~ 9 _ ( x _ 2 ) 2 , to find ar. 
 
 a? — V# 2 — 9 
 
 8. Given x 2 — xy = 48 and xy — y 2 = 12, to find x and y. 
 
 9. The number of permutations of n things taken r to- 
 
 gether is equal to 10 times the number when taken 
 r — 1 together; and the number' of combinations of 
 n things taken r together is to the number when 
 taken r — 1 together as 5 to 3 ; required the value 
 of n and r. 
 
 3 4- 2x 
 10. Expand "f into a series of ascending powers of x, 
 
 by the method of indeterminate coefficients. (Four 
 terms of the series will be sufficient.) 
 
 X. 
 
 Entrance Examination, Sept., 1883. 
 (State what text-book you have studied, and to what extent.) 
 
 1. Given x ~~ y ^ = a and x ; y ~^ = b to find x and y. 
 
 x — y — 1 x-{-y — 1 * 
 
 2. Simplify (i.) V27 + 2 Vi8 + 3 VlOa 
 
 (ii.) (V^b)\V^W)\ 
 ,..., x 2p(q - 1) ~y 2q(p - 1) 
 ( 1U ') x p(q ~ 1} + y q{ 2- l) 
 
ALGEBRA. 85 
 
 Form an equation whose roots shall be 2 and — 3. Re- 
 solve x 2 — 3x-\-4. into two factors. 
 
 Given - 4- - = 5 and -= + - » = 13, to find x and y. 
 x ' y x 2 y 
 
 _ n . 3x+V4:X — x' 2 
 
 0. Given . = 2, to find x. ■% 
 
 Sx-V^x-x 2 
 
 6. To deduce a formula for the sum of a geometric pro- 
 
 gression in terms of the first term, the ratio, and the 
 number of terms. 
 
 7. Having 10 different letters, how many sets of two each 
 
 can you form of them, differing by at least one 
 letter ? 
 
 8. Expand into a series of ascending powers 
 
 1 — Ax -j- x 
 of x by the method of indeterminate coefficients. 
 
 (Four terms of the series will suffice.) 
 
 9. Express log \p— £- in a form adapted to computation. 
 
 10. To deduce a formula for the amount of a given sum 
 of money for a given time at a given rate of com- 
 pound interest. 
 
 XI. 
 
 Amherst College, Amherst, Mass. 
 
 Examination for Admission, June, 1883. 
 
 1. Find the value of 6a-[4Z>- {4a-(6a-45)}]. 
 
 2. Divide a~ Zn - b 6n by ar n - b 2n . 
 
 3. Show that a = 1 ; also, that a~ m = 1 -v- a m . 
 
 4. Resolve a 4m — ¥ m into its prime factors. 
 
86 EXAMINATION PAPERS. 
 
 5. Find the G.C.D. of a 4 - V and a 3 + a 2 b - ab 2 - b\ 
 
 6. Given ?>ax—2bx — \c — \mx=%c-\-\mx—ri—bx-\-2ax, 
 
 to find x. 
 
 7. Divide the number a into two parts, such that the sec- 
 * ond part shall equal m times the first part plus n. 
 
 8.'3y— 2a? = 9; 7x + y = 26 ; find x and y. 
 
 9. Multiply Va + c by Va+~c. 
 
 10. Zx 2 -4* = 15. Find*. 
 
 11. Expand (1 -J- # 2 ) 7 by the binomial formula. 
 
 12. Find the (2n) th term of the series 1, 3, 5, 7, 
 
 XII. 
 
 Williams College, Williamstown, Mass. 
 
 Entrance Examination, June, 1883. 
 
 1. Divide * 2 + i + 2 by x + -- 
 
 x 2 J x 
 
 2. Add the fractions and 
 
 2a-2b 2b-2a 
 
 3. Simplify 1 
 
 t+\ 
 
 f 
 
 4. The sum of two numbers is 5760, and their difference is 
 
 equal to one-third of the greater. Find the numbers. 
 
 5. -+| = 1; ~- + £ = !• Findxandy. 
 a b 3a 66 6 
 
 6. Solve the equation V3 x -\- 4 -f V3 ^ — 5 = 9. 
 
ALGEBEA. 87 
 
 4. Solve 3^- 8;r ~ 19 -8=^^+ 5:r ~ 88 + 10. 
 2 4 3 
 
 XIII. 
 
 Tufts College, College Hill, Mass. 
 
 Examination for Admission, June, 1881. 
 
 1. Divide 2 a m+l — 2 a n+1 — a m+n + a 2n by 2 a — a n . 
 
 2. Find the G.C.D. of ah + am, 6n -f- mn, and b 2 n — m 2 n. 
 
 4 1 
 
 3. Amplify _£ 
 
 3a?- 
 
 _ 2 
 
 5. Solve V4 -f a; = 4 — V#. 
 
 6. Solve I + ? = iland? + - = -- 
 
 x y 15 x y b 
 
 7. There are three numbers whose sum is 324 ; the second 
 
 exceeds the first as much as the third exceeds the 
 second; the first is to the third as five to seven. 
 What are the numbers ? 
 
 x h 
 
 8. - + - = c. Find the values of x. 
 a x 
 
 XIV. 
 
 Trinity College, Hartford, Ct. 
 
 Examination for Admission, June, 1883. 
 
 (One problem may be omitted in each of the three divisions indi- 
 cated by the letters A, B, C.) 
 
 A. 
 
 1. Find the G.C.D. of 2a; 2 + x- 1, x 2 + 5x + 4, anda^ + l. 
 
 o Q1 .i .• 6a;+7 2a;-2 2a; — 1 
 Z. bolve the equation — = . 
 
 H 15 7ar-6 5 
 
88 EXAMINATION PAPERS. 
 
 3. Two workmen -together finish some work in 20 days ; 
 
 but if the first had worked twice as fast and the 
 second half as fast, they would have done it in 15 
 days. How long would it take each alone to do the 
 
 work ? 
 
 B. 
 
 4. Multiply 2V Z ^3-3V = 2 by 4V^3+6V :r 2; di- 
 
 vide V— 5 by V— 1. Explain the process in each 
 case. 
 
 5. Solve the equation V#— 3 — V# — 14 — V4# — 155 == 0. 
 
 Give and explain the rule for solving a quadratic 
 equation. 
 
 6. Solve the equation _^ + 1 = _^L_ + -^ ; 
 
 also, x 4 + 4* 2 = 117. 
 
 C. 
 
 7. Find two numbers such that their product is 96, and 
 
 the difference of their cubes is to the cube of their 
 difference as 19 to 1. 
 
 8. In an arithmetical progression, a = 3, Z=42£, d = 2£) 
 
 find n and s. Explain the rule for the sum of a 
 geometrical progression. 
 
 9. Expand {a — b) n and ( I + 3 y J by the binomial theorem. 
 
 XV. 
 
 Wesleyan University, Middletown, Ct. 
 
 Examination for Admission, June, 1883. 
 
 1. When a = 1, b = 0, c— % find the value of 
 
 (Sa + 2)(2a-\-b)-a\2c-a[(3a 2 -\-6) + c]\. 
 Multiply a*a* + 2ai3* by 2tfx\-a\x\ 
 
ALGEBRA. 89 
 
 2. Factor ^-64; a 2 Z> 2 -3a& -4. # 
 
 3. Gl ven 7 g_l)-3(2 y +8) = 0;" 2 -^-™£±9 = 0; 
 
 to find x and y. 
 
 4. Solve at least two of the following : 
 
 (i.) 6a; 2 -13^ + 6 = 0; 
 
 (ii.) 3^a/^+-^= 16 ; 
 (-)^ + ^ = 8; i + i=| 
 
 How do you " complete the square " ? 
 
 re- 
 
 From a-vl z take & a| — , and express the 
 
 + 
 suit in its simplest form. 
 
 6. Write the repeating decimal 0.3 as the sum of a geomet- 
 rical progression. Find the limit of the sum. 
 
 XVI. 
 
 Cornell University, Ithaca, N.Y. 
 
 Entrance Examination, June, 1882. — Elementary Algebra. 
 
 1. Define : known and unknown quantities, positive and 
 negative quantities, addition, a common multiple of 
 two or more numbers, a radical, an equation, a 
 theorem. 
 
 i* into three prime factors. 
 
 3. Eeduce the fraction ~j~ $ ' ^± JL to an equivalent 
 
 ■Vx — y 
 fraction having a rational denominator. 
 
 4. Divide x + y + z — 3 ~Vxyz by a£ + y$ + zi 
 
90 • EXAMINATION PAPERS. 
 
 5. For $8 1 can buy 2 pounds of tea, 10 pounds of coffee, 
 
 and 20 pounds of sugar, or 3 pounds of tea, 5 pounds 
 of coffee, and 30 pounds of sugar, or 5 pounds of tea, 
 5 pounds of coffee, and 10 pounds of sugar. What 
 are the prices ? 
 
 r> a i x-u o,x — b , a bx bx — a 
 
 6. Solve the equation ■ f- - = — . 
 
 H 4 3 2 3 
 
 7. Solve the equation x -f 5 + V^+5 = 6, giving all the 
 
 roots. 
 
 8. Solve the equation x + a + x ~ 2a =l, and get the 
 
 ^ x—2a x+a 8 
 
 sum and the product of the two roots. 
 
 XVII. 
 
 Entrance Examination, June, 1882. — Advanced Algebra. 
 
 1. Prove the formula for the development of (a + x) n ; and 
 
 from this formula get the development of (l-f-'tf-f-rc 2 ) 3 , 
 and four terms of (a 2 — x 2 )~%. 
 
 2. Prove the formula for the sum of a geometrical progres- 
 
 sion, the first term, the ratio, and the number of 
 terms being given. From this formula obtain an 
 expression for the amount of a deferred annuity at 
 compound interest ; the annual payment, rate, and 
 time being given. 
 
 3. By the method of differences, find log 24, by continuing 
 
 the series : log 20 = 1.3010, log 21 = 1.3222, 
 log 22 = 1.3424, log 23- 1.3617. 
 From the same data find log21J'by interpolation. 
 
 4. Prove that log a 5 X log 6 # = log a #. 
 
 5. By the method of undetermined coefficients prove that, if 
 
 y = x + x 2 -f x 3 -}- , then also <c = y — y 2 + y 3 — 
 
 6. By continued fractions, find five successive approxima- 
 
 tions to the value of V2. 
 
ALGEBRA. 91 
 
 7. Depress the equation x*-\- 2x 3 + x 2 = 35 a; + 74, by re- 
 
 moving a commensurable root, and then find an in- 
 commensurable root correct to two decimal places. 
 
 8. Prove that, if the coefficients . of an equation with one 
 
 unknown quantity be real, any imaginary roots 
 enter it in pairs. 
 
 XVIII. 
 
 Stevens Institute of Technology, Hoboken, N.J. 
 
 Specimen Entrance Examination Paper. 
 
 1. Multiply V2a by -VSa. 
 
 2. Divide 1 — x by l-\- x to four terms. 
 
 3. Divide V3a by V2o?. 
 
 4. Solve l--gg-ng = 0andq- 2 y - 3 ==l. 
 
 by —c 1 
 
 Sx — 2y 
 
 5. Solves— — = 2: 4x — dy = 2. 
 
 oi/ — x 
 
 6. Solve -\/a — x — Vo — x = -y/c — x. 
 
 7. Solve Zxy — by 2 = 2; bxy + Sx 1 = l. 
 
 8. Solve * - = a; bx — *—= — - = ay. 
 
 y x 5 * • 
 
 9. Establish the equation for the permutations of n things 
 
 taken r at a time. 
 
 10. Prove that a proportion taken by division is a true 
 
 proportion. 
 
 11. If x varies as a-\-by, and when x = l, y = 2; and 
 
 x = 2, y = — 5; show that 7x = 9 — y. 
 
 12. In an arithmetical series, given the common difference, 
 
 first term, and number of terms, to find the sum and 
 the last term. 
 
 13. In a geometrical progression, given the first term, 
 
 number of terms, and the last term, to find the sum 
 of the terms. 
 
92 EXAMINATION PAPERS. 
 
 XIX. 
 
 Madison University, Hamilton, N.Y. 
 Entrance Examination, June, 1882. 
 
 1. Define : (i.) positive and negative quantities ; (ii.) sys- 
 
 tem of notation ; (iii.) similar terms ; (iv.) ratio ; 
 (v.) compound ratio ; (vi.) proportion. 
 
 m t m t m t 
 
 2. Multiply a*h~~* — a s *&~ s + l by aF l b~ r ° + l. 
 
 3. Eationalize the denominators of the following fractions : 
 
 1 aT 3 
 
 (2f £ (3)H(5) § 
 
 4. Given — == ~j~ — r~, to find the value of x. 
 
 2--M 40 + a* 
 
 5. Four given numbers are represented by a, b, c, and d ; 
 
 what quantity added to each will make them pro- 
 portional ? 
 
 6. Suppose a body to move eternally in this manner: viz., 
 
 20 miles the first minute, 19 miles the second min- 
 ute, 18^- miles the third minute, and so on in geo- 
 metrical progression ; what is the utmost distance it 
 can.reach ? 
 
 7. A hare, 50 of her leaps before a greyhound, takes 4 leaps 
 
 to the greyhound's 3 ; but 2 of the greyhound's 
 leaps are equal to 3 of the hare's. How many leaps 
 must the greyhound take to catch the hare ? 
 
 XX. 
 
 Vassar College, Poughkeepsie, N.Y. 
 
 Specimen Examination Paper for Admission. 
 
 1. Factor the following expressions : 
 
 9#* — y 2 ; ra 2 — 2ran + n 2 ; n s — n ; m i —n i ; x 3 + a s . 
 
ALGEBRA. 93 
 
 2. What is the rule for transposing a term from one side 
 
 of an equation to the other ; what is the principle ? 
 
 3. Solve the equation 
 
 a a-\-b a — b 
 
 27a~ 3 b 20 
 
 4. Find the fifth power of 2 a 2 ; the fourth root of • 
 
 lod u r 
 
 5. Square x ; cube 3 -f- V2". 
 
 6. A and B engage to mow a field. A alone can mow it 
 
 in b days, and B alone in c days. In what time can 
 both together mow it ? 
 
 7. Solve the equation V# + 4 — V# = V# + f . 
 
 8. Solve the the equations 
 
 aj»-^ = 2J5; ^ 2 + ^ + y 2 = 43. 
 
 9. Find four values of x in the equation x 2 + — = a 2 -f — 2 - 
 
 10. Form the equations whose roots are a, —a, and b. 
 
 11. Deduce the formula for the sum of n terms of a 
 
 geometrical progression. 
 
 XXI. 
 
 College of New Jersey, Princeton, N.J. 
 
 Examination for Admission, Sept., 1883. 
 
 1. What text-book have you used ? 
 
 2. Simplify the expression i 5 Ja* + P 
 
 b 
 
 a 
 
94 EXAMINATION PAPERS. 
 
 3. Solve the following equations : 
 
 5-6g I ^ + 14 - 17-3:r 4^ + 2 
 3 5 3 ' 
 
 2U+V*/ *-2 2* 3' 
 ^ + a^ = 6. 
 
 4. Solve the simultaneous equations : 
 
 ""Iff. V — i — a . 
 
 5. Divide x-^-.y + z — Z&ytzt by a£ -f- yi -}- zi. 
 
 6. Find the square root of 
 
 16a; 4 - 16a^ 2 + ISVx 1 + 4a 2 6 2 - 8a6 3 + 46 4 . 
 
 XXII. 
 
 John C. Green School of Science, Princeton, N.J. 
 
 Entrance Examination, Sept., 1883. 
 
 1. What text-book have you used, and how much of it have 
 
 you studied ? 
 
 2. Solve the following equations : 
 
 (3a — x)(a + 2x) = (5a + x)(a — 2x); 
 
 l-^zl-*Lz|; (*»-5) a + 29(^-5) = 96. 
 
 3. Solve the following pairs of simultaneous equations : 
 
 ^-y-31; 57+^ = 33; 
 h J ' 5 
 
 3» + ^2 = 85 ; ay = 42. 
 
 4. Form the quadratic equation whose roots are -| and -§. 
 
 5. Add v% \^, and \f. 
 
 6. Find the square root of# 4 +— - + 4a£— a? + 4a£— 2 a: - *. 
 
ALGEBRA.. 95 
 
 XXIII. 
 
 University op Pennsylvania, Philadelphia, Pa. 
 
 Examination for Admission, June, 1882. 
 
 1. Divide a; 6 - 9a: 5 -f 25a; 4 - Sx 3 - 98 a; 2 + 156 a; - 72 by 
 
 x 3 — 2a? — Ax -{-8 by synthetic division. 
 
 2. Simplify - — — ^ . + — -^ - + - — rr- 
 
 (a— b)(a— c) (b — a)(b — c) (c—a)(c—b) 
 
 3. Simplify 
 
 • 2^1^-^i7-2^^+2^4¥8-2^+3^875. 
 
 4. Find the difference between 2 + 1 -f I + to infinity, 
 
 and i + £ + I + to 12 terms. 
 
 5. Find the square root of 
 
 a; 8 -12^+62^ 6 -180^4-321^-360^+248^-96^+16. 
 
 6. Find the value of x from V2aT+6 + V2a;-5 = 11. 
 
 7. Find x and y from 
 
 x , 3 7 , 2 3 1 
 
 and 
 
 a; + 2 2/-3 12 a; + 2 y-3 12 
 
 8. Solve the equations x-\-y — z = 4, 2a;— 3y-f 4z = 20, 
 
 and 5 a; — 6y + 3 z = 10. 
 
 9. Find a; from \ = 1+1+1. 
 
 a-\- b + x a o x 
 
 10. Solve the equations #*+ 10 xy = 11 and 5xy — 2>y 2 = 2. 
 
 XXIV. 
 
 University of Michigan, Ann Arbor, Mich. 
 
 Examination for Admission, June, 1883. 
 
 Define exponent. Illustrate the significance of fractional 
 and negative exponents. Write ^/ax 2 y~ m without 
 using the radical sign. 
 
96 EXAMINATION PAPERS. 
 
 2. Define elimination. What method do you prefer, and 
 
 why ? Apply to the equations 
 a , b c , d 
 
 x y x y 
 
 3. Expand (a 2 + x 2 )% by the binomial formula. 
 
 4. Solve the equation 3V5-4 = 15 + VSs 
 
 2 + Va? 40 + Val 
 
 5. Produce the formulae for last term and swra in arith- 
 
 metical and geometrical progression. 
 
 6. Simplify ^ EL, * fLL.. 
 
 r J a; a; 2 
 
 7. Rationalize the denominator of 
 
 •vV + a?+r+ yV + a: — 1 
 V^ 2 + ^ + l- V^ + a;-l 
 
 8. Solve the simultaneous equations 
 
 4(^ + 2/) = 3o;y; # + y + a; 2 + y 2 = 26. 
 
 9. Solve the simultaneous equations 
 
 a; 2 + n*y + y 1 = 52 ; #?/ — x 2 = 8. 
 10. Define logarithm, mantissa, characteristic. How can 
 you extract roots by logarithms ? 
 Given log x = 2.301030, what is loga;i? 
 
 XXV. 
 
 Lake Forest University, Lake Forest, III. 
 
 Entrance Examination, June, 1882. 
 
 1. Define coefficient, exponent, similar quantities, monomial, 
 
 binomial, equation. 
 
 2. Find the G.C.D. of 
 
 a; 4 + 2a; 2 + 9and 7x z - 11a* + 15a? + 9. 
 
ALGEBRA. 97 
 
 3. Find the value of x in 
 
 Sx — 1 . 5 — x 2^— -4 __g ff-f 2 
 
 7^4 12 ~ 28 ' 
 
 4. Explain the rule for subtraction, showing why the 
 
 signs of the subtrahend are changed. Illustrate by 
 diagram or numbers. 
 
 5. A man rows a boat with the tide 8 miles in 48 minutes, 
 
 and returns against a tide two-fifths as strong in 80 
 minutes, what is the rate of the stronger tide? 
 
 6. The product of two numbers is 702, and their sum is 60. 
 
 Find the numbers. 
 
 7. Factor x*-2x-3 and ^ -^- 13a; + 24. 
 
 a , -y/a—x Va — # ,- 
 
 o. bolve = -y/x. 
 
 x a 
 
 9. Solve a; -f 5V37-z=:43. 
 10. What number added to its reciprocal makes 2.9? 
 
 XXVI. 
 
 Education Department, Ontario. 
 
 Examination of Third Class Teachers, July, 1883. 
 
 1. Divide 
 
 (i.) (a-b)c 3 + (b-c)a* + (c-a)b z by 
 (a — b)(b — c) (c — a)) 
 
 ^ £±£_£±£ h i_i. 
 
 x 3 ^ x 2 ^ x y 
 
 2. "What must be the values of a, b, and c, that x 3 -f- ax 2 
 
 -\-bx-\- c may have x—1, x—2, and x — 3 all as 
 factors ? 
 
 3. Find the H.C.F. of 
 
 (i.) 3x*-4:X 3 + l{m&4:X*-5x i -x 2 + x + l; 
 (ii.) 8x s -if+27z 3 +l8x!/z and 4o*+12a?2+ 9z 2 -y\ 
 
98 EXAMINATION PAPERS. 
 
 4. Simplify 
 
 (n /4a» -A/ 2x {\,(Z* {\f 4s»+2sy 1 \, 
 W Vy 2 A2^~y / W A^+^y+y 2 /' 
 
 rin 3»+(q + 6)3» + (a6 + l)tt+6 
 
 k ' } bx^ + (ab-{-V)^+{a + b)x+l 
 
 5. Find the value of x that will make <^ + ^+ad+bc 
 
 independent of c and d. 
 
 6. (i.) If « + 6 + c = 0, then -I + l+-L= ji+l+I} 2 . 
 
 a 2 o 2 c 2 C a 6 c ) 
 (ii.) If x = a 2 + b 2 -f c 2 and y = «6 -f- 5c + ra, 
 
 then rr 3 + 2tf - Sxf = (a 3 + 6 3 + c 3 - Zabc) 2 . 
 (iii.) If 2a = y -j-z, 25 = 2; + a:, 2c = .T + y, express 
 (a + 5 + c) 3 - 2 (a + 5 + c) (a 2 + 6 2 + c 2 ) 
 in terms of or, y, and 2;. 
 
 7. Find a value of a which will make the quantities 
 
 (a+b)(a + c) and (a + c)(a + d) 
 
 a-\-b -{- c a-\- c-\- d 
 
 equal to one another. 
 
 8. Solve the equations 
 
 (i.) V^r + 3 + Vx + 2 = 5; 
 
 ,.. s b—x . b — 2x . ar-f-1 2-\-5x n 
 
 (11. ) ! ! = : 
 
 V ; 3 ^ 4 ^ 3 2 
 
 (iii.) O + a + $)(<? + <*) = <> +c + d)(a ■+ 5), 
 
 where c? + c? is not equal to a -f &. 
 
 9. One side of a right-angled triangle exceeds the other 
 
 by 3 feet, neither being the hypotenuse, and its area 
 is 18 square feet. What are the sides ? 
 10. A cistern with vertical sides is h feet deep. Water is 
 carried away from it by one pipe |- as fast as it is 
 supplied by another. Find at what point in the 
 side the former pipe must be inserted that the cis- 
 tern may fill in twice the time it would did water 
 not flow from it at all. 
 
ALGEBRA. 99 
 
 XXVII. 
 
 University of Toronto, Toronto, Ont. 
 
 Junior Matriculation. — Annual Examination for Honors in Algebra, 
 
 1883. 
 
 1. Find the product of (a + b), (a 2 + oh + b 2 ), (a - b), 
 
 and (a 2 — ab-\- b 2 ). 
 
 2. If a and b are positive integers, show that 
 
 x a x x h = x a+h . 
 
 3. Prove the rule for finding the G.C.M. of two quantities. 
 Find the G.C.M. of 6^+ Wx 4 y- 4^V- lO^yz 2 and 
 
 9x 3 y - 27 x l yz — 6xyz 2 + 18yz 3 . 
 
 4. State the rule for extracting the square root of a com- 
 
 pound quantity. 
 Extract the square root of x 2 -f- y. 
 
 5. Solve the following equations : 
 
 (i.) 3a?-M = li; 2y + 3z = 16, 5> + 4y = 35; 
 
 (ii.) 
 
 x -\- a x -\-b 
 x — a x— b 
 
 (in.) _ + _ = 2+- 
 ax x 
 
 6. When are quantities said to be in geometrical progres- 
 sion, when in harmonical progression, and when in 
 arithmetical progression ? (i.) Find two harmonical 
 means between a and b. (ii.) The first term of a 
 geometric series is •£■, the ratio J, and the number of 
 terms is 6 ; find the sum of the series. 
 
100 EXAMINATION PAPERS. 
 
 7. Show that the number of combinations of n things taken 
 
 r together is 
 
 n(n—I)(n — 2) (n - r -f 1) 
 
 1x2x3 r 
 
 How many words of four letters can be formed out of 
 the first 13 letters of the alphabet, having one vowel 
 in each word ? 
 
 8. Expand to five terms (a -f 5)~*. 
 Show that (ji|Y 
 
 -l + ^ + |(^ 2 + ^) + |(^ + ^) + T ^(^+^) + 
 
 9. A number consists of two digits : when the number is 
 
 divided by their sum the quotient is 4, and when 
 divided by their difference the quotient is 12'; find 
 the number. 
 10. The crew of a boat rowed six miles down a river, and 
 half-way back again, in 2 hours. Supposing the 
 stream to have a current 2\ miles an hour, find at 
 what rate they would row in still water. 
 
 XXVIII. 
 
 College of Ottawa, Ottawa, Can. 
 
 Matriculation Examination. Session 1882-83. 
 
 1. Translate the following into common language : 
 
 l + 2a , + * 
 
 2. Divide (i.) 2a 2 b + b d + a 3 -f 2ab 2 by a 2 + b 2 + ab ; 
 
 (ii.) 12w~ 2 y~ 4 by —4 xy 2 . 
 
 3. Find the prime factors of 25<? 2 c? 4 — 9a 6 c 4 . 
 
 Find the G.C.D. and L.C.M. of 9mx 2 —6mx + m and 
 9 nx % — n. 
 
ALGEBRA. J.01 
 
 4. Find the sum of 
 
 a-\- b , b-\- c , 
 
 ~r 7 rz 7T ~r 
 
 (b — c)(c — a) (c — a) (a—b) (a — b)(b — c) 
 Find the algebraic sum of 
 
 __3 7 4-20a; 
 
 \-2x 2x+l 4a*- 1 
 
 r- a , 3a; — 3 , A 20-x 6a; — 8 , 4a;-4 
 
 5, Solve, 5~ + ^— 2 -^- + -^- 
 
 Find a;, y, and 'z in the following equations : 
 
 # + 2/ + ^ = 24 ; x — ?/ -f z = 8 ; a; + 3/ — z = 6. 
 
 6. Raise — — - — %- to the cube. 
 
 7. Give the simplest form of "\3V3. 
 
 8. Show why the square may be completed in the quad- 
 
 ratic 3 a; 2 — 7 a; = 20, by the same rule as in x 2 + 2x 
 = 24, without introducing fractions. 
 
 9. Given x 2 - 2xy = 12 and x 2 -y 2 = 12, to find x and y. 
 
 10. A boy being asked how many sheep his father had, 
 replied that f of \ the flock would be 25 less than 
 the whole flock. How many sheep had his father ? 
 
 XXIX. 
 
 College of Ottawa, Ottawa, Can. 
 
 Matriculation Examination, Session 1883-84. 
 
 1. Clear away the parentheses, and reduce the following 
 expression : 
 
 a + b-(2a-Sb)~4:(5a + 7b)~(-lSa + 2b) 
 + S{a-e>(b-a)l 
 
102 EXAMINATION PAPERS. 
 
 2. Give the three formulas for the expansion of (a -f- b) 2 , 
 
 (a — b) 2 , and (a + b)(a — b), and give an example 
 for each formula. 
 
 3. Divide 5a?- 3 - 4 a; 2 + # 4 + # 3 by -3 + x 2 -2x. 
 
 4. Find the G.C.D. and the L.C.M. of the three following 
 
 expressions : 
 
 (2ar-4)(3a?-6); (s-3)(4a;-8).; (2a?-6)(5a?-10). 
 
 wi 2 + w 2 
 
 . Simplify , , X 
 
 n m 
 
 6. Solve the equations 
 
 2x+4y-Sz = 22; 4:X-2y+5z = 18 ; 5x+7y-z = 63. 
 
 7. Extract the square root of 
 
 15a 4 b 2 + a 6 - 6 a 5 5 - 20a 3 5 8 + b« + 15a 2 6 4 - 6aZ> 5 . 
 
 8. Convert V I into such an expression, not a decimal, as 
 
 shall not necessitate two extractions in finding the 
 cube root of f . 
 
 9. Solve the following equation : I x 2 — lx + 20 1 = 421 . 
 
 10. The hypotenuse of a right-angled triangle is 20 feet, 
 
 and the area of the triangle is 96 square feet. Find 
 the length of the legs. 
 
 11. Find the tenth term, and the sum of ten terms, of the 
 
 series 1, 4, 10, 20, 35. 
 
 12. Develop — into an infinite series by the method 
 
 of undetermined coefficients. 
 
 13. Find the value of x in the equation 5 X == 30. 
 
ALGEBRA. 103 
 
 XXX. 
 
 McGill University, Montreal, Can. 
 
 School Examination, June, 1883. 
 
 1. Multiply 1 + 2a? — a?— ?# 3 by itself, and find the value 
 
 of the result if 1 — 2x = 3. 
 
 2. Find the remainder when a 5 -4a 3 6 2 -8a 2 6 3 -17a& 4 -156 5 
 
 is divided by a 2 — 2 ab — 3 b 2 . 
 
 3. Simplify ix(x+V)\x+2-l(2x+l)\' t Zg^g+l. 
 
 4. Reduce the following fractions to their lowest terms : 
 
 a 2 x-{-a 3 ( _ (#*— q 4 )(ff— a) t 1-f-^ 3 
 
 ax 2 - a 3 '' {x 2 +a 2 -2ax){ax+x 2 Y l+2x+2x , +a? 
 
 5. Find the square root of 
 
 ^ 4 +2^ 3 -^ + landof 4 ^- 4a? + l . 
 4 9x 2 + 6a; + l 
 
 6. Solve the equations 
 
 (i.) 2x — - = 18 ; (ii.) (m + w) (m — x) = m (n—x) ; 
 (iii.) 2^-^ = 4; 3y + ^ = 9. 
 
 7. If aa^ -\-bx-\-c becomes 8, 22, 42, respectively, when x 
 
 becomes 2, 3, 4, what will it become when x = — \ ? 
 
 8. Find two numbers which produce the same result, 7, 
 
 whether one be subtracted from the other, or the 
 latter be divided by the former. 
 
 9. In a certain school there are 6 boys to every 5 girls ; if 
 
 there were 2 boys less and 2 girls more, there would 
 be the same number of each. Find the number. 
 
 10. Any odd number may be represented by 2v+l. Prove 
 that the difference of the squares of any two odd 
 numbers is exactly divisible by 8. 
 
104 EXAMINATION PAPERS. 
 
 XXXI. 
 
 University of Cambridge, Eng. 
 
 Second General Examination for the Ordinary B.A. Degree, Nov., 1880. 
 Time allowed, 3 hours. 
 
 , I a1 ,, ,. r , x-2 a; + 23 10 + a; 
 
 1. Solve the equations (i.) x -— — = — ~ — ; 
 
 o 4 5 
 
 (ii.)-2_=-$-_; (iii.) 1 + 1 = 43, 2 + | = 42. 
 
 x—b x—a 9 8 8 9 
 
 2. Solve the quadratic ax 2 + &£ + c = 0, and determine 
 
 the condition that its roots may be equal. 
 If a,* ft be the roots, form an equation whose roots are 
 
 - and -• 
 a jS 
 
 3. Solve the equations 
 
 r v Sx + 2 , 3a; — 2 4a; 2 + 12a; + 2 
 
 (l.) ! ■ ! — ; 
 
 k ; x-3 x+S x 2 -9 
 
 (ii.) Vx + Va + x — —z I 
 
 (iii.) (* + 2y) (2a; + y) = 20, 4ar (a; + y) = 16 - ?/ 2 . 
 
 4. The first term of an arithmetical progression of n terms 
 
 is a, and the last term I. Find the sum, and also 
 the common difference. 
 If n be odd, and the sum of the even terms be sub- 
 tracted from the sum of the odd, show that the 
 
 result is • 
 
 2 
 
 5. Find the sum of n terms of a series in geometrical pro- 
 
 gression. 
 If the sum of a geometrical series to infinity be n times 
 
 the first term, show that the ratio is 1 . 
 
 n 
 
 6. Find the sum of the series : 
 
 (i.) 2 +2£ + 2* + , to 12 terms; 
 
 (ii.) i— I + f — , to 8 terms; 
 
 (iii.) 31 + 21 + H + , to infinity. 
 
ALGEBRA. 105 
 
 7. Show that a ratio of greater inequality is increased by 
 
 taking the same quantity from both its terms. 
 Show that the ratio a — x : a -j- x is greater or less than 
 the ratio a 2 — x z : a 2 + x 2 , according as the ratio a : x 
 is one of less or greater inequality. 
 
 8. Define proportion. When are quantities said to be in 
 
 continued proportion ? 
 If a, b, c, d be in continued proportion, show that 
 f a -b V__ a 
 [b - c) d 
 
 9. When is one quantity said to vary directly and when 
 
 inversely as another? 
 The volume of a sphere varies as the cube of its radius : 
 if three spheres of radii 9, 12, 15 inches be melted 
 and formed into a single sphere, find its radius. 
 .10. A and B start simultaneously from two towns to meet 
 one another. A travels 2 miles per hour faster 
 than B, and they meet in 7 hours ; if B had trav- 
 elled 1 mile per hour faster, and A at only half his 
 previous pace, they would have met in 9 hours. 
 Find the distance between the towns. 
 
 11. A wine-merchant buys spirit, and after mixing water 
 
 with it, sells the mixture at two shillings per gallon 
 more than he paid for the spirit, making 23 1 per 
 cent on his outlay : if he had used double the quan- 
 tity of water he would have made 37 ? per cent ; 
 what proportion of water was there in the mixture ? 
 
 12. Two elevens, A and B, play a cricket match. A's first 
 
 innings is the square of the difference of B's two 
 innings, and A's second one-third the sum of B's 
 two innings ; A scored 60 more their first innings 
 than in their second, and lost the match by one run. 
 What were the respective scores, B having first 
 innings ? 
 
106 EXAMINATION PAPEES. 
 
 XXXII. 
 
 University of Cambridge, Eng. 
 
 Second Previous Examination, Dec, 1880. — Time allowed, 2£ hours. 
 
 1. Define coefficient, term. 
 
 Find the coefficient of x in the expression 
 
 ^_|2a-6( C -*)j. 
 
 2. Find the continued product of 
 
 x 2 + 3x + 2, x 2 — 5# + 6, ^ + 2^ — 3, 
 
 and multiply together 
 
 ^ + (V2-l)o; + l, * 2 -(V2 + l> + l. 
 
 3. Divide x*-(b-2)x i -(2b-l)x' i -(b 2 +2b-8)x+3b+Z 
 
 by af + Zx + b + l. 
 
 4. Simplify (i.) -X--J 
 
 (ii.) 
 
 x — 2 x' 2 — 3# + 2 r 5 — 4# + 3 
 l+x 1+x 2 
 1 + x 2 1 + a 3 
 1 + x l 1 + x 3 ' 
 
 1 + a 8 1 + ** 
 
 If a measures both 6 and c, prove that it will measure 
 
 the sum of any multiples of b and c. 
 Find the G.C.M. of 1 + x + x z - x> and 1 - **- x e + x\ 
 Solve the equations 
 
 — 5 x 
 
 I 
 
 1 
 
 , 3x — 5 # + l_o 
 
 (lL) 2<>+3)~30 + 2) ' 60+1)' 
 (m.)i(x + i/\=i(x-y\ 3z+lly = 4; 
 
 (iv.) 3* 2 + l = ?~; 
 
 f , x+a . b x 2 + ab . 
 
 (v.) — -7-7 + -= ' 
 
 a + 6 a ax 
 
 (vi.) 0+1) (y + 2) = 10, zy = 3. 
 
ALGEBRA. 107 
 
 7. If a + b = 1, prove that (a 2 - b 2 ) 2 = a 3 + b* - ab. 
 
 8. If j = -, prove that each of these fractions is equal to 
 
 b a 
 a-\- one 
 b-\-md 
 If a + b, b-\-c, c-\-a are in continued proportion, prove 
 that b -\- c, c -\- a, c — a, a — b are proportionals. 
 
 9. When is one quantity said to vary as another ? 
 
 If - + - varies inversely as x + y, prove that x 2 + y 2 
 x y 
 
 varies as xy. 
 
 XXXIII. 
 
 University of Cambridge, Eng. 
 
 General Examination for the Ordinary B.A. Degree, June, 1881. 
 Time allowed, 3 hours. 
 
 1. Prove that (x + 4) 3 - (x + l) 3 = 9 (x + 1) (x + 4) + 27. 
 
 2. Simplify 
 
 (l0 (o + a +c) A + i + iW* + «0(« + *)(« + »). 
 
 w v ' \a x bej abc 
 
 r . x a 2 6 2 
 
 6 2 - a 2 - 6 2 + 1 
 
 a& — a — b -f- 1 
 
 3. Find the G.C.M. of ^-4^ 2 +2a;+3 and 2x i -x 2 -5x-3. 
 
 4. Prove that, if on and n be positive integers, 
 
 (ar) n = (a n )"\ 
 Prove that (V3) 3V3 - (3 V3) V3 . 
 6. Solve the equations 
 
 (i.) i(5* + l) + ^3 = z + J| ; 
 
 (n.)-i- + r J- = ?L±l. 
 
 (iii.) ax -f 6y = 2, ab (x + y) = a + b. 
 
108 EXAMINATION PAPERS. 
 
 6. Show that the product of the roots of the equation 
 
 ax 2 + bx -f- c = is — 
 a 
 
 Prove that the difference of the roots of the equation 
 x 2 + px + q = is equal to the difference of the 
 roots of the equation x 2 + Spx + 2p 2 -f q — 0. 
 
 7. Solve the equations 
 
 (i.) (x-2) 2 + (x + b) 2 = (x+7) 2 ; 
 (ii.) ax 2 -{- 2bx = a — 2b ; 
 (iii.) V# + 2/ = Vy + 2, x — y = 7. 
 
 8. Find the sum of w terms of a G.P. of which the first 
 
 and second terms are a and b. 
 If each term of a G.P. be squared, prove that the new 
 series will also form a G.P. 
 
 9. If f = 4 prove that each of these fractions is equal to 
 
 pa + qc 
 pb-\-qd 
 
 10. What is meant by saying that A varies as B ? 
 
 If the volume of a cone varies jointly as its height, 
 and the square of the radius of its base, show that 
 if the heights of three cones of equal volume are in 
 continued proportion, so also are the radii of their 
 
 11. Find a fraction such that the denominator exceeds the 
 
 square of half the numerator by unity, and the prod- 
 uct of the sum and difference of the numerator 
 and denominator is 64. 
 
 12. A vessel is half full of a mixture of wine and water. 
 
 If filled up with water, the quantity of water bears 
 to that of wine a ratio nine times what it would be 
 were the vessel filled up with wine. Determine the 
 original quantities of wine and water. 
 
ALGEBRA. 109 
 
 XXXIV. 
 
 University of Cambridge, Eng. 
 
 Previous Examination, June, 1881. — Time allowed, 2\ hours. 
 Elementary Algebra. 
 
 1. Simplify - of - (x - b) 2 - (a - b) (a + b — 2>x) ; 
 
 and find the value of 
 
 S(a-b)(a + 2b) + 2(a-2b)(2a-b) + 2(2a-by, 
 
 when a = and b = — 2. 
 
 2. Divide 7a 3 - 22a 2 6 + 4a6 2 - 36 3 by a -36. 
 
 3. Resolve into factors 
 
 a; 2 -2a;-255; 21a^-13a;y-20y 2 ; (o; + 2y) 3 -y 3 . 
 
 4. Simplify 
 
 (i) a 2 - 16a;— 17 . 
 lLj a; 2 - 22a; + 85' 
 
 5. Find the L.C.M. of 
 
 (a* - tf), (2x 2 - Zxy ; + y 2 ), and (a; 3 + x 2 y + xy>). 
 
 6. Solve the equations 
 
 00 ~2~ + ~^ 5~ + 5 ' 
 
 (ii.) 2a?(7a?-10) = 13(a?-l); 
 
 (iii.) 7 + ^|+ 4 = 0; 
 
 x — 6 (x— l)(x— 3) 
 
 (iv.) 5abx + 2y=ie>b, 3abx + 4:y=18b ; 
 (v.) a; 2 + 3a;y = - 8, y 2 -xy =12. 
 
 7. Extract the square root of 
 
 9 a; 4 - 6a; 3 + 43a; 2 - 14a; + 49. 
 
 8. Prove that (a m ) n = a mn . 
 
 a rr (a p - q ) p + q X (a q ) q+r 
 
 bimplifv * —, — r — - — 
 
 1 J (a p ) p - q 
 
110 EXAMINATION PAPERS. 
 
 9. If a and ft be the roots of the equation x 2 —px + q = 0, 
 then will p — a + ft and q = a/3. 
 Form the equation whose roots are 27 and — 13. 
 
 10. When are three quantities said to be in continued pro- 
 
 portion ? 
 Show that if x, (x-\-y), and (x -f- 2y -f z) be in con- 
 tinued proportion, then x, y, z will also be in con- 
 tinued proportion. 
 
 11. Prove that if x oc y and y cc z, then will x cc z. 
 
 Given that x varies inversely as (y* — 1), and is equal 
 to 24 when y = 10 ; find x when y = 5. 
 
 XXXV. 
 
 University of Cambridge, Eng. 
 
 Previous Examination, June, 1881. — Time allowed, 2J hours. 
 Higher Algebra. 
 
 1. If the first two terms of an arithmetical progression 
 
 are given, find the sum of the first n terms. 
 The sum of n terms of an arithmetical progression, 
 whose first two terms are 43, 45, is equal to the sum 
 of 2n terms of another progression, whose first two 
 terms are 45, 43 ; find the value of n. 
 
 2. Find the sum of n terms of a geometrical progression 
 
 whose first term and common ratio are given. 
 The sum of 2w terms of a geometrical progression, 
 whose first term is a and common ratio r, is equal to 
 the sum of n terms of a progression, whose first 
 term is b and common ratio r 2 ; prove that b must 
 be equal to the sum of the first two terms of the 
 first series. 
 
ALGEBRA. Ill 
 
 3. Sum the following series to 12 terms : 
 
 (i.) !-!-¥- 
 
 (ii.) ! + * + «+ 
 
 (iii.) 1-1.2 + 1.44- 
 
 How many strokes are struck in a week by a clock 
 that tells the hours? 
 
 4. If the sum of the first n terms of a series be 32 n 2 , find 
 
 the rth term. 
 
 5. For what values of m is x m -f- y m divisible by x -f- y ? 
 Divide a? + Sa 2 b + 3ab 2 + b 3 + c* by a + b + c. 
 
 6. A clock gains 4 minutes per day ; what time should it 
 
 indicate at 6 o'clock in the morning in order that it 
 may be right at 7.15 p.m. on the same day ? 
 
 7. The first four nights of the boat-races both divisions 
 
 rowed, and 32 bumps in all were made. The greatest 
 number on one evening, in the first division, was 
 reached twice, and was equal to the least number in 
 the second division, which also occurred twice. This 
 number is the middle one of five consecutive num- 
 bers, of which the first two represent the number of 
 bumps the other two nights in the first division, and 
 the last two represent the other bumps of the second 
 division. How many bumps were made in the first 
 division ? 
 
 8. Define the logarithm of a number to a given base. Prove 
 
 ivyi 
 
 log a - = log a m — \og a n; 
 log a Mog 6 a = l. 
 
 9. Find the values of 
 
 log, a", log 343 -v/49, log 3 0.027. 
 
112 EXAMINATION PAPERS. 
 
 10. Having given log 2 =0.3010300; 
 log 3 =0.4771213; 
 log 4.239 = 0.627263; 
 Jog 4.24 =0.627366; 
 2 128 3i 
 
 find the value of 
 
 10 3 
 
 XXXVI. 
 
 University of Cambridge, Eng. 
 
 Second General Examination for the Ordinary B.A. Degree, 
 Nov., 1881. — Time Allowed, 3 hours. 
 
 1. Solve the equations 
 
 M 4a; 2 +13a;+14 4a; -f- 5 
 
 ( _2^+ll__^ _J 1_ 
 
 v J x*-x+l a?+l x+S x-S 
 (iii.) a;-lly = l, Illy -9a; = 99. 
 
 2. Find two consecutive numbers, such that the fourth 
 
 and eleventh parts of the less together exceed by 1 
 the fifth and ninth parts of the greater. 
 
 3. A certain number of two digits is multiplied by 4, and 
 
 the product is less by 3 than the number formed by 
 inverting its digits ; if it be multiplied by 5, the 
 tens' digit in the product is greater by 1, and the 
 units' digit less by 2 than the units' digit in the orig- 
 inal number: find the number. 
 
 4. Solve the equations 
 
 (i.) 3a; 2 -lla;-4 = 0; 
 (ii.) V7ar+1 = 3+ V2a?-1; 
 (iii.) x 2 + xy - 2 if = - 44, xy + Zy 2 = 80. 
 
ALGEBRA. 113 
 
 5. If the greater sides of a rectangle be diminished by 3 
 
 yards, and the less by 1 yard, its area is halved. 
 If the greater be increased by 9, and the less dimin- 
 ished by 2, the area is unaltered ; find the sides. 
 
 6. If the number of pence which a dozen apples cost is 
 
 greater by 2 than twice the number of apples which 
 can be bought for Is., how many can be bought for 
 9 s. 
 
 7. Define ratio. If a be less than b, show that a : b is a 
 
 less ratio than a + 1:5 + 1. What is the least 
 integer which must be added to the terms of the 
 ratio 9 : 23, so as to make it greater than the ratio 
 7:11? 
 
 8. The first and fourth terms of a proportion are 5 and 
 
 54; the sum of the second and third terms is 51; 
 find them. 
 
 9. If A varies directly as P, inversely as Q, and directly 
 
 as P, and, if when P= a, Q = b, P = c, A = abc, 
 
 find A when P=> Q = ™ R = a ±- 
 abc 
 
 10. Find the sum of n terms of a geometrical progression, 
 
 of which the first term is a and the common ratio r. 
 Sum: 
 
 64 + 641 + to 29 terms in arithmetical progression; 
 
 64 + 96 + to 7 terms in geometrical progression. 
 
 11. The common difference of an arithmetical progression 
 
 is 2, and the square roots of the first, third, and 
 sixth terms are in arithmetical progression; find the 
 series. 
 
 12. The sum of four numbers in geometrical progression 
 
 is 170, and the third exceeds the first by 30 ; find 
 them. . 
 
114 EXAMINATION PAPERS. 
 
 XXXVII. 
 
 University of Cambridge, Eng. 
 
 Second Previous Examination, Dec, 1881. — Time Allowed, 2\ hours. 
 
 1. Simplify 6(a-2b)(b-2a)- (a-Sb) (46- a) -12ab, 
 
 and from the sum of (2 a — b) 2 and [a — 2 b) 2 take 
 the square of 2 (a — b). 
 
 2. Define multiplication, product, and coefficient. 
 Divide 14 a 4 + 15 a 3 b + 33 a 2 b 2 + 36 a£ 3 + 28 b* 
 
 by 7a. 2 -3a£+146 2 . 
 
 3. Find the value of (a-b) 2 +(b-c) 2 +(a-b)(b-c) + bc 2 
 
 when a = 1, 5 = — 2, c = J. 
 
 4. Eesolve into the simplest possible factors : 
 
 (i.) Sof + bxy-by 2 ; 
 
 (ii.) x^-lSx'y + ^xy 2 ; 
 (iii.) (a + 2b + 3c) 2 -4(a + b-c) 2 ; 
 (iv.) 81a; 4 -625y 4 . 
 
 5. Define the highest common factor of two algebraical 
 
 expressions. 
 Find the highest common factor of 
 
 7a 8 - 10a*- 7*+ 10 and 2x 3 -x 2 -2x-{-l. 
 
 6. Reduce to simple fractions in their lowest terms : 
 
 (1 * ; x 2 + bxy + 6y 2 ' x 2 + xy-2y 2 ' 
 
 (" \ x ~ a | a 2 + 3 q£ . x -f- a t 
 x + a a 2 — x 2 x — a' 
 ab 1_1 
 
 ,... x a + 6 a 2 6 2 
 
 ^•)t-^^ x t~t- 
 
 ° + a 2 - b* a b • 
 
ALGEBRA. 115 
 
 7. Solve the equations 
 
 {l ' } 5 9 7' 
 
 (^1 + 1=14 | + |=24; 
 
 (iii.) bx 2 - 17 x + 14 = 0; 
 
 (iv.) x 2 +y 2 = ba 2 +5b 2 +Sab, xy=2a 2 +2b 2 +5ab. 
 
 8. Find the value of x m X x n , when m and w are positive 
 
 integers. 
 Simplify a 2p+ * X a p+49r -f- a*"*. 
 
 9. Define the antecedent and consequent of a ratio. 
 
 If 7 (x — y) = 3 (x + y), what is the ratio of x to y? 
 
 10. Show that if a:b::c:d::e:f, then will 
 
 a + 3c + 2e:a-c:: 6 + 3 d + 2/ : £ — /. 
 
 11. Find two numbers such that their sum, their differ- 
 
 ence, and the sum of their squares are in the ratio 
 5:3:51. 
 
 12. Prove that if x varies as - -f -, and is equal to 3 when 
 
 V * 
 y = 1 and 2 = 2, then xyz = 2 (y -j- z). 
 
 XXXVIII. 
 
 University of Cambridge, Eng. 
 
 General Examination for the Ordinary B.A. Degree, June, 1882. 
 Time Allowed, 3 hours. 
 
 1. Simplify a ^ b 
 
 a-\-b a — b 
 
116 EXAMINATION PAPERS. 
 
 2. Solve the equations 
 
 w 4 v ; 7 3 , 
 
 (ii.) 27^-24^-16 = 0; 
 
 (iii.) 2y + --4 = 5y + - + 2 -- + 4. 
 a; a? x 
 
 i 
 
 3. What is the meaning of the expressions x~ n , x p ? 
 
 a . ,., b^Xc"** 2 * /iV M+1) 
 S-phfy -p^- X y • 
 
 4. Solve the equations 
 
 (i.) x 2 — {a — b) x -\- (a — b -\- c) c = 2cx -\- ab ; 
 (ii.) Vl4rr + 9 + 2V^ + l + V3# + l = 0; 
 (iii.) 3#-2Va^ + 9 = 0, 5V^-3Vy-3 = 0. 
 
 5. Show thai^the sum of the roots of the equation 
 
 ax 2 — bx + c = is ~ 
 If a and ft are the roots of the above equation, form the 
 
 equation whose roots are , 
 
 , a P 
 
 6. When is one quantity said to vary as another ? 
 
 If ax + by + 1 = 0, where a and b are constant, and x 
 and y are variable, and if the values of x are 2 and 
 — 9 when the values of y are 1 and — 4, respectively, 
 what will be the value of x when y is zero ? 
 
 7. Define a geometrical progression. 
 Find the geometrical mean of 
 
 9# 2 — 12;z + 4 and 4.r 2 - 12 x -f- 9. 
 
 8. The first term of an arithmetical progression is 38, 
 
 and the fourth term is 86 ; find the sum of the first 
 twelve terms. 
 The first term of a geometrical progression is 27, and 
 the third term is 48 ; find the sum of the first six 
 terms. 
 
ALGEBRA. 117 
 
 9. Find, to four places of decimals, the sum to infinity of 
 the series 1 -\ (- -i- + 
 
 vs 
 
 10. The perimeter of a rectangular field is 306 yards, and 
 
 the diagonal is 117 yards. What is the area? 
 
 11. The expenses of a tram-car company are fixed, and 
 
 when it only sells threepenny tickets for the whole 
 journey it loses 10 per cent. It then divides the 
 route into two parts, selling twopenny tickets for 
 each part, thereby gaining 4 per cent, and selling 
 3300 tickets every week. How many persons used 
 the cars weekly under the old system ? 
 
 12. The price of a passenger's ticket on a French railway 
 
 is proportional to the distance he travels ; he is 
 allowed 25 kilograms of luggage free, hut on every 
 kilogram" beyond this amount he is charged a sum 
 proportional to the distance he goes. If a journey 
 of 200 miles with 50 kilograms of luggage cost 25 
 francs, and a journey of 150 miles with 35 kilograms 
 cost 16£ francs, what will a journey of 100 miles 
 with 100 kilograms of luggage cost? 
 
 University of Cambridge, Eng. 
 
 Previous Examination, June, 1882. — Time Allowed, 2\ hours. — 
 Elementary Algebra. 
 
 1. Find the value of 
 
 (Za-bb)(a-c) + c{2a-c(Za-b)-b 2 (a-c)\, 
 when a = 0, 6 = 1, c = — -J. 
 
 2. Prove that 
 
 Multiply 9 a* xi + 6 aA a$ -f 4 a* by 9 a — 6 a& xi + 4 a* x. 
 
118 EXAMINATION PAPERS. 
 
 3. Resolve into their simplest possible factors : 
 
 \0x 2 +x-2; tf 2 +4;r-4y 2 +4; a?-3a 2 b + 3ab 2 -b*+c\ 
 
 4. What is meant by a common multiple of two quanti- 
 
 ties? Prove that the sum of two quantities is a 
 multiple of any of their common measures. 
 Find the L.C.M. of 
 
 x 3 — 3x 2 y-{-3xy 2 — 2y 3 , x s —x 2 y—xy 2 —2y 3 , and x^x^-j-y*. 
 
 5. Prove the rule for the multiplication of a fraction by 
 
 an integer. 
 Reduce to fractions in their lowest terms : 
 1 
 
 (i) 
 
 2a-3Z> 
 
 2a-Zb- . 2a-db 
 
 { J \x + y x't + tf J ^ Xy^x 1 J 
 
 6. Prove that a quadratic equation cannot have more than 
 
 two roots. 
 Solve the equations 
 
 (ii.) a (2x-y)+b(2x+y) = c(2x-y)+d(2x+y) = 1 ; 
 (iii.) x 2 - 11^-42 = 0; 
 
 (i v .) 1 |_6 = -JL__i-_ 2_. 
 
 V ; 0-1)0-2) s-2 ar-1 
 
 7. A certain number, consisting of two digits, becomes 
 
 110 when the number obtained by reversing the 
 digits is added to it ; also the first number exceeds 
 unity by five times the excess of the second number 
 over unity. What is the number ? 
 
 8. Define a third proportional to two quantities. Having 
 
 given a third proportional to a and b, and also to b 
 and a, determine a and b in terms of them. 
 
ALGEBRA. 119 
 
 If a:b : : c: d, and x be a third proportional to a and c, 
 and y to b and c?, prove that the third proportional 
 to x and y is equal to that to a and e?. 
 
 9. "When is a quantity said to vary inversely as another ? 
 If a and b each vary inversely as c, prove that the 
 sum of any given multiples of a and b varies in- 
 versely as any given multiple of e. 
 
 Given that x — y varies inversely as z-f -, and x-\-y 
 
 1 . z 
 
 inversely as z ; find the relation between x and z, 
 
 provided that x — 1 and y = 3, when z = ^. 
 
 XL. 
 
 University of Cambridge, Eng. 
 
 Previous Examination, June, 1882. — Time Allowed, 2} hours. 
 Higher Algebra. 
 
 1. A has twice as many pennies as shillings ; B, who has 
 
 8d. more than A, has twice as many shillings as 
 pennies ; together they have one more penny than 
 they have shillings. How much has each ? 
 
 2. A man can walk a certain distance in 4 hours ; if he 
 
 were to increase his rate by one-fifteenth, he could 
 walk one mile more in that time. What is his rate? 
 
 3. Solve the equations 
 
 (i.) V5^ + l = 2+V^Tl. ' 
 (ii.) x 2 -2y 2 = 7, 2x + y = 7. 
 
 4. A man buys a number of articles, for £1, and makes 
 
 £1 Is. Qd. by selling all but two at 2d. apiece 
 more than they cost, How many did he buy ? 
 
120 EXAMINATION PAPEES. 
 
 5. Find the sum of n terms of the progression 
 
 a + ar -{- ar 2 -{- 
 
 Find the sum of 10 terms of the progression 
 64 + 96 + 144 + 
 
 6. The fifth term of an arithmetical progression is 81, and 
 
 the second term is 24 ; find the series. 
 
 7. Find an arithmetical progression whose first term is 3, 
 
 such that its second, fourth, and eighth terms may 
 be in geometrical progression. 
 
 8. If a : x : : b : y : : c : z, prove that 
 
 la 3 + mb 2 y + ncz 2 pa + qb + re 
 la 1 x + mby 2 + nz 3 px + qy + rz 
 
 9. Prove that log a (pq) = log a p + log a q. 
 
 10. Having given log 10 2 = 0.3010300, find the logarithm 
 to base 10 of 25, 0.03125, and (0.025)i 
 
 XLI. 
 
 Univeesity of Oxfoed, Eng. 
 
 Local Examination, Junior Candidates, May, 1880. — Time Allowed, 
 2} hours. 
 
 No credit will be given for any answer, the full working of which 
 is not shown. 
 
 I. Algebea. 
 
 t v jx-l i ,a 2 -c-3ac(b-2c) L T+b 
 
 1. Find the value of j— ; — prt L + \2a 7"' 
 
 when a = 1, b — 0, and c — — \. 
 
 2. Multiply x* — ax z + a?x — a* by x 2 + ax + a 2 ; also di- 
 
 vide p* - 9pq" + 18^ 4 by p 2 - Zpq + 3 q 2 . 
 
ALGEBRA. 121 
 
 3. Simplify 
 
 (i) ( 2x +y _ v V x X +A - 
 
 \ y 2x + y)\x + y x J' 
 
 (^ _i oS + 2 , s + 3 
 
 V ' ; x + 1 x 2 -l^ {x-yY 
 
 4. Find the G-.C.M. of^ 3 -6^-4and Sx 3 — 8x + 8 
 
 the L.C.M. of (3 a 2 - S"ab) 2 , 18 (a? b 2 - ab 4 ), and 
 24(W-6 6 ). 
 
 5. Solve the equations 
 
 (i.) 5^3_, (; ,_ 24) = 2^1 + 1| 
 
 (iL) S= 2 2 lSV 6( * + «= n(y+5) - 
 
 II. Higher Algebra. 
 
 6. Solve the equations 
 
 rn 2* 10^+1_ 3 . 
 
 (ii.) a(a-6)a; 2 + 6(> + 6);r-26 2 =:0; 
 (iii.)V + 2ay=|, 0^-4^=1^ 
 
 7. The sum of 2 numbers is 35 ; and their difference ex- 
 
 ceeds one-fifth of the smaller number by 2 ; find the 
 numbers. 
 
 8. After £12 have been divided equally among a certain 
 
 number of men, an additional shilling apiece is given 
 to them ; and it is then found that each possesses as 
 many shillings as there are men. Find the number 
 of the men. 
 
 9. Prove that if b be a mean proportional between a and 
 
 c, thena 2 + 2b 2 :a::b 2 + 2c 2 :c. 
 10. Sum to 6 terms the series J + 1\ + 3-J- -f Also in- 
 sert 12 arithmetic means between —\ and 5. 
 
122 EXAMINATION PAPERS. 
 
 XLII. 
 
 University of Oxford, Eng. 
 
 Local Examination, Senior Candidates, May, 1880. — Time Allowed, 
 2 £ hours. * 
 
 No credit will be given for any answer, the full working of which 
 is not shown. . 
 
 Candidates are reminded that in order to pass in mathematics they 
 must satisfy the Examiners in the first part of this paper. 
 
 I. Algebra to Quadratic Equations. 
 
 1. If m = ■ an( i n = ^ > find the value of 
 
 2 2 
 
 m 2 -f n\ 
 
 2. Find the G.C.M. of 2^+3^-7^-10, 5-9^-4^+4^; 
 
 and the L.C.M. of a 2 + ab, b 2 -f- ab, ab. 
 
 3. Simplify -J- -?+l 
 
 x—1 X X + 1 
 
 4. Find the square root of 4a; 4 - 20# 3 + 13^ + 30# -f 9. 
 
 5. Multiply xi + 2 a* + 2 by xi - 2x1 + 2, and divide 
 
 a 2 b~ 2 + b 2 a~ 2 -f 1 by ai" 1 + 6a" 1 - 1. 
 
 6. Solve the equations 
 
 i; i-s 20' 
 
 (ii.) * = (2-ar)(2 + a0; 
 /••• N a x b x 
 
 (ill.) - _!_- =- 4-- • 
 
 • v ; x^a x^b' 
 (iv.) x 2 + y 2 = 169 = hx + 12y. 
 
 7. Two rectangular fields each contain one acre ; one of 
 
 the fields is four poles shorter and two poles broader 
 than the other. Find the length and breadth of 
 each field. 
 
ALGEBRA. 123 
 
 II. Higher Algebra. 
 
 8. A spends £a in buying a number of articles, all at the 
 
 same price ; B spends £ b in the same way, except 
 that he buys n more articles than A buys, and pays 
 £c less for each. Find an equation to determine 
 the number of articles bought by A. 
 
 9. Solve the equations 
 
 (i.) 4*+l = 5x2*- 1 ; 
 (ii.) x* - Sx* - 2x 2 - Sx + 1 = ; 
 (iii.) Sx + 4y = 23 (in positive integers). 
 
 10. Prove the formulae for finding the nth. term and the 
 
 sum of n terms of an arithmetic progression, the 
 first term and the common difference being known. 
 A man pays his gardener 15 s. a week for the first fort- 
 night ; at the end of the first and of every succeed- 
 ing fortnight he raises the wages 6 d. per week. 
 What will the gardener have received in all at the 
 end of fifty weeks ? 
 
 11. Find the cost of an annuity of £A per annum, to be 
 
 paid quarterly, and to continue for p years ; the first 
 payment to be made at the end of the first quarter, 
 reckoning compound interest, at the rate of £r per 
 cent per annum, to be due at the date of each quar- 
 terly payment. 
 
 12. Enunciate the binomial theorem. 
 
 Show that the coefficient of the middle term in the ex- 
 pansion of (1 + x) 2n is the sum of the coefficients of 
 the two middle terms in the expansion of (l+^) 2w_1 . 
 
 13. Prove that 
 
 1 , 1 
 
 00 1 +,T+r^+ =1 +,t+^+ 
 
 x . x 
 
 Li |2 ; ii [2 
 
 /•• n i ll -\- x x , x z . x h , 
 
124 EXAMINATION PAPERS. 
 
 XLIII. 
 
 University of Oxford, Eng. 
 
 Local Examination, Junior Candidates, June, 1881. — Time Allowed, 
 1\ hours. 
 
 No credit will be given for any answer, the full working of which 
 is not shown. 
 
 I. Algebra. 
 
 1. Find the value of a 3 + b 3 - c 3 + 3 abc : 
 
 (L) when a = J, £ — 3, c = $; 
 (ii.) when c = a + b. 
 
 2. Multiply together x 2 — 7a; +,6, x 2 + 7x — 18, x 3 — 1, 
 
 and express the result in simple factors. 
 
 3. Find the G.C.M. of 2x*+ 1 x 2 + 10a; + 5 and x 3 + Sx> 
 
 + 4^ + 2,andtheL.O.M.of6^y 2 (^ + 2/),3^ 3 (^-3/) 2 , 
 and 4 (x 2 — y 2 ). 
 
 4. Simplify 
 
 (^ 2 + 2/ 2 ) ^(^ 2 + 3/ 2 ) ay y 
 
 5. Solve the equations 
 
 ,. v I — rg . 2_ff . #— 1 
 
 W 2 + f 4 + 3 ' 
 (ii.) 9^-8y = l, 12s — 10y = l. 
 
 6. Into a cistern one-third full of water 31 gallons are 
 
 poured, and the cistern is then found to be half full ; 
 find its capacity. 
 
ALGEBRA. 125 
 
 II. Higher Algebra. 
 
 7. Solve the equations 
 
 (i.) x = ^+^; 
 K J 64^49' 
 
 (ii.) ax 2 = cx — bx 2 ; 
 
 v J a + b 
 
 (iii.)^+*±2 = o,*j, = i. 
 
 8. A person bought a certain number of sheep for £210. 
 
 He lost 10, and to make up the deficiency sold the 
 remainder at 10 s. profit per head. How many did 
 he buy ? 
 
 9. Prove the rule for the summation to infinity of a geo- 
 
 metrical progression ; and sum to n terms and to 
 infinity Si + 5 + 3 + 
 
 10. The seventh term of an arithmetical progression is 1 ; 
 
 and the sum of twenty-five terms is zero. Find the 
 progression. 
 
 11. If a : b : : c : d, prove that a-\-b:a — b \\ c-\- d:c — d. 
 
 12. If 2x + Zy :2x-Zy ::2a 2 + U 2 :2a 2 -Zb 2 , then x 
 
 has to y \ho, duplicate ratio that a has to b. 
 
 XLIV. 
 
 University of Oxford, Eng. 
 
 Local Examination, Senior Candidates, June, 1881. — Time Allowed, 
 2\ hours. 
 
 No credit will be given for any answer, the full working of which 
 is not shown. 
 
 I. Algebra to Quadratic Equations. 
 1. Prove that 
 
 (a+b)(a+x)(b + x)-a(bi-xy-b(a+xy=(a-b) 2 x, 
 and divide a 3 — b 3 by a* — 2ab* + 2ahb — M. 
 
126 EXAMINATION PAPERS. 
 
 2. Resolve into component factors 
 
 (i.)63aty-28ay ; (ii.) a 5 - a 4 b - ab* + b*. 
 Find the remainder when a n + b n is divided by a — b. 
 
 3. Find the G.C.M. of 
 
 x 4 -6^ 3 +13^ 2 -12.r+4 and ^ 4 - 4 ^+8^-16^+ 16, 
 and the L.C.M. of x? — if, ar'-J-y 3 , x 3 — xy 2 , and 
 
 4. Simplify the fractions 
 px a; + 2 ^+1 2 
 
 (ii.) 
 Cm.) 
 
 ar+1 
 
 1 
 
 x + 
 
 1 
 
 2 
 
 x -f4' 
 
 1 
 
 I- 1 
 
 1 
 x* 
 
 1 
 a? 
 
 1 -^ 
 
 a; 
 
 *-fcl 
 
 Va; + Va; — 
 
 1 
 
 V# — Va- — i 
 
 Var — Va? — 1 Va- + Va;— 1 
 
 5. Solve the equations 
 
 (i.) iK2^-32)-(^-hl6)}= 1 i r S(^_20)-(2^-ll)S; 
 (ii.) (a;+5)(y+7) = (ar+l)(y-9)-j-112, 2*+5 = 3y-4; 
 
 V } a^ + 5 
 
 6. A person invests £500, part of it at 5 per cent and 
 
 the remainder at 3 per cent ; and he thus gets 4? 
 per cent on the whole. How much does he invest 
 at each rate of interest? 
 
 7. Find the square root of 9-24ar-68a7 2 +112a; 3 +196^ 4 . 
 
 II. Higher Algebea. 
 
 8. Prove that a ratio of greater inequality is diminished, 
 
 and a ratio of less inequality is increased, by add- 
 ing the same number to each of its terms. 
 If a: a — b :: c : c — d, then a-\-b : b ::<? + d:d. 
 
ALGEBRA. 127 
 
 9. Prove that the geometrical mean between two numbers 
 is also the geometrical mean between the arithmeti- 
 cal and harmonical means. 
 Sum the series : 
 
 (i.) 25i+24+22|+21 + to 15 and to 21 terms ; 
 
 (ii.) 4j*g-+2f +1-J-J • to n terms, and to infinity. 
 
 10. Prove that the number of combinations of n things 
 
 taken r together is equal to that of n things taken 
 n — r together, and greater than that of n — 1 things 
 taken r — 1 together. 
 How many different numbers can be made with all or 
 any of the figures of the number 1881 ? 
 
 11. Employ the binomial theorem to expand (a — x) n ; also 
 
 (a + x)% in ascending powers of x to 5 terms. 
 
 12. If a, ft are the roots of the equation ax* + bx-\- c = 0, 
 
 find the values of a+/J and a 3 +/? 3 in terms of a, b, c. 
 
 13. Solve the equations 
 
 (i.)* 3 -i + 7(* 3 + l) = 0; 
 (ii.) 3^ 2 + y 2 = 3^+7 = 19. 
 
 XLV. 
 
 University of Oxford, Eng. 
 
 Local Examination, Junior Candidates, June, 1882. — Time Allowed, 
 2$ hours. 
 
 No credit will be given for any answer, the full working of 
 which is not shown. 
 
 I. Algebea. 
 
 1. Find the value of 
 
 x 2 — y l 
 x 2 + y l 
 
 /• \ __r a-\-b a — 
 
 (i.) when x = -J-, y = ~2 
 
 (ii.) when a? = J, y = - \. 
 
128 
 
 EXAMINATION PAPERS. 
 
 2. Square 2b — 3c, and find the product of a-j-2b — 3c 
 
 and a — 2b + 3 c. 
 
 3. Find the G.C.M. of 2 a 5 6 + 2 a 2 6* and 4a 5 + 4 a 3 6 2 + 4 a£ 4 ; 
 
 and the L.C.M. of 5x(x 2 + 2x+l), 10s* (a*— 1), 
 and 15(>+ 1) (* 2 - 2 ^ + 1 )- 
 
 4. Simplify 
 
 (i.) 
 (ii.) 
 
 1 
 2 
 
 2x 
 
 5- 
 
 x _ 
 
 .5-* 1 
 
 x — 
 
 ^2 
 
 2 
 = 4, 
 
 ar-2 2 
 
 2y + | = 
 
 1. 
 
 A and B set out at the same time from the same spot 
 to walk to a place 6 miles distant and back again. 
 After walking for 2 hours, A meets B coming back. 
 Supposing B to walk twice as fast as A, and each 
 to maintain uniform speed throughout, find their 
 respective rates of walking. 
 
 II. Higher Algebea. 
 Solve the equations 
 
 « 1[ 
 
 (iii.) 
 
 (x + 2)(2x- 
 
 x + y .,; 3 ^ 
 
 1) 0-2)0 
 -9y 2 =16 
 
 (2o;+l)J : 
 
 Ax 2 
 
 x — y 
 
 8. Show that the sum of any two consecutive whole num- 
 bers is equal to the difference of their squares. 
 
ALGEBRA. 129 
 
 9. Find the sum to n terms of an arithmetical progres- 
 sion, the first term and the common difference being 
 given. 
 What is the amount of a debt which can be discharged 
 in two years by the payment of 10 s. the first month, 
 £1 the second, 30s. the third, and so on, no interest 
 being exacted ? 
 
 10. Sum to n terms, and to infinity, the series |- — -J- -f f — 
 
 11. If Oy : b x : : a* : b 2 , prove that 
 
 (a, + a 2 ) 2 : (b x + b 2 f : : a, 2 + a* : W + W. 
 
 12. Show that if b is a mean proportional between a and c, 
 
 then (a 2 + V) (b 2 -f c 2 ) = (ab + be) 2 . 
 
 XL VI. 
 
 University of Oxford, Eng. 
 
 Local Examination, Senior Candidates, June, 1882. — Time Allowed, 
 2\ hours. ' 
 
 No credit will be given for any answer, the full working of 
 which is not shown. 
 
 Candidates are reminded that in order to pass in mathematics, 
 they must satisfy the examiners in the first part of this paper. 
 
 I. Algebra to Quadratic Equations. 
 
 1. Prove that (a + 2 bf = a? + 2 b z + 6 b . (a + b)\ and find 
 
 the quotient a 2 + aU + b -f- a + ah bi -f b?. 
 
 2. Simplify 
 
 I+-7- 1+ 1 ' 
 
 y + z z + x x + y 
 
 &-*)' , r (c-a)V . 
 
 (iii) ^-^ J ' ('-")' -.- ( a ~*) 
 
130 EXAMINATION PAPERS. 
 
 3. Find the G.C.M. of 3^ + 17a; 2 + 22* + 8 and 6x 3 
 
 + 25# 2 + 23* + 6, and the L'.C.M. of (V - yj, 
 x z + y 3 , x s — y 3 , and (x 2 — y 2 ) 2 . 
 
 4. Extract the square root of 
 
 (i.) 7-4 V3; 
 (ii.) x*(x z + 2) + 2* 2 (* 2 + 1) - *(* - 2) + 1. 
 
 5. Solve the equations 
 
 (i.) ll*-5(*-6)-6(3*-ll) + 9(*-7) = 0; 
 
 ^ ; 8^6 12^4 
 (iii.) (3 x - 6) 2 + 7 x* - 256 = 0. 
 
 6. My income of £240 is derived from two sums invested 
 
 at three and nine per cent respectively ; but, if the 
 rates of interest were interchanged, my income would 
 be doubled ; find the sums invested. 
 
 II. Higher Algebea. 
 
 7. If a : b : : c : d, prove that 
 
 (i.) a 3 + d 3 >b 3 + c z ] 
 (ii.) Sa+2b:Sa — 2b::Sc + 2d:Sc — 2d. 
 
 8. Find an expression for the sum of n terms of a geo- 
 
 metrical series, and explain the expression " sum to 
 infinity." 
 Find the fourth term in 
 
 (i.) 2 + 21 + 3 + 
 
 (ii.) 2 + 21 + 31+ 
 
 (iii.) 2 + 21 + 31+ 
 
 9. Find the number of permutations n P r of n things taken 
 
 r together. If "P 4 : —*jp« : : 3 : 22, find n. 
 
 10. Show that the number of terms in the expansion of 
 (a + x) n is n + 1, if n is a positive integer. 
 Apply the binomial theorem to find (10.001) 7 to five 
 places of decimals. 
 
ALGEBRA. 131 
 
 11. If a, /3 are the roots of ax 2 + bx + c = 0, find the equa- 
 
 tion whose roots are 
 
 (i.) a\ F; (ii.)«(l+/B),/8(l+a). 
 
 12. Solve the equations 
 
 (i.) 144 a* - 1 + 6 -Vdx 2 - x = 16 a? ; 
 (ii.) x 2 (s»-y ») = 25, y»(s» + y 2 ) = 19| ; 
 
 (iii.) </U-X+VU + X = 4:. 
 
 XL VII. 
 
 University of Oxford, Eng. 
 
 .Krsi Examination of Women, May, 1880. — 2$me Allowed, 2\ hours. 
 
 1. Find the value of — - — ^— — when a = l,b — — %,c = Q. 
 
 a 3 + 46 3 
 
 2. Take a + 2b + Sc — 4d—5e from 3ct-46 + c — c7 + e. 
 
 3. Multiply a 3 - a 2 6 + ah 2 -b 3 by a -f- b, 
 
 and divide a 4 -f a 2 5 2 + & 4 by a 2 + a£ + b 2 . 
 
 4. Find the H.O.F. of a 4 + 5 a 2 - 6 and a 4 + 5 a 3 + 4, and 
 
 the L.C.M. of 12(a 3 -6 3 ), 15(a 3 +6> 3 ), 20ab(a 2 -b 2 ). 
 
 5. Simplify " a + 4* + 3 
 
 1 J a 2 -a- 2 
 
 6. Extract the square root of a 4 — 2a 3 + 2a 2 — a + J. 
 
 7. Solve 
 
 (i.) 70-1) -60-2) = 3(^-3); 
 
 (ii.) 1 + 1+1=*- 7; 
 
 (iii.) (x - 3) - 13) == (a - 4) - 9) ; 
 (iv.) 8rs + 3y=74, 9^-2y = 51. 
 
 8. I have in my purse £1 13 s. 9 c?. made up of a certain 
 
 number of pence, twice the same number of farth- 
 ings, and thrice the same number of fourpenny 
 pieces. Find the number of each coin. 
 
132 EXAMINATION PAPERS. 
 
 9. A is thrice as old as B. Seven years ago A was four 
 times as old as B. Find their ages now. 
 
 10. A and B play at cards. A wins six shillings, and finds 
 he has thrice as much as B. The game is continued 
 till A finds he has lost twenty-four shillings, and 
 then has a third of what B has. With what sum 
 did each begin ? 
 
 XLVIII. 
 
 University of Oxford, Eng-. 
 
 Second Examination of Women, May, 1880. — Time Allowed, 2 hours. 
 
 1. Divide a* — a~* by oh — a~%. 
 
 2. Simplify 
 
 (6a 2 -a-2)(8a 2 --10a + 3)(12a 2 + 17a + 6) , 
 (8a 2 - 2a - 3) (12a 2 + a - 6) (6a 2 + a - 2) ' 
 6a 2 -17a+12 . 27aM-18a-24 , 25a 2 -25a+6 
 12a 2 -25a+12 i_ 12a 2 +7a-12 + 20a 2 -23a+6* 
 
 3. Extract the square root of 
 
 (i.) 4a 2 +- 2 -ll-6a- 1 + 4a; 
 a 
 
 (ii.) 41-12V5. 
 
 4. Solve the equations 
 
 v x + b x-\-a 
 (ii.) ^-7*4-10 = 0; 
 (iii.) V* - 2 -f- s/x + 3 = V±x -j- 1 ; 
 (iv.) x + y = 8, ay =15. 
 
 5. If a : b : : e : d, prove that a -\-b : a — b : : c + d : c — d. 
 
ALGEBRA. ISi 
 
 6. £21 18 6-. is divided equally among a certain number 
 
 of persons. If each received a penny less, each 
 would have had as many pence as there were per- 
 sons. Find the number of persons. 
 
 7. A steamer takes 2 hours 24 minutes less time to travel 
 
 from A to B than from B to A. The steamer trav- 
 els at the rate of 16 miles an hour, and the stream 
 flows at the rate of 6 miles an hour. Find the dis- 
 tance of A from B. 
 
 XLIX. 
 
 University of Oxford, Eng. 
 
 First Examination of Women, June, 1882. — Time Allowed, 2 \ hours. 
 
 1. Find the value of x{y-\-z)-\-y[x— (y-fz)] — z[y— x{z— x)] 
 
 when x = 3, y = % 2 = 1. 
 
 2. Subtract 2 a* + fa* — x + \ from | x* + x 2 - \ x - f . 
 
 3. Multiply l + 2x+3y+4:X 2 -6xy + 9y 2 by l-2x-Sy. 
 
 4. Divide x* --^x 3 + x 2 + %x -2 by #-■§. 
 
 5. Kesolve into factors 
 
 (i.) 4^-36^ 2 2/ 2 ; 
 (ii.) (2x-3y) 2 -(x-2y)\ 
 
 6. Find the G.C.M. of x 3 - 3x + 2 and x 3 + 4# 2 - 5. 
 
 7. Find the L.C.M. of 
 
 12(1 -x 2 ), 15(1 -a;) 2 , and 20(a? + s»). 
 
 8. Simplify 
 a 2 h _ c 2 d _ 4:ab _ bed — cd 2 
 
 W-x^x^x^ 
 
 cd U 2 ~Ud 
 
 W V v l + 4^" r l-4^ ' Vl-4a? 1 + 4^ 
 
134 EXAMINATION PAPEES. 
 
 9. Solve the equations 
 
 W 3^ 9 27 11 
 
 fin x+a a- x+h -2- 
 
 2*-_3 2y+_3 = 
 
 y x 
 
 10. A person walks a certain distance at the rate of 3? 
 
 miles an hour, and finds that if he had walked 
 4 miles an hour, he would have gone the same dis- 
 tance in less time by one hour; what is the distance? 
 
 11. Find two numbers such that, if half the first be added 
 
 to the second, or \ of the second be added to the 
 first, the sum will in either case be 30. 
 
 12. Find the square root of 
 
 a 6 - 4a 5 + 8a 4 - 10a 3 + 8a 2 -4a + 1. 
 
 L. 
 
 Univeesity of Oxfoed, Eng. 
 
 Second Examination of Women, June, 1882. — Time Allowed, 2 hours. 
 
 1. Find the value of (a + &-c) 2 +(Z>+c-a) 2 + 0+a-6) 2 
 
 when a = 2, b = 3, c = — £. 
 
 2. If a + b + c =p, be + ca + oh = q 2 , and abc = r 3 , prove 
 
 that a 3 + b z + c 3 =p z - Spy* + 3 r 3 . 
 
 3. Find the G.C.M. of - 
 
 2 a* + x 2 _ 12x + 9 an( j 2a: 3 -7x 2 + l2x- 9, and 
 
 1 ; 1 
 
 2s 8 - 7a? + 12a? -9 $af+y— 1&* + 9 
 
 . A , , . ii rr — a x4-a , 2a; (3 a — a;) 
 
 4. Add together -; — - — ^, - f c- 2 , and 
 
 simplify 
 
 (x -f- a) 2 ' (a; — a) 2 ' (a; — a) (x + a) 2 
 
ALGEBRA. 135 
 
 5. Find the square root of 
 
 x 6 + aV + 2a? x* + la'x 2 - d'x + a\ 
 
 6. Solve the equations 
 
 (,) ' • 2 1 
 
 2a; -1 4:x-'d x— 1' 
 
 (ii.)V53* + a?-16 = 3a?-2; 
 (iii.) (a: — y) (a? — 3y) = 24, a; — 2y = 5. 
 
 7. There are 250 flowers in a conservatory ; the number 
 
 of geraniums is five times the number of roses, and 
 is less by 30 than the number of other flowers. How 
 many roses, geraniums, and other flowers respec- 
 tively are there in the conservatory ? 
 
 8. Find the value of x in each of the following propor- 
 
 tions : 
 
 (i.) 1 : x : : 2 : 3 ; (ii.) 1 : 2 : : 3 : x ; (iii.) 4 : x : : x : 9. 
 
 9. Two persons, A and B, start at the same time by their 
 
 watches, from two places 24 miles apart, and drive 
 towards each other at rates which are as 2 : 3 ; but 
 in consequence of A's watch being \ hour too fast, 
 and B's i hour too slow, they meet half way. At 
 what rate does each drive ? 
 
 LI. 
 
 University of Oxford, Eng. 
 First Examination of Women, June, 1883. — Time Allowed, 2\ hours. 
 
 1. Evaluate (x - yf + (y - zf + (z - x)\ when x = 3%, 
 
 y = 2* jZ =tt. 
 
 2. From the sum of ^(2a?-3y-j-4z) and ^ (4a; + 3?/ -llz) 
 
 subtract -fa(8x — 9y -f- 62). 
 
136 EXAMINATION PAPERS. • 
 
 3. Multiply x 2 + y 2 + a (x — y) by xy — a (x + y) + a*. 
 
 4. Divide a* + Sy 3 - 125 z 3 + 30#yz by a; -f 2y — 5 z. 
 
 5. Express in factors 
 
 (i.) 7 a* -77a; -182; 
 (ii.) 20 a; 4 - 60 ar 5 ?/ + 45 a; 2 y 2 . 
 
 6. Find the G.C.M. of a*+lla; + 30, 9a; 3 + 53a; 2 -9a;-18. 
 
 7. Find the L.G.M. of 
 
 15a; 2 (a 2 -2aa;+a; 2 ), 21a 2 (a 2 +2aa;+a; 2 ), 35 aa<a 2 - a; 2 ). 
 
 8. Simplify a ~ b a + b and find the value of 
 
 1 
 
 a—b a-\-b 
 a 
 
 • a—b a—b 
 
 x + 2 :x-2 4 
 
 when x= i. 
 
 2-x 2 + x 4:-X 2 
 9. Solve the equations 
 
 (i.) f(*-l) + 3(|-9)-(*-13) = ll; 
 
 (ii.) JL + 1 ^ = 0; 
 
 x—a x—b x—a—b 
 
 ,... N 5a: + 7y 7a;-J-5y Q 
 (m.) — 1^ = — ^-^ = 8 - 
 
 10. The sum of three consecutive whole numbers exceeds 
 
 the greatest of them by 19 ; what are the numbers? 
 
 11. Find the fraction which is such that if 3 be subtracted 
 
 from the numerator, and ,5 added to the denomina- 
 tor, the value is \ ; but if 5 be subtracted from the 
 numerator, and 3 added to the denominator, the 
 value is -J. 
 
 12. Extract the square root of 
 
 36a; 2 -I20ax- 12a 2 x + 100a 2 -f- 20a 3 + a\ 
 
ALGEBRA. 137 
 
 LII. 
 
 University of Oxford, Eng. 
 
 Second Examination of Women, June, 1883. — Time Allowed, 2 hours. 
 
 1. Divide x* — if — xy (x* -\-2f) by (x -\- y) (x — y) — xy, 
 
 and verify the result when x = 2y = |. 
 
 2. Prove that (cy — hzf -f (az — ex) 2 -f (bx — ay) 2 = x 2 -\-y 2 
 
 + z 2 - (ax + by + cz) 2 if a 2 + b 2 + c 2 = 1 . 
 
 3. Find the G.C.M. and the L.C.M. of 
 
 7x 3 -2x 2 -b and 7* 3 + I2x 2 + 10 # + 5. 
 
 4. Simplify 
 
 *0+y) O^'V + zy+y*) ' O+y) 2 ' 
 (ii.) t . * ■ 2ab . 
 
 5. Find the square root of 
 
 (^ + y 2 )(^ + 2 2 ) + 2^(^ + 7/ 2 )(3/ + 2 )+4^y Z . 
 
 6. Solve the equations 
 
 r , x — 3 2x — l . 7x — 11 fi 
 
 (l - } -2f 3T + -5|- = 6 ' 
 
 '..\ 5(Sx 2 -l) 2 
 
 (iii.) ^ 2 + y 2 = 9, a;-3y + 3 = 0. 
 
 7. A number of men are employed at 13 s. 6d. per week 
 
 each, and women at 10s. 6d. ; the number of women 
 exceeds that of the men by 6, but the men's wages 
 amount to 27s. per week more than the women's; 
 how many of each sex are employed ? 
 
138 EXAMINATION PAPERS. 
 
 8. Find the mean proportional between *£ and f ; and 
 
 the fourth proportional to 9, 11, 27. 
 If the first two terms of a proportion are 3 and 2, and 
 the third term exceeds the fourth by 5, what are 
 the third and fourth terms ? 
 
 9. Divide £500 among three persons, so that the share of 
 
 the first may be to that of the second as 8 : 17, and 
 the share of the third to the other two together as 
 3:2. 
 
 LIII. 
 
 University of London, Eng. 
 
 Matriculation Examination, Arithmetic and Algebra, June, 1877. 
 
 1. Divide ff + 6 f- 7 2V h J 5 f- 3 T + TT> and multiply 
 
 the result by 6^ + $£> - 1% 
 
 2. Find what fraction of a guinea is equal to the differ- 
 
 ence between f of a crown and |f of a shilling. 
 
 3. Calculate to five places of decimals the fraction 
 
 3.70271 x 0.64732 
 0.043679 
 
 4. Reduce the circulating decimal 1.52372 to a vulgar 
 
 fraction in its lowest terms. 
 
 5. Extract the square roots, to five places of decimals, of the 
 
 numbers, 3.9726523, 0.39726523, and 0.039726523. 
 
 6. Simplify the algebraic expression, 
 
 _b_ 1 
 
 Sax— 5 by ax 2 by' 1 
 
 T~ V ~Zax-2>by k 
 
 7. Divide 15a; 5 - 17a?*- 24a 3 + 138a; 2 - 130a; + 63 by 
 
 5 x 3 + 6 x 2 — 9 x -f- 7, and verify the result. 
 
 o -^ xi . -oX 2 — xy-\-y 2 a; 3 + ?/ v ,, x u 
 
 8. Prove that if -= y y . = . \ % X - then - = - 
 
 u 2 — uv-\-v 2 u ij rv 6 y y v 
 
ALGEBRA. 139 
 
 9. Given the first term, the middle term, and the number 
 of terms in an arithmetical progression ; find the 
 sum of the series. Has this problem a meaning, if 
 the number of terms is even ? 
 
 10. Find to n terms and to infinity the sum of the geomet- 
 
 rical progression in which the fourth term is 1 and 
 the ninth term is -g-j-g-- 
 
 11. Solve the equations 
 
 r , 1-Sx . 3^ + 1 ^ 2 
 ^ 2^2 1-3*' 
 
 (ii.) 2x-4:7/ = 7, Sx + 7y=19. 
 
 12. A tourist, having remained behind his companions, 
 
 wishes to rejoin them on the following day. He 
 knows that they are 5 miles ahead, and that they 
 will start in the morning at eight o'clock, and will 
 walk at the rate of 3 4 miles an hour. "When must 
 he start, in order to overtake them at one o'clock, 
 ' p.m., walking at the rate of 4 miles an hour, and 
 resting once for half an .hour on the road ? 
 
 LIV. 
 
 Science Schools and Classes, England. 
 
 Mathematics, First Stage, May, 1880. 
 
 Not more than three questions are to be answered. The number 
 of marks assigned to each question is given in brackets. 
 
 1. Find the value, when x = 5 and y = 3, of 
 
 x* - 4 x 1 y + 6 x 2 y 2 - 5 xy* + 2 y 4 ffi -, 
 
 2x' - &x 3 y + 6x 2 y 2 - im? + y* L " J 
 
 2. Multiply a 3 — x' A by a 2 — x 2 , and divide the product by 
 
 (a-x) 2 . [8.] 
 
 3. Simplify (a- b)(b + c) (c + a) + (b -c) (c + a) (a + b) 
 
 4- (c — a) (a + b) (b + c), and find its value when 
 = 1, 6=~3, and c = ~2. [10.] 
 
140 EXAMINATION PAPERS. 
 
 4. Solve the equations " [12.] 
 
 (,) 7(. + !)-5.,(J- + i) = 4. [6.] 
 
 (ii.) 0.5^+0.073/ = 0.93, 0.03a;-0.4y = 0.46. [6.] 
 
 5. The rent of a shop is -J- of the rent of the whole house 
 
 of which it is a part. Being separately rated, its 
 occupier pays £10 15 s. Od. a year less in rates than 
 the occupier of the rest of the house. The rates are 
 3s. Id. in the pound. What is the rent of the 
 whole house? [14.] 
 
 6. Find, as a fraction in its lowest terms, the value of 
 
 1 1 
 
 # 3 -3* 2 -15a; + 25 x s +7x i +5x-25 
 
 [12.] 
 
 LV. 
 
 Science Schools and Classes, England. 
 
 Mathematics, Second Stage, May, 1880. — Arithmetic and Algebra. 
 
 Not more than' three questions are to be answered. The number 
 of marks assigned to each question is given in brackets. 
 
 1. Assuming that a franc is worth 9.504c?., and a hard 
 
 dollar 50.49 d., what is the smallest sum in francs 
 that can be exactly paid in hard dollars ? [16.] 
 
 2. Show how to find the square root of a vulgar fraction, 
 
 so as to make sure of obtaining it in a finite form, if 
 it has one. 
 Ascertain whether the square roots of the following frac- 
 tions are finite or not : jtffo Ifflfc 2flL } J fa. [18.] 
 
 3. Solve two of the following sets of equations : [20.] 
 
 (i.) V* 2 + a % = —?~ [10.] 
 
 V* 2 +a 2 a 
 
ALGEBRA. 141 
 
 (ii.) x + y =7, **+ f = 133. __ [10.] 
 
 (iii.) xy J r ^/x J r y = ll, 2xy—^Jx J r y = lo. [10.] 
 
 4. What is the least integral multiplier which will make 
 
 17a 5 — 68a;*y+ 102 ary ■- 68 ary + VJxtf a com- 
 plete cube ? _ [20.] 
 
 5. A rectangular plot of ground measures 42 acres, and 
 
 its diagonal is 1243 yards long. "What are its sides ? 
 
 [22.] 
 
 6. Two boys start at the same instant from the same cor- 
 
 ner of a square, the length of one of whose sides is 
 200 yards, and they run round it in opposite direc- 
 tions : one (A) runs at the rate of 100 yards in 15 
 seconds, and loses 2 seconds in turning a corner ; 
 the other (B) runs at the rate of 100 yards in 16 
 seconds, arid loses 1 second in turning a corner. 
 Where do they meet? [26.] 
 
 LVI. 
 
 Science Schools and Classes, Eng. 
 
 Mathematics, First Stage, May, 1881. 
 
 Not more than three questions are to be answered. The number 
 of marks assigned to each question is given in brackets. 
 
 f , fz-xY , z 2 , a' + bW zd V 
 
 1. Show that 
 
 x* 
 
 a* 
 
 are identical expressions ; that is to say, that the one 
 
 can be deduced from the other. Find their values, 
 
 when x — &, z = 4, and a = b == 5. [8.] 
 
 2. Simplify J — ) f_J [ J , and find its value when 
 
 a = 7, J = 3, c = 2, m = 2, w=l. [8.] 
 
142 EXAMINATION PAPERS. 
 
 3. Find the Q.C.M. and L.C.M. of 2x* + 9a; 2 + 5 a; -f 12 
 
 and 2a; 4 + 4a; 3 + 13a; 2 +11 x +12. [12.] 
 
 4. Solve the equations [1-] 
 
 (i.) (2 + Zx) (i - g = (3 - 2*) ( 6 - ^ : [0.] 
 
 (ii.) (x+, J y-(x- 1/ f = 3o2,x(y+b) = U3. [C] 
 
 5. Of two squares of carpet, one measures 44 feet more 
 
 round than the other, and 187 square feet more in 
 area. What are their sizes? [12.] 
 
 6. Oranges are bought for half-a-crown a hundred ; some 
 
 are sold at 3 s. 6 c?. a hundred, and the rest at 2 s. 
 10^ d. a hundred. The same profit is made, as if 
 they had all been sold at 3 s. 1-J-c?. a hundred. Of a 
 thousand oranges sold, how many fetch 3 s. 6d. a 
 hundred? [14.] 
 
 LVII. 
 
 Science Schools and Classes, Eng. 
 
 Mathematics, First Stage, May, 1882. 
 
 Not more than three questions are to be answered. The number 
 of marks assigned to each question is given in brackets. 
 
 -1. Divide x 3 + 8f- 27z 3 + 18xyz by x + 2y-3z, and 
 test your answer by substituting x = 5, y = — 4, 
 z = 3 in the dividend, divisor, and quotient. [12.] 
 
 2. Show that the product of 1 , 1 -| — -, and 
 
 . 0/ ,. 4ab . (x-\-a)(x — b) 
 x4-2(a — o) — is- P • 
 
 v J x x 
 
 Write down the value of each factor and of the product, 
 when x = 3 a = 35. • [12.] 
 
 3. Reduce to its lowest terms — — . ,,, — — • [12.] 
 
 x — or — 7a; 2 + x + 6 
 
ALGEBRA. 143 
 
 4. Divide x 2 -\-px-\- q by x — a, and find the relation that 
 
 must hold good between a, p, and q, when the divi- 
 sion can be performed without leaving a remainder. 
 
 [12.] 
 
 5. Solve the equations [12.] 
 
 (i.) i(0.75-*) + H0.47 + 2*) = (3- T V)*. [«J 
 (ii.) *.j-X—"f— Wl ["6.1 
 
 6. A man has 1000 apples for sale ; at first he sells so as 
 
 to gain at the rate of 50 per cent on the cost price ; 
 when he has done this for a time the sale falls off, 
 so he sells the remainder for what he can get, and 
 finds that by doing so he loses at the rate of 10 per 
 cent; if his total gain is at the rate of 29 per cent, 
 how many apples did he sell for what he could get ? 
 
 [14-] 
 
 LVIII. 
 
 Science Schools and Classes, England. 
 
 Mathematics, Second Stage, May, 1882. — Arithmetic and Algebra. 
 
 Not more than three questions are to be answered. The number 
 of marks assigned to each question is given in brackets. 
 
 1. Given V5 = 2.236068, express — ^ and — as 
 
 V20 V5~+2 
 
 decimals, true to the fifth place. [18.] 
 
 2. Solve the equations [25.] 
 
 (i.) x*- 2a?- 3 07+ 4 = 0; 
 (ii.) x 2 + y = 8, 3^- + 2y=-7. 
 
 3. A farmer sold 7 oxen and 12 cows for £250. He sold 
 
 3 more oxen for £50 than he did cows for £30. 
 Required the price of each. [25.] 
 
144 EXAMINATION PAPERS." 
 
 4. What is the term which must be added to 
 
 9a? 4 +12^ 3 +20^ + 25 to make it a complete square? 
 
 t 25 -] 
 
 5. Find (x-\-2y)\ and obtain the sixth root of 
 
 6,321,363,049. [25.] 
 
 6. It is known that the volume of a cylinder varies as 
 
 the base and height jointly. If the volume of the 
 first of two cylinders is to that of the second as 
 11:8, and the height of the first is to that of the 
 second as 3:4, and if the base of the first has an 
 area of 16.5 square feet, what is the area of the 
 base of the second ? [25.J 
 
 LIX. 
 
 Science Schools and Classes, England. 
 
 Mathematics, First Stage, May, 1883. 
 
 • Not more than three questions are to be answered. The number 
 of marks assigned to each question is given in brackets. 
 
 1. Explain why the product is a 12 , when a 5 is multiplied 
 
 by a 7 , and why the quotient is a 5 when a 8 is divided 
 by a\ [8.] 
 
 2. Obtain (ct* + 3 x + 15) 2 - (x z — 3 x + 15) 2 in its simplest 
 
 form, and find its value when 2x = — 5. [12.] 
 
 3. Simplify the expressions [12.] 
 
 r s x — a a — b 
 
 (1) — x — ; 
 
 a b x 
 
 <«>(I + eM°-9- 
 
 Find the G.C.M. of x* - 5x z - 6x 2 + 35^; - 7 and 
 3 x z — 23 x 2 -j- 43 x — 8, and write down these expres- 
 sions in factors. [12.] 
 
ALGEBRA. 145 
 
 5. Solve the equations [12.] 
 
 W * 0.7 ~ 0.35 ' 
 
 W 4 "" 5 " 5' 4^5 12 
 
 6. A sum of £23 14s. is to be divided between A, B, G ; 
 
 if B gets 20 per cent more than A and 25 per cent 
 more than C, how much does each get? [16.] 
 
 LX. 
 
 Civil Service of Great Britain. 
 
 Competitive Examination of Candidates for Inspectorships of National 
 Schools, Ireland, 1878. — Time Allowed, 3 hours. 
 
 1. Find the value of ,. , when a = 2, 6 = 4, 
 
 b 3 — b 2 -\-b — a 
 
 and the value of (1 -f a)i X (1 — 6)~* when a = £, 
 b = l 
 
 2. Divide 6^ 3 + ^ 2 + a:4-2 by %&— x+1. 
 
 Find the first four terms in the quotient obtained when 
 1 + x is divided by 1 — 2x. 
 
 3. Simplify the expressions 
 
 (i.) (16 a 6 b 2 )h x (aWy x (2 at 5)4 ; , 
 (ii.) il+a-\-b) 3 +(l+a-b) 3 4-(l-a+bY+(l-a-b) 3 . 
 
 4. Multiply a 3 b~ 3 - ab~ l + a~ 1 b-a,- 3 b 3 by ab~ l + cr 1 5, 
 
 and divide (x 2 y mn f l b,y {aTf)"". 
 
 Sx 3 — X 2 07 1 
 
 5. Reduce the fraction -— — to its lowest 
 
 . , 3x° — 4:x 2 — x + 2 
 
 terms. , ' 
 
 6. Find the L.C.M. of 
 
 (i.) 7a 3 bx 2 , 3 ax 3 , 6ab 3 ; 
 
 (ii.) dx 2 T x-2, 2x 2 + x-3. 
 
146 EXAMINATION PAPERS. 
 
 7. Find the value of a which will make x* — x s — x 2 — ax 
 
 divisible, without remainder, by x 2 -f- x. 
 
 8. Solve the equations 
 
 v ; x 2x 3x' K J 1-a? 2-s 
 
 9. Extract the square roots of 
 
 (i.) 9 - 4V2 ; (ii.) x 2 - 6x + 13 - 122T 1 + 4ar 2 . 
 
 10. Solve the equations 
 
 (i.) Vx-VT+x^ = ~; 
 n vx 
 
 (ii.) L-^ x «z^ = i ; 
 
 1 -f- aa; a -}- a; 
 (iii.) .r 2 + xy = 32, y 2 — xy = 18. 
 
 11. A train is timed to travel between two stations, A and 
 
 C, at 45 miles an hour. It gravels from A to an 
 intermediate station, B, at the rate of 40 miles per 
 hour, and the speed is then increased to 50 miles an 
 hour. The train arrives punctually at the time 
 appointed. Compare distances of B and C from A. 
 
 LXI. 
 
 Civil Service of Great/Britain. 
 
 Open Competitive Examination for Clerkships of the Superior Class in 
 the India Office, 1879. — Time Allowed, 3 hours. 
 
 Full marks may be obtained by doing less than the whole of 
 this paper. 
 
 1. If 77i and 7i be positive integers, prove that a m Xa n = a m+n ; 
 and, assuming this formula to hold good for all 
 
 771 
 
 values of the indices, deduce the meanings of o», a , 
 and a~ n . 
 
ALGEBRA. 147 
 
 2. Divide 
 
 x i +y i +z 4 -2(x 2 y 2 +y 2 z 2 +z 2 x 2 ) by x 2 +2yz-f-z 2 , 
 and multiply 
 
 2a^+-Ta*yt-fl% by aty" 1 — a%-3 + a*3T 4 . 
 
 3. Prove that the L.C.M. of any two algebraical expres- 
 
 sions is equal to their product divided by their 
 G.C.M. 
 Find the G.C.M. and L.C.M. of the expressions 
 (x 2 + b 2 ) c + (b 2 + c 2 ) x and (x 2 -b 2 )c + (b 2 - c 2 ) x. 
 
 4. Simplify the following expressions : 
 
 i+ ^ 
 
 '4x 2 — 6#y 
 r - \ f CL + V a-\-2x 
 
 a 2 + 2a# — ay — 2xy a 2 -\- ay — 2ax — 2xy 
 a 2 -\- 2ax -\- ay -\- 2xy _ 
 3 a 2 + a?/ -{-2 ax — 2xy' 
 
 ,... s 2 Va — 6 . . 3\A 
 
 ( m ~ / ? — ^ , , + 
 
 5. Find the condition that the roots of the equation 
 ax 2 -{- bx -\~ c = may be real, and show that the 
 roots of the equation (x -f p) (x -\- q) = pqx 2 will 
 always be real if p and q are real. 
 
 6. 
 
 Solve the equations 
 
 
 r n # + 2a 3 n 
 
 
 (ii.) 6^ 2 + l") + 5^ + ^ = 38; 
 
 
 (iii.) t r 3 y — a; 4 = 27, #?/ 3 — ^ 3 y = 84, 
 
148 EXAMINATION PAPERS. 
 
 7. A man, who drives twice as fast as he can walk, finds 
 
 that it takes him 9 hours to drive to a certain town 
 and to walk back, and that when he has accom- 
 plished half of the return journey, he meets a man 
 who set out to walk from the same place an hour 
 and a half later, and is travelling a mile and a half 
 per hour more slowly than himself. Find the dis- 
 tance of the town from the starting point, and the 
 rate at which each man walks. 
 
 8. Three lamps of equal brilliancy are placed in three 
 
 different corners of a square room. Compare the 
 intensities of light at the fourth corner and at the 
 centre of the room, assuming that the illumination 
 from a source of light varies inversely as the square 
 of its distance. 
 
 9. Show how to insert any number of harmonic means 
 
 between two given quantities. 
 If 1p and 3 q be the ^>th and ^th terms, respectively, 
 of an harmonic series, prove that the (p + ^)th term 
 will be 6(p — q)> 
 
 10. Prove that the number of permutations of n things 
 
 taken r together is n(n— l)(n — 2) (n — r+1). 
 
 In how many ways can 24 ships belonging to 4 differ- 
 ent nations be arranged in 4 lines, each consisting of 
 6 ships of the same nationality ? 
 
 11. Expand (a 2 — x 1 ) ~% to five terms, and show that the 
 
 / 1 Y" 
 
 middle term in the expansion of ( x -j- — J is equal to 
 
 1x3x5 (2n-l) \ ■' 
 
 1x2x3 n 
 
 12. Express the fourth root of 89-28VI0 as the differ- 
 
 ence of two surds, and extract the fifth root of 
 99,999 correct to 15 places of decimals. 
 
ALGEBRA. 
 
 149 
 
 LXII. 
 
 Civil Service of Great Britain. 
 
 Open Competitive Examination for Admission to the Royal Indian 
 Engineering College, July, 1879. — Time Allowed, 3 hours. 
 
 1. Find the value of 2±$ + b±o + e_±d + d+a when 
 
 o — c c — d a — a a— b 
 a = 6, 6 = 4, c = 3,(*=l; and of a; 3 -4a; 2 +6a;~4, 
 when a; = 1 + V— 1. 
 
 2. Multiply a?-(2+V8)y-|-(l-f-VS)2 by s-(2- V3)y 
 
 + (1— V3)z; and divide a: 2 — 4ar?/-f-?/ 2 by a; — (2-fV3)y. 
 
 3. Explain the law of indices in the multiplication and 
 
 division of algebraic quantities. 
 Multiply x§ + a$ -f #* by rci — afi ; and reduce to its 
 simplest form (zk ><: a;"*) - * ■+- x~*. 
 
 4. Simplify the following expressions : 
 gv 25a; 4 + 5s*<-ar-l 
 
 (ii.) 
 
 20 a; 4 + a; 2 - 
 + 
 
 a — a; ft -f- # ft 
 
 ft 2 + a; 2 
 
 a 2 + a* 
 
 ft 4 + a; 4 
 (ft 2 + a* 2 ) 2 
 
 5. Solve the equations 
 
 (i.) Va^+V4+^ = JL; 
 
 (ii.)^_V^±iV^2a; = 
 
 x + l \fx—\ 
 
 (iii.) 
 
 20 
 
 =11; 
 
 x—l 2\a7+l, 
 (iv.) a; 2 + y 2 = 8, 2xy-tf = ±. 
 
150 EXAMINATION PAPERS. 
 
 6. A number having two digits is to the number formed 
 
 by inverting the order of the digits as 8 to 3, and 
 the sum of the two numbers is 99. Find them. 
 
 7. Prove the rule for finding the G-.C.M. of two algebra- 
 
 ical quantities. 
 Find the L.C.M. of 
 
 a 2 -ab-\-b 2 , a 2 + ab + b 2 , a z -b\ a 3 +b\ and (a 2 -bj. 
 
 8. Find the square roots of 
 
 81a; 4 +108^-24^ + 4 and 327-87VT5. 
 
 9. If p, q, and r are the arithmetic, geometric, and har- 
 
 monic means of two algebraical quantities, show that 
 
 (i.) pr = q 2 ; (ii.) p -| = Vp 2 - q 2 . 
 
 10. Sum to n terms, and where possible to infinity, 
 
 (i.) 3, 21 If 
 
 (ii.) 3, % Iff 
 
 1 11 
 
 (iii.) 
 
 2x3 3x4 4x5 
 
 11. Distinguish between permutations and combinations, 
 
 and find the number of the latter that can be formed 
 from n things taken r together. 
 In how many ways can a guard of 8 soldiers be selected 
 from a company of 25, and in how many of them 
 will two particular men be on guard together ? 
 
 12. Find the sixth term of ( 3 x — ^ J , and the two middle 
 
 terms of {a — x) 9 . ^ ' 
 
 13. If 2_+te = *±^ = £+^, show that 
 
 b -\- cy c-\-ay a-\~by 
 a 3 + & 3 +c 3 -3afo = 0. 
 
THIS BOOK IS DUE ON THE EAST DATE 
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 APR 23 1946 
 
 MAY 21 1936 QgfStftt69l* 
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 UNIVERSITY OF CALIFORNIA LIBRARY