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ALGEBRA 
 
 EXAMINATION PAPERS y . v 
 
 FOR ADMISSION TO 
 
 Harvard, Yale, Amherst, Dartmouth, Brown, 
 
 AND TO THE 
 
 MASS. INSTITUTE OF TECHNOLOGY, 
 
 June, 1878, to Sept. 1889 inclusive. 
 
 EDITED BY 
 
 WILLIAM F. BRADBURY, A.M., 
 
 HEAD MASTER OF THE CAMBRIDGE LATIN SCHOOL, AUTHOR OF 
 ELEMENTARY ALGEBRA, GEOMETRY, ETC. 
 
 BOSTON: 
 
 THOMPSON, BROWN, AND COMPANY, 
 
 23 Hawley Street. 
 
M<n 
 
 
 
 XhhJ 
 
 Copyright, 1889, 
 By William F. Bradbury. 
 
 iHitforattg Pras: 
 
 JOHS WlLSOH AND Soij, CAMBRIDGE. 
 
PEEFACE. 
 
 That any single work on Algebra will contain all that 
 may be demanded for admission to all the colleges is 
 hardly to be expected. Yet many secondary schools are 
 preparing pupils for several colleges, and to secure good 
 results teachers must know the demands of all. For sev- 
 eral of the colleges a very thorough training is necessary, 
 and success cannot be secured without careful preparation 
 and a great deal of practice in written work. For Har- 
 vard the requirement in Algebra was very greatly in- 
 creased about 1878, and since that time it has stood at 
 this high standard. 
 
 Any applicant who undertakes these Harvard papers 
 without having been thoroughly trained by written exam- 
 inations — limited in time as at the actual examination 
 — is almost sure to fail. 
 
 This book contains the admission papers in Algebra 
 of several leading colleges, and thus affords an oppor- 
 tunity for a comparison of requirements in Algebra that 
 has never been given before. Here are over a thousand 
 examples that have been given as tests by these colleges. 
 No better list for practice can be found. 
 
IV PREFACE. 
 
 It should perhaps be said that in 1887 and 1888 Dart- 
 mouth had no printed examinations for admission, — as 
 she received most of her candidates on certificate, — and 
 that the papers inserted for those years were given to the 
 classes after a few weeks' work in the College. The pa- 
 pers for Yale for September, 1884, and September, 1888, 
 the compiler has been unable to obtain. 
 
 W. F. B. 
 
 Cambridge, October, 1889. 
 
CONTENTS. 
 
 Page 
 
 Harvard Papers, 
 
 June 1878 to Sept. 1889 1-21 
 
 Yale Papers, 
 
 June 1878 to Sept. 1889 22-38 
 
 Amherst Papers, 
 
 June 1878 to Sept. 1889 39-57 
 
 Dartmouth Papers, 
 
 1878 to 1889 58-65 
 
 Brown Papers, 
 
 1878 to Sept. 1889 66-77 
 
 Institute op Technology Papers, 
 
 June 1878 to Sept. 1889 78-99 
 
EXAMINATION PAPERS IN ALGEBRA FOR ADMIS- 
 SION TO HARVARD COLLEGE. 
 
 June, 1878. 
 
 1. Two workmen, A and B, are employed on a certain job at 
 different wages. When the job is finished, A receives $ 27, and 
 B, who has worked three days less, receives f> 18.75. If B had 
 worked for the whole time, and A three days less than the whole 
 time, they would have been entitled to equal amounts. Find 
 the number of days each has worked, and the pay each receives 
 per diem. 
 
 2. Find the value of x from the proportion 
 
 /10 tya 2 y / 5a^a 2 ' 96~ 3 
 
 V 3^/6V : "-Vl^ 1 : y-5 ' 
 
 Express the answer in its simplest form, free from negative and 
 fractional exponents. 
 
 3. Simplify the expression 
 
 x 2 -\- y 2 x 2 — y 2 
 x 2 — y 2 x 2 -\- y 2 
 
 X — V i xJ rV 
 x -\- y 'a; — y 
 
 1. "Write out the first five terms and the last five terms- of 
 (x — y) 13 . 
 
 5. Find the value of x from the equations 
 a x -\- b y = I, 
 cy -j- dz =m, 
 ex -\- fz7=zn. 
 
I ELEMENTARY ALGEBRA. 
 
 6. Find the greatest common divisor and the least common 
 multiple of 6 x 2 -\- 7 x — 5 and 2 X s — x 1 + 8 x — 4. 
 
 7. Solve the equation 
 
 s-f-13a-f36 a — 26 _ 
 5cs — 36 — a; a;-|-26 
 
 September, 1878. 
 
 1. Three men, A, £, C, are tried on a piece of work. It is 
 found that A and B together can do a certain amount in 12 hours ; 
 B and C can do the same amount in 8 hours and 24 minutes ; 
 and G and A can do the same amount in 9 hours and 20 min- 
 utes. Find the time which each man would require to do the 
 same amount singly. 
 
 2. Simplify 
 
 _ fa — -26 3x(a—b) \ 
 ~ \x+b a; 2 — 6 s / 
 
 '+36 2 a + 36a; 2 -f X s ^ (» + 6) 2 
 
 a? — 6 s "*" a^+^aT-flT 2 
 
 3. Write out the first five terms and the last five terms of 
 (x — yfK 
 
 4. Solve the equations, \ (x — y)=x — 4, xy = 2x-\-y-\-2. 
 
 5. Solve the equation, x -| — = 1 -| j-r^ — ■ 
 
 a 2 + 36 2 
 
 6. Find the value of x from the proportion, 
 
 3a _1 / a y _ If 6 V 
 
 2 6 2 : 126^5 l^W - * : V Vv^V • 
 Find a result free from fractional and negative exponents, and 
 in the most reduced form. 
 
 7. Find the greatest common divisor and the least common 
 multiple of 4a; 4 — x 2 — Gx — OandSa; 4 — 2a; s — Ix 2 — Gx — 9. 
 
EXAMINATION PAPEKS. 
 
 June, 1-879. 
 1. Solve the equation, 
 2ax — 46 bx — a 2abx 
 
 bx — a 2ax — b~ 2abx 2 — (2 a 2 -\- b 2 ) x -\- a b ' 
 
 Reduce the answers to their simplest forms. 
 
 2. Solve the equations, 
 
 *_1 + *1±1 =0 , 
 x ' x-\-y ' 
 
 4x-\-3y = 1. 
 State clearly what values of x and y go together. 
 
 3. Find the value of x from the proportion 
 
 3 6 12/a 1 
 
 TV" 
 
 6 18 . _ 2(3ac) 2 . 2 y^a 
 
 : x 
 
 c" fyib'c 6 ) be 
 
 4. Simplify the fraction 
 1 
 
 x — y x 2 — y 2 x 2 -\- y 2 
 
 x-\-y 
 
 
 5. Find the greatest common divisor of 
 
 2* 2 — 3x +1 and 2 a; 2 — x— 1. 
 
 6. Put the following question into equations : — 
 
 A and B walk for a wager on a course of one mile (5280 feet) 
 in length. At the first heat, A gives B a start of 45 seconds, 
 and beats him by 110 feet. At the second heat, A gives B a 
 start of 484 feet, and is beaten by 6 seconds. Required, the 
 rates at which A and B walk. 
 
4 ELEMENTARY ALGEBEA. 
 
 September, 1879. 
 
 1. Several friends, on an excursion, spent a certain sum of 
 money. If there had been 5 more persons in the party, and 
 each person had spent 25 cents more, the Dill would have 
 amounted to $ 33. If there had been 2 less in the party, and 
 each person had spent 30 cents less, the bill would have amounted 
 to only $ 11. Of how many did the party consist, and what did 
 each spend ? Find all possible answers. 
 
 2. Solve the equations, 
 
 2z + 4y + 270 = 28, 
 7 x — 3y — 15z=3, 
 9w— lOy— 33j=4. 
 
 3. Solve the equation, 
 
 a;_|_3& 36 _ a-1-36 
 
 8a 2 — 12a& "■" 4 a 2 — 9 6 2 — (2 a + 3 b) (x — 3 b) ' 
 Eeduce the answers to their simplest forms. 
 
 4. Calculate the sixth term of 
 
 Eeduce the answer to its simplest form, cancelling all common 
 factors of numerator and denominator, performing the numerical 
 multiplications, and giving a result which has only one radical 
 sign and no negative or fractional exponents. 
 
 5. Simplify the fraction 
 
 2x + y s 4z 2 -f-y e 
 2x — if' 4a a --.y 6 
 2 a; — .y 8 8 a; 8 — / 
 
 2x-\-y i 8 *»-}-/ 
 
 6. Find the greatest common measure and the least common 
 multiple of 
 
 4 x i _|_ 1 4 a x * _ 1 8 a 9 x 2 and 24 a x* + 30 a 8 * -)- 1 26 a 4 . 
 
EXAMINATION PAPEES. 
 July, 1880. 
 1. Eeduce to its simplest form 
 
 .+ ■ - 
 
 . x — 2 
 
 2. Divide Qx m + Sn — 19a m + 2n -f 20z" 1 + » — 7 x m — ix m ~' ' 
 by 3 a 2 " — 5x" -\- 4. 
 
 (2 a/ a 6 aV^V 1 
 — -^ — — J , reducing it 
 O (A J 
 
 to its simplest form. 
 
 4. Find the greatest common measure and the least common 
 multiple of 2 x 6 — 11 x 2 — 9 and 4a: 6 + 11 a 4 + 81. 
 
 5. A man walks 2 hours at the rate of 4^ miles per hour. 
 He then adopts a different rate. At the end of a certain time, 
 he finds that if he had kept on at the rate at which he set 
 out, he would have gone three miles further from his starting- 
 point ; and that if he had walked three hours at his first rate and 
 half an hour at his second rate, he would have reached the point 
 he has actually attained. Find the whole time occupied by the 
 walk and his final distance from the starting-point. 
 
 + 
 
 6. Solve the equation 
 
 a _ b(Zx + 1) _ 1 
 
 b(2x — 1) a (a* — 1) — (2a; — l)(x+l) ' (2x— l)(a— 1) 
 
 Eeduce the answers to their simplest forms. 
 
 September, 1880. 
 
 1. Eeduce to its simplest form as one fraction 
 
 f x + y , x* -\- y 2 \ _^ f x — y _ x* — y 8 \ 
 \^_y1-*« + y»; • \x + y x* + f/ ») 
 
6 ELEMENTARY ALGEBRA. 
 
 2. Find the greatest common measure and the least common 
 multiple of 
 
 a; 6 +3a; 5 -f-3a; 4 -f 9a; 8 — 4a; 2 — 12a; and a; 6 + 3a: 6 — X s — 3a; 2 . ' 
 
 / a ,j a \ w 
 
 3. Find the sixth term of ( ^ 2 6 *J 6 s \ , reducing the 
 
 literal part of the term to its simplest form, and the numerical 
 part into its prime factors. 
 
 4. A reservoir, supplied by several pipes, can be filled in 15 
 hours, every pipe discharging the same fixed number of hogs- 
 heads per hour. If there were 5 more pipes, and every pipe 
 discharged per hour 7 hogsheads less, the reservoir would be 
 filled in 1 2 hours. If the number of pipes were 1 less, and every 
 pipe discharged per hour 8 hogsheads more, the reservoir would 
 be filled in 14 hours. Find the number of pipes and the capacity 
 of the reservoir. 
 
 5. Solve the equation 
 
 2a + l _ 3a-l-l _l/l_2\ 
 b a x\b a J 
 
 Reduce the answers to their simplest forms. 
 
 June, 1881. 
 
 1. What are the factors of x i -\-rf1 Eeduce to its simplest 
 
 x* — y 2 y 
 form the product of -3 — r — 3 and — ; ; 
 
 V x 
 
 2. Solve the equation - — ; ; — = 1 -] 1 
 
 1 -\- a -\-x ax 
 
 3. Find the square root of 
 
 1 + */Wx^ -f 10*" 1 + 12 Va^ + 9 a!"". 
 
EXAMINATION PAPERS. 7 
 
 4. What is meant by the expression a$ ? 
 
 5. Solve the equation \/x — 8 — \/x — 3 = \/x. 
 
 6. A man rows down a stream, of which the current runs 3J 
 miles an hour, for If hours. He then rows up stream for 6£ 
 hours, and finds himself two miles short of his original starting- 
 place. Find his rate and the distance he rowed down stream. 
 
 7. Find the 4th and the 14th term of (2 a — b) le . 
 
 September, 1881. 
 
 1. Solve the equation 
 
 4 a 2 4 a 2 - 
 
 -b 2 V 
 
 x + 2 ' x (x 2 - 
 
 — 4) x — 2 
 
 2. Multiply 
 
 x + y x — y 
 x — y x-\-y 
 
 x?-f } 2y 
 
 3. Solve the equations 
 
 
 m , n 
 
 \-- = a ; 
 
 x y 
 
 x ' y ~ 
 
 4. Find the greatest common measure of 
 
 x 4 — 115 sc +24 and 24x 4 — 115a; 3 + 1. 
 
 5. A man bought a number of railway shares when they were 
 at a certain rate per cent discount for $ 8500 ; and afterwards, 
 when they were at the same rate per cent premium, he sold all 
 but 20 of them for $ 9200. How many did he buy, and what 
 did he give for each of them 1 
 
 6. Find the last four terms of 
 
 (J — 2$) 
 
 20 
 
 reducing the numerical part of each term to its prime factors. 
 
8 ELEMENTARY ALGEBRA. 
 
 June, 1882. 
 
 1. Simplify 
 
 a 2 — be , b 2 + ca , c 2 + aft 
 
 (a — b) (a — c) "i" (b + c) (ft — a) "+" (c — a) (c + ft)' 
 
 2. A man bought a certain number of sheep for $ 300 ; he 
 kept 15 sheep, and sold the remainder for $ 270, gaining half a 
 dollar a head. How many sheep did he buy, and at what 
 price 1 
 
 3. Find the greatest common divisor of 
 
 2x 5 — 11a; 2 — 9 and 4a: 5 + lla: 4 + 81. 
 
 ic-,., .- (4 a 2 — ft 2 ) (x 2 +1) 
 
 4. Solve the equation — „ , „ — - = 2x. 
 
 ia 2 -\-b 2 
 
 Eeduce the answers to their lowest terms. 
 
 5. Find the square root of 
 
 a; 3 +2a;i— 3 x 2 — 4 sc« -f 4 x. 
 
 6. A and B can do a piece of work in 18 days ; A and C can 
 do it in 45 days ; B and C in 20 days. Find the time in which 
 A, B, and C can do it, working together. 
 
 x — 3a-\- 
 
 Sbptember, 1882. 
 4 a 2 
 
 1. Simplify " + a: , 
 
 a -\- x 
 
 2. Solve the equations < 
 
 (- + - = 4 
 
 3. Find the factors of the least common multiple of 
 3z 6 + 2a; 4 + x 2 and 3a: 4 + 2a: 8 — 3a: 2 + 2* — 1. 
 
EXAMINATION PAPERS. 9 
 
 4. Solve the equation 
 
 (3 -f 4 2 ) (x 2 — x + 1) = (36 2 + 1) (a; 2 + x + 1). 
 
 5. Find the terms which do not contain radicals in the de- 
 velopment of 
 
 (VK-*D*. 
 
 6. .4 hires a certain number of acres for $ 420. He lets all 
 but four of them to B, receiving for each acre $2.50 more than 
 he pays for it. The whole amount received from B is $420. 
 Find the number of acres. 
 
 7. Which is the larger, <\/l0 or *\/46 1 Give the reason for 
 your answer. 
 
 June, 1883. 
 1. Solve the equation 
 
 1 4 ace 2 — 3b (x — 2) 
 
 x 
 
 2a(x 2 + l) + 36 ' 
 
 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. 
 
 3. Find the prime factors of the coefficient of the 6th term 
 of the 19th power of (a — b). What are the exponents in the 
 same term, and what is the sign 1 
 
 4. Reduce the following fraction to its lowest terms : 
 
 x * _|_ 2 cc 2 -f- 9 
 
 a;*_4a; 3 -f- 10a; 2 — 12 as + 9 
 
10 ELEMENTARY ALGEBRA. 
 
 5. Prove that, if a : b = e : d, 
 
 a-\-b a — b a b 
 
 c -f- d c — d c d 
 
 6. Solve the equations 
 
 x y = 4 — y 1 , 
 2x 2 — y 2 =17. 
 
 Find all the answers, and show what values of x and y belong 
 together. 
 
 September, 1883. 
 
 1. A man setting out on a journey drives, at the rate of a 
 miles per hour, to the nearest railway station, distant b miles 
 from his house. On arriving at the station, he finds that the 
 express-train for his place of destination has left c hours before. 
 At what rate should he have driven in order to reach the station 
 just in time for the express-train 1 
 
 Having obtained the general solution of this question, find 
 what the answer becomes in the following cases : 
 
 (1) when c = ; (2) when c = - ; (3) when c = • 
 
 In case (2), how much time does the man have to drive from 
 his house to the station ? 
 
 In case (3), what is the meaning of the negative value of c 1 
 
 2. Solve the equation 
 
 (2 x — 1)* — (3 x -f- l)i = (* — 4)i. 
 
 3. Solve tlje equation 
 
 ax __ x ( x ~ 3 l \ _ 2 
 
 a 2 x — 2 a \a 2 x — 2 x) 2x — a 2 x 2 ' 
 
 reducing the answers to their lowest terms. What do the 
 answers become, if a = — 1 ? 
 
 4. Reduce the following fraction to its lowest terms : 
 
 6 x s — 2 x* — 1 1 x s -f- 5 x 2 — 1 x 
 9* 6 -f 3a; 4 — lla; 8 4- 9* 2 — 10x" 
 
EXAMINATION PAPERS. 11 
 
 5. What is the value of a"1 of ar *"? Give the reasons. 
 
 6. Solve the equations 
 
 xy -{- & = 0, 
 9x 2 — y 2 =7. 
 
 Find all the answers, and state what values of x and y belong 
 together. 
 
 June, 1884. 
 
 1. A landowner laid out a rectangular lot containing 1200 
 square yards. He afterwards added 3 yards to one dimension 
 of his lot, and subtracted \\ yards from the other; thereby 
 increasing the area of his lot by 60 square yards. Find the 
 dimensions of the lot before and after the change. 
 
 How do you explain the negative solution 1 
 
 2. Solve the equation 
 
 x-\- 1 2 t + 2 
 
 c ex ax — bx ' 
 
 reducing the answers to their simplest forms. 
 
 3. Solve the equations 
 
 x 2 + y 2 — 52 ; xy + U = 0. 
 Find all the sets of answers, and state which answers belong 
 together. 
 
 4. Multiply «3 — ai -\- 1 — a~ i -\- a~i by aM -j- 1 -f- a" *■ 
 Simplify the following expression : 
 
 V[** ■ •($! 
 
 5. Prove that if the corresponding terms of two proportions 
 be multiplied together, the result is a proportion. 
 
 6. Find the greatest common divisor of 
 
 9a; 5 — 7a; 8 -f 8x 2 -\- 2x — 4 and 6a: 4 — 7ce 8 — 10a; 2 + 5x -f 2. 
 
12 HAKVAKD EXAMINATION PAPERS. 
 
 September, 1884. 
 
 (Time allowed, 1J hours.) 
 
 1. Solve the equations 
 
 2 x 1 + 3 x y - 3 if + 12 = 0, 
 3a; + 5y + l = ; 
 and state what values of x and y belong together. 
 
 ocn ±i, i.- a— c x — a 3 6 (a; — c) 
 
 z. bolve the equation = ^— ; — , 
 
 x — a a — c (a — c) (x — a) 
 
 reducing the results to their simplest form. 
 
 3. Find the sixth term of the 19th power of f j/x 2 — ^-\ , 
 reducing the result to its simplest form. ' 
 
 4. Find the greatest common divisor of 
 
 2x 4 -3x 8 + 2x 2 — 2x -3 and 4 a: 4 + 3 a; 2 + 4 a; - 3. 
 
 5. Solve the equation 
 
 V7 - x + \/3 x + 10 + «</x + 3 = 0. 
 
 6. A vessel is half full of a mixture of wine and water. If 
 filled up with wine, the ratio of the quantity of wine to that 
 of water is ten times what it would be if the vessel were filled 
 up with water. Find the ratio of the original quantity of 
 wine to that of water. 
 
 June, 1885. 
 
 (Time allowed, 1| hours.) 
 
 1. Three students, A, B, and C, agree to work out a series 
 of difficult problems, in preparation for an examination ; and 
 each student determines to solve a fixed number every day. 
 A solves 9 problems per day, and finishes the series 4 days 
 before B; B solves 2 more problems per day than C, and 
 
ALGEBRA. 13 
 
 finishes the series 6 days before C. Find the number of 
 problems, and the number of days given to them by each 
 student. 
 
 2. Solve the following equation, reducing the answers to 
 their simplest form : 
 
 2 
 
 / a(l + 2x) __ b (3 x - 1) \ 
 \b(l + 3x) a(2x + l)J 
 
 = 0. 
 
 1 + 3* 
 
 V3 1 
 
 3. Solve the equation — — 
 
 </2 x - 1 - Vx - 2 Vx^l 
 
 4. A certain whole number, composed of three digits, has 
 the following properties : 10 times the middle digit exceeds 
 the square of half the sum of the digits by 21 ; if 99 be added 
 to the number, the order of the digits is inverted ; and if the 
 number be divided by 11, the quotient is a whole number of 
 two digits, which are the same as the first and last digits of 
 the original number. Find the number. 
 
 5. Given * + 6 _ y = 8 ; find the value of 1 ^~ 3y . 
 
 7 x — 2y 2x — y 
 
 6. Find the greatest common divisor of 
 3a; 4 -a: 8 -2a; 2 + 2a;-8 and 6z 8 + 13 a; 2 + 3 x + 20. 
 
 7. Find the square root of 
 
 4 - 12 x + 5 x 2 + 26 x s — 29 x 4 - 10 x*> + 25 x\ 
 
 September, 1885. 
 
 (Time allowed, 1J hours.) 
 
 1. A certain manuscript is divided between A and B to be 
 copied. At A's rate of work, he would copy the whole manu- 
 script in 18 hours ; B copies 9 pages per hour. A finishes his 
 portion in as many hours as he copies pages per hour ; B is 
 occupied 2 hours more than A upon his portion. Find the 
 number of pages in the manuscript, and the numbers of pages 
 in the two portions. 
 
14 HARVARD EXAMINATION PAPERS. 
 
 2. Solve the following equation, reducing the answers to 
 their simplest form : , „ ,„ 
 
 b-(x ) 
 
 h( x ~ a ) _ 1 _ 2 ^ « / _ 
 
 b (x + a) a (x + a) 2 
 
 3. Solve the equations 
 
 SVx + 2-Vy _ x*+l _ y 2 -64 . 
 
 4^-2^/2/ ' 16 * 2 ' 
 
 finding all the values of x and y, and showing which values 
 helong together. 
 
 4. Two casks, of which the capacities are in the ratio of 
 a to b, are filled with mixtures of water and alcohol. If the 
 ratio of water to alcohol is that of m to n in the first cask, and 
 that of p to q in the second cask, what will be the ratio of 
 water to alcohol in a mixture composed of the whole contents 
 of the two casks ? Reduce the answer to its simplest form. 
 
 What does the answer (in its simplest form) become, if 
 m = q = ? and what is the simplest statement of the 
 question in this case ? 
 
 5. Find the 10th term of (x - yf ; of (~ - ^-Y- 
 
 The numerical coefficients are not to be computed, but expressed 
 in terms of their prime factors ; the literal parts are to be reduced to 
 the simplest form. 
 
 Note. The above five questions constitute the paper; and all applicants 
 are expected to do them if possible. The following question is not required, 
 and is not necessary to make a perfect exercise; but it may be added^ at 
 the discretion of the student, and will be counted to improve the quality 
 of an imperfect exercise. 
 
 6. Reduce to its lowest terms 
 
 6a; 5 — 9x* + Ux* + 6x i — 10x 
 lx 6 + 10 x* + 10 x a + 4 a; 2 + 60 x ' 
 
ALGEBRA. 15 
 
 June, 1886. 
 
 (Time allowed, 1J hours.) 
 
 1. A boat's crew, rowing at half their usual speed, row 
 three miles down a certain river and back again, in the middle 
 of the stream, accomplishing the whole distance in 2 hours 
 and 40 minutes. When rowing at full speed, they go over 
 the same course in 1 hour and 4 minutes. Find (in miles per 
 hour) the rate of the crew when rowing at full speed, and the 
 rate of the current. 
 
 (Notice both solutions of this problem.) 
 
 2. Solve the equation 
 
 3 V* 3 + 17 + Vx s + 1 + 2 \/5x i + 41 = 0. 
 Substitute the answers, when found, in the equation, and show 
 in what manner the equation is satisfied. 
 
 3. 
 
 Solve the 
 
 ! equations 
 
 
 
 
 4 
 
 x + - 
 
 X 
 
 S|=^ + '>. 
 
 X 
 
 + 3* 
 
 4. 
 
 Solve the 
 
 •■ equation 
 
 
 
 
 
 (a + 2 b) x a 
 
 2 
 
 4 5 2 
 
 
 a — 2 b a — 
 
 2b 
 
 X 
 
 and reduce the answers to their simplest form. 
 
 5. Find the greatest common divisor and the least common 
 multiple of 4a; 8 — 4a; 2 — 5x + 3 and 10a; 2 — 19a; + 6. 
 
 6. Find the 6th and the 25th terms of the 29th power of 
 (x — y) ; reducing the numerical coefficients to their prime 
 factors, and not performing the multiplications. 
 
 (3/ jn 
 
 "X _ 2~ ) ' 
 reducing exponents to their simplest form, and combining 
 similar factors. 
 
16 HARVARD EXAMINATION PAPERS. 
 
 Septbmbeb, 1886. 
 
 (Time allowed, 11- hoars.) 
 
 1. Solve the equation 
 
 ^[2b(x + l)f a /l 5ax-4:b \ =0 _ 
 4 b x 3 + 5 a x \x 4 b x 2 + 5 a) ' 
 
 and reduce the answers to their simplest forms. 
 
 2. Solve the equation x~" — x* — 1 (x s + 1). 
 
 3. A and B have 4800 circulars to stamp for the mail ; and 
 mean to do them in two days, 2400 each day. The first day, 
 A, working alone, stamps 800 circulars, and then A and B 
 together stamp the remaining 1600 ; the whole job occupying 
 3 hours. The second day, A works 3 hours, and B 1 hour ; 
 but they accomplish only ^ of their task for that day. Find 
 the number of circulars which each stamps per minute, and 
 the length of time that B works on the first day. 
 
 4. Find the value of x from the proportion 
 
 5ac */~n i 4 /9^ 3a 2 . /3c 
 
 and express the answer with the use of only one radical sign. 
 
 5. Given the three expressions 
 
 2x 4 + a: 8 - 8a; 2 - x+ 6, 
 4a: 4 + 12a; 8 - a 2 -27 a; -18, 
 4 x* + 4x 8 -17x*- 9 x + 18 ; 
 find the greatest common divisor and the least common mul- 
 tiple of the first two of these expressions ; also those of the 
 whole group of three. 
 
 June, 1887. 
 
 (Time allowed, 1 hour.) 
 
 1. Solve the following equation : 
 
 Vx — 3+ a/3 x + 4 + VaT+2 = 0. 
 Find two answers, and verify the positive answer by showing 
 that it satisfies the equation. 
 
ALGEBRA. 17 
 
 2. A broker sells certain railway shares for $ 3240. A few 
 days later, the price having fallen $ 9 per share, he buys, for 
 the same sum, 5 more shares than he had sold. Find the 
 price and the number of shares transferred on each day. 
 
 3. Solve the following equation, finding four values of x : 
 
 x i + (2 a 2 + 3 a b - 2 6 2 ) 2 = 5 (a 2 + b 2 ) x\ 
 
 4. Reduce the following expression to its simplest form as 
 a single fraction : -, _ » .. _ 
 
 1 + x s ~ 1 + x 
 
 1 + X 2 1 + X 
 1 - X 2 + 1 - X 
 
 September, 1887. 
 
 (Time allowed, 1 hour. ) 
 
 1. Solve the following equation, finding four values of x : 
 
 , , w ja a 2 (z + «) b\x-b) 3a 2 b 2 
 
 (x + a) (sb — o) r-^ = ; 
 
 v ' y ' x + b x — a (x-a)(x + b) 
 
 2. At 6 o'clock on a certain morning, A and B set out on 
 their bicycles from the same place, A going north and B 
 south, to ride until 1£ p. m. A moved constantly northwards 
 at the rate of 6 miles per hour. B also moved always at a 
 fixed rate ; but, after a while, he turned back to join A. Four 
 hours after he turned, B passed the point at which A was 
 when B turned; and, at 1£ p.m., when he stopped, he had 
 reduced, by one half, the distance that was between them at 
 the time of turning. 
 
 Find B's rate, the time at which he turned, the distance 
 between A and B at that time, and the time at which B would 
 have joined A if the ride had been continued at the same rates 
 of speed. Find the answers for both solutions. 
 
 3. Find the sixth term of each of the following powers : 
 
 „ f &a? b ^ 
 
 (x-yY; { —^ 
 
 Vb 
 
18 HARVARD EXAMINATION PAPERS. 
 
 4. Reduce the following fraction to its lowest term : 
 6 a; 4 -13 a; 3 + 3 a 2 + 2 a: 
 6a; 4 - 9 x 3 + 15 x 2 - 27 x - 9 ' 
 
 June, 1888. 
 
 (Time allowed, 1 hour.) 
 
 1. Eednce the following expression to its simplest form as 
 
 a single fraction : „ „ . 
 
 ° 1 — x z I x \ 
 
 1 + y VI + x ~ 1 
 
 _ I 1 a; 2 + y 2 - x + y \ ' 
 
 \l-y 1-f I 
 
 2. Solve the following equations, finding, and reducing to 
 their simplest forms, two sets of values of x and y : 
 
 (x + 3y):(2x-y)=Q > -^):l, 
 
 x 2 = l{xy + 3ay + 18 a 2 ). 
 What are the answers, when a = 2 and b = — 3 ? 
 
 3. Two travellers, A and B, go from P to Q at uniform but 
 unequal rates of speed. A sets out first, travelling on foot at 
 the rate of 20 minutes for every mile. B follows, going 1 
 
 p Q 
 
 mile while A traverses the distance -^-- . B overtakes and 
 
 oO 
 
 passes A, 8 miles from P; and when B reaches Q, he is 9 
 miles ahead of A. Find the distance P Q, and B's rate of 
 speed in minutes to the mile. 
 
 (Obtain two solutions.) 
 
 4. Two men, working separately, can do a piece of work in 
 x days and y days, respectively ; find an expression for the 
 time in which both can do it, working together. 
 
 A is 20 years old, and B is — 2 j'ears older ; what is the 
 age of B ? 
 
 What are the values of x which satisfy the equation a; 2 — 3i? 
 
ALGEBRA. 19 
 
 5. Write out (cc — y) n . 
 Find the square root of 
 
 4 a; 6 - 12 a; 5 + 5 x* + 26 x s - 29 x 2 - 10 x + 25. 
 
 
 
 September, 
 
 1888. 
 
 
 
 
 
 (Time allowed, 1 hour ) 
 
 
 
 1. 
 
 Reduce the following expression to its lowest terms 
 
 as a 
 
 ,ing 
 
 le fraction : 
 
 2x 
 "3" 
 
 
 
 
 
 1 
 
 2 x 3 + 11 x 2 
 
 -43 a: 
 
 -24 
 
 
 
 X 
 
 14 x s - 31 a; 2 
 
 -31* 
 
 - 6 
 
 
 2. Solve the following equations, finding, and reducing to 
 their simplest forms, two sets of values of x and y : 
 
 0. 
 
 a 2 - b 2 ) 
 
 (b — a)x \(a + b) y 
 What are the values of x and y, if a = 3 and J = — 1 ? 
 
 3. Tristram is ten years younger than Launcelot ; and the 
 product of the ages they attained in 1870 is 96. Find the 
 ages they attain in 1888. 
 
 (Two solutions.) 
 
 4. A sum of $ 100 is put at compound interest at 4 per cent 
 per annum for x years ; find a formula for the amount. 
 
 5. Write out the first five terms and the last five terms 
 of {x-y)*\ 
 
 Find and reduce to its simplest form the fifth term of 
 
 ft, 3&-v> 
 
 */0 
 
20 ALGEBRA. 
 
 June, 1889. 
 
 (Time allowed, 1 hour.) 
 
 1. Find the greatest common divisor and the least common 
 multiple of 
 
 6a: 4 — 5x a — 10 x* + 3x - 10 and 4a; 3 — 4a; 2 - 9a: + 5. 
 
 2. Solve the following equations, finding and reducing to 
 their simplest forms two sets of values of x and y : 
 
 l _ I y _ x + Qa \ _ 
 
 x + y \a{x — y) x 2 — i/J ' 
 y : (7 x - 2 y) = {b - a) : (2 a - 9 b). 
 
 What do the answers become, when a — 6 and b = — 2 ? 
 
 3. A certain librarian spends every year a fixed sum ■ for 
 books. In 1886, the cost of his purchases averaged two dol- 
 lars per volume ; in 1887, he bought 300 more volumes than 
 in 1886 ; and in 1888, 300 more volumes than in 1887. The 
 average cost per volume was thirty cents lower in 1888 than 
 in 1887. Find the number of volumes bought each year, and 
 the fixed price paid for them. (Obtain two solutions.) 
 
 4. Find the fourth term of (a; — y) 2 ' ; 
 
 / b 5 4,-2X27 
 
 5. Solve the equation 
 
 *J(x + a) + ^/x + ^/{x — a) = 0. 
 
 Explain the possibility of satisfying this equation, the con- 
 necting signs being both plus. 
 
 September, 1889. 
 
 (Time allowed, 1 hour. ) 
 
 1. Reduce the following expression to its simplest form as 
 a single fraction : — 
 
 a + b 
 
 2 T a + bj 
 
HARVARD EXAMINATION PAPERS. 21 
 
 2. Find the second term of (x — y) 61 ; 
 
 Extract the square root of 4 a; 4 — 12 x s — 11 a; 2 + 30 x + 25. 
 
 3. The distances traversed in any given time by two cou- 
 riers, who are travelling on the same road in the same direc- 
 tion, are to each other in the ratio of p to q. The second 
 courier passes a given point on the road n hours later than 
 the first. How many hours after he passes this point will the 
 second courier be with the first ? 
 
 In what case is the answer negative, and what is the inter- 
 pretation of this result ? 
 
 4. A certain principal (a; dollars), at simple interest at y 
 per cent per annum for two years, earns a certain interest 
 
 \-j?d). If the principal had been (x — 20) dollars, and at 
 
 compound interest at y per cent per annum, compounded an- 
 nually, it would have earned the same interest in the same 
 time. If the principal had been (a; + 80) dollars, and at 
 simple inteiest at (y — \) per cent per annum, the interest 
 for two years would have been $1 less than that actually 
 earned. Find the values of x and y. 
 
 5. Solve the following equations, finding and reducing to 
 their simplest forms two sets of values of x and y : — 
 
 x + y : x — y = a : b, 
 
 What do the answers become, when a = — 3 and b — 1 ? 
 
 a; 
 
22 ELEMENTAKY ALGEBRA. 
 
 EXAMINATIONS FOR ADMISSION TO YALE 
 COLLEGE. 
 
 June, 1878. 
 
 ft I ft f* -^ - ft IT i 7"" 
 
 1. (a) Reduce — — — 3 5 to its lowest terms. 
 
 a/6 \Th 
 
 (b) Multiply a- e b 2 by -f= ; and divide a~ 6 b 2 by ~. 
 
 2. Solve the equations : 
 7 x — 6 x — 5 x 
 
 («) 
 
 .35 6a;— 101 5 
 
 3. (a) Solve the equation — — - -\- 7f = 8. 
 
 Jo 
 
 (6) It is required to find three numbers such that the product 
 of the first and second may be 15, the product of the first and 
 third 21, and the sum of the squares of the second and third 74. 
 
 4. Find the sum of n terms of the series 1, 2, 3, 4, 5, 6, &c. 
 
 5. By the binomial theorem expand to five terms (a 3 — b s )~i. 
 
 September, 1878. 
 1. Find the value of each of the following expressions : 
 
 (°) tj—x — iiXii + i ); 
 
 w 1 -\-y x-\-x 2 \ '1 — as/ 
 
 (b) (a-« -»§)•; 
 
 (c) 3Vf + 2VA + 4VA- 
 
EXAMINATION PAPEES. 23 
 
 2. (a) --^ = a;— \-~=b; \~-=o. 
 
 s ' x y z x y z x ' y z 
 
 Find x, y, and z. 
 
 (b) Solve the equation 
 
 17 — 3* 4x4-2 K /„ 7a:+14\ 
 
 —5 j^ = 5 -\ Gx — iH- 
 
 3. Solve the equations 
 
 . . 10 10 3 
 
 x sc — j— 1 x-\-2' 
 (b) 2a;3-f 3a:J = 2. 
 
 4. (a) Find the sum of 13 terms of the series 2\, 2£, 3£, etc. 
 (b) Find the value of 1 + i + tV + Ai etc -> to infinity. 
 
 5. By the binomial theorem expand to five terms (a 2 -\- a; 2 ) - *. 
 
 June, 1879. 
 
 1. Divide (3 a — b) bj a-\-b-\ and simplify. 
 
 1 ,a — b 
 
 ^ a + b 
 
 2. (a) Find the sum and difference of \/l8a 3 & 3 and \/50 a s 6*. 
 (S) Multiply 2 V3 — V 11 ^ by 4 a/3 — 2 V 13 ^- 
 
 . x — 1 . 23 — a; „ 4 + a; 
 
 3. Solve the equation ■ — 1 ^ — = 7 j — . 
 
 4. Solve the equation 
 
 x- 
 
 3 a; — 4 7 
 
 a; 
 
 — 2 a;— 1 20 
 
 5. The sum of an arithmetical progression whose first term 
 is 2 and last term 42, is 198; find the common difference and 
 the number of terms. 
 
 6. Expand to four terms, by the binomial theorem, (a s —b*)i. 
 
24 ELEMENTARY ALGEBRA. 
 
 September, 1879. 
 
 1. Add together ^j^ , ia . { J_^ 2 j { J +af) - 
 
 2. (a) Multiply together £ y' 3, £^/3, and J^/3. 
 (6) Divide 9m s («- 6)i by 3 m (a — 6)i. 
 
 3. Solve the equations 
 
 , > „ x — 4 5a; -J- 14 1 
 
 <» i + i = - 5 1 + 1 = 6; i + i = a. 
 
 x y x z y z 
 
 4. Solve the equation 
 
 15 72 — 6a; 
 
 = 2. 
 
 a; 2a; 2 
 
 5. Find the sum of 20 terms of the series 1, 4, 10, 20, 35, <fcc. 
 
 6. By the binomial theorem expand to 4 terms 
 
 (a) (l — b)-i; (6) (a 2 — x*)i. 
 
 July, 1880. 
 
 , > TV . , a + 1 . a — 1 a + 1 a — 1 
 
 1. (a) Divide — — - -\ j— - by — '—- — - , and re- 
 
 w a — 1 r a + 1 J a — 1 a + 1' 
 
 duce the quotient to its simplest form. 
 (5) Find the greatest common divisor of 
 
 x 4 — 6a; 2 — 8a; — 3 and 4a; 8 — 12a; — 8. 
 
 2. (a) Find the sum of 6 V^ 2 , 2 -^2a, and v'sT*. 
 (J) Reduce to its simplest form the product 
 
 (x-l-V^2)(x-l + V^2)(x-2 + V^3)(x~2-V^S). 
 
EXAMINATION PAPERS. 25 
 
 3. Solve the equations 
 
 (a) i(2x — 10) — ^(3* — 40) = 15 — fc(57 — x) ; 
 
 . . a; a: 2 + 1 
 
 4. Four numbers are in arithmetical progression ; the product 
 of the first and third is 27, and the product of the second and 
 fourth is 72. What are the numbers 1 
 
 5. By the binomial theorem expand to 4 terms, 
 
 (a) (1-6)" 3 ; (6) (z 2 - *■)». 
 
 September, 1880. 
 
 1. (a) Eequired, in its simplest form, the quotient of 
 
 a 4 — x* a 2 x -f- a; 3 
 
 a 2 — 2 as; -|- a 3 ' a 3 — x 3 
 
 (b) Find the greatest common divisor of 
 
 6a: 2 — 17a: + 12 and 12a: 2 — 4a: — 21. 
 
 2. Find the sum of 
 
 vT6, #81, — #=512, #192, — 7 #9. 
 
26 ELEMENTAKY ALGEBKA. 
 
 3. Solve the equations 
 
 (a) 5 x _2£zzi + 1=3a . + ^±2 + 7i 
 (6) 3a* 4- 10a: — 57 = 0; 
 
 K) 3 10 ^ 6 5 
 
 4. Find three geometrical means between 2 and 162. 
 
 5. By the binomial theorem expand to 4 terms, 
 
 (a) (l+a)»; (6) (g . _!. ^ 
 
 July, 1881. 
 
 1. Free from negative exponents (4a _3 6 2 x -4 ) -4 . 
 
 a 2 2x 15 
 
 2. Eeduce to lowest terms -=—. — — — : — ^-r • 
 
 or -j- 10x 4- 21 
 
 3. Factor r? — 2 ra 2 + »; a; 8 — 1 ; ar* — »V; *" 4" /■ 
 
 2 
 
 4. Make denominator rational of 
 
 V» — \/2 
 
 5. Multiply V* — 2 + V 1 ^ by \G + 2 — V^~3. 
 
 „ „ , 5 3* + l 1 
 
 6. Solve J— = -■ 
 
 x ;r 4 
 
 7. Solve a; 2 — a;?/ = 153; a? + V = !• 
 
 8. By the Binomial Theorem expand to four terms 
 
 9. Sum the infinite series 1 4- x H h &c. 
 
 2 4 
 
 \/» — x 
 
 i 
 
EXAMINATION PAPEES. 27 
 
 September, 1881. 
 
 1. Find the greatest common divisor of 
 
 a; 2 — 16 and a 2 — x — 20. 
 
 2. Factor x*y + xy\ x 8 — if, rfx* -4- 8 m' -f 16 s, 
 X s + 4a; 2 -+- 4x -4- 2 sb + 4. 
 
 3. Simplify ^ x + y . 
 
 a; ?/ 
 
 * — y * + y 
 
 2 2 
 
 4. Solve a; = ■ - ■_ -| ■ . 
 
 x -\- v 2 — a; 2 x — V^ — x 3 
 
 5. Add tyU, \^54, and \Vl28. 
 
 6. Multiply 2 — \/^3 — 3 V^~2 by 4 \/^3 -f 6 V^^ 
 
 x 
 
 8 
 
 8. Expand (« s — a)4 to four terms. 
 
 9. Given, in arithmetical progression, the first term, common 
 diiference, and sum of series ; find last term. 
 
 June -July, 1882. 
 1. Factor a 8 — 4a 2 6+4a* 2 , 4<cY— 9sbV, ft 2 — 4& 2 +46c— c 4 . 
 
 2. Solve a 8 = 21 -f- V« 2 — 9- 
 
 3. Find the continued product of 
 
 * — (2+VS), a — (2— a/3), * — (3— \/^l), x— (3+\^=T). 
 
 4. Divide 50 into two parts, such that the greater increased 
 by 3, shall be to the less diminished by 3, as 3 to 2. 
 
 f x* -4- tf 9 = 25 ) 
 
 5. Given ] ; „, f , to find « and tf. 
 
 C 2xy == 24 > 
 
28 ELEMENTARY ALGEBRA. 
 
 6. Sum the infinite series 1, £, £, etc. 
 
 § x |9 
 
 7. Eesolve — s ; — — into partial fractions. 
 
 x 2 — 8a; -J- 15 
 
 8. Expand by the binomial theorem to 3 terms - V^ 2 — a' 
 
 9. Revert the series y = * + x 2 4" x" 4" **• 
 
 September, 1882. 
 
 1. Factor x 8 — x 2 y 6 , x 2 -\- x -\- \, x*z -f- 2a; 8 2/ 8 « + % f^ 
 
 2. Multiply x—2 \/5 + 3 ^/—S by * — 2 y^— 3 V— 5. 
 
 3. Find x from the equation r = 7 ::• 
 
 ^ x — 2 a; — 4 as — o a; — o 
 
 4. Find a; from (\/x — \/b)l = nxi. 
 
 x i 
 
 5. Find x from - K =2x. 
 
 x — 20 
 
 6. The sum of the first and second of four iranibeTS in geo- 
 metrical progression is 15, and the sum of the third and fourth 
 is 60. Required the numbers. 
 
 7. Expand „ into an infinite series. 
 
 2>x — x 1 
 
 8. How many combinations can be made of 8 letters taken 
 5 at a time 1 
 
 9. Extract the square root of 6 -)- y^O- 
 
EXAMINATION PAPERS. 29 
 
 June, 1883. 
 
 1. Reduce the following expression to its simplest form, 
 1,1.1 
 
 x (x — a) {x — b) a (a — x) (a — b) b (b — x) (b — a) ' 
 
 2. Resolve y 9 — b 9 into three factors. 
 
 3. Change x y~* — 2 xi y~ x z~^ -4- « _1 to an expression which 
 will contain no negative exponents. 
 
 T „a-\-b-\-c-{-d a — b A- c — d 
 
 4. If — \—r- ! — , = =— ^ 1 — , , prove by the 
 
 a -\- b — ■ c — d a — b — c -\- d 
 
 principles of proportion that - = - • 
 
 5. Find the value of 2 a y/1 -f- x 2 when 
 
 6. Given (7 — 4 \/H) ^ + (2 — v^) x = 2, to find x. 
 
 7. The sum of two numbers is 16, and the sum of the recip- 
 rocals is J. What are the numbers 1 
 
 8. Compute the value of the continued fraction 
 
 1 
 
 • + - ' 
 
 i+ ' 
 
 * + t 
 
 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. 
 
30 ELEMENTARY ALGEBRA. 
 
 September, 1883. 
 
 1. Divide 
 
 c — b _ c 2 — b 2 c+l c 3 + b a 
 
 2. Find the value of — to three decimal places. 
 
 V3+1 
 
 3. Given \/xl = 2 <\/2, to find a;. 
 
 4. Find the — f th power of 256 xi y~*. 
 
 5. A grocer has two sorts of tea, one worth a cent3 a pound, 
 the other b cents a pound. How many pounds of each sort 
 must be taken to make a mixture of m pounds worth c cents a 
 pound ? 
 
 6. Reduce -r—_ — ■. .-*- ^ ; to its simplest form. 
 
 b m + i b m ~i 
 
 7. Solve the equation 
 
 4 \/x + Vs = 21. 
 
 8. Find the cube root of 
 
 a* — 6 a; 8 + 3 a; 4 + 28 a; 3 — 9 a; 2 — 54 a; — 27. 
 
 9. In the proportion 
 
 a 2 — b 2 c 2 — d? „ , a c 
 
 prove that - = - • 
 
 10. Insert three arithmetical means between — 9 and 18. 
 
 11. Write the eleventh term of (a — b) 12 . 
 
 June, 1884. 
 
 1. Reduce the following fractions to their lowest terms : 
 
 a?c + abc-\- i 2 c_ b 2 + 4y 2 
 liT+aW + b* ' 6 6 -f 64 j/ 6 ' 
 
EXAMINATION PAPERS 
 
 x -\- 1 /x 1 
 
 2. Reduce 4/ — ■ — - to its simplest form. 
 
 x — l " r-|-l r 
 
 3. Solve the equations 
 
 . 2(a» + 6 2 ) 4ab 
 
 4. Multiply * — VF + 1 — V— 10 — 2y5 by 
 
 x — V5 + 1 -f V— 10 — 2^/5. 
 
 5. Find the number whose cube root is one fifth of its square 
 root. 
 
 6. Find x from the equation Vl +■« — •«* — 2 (1 — |-;r — £ 2 )= J. 
 
 7. A and B can do a piece of work together in 8 days. A 
 works alone 4 days, and then both finish it in 5 days more. 
 In what time could each have done it alone ] 
 
 8. A traveller has a journey of 132 miles to perform. He 
 goes 27 miles the first day, 24 the second, and so on, travelling 
 3 miles less each day than the day before. In how many days 
 will he complete the journey 1 
 
 9. The. ratio of the circumference of a circle to its diameter 
 is 3.141592. Find by continued fractions three approximate 
 values. 
 
 10. Expand by the method of undetermined coefficients to 
 four terms V a — ' *"• 
 
32 YALE EXAMINATION PAPERS. 
 
 1. Given 
 
 June, 1885. 
 5a; + 2 /_ 3a;-l\ 3a; + 19 (x + 1 
 
 3 
 
 to find x. 
 
 ('-^X-^-^M 
 
 „,..,., b — y ca + ay b'' + y e ,5 
 
 2. Multiply . , ■ / , , 73 ' ■ , ,, , , » and " ' 
 
 r a 3 + y 3 P — by b' + y c 
 
 3. Multiply x — \ (1 - V 17 ^) by a; - J (1 + y^l). 
 
 4. Divide a; 2 ?/~ a — 2 + a; -2 ?/ 1 ' by x%y~% — x~% y*. 
 
 5. Given 91 a: 2 — 2x = 45, to find both values of x. 
 
 7 4 
 
 6. Given \- ■ — =: 4, 
 
 1 2 
 
 1 = 1, to find x and y. 
 
 V x Vv 
 
 7. Expand by the Binomial Theorem to five terms (1 + a)**. 
 
 8. In Arithmetical Progression, given d = the common 
 difference, a = the first term, and s = the sum of series ; 
 derive the formula for I = the last term. 
 
 „ T . ya^—bx + *Jc — mx \/a — bx — a/c — mx 
 
 9. If = ■ , prove 
 
 y a — 6 a: + y re x — c? y a — 6 a; — y w x— d 
 
 ft ^ ;£ 
 
 by using the principles of proportion that = 1. 
 
 nx — d 
 
ALGEBRA. 33 
 
 September, 1885. 
 
 a 2 1 
 
 b a a 
 
 1. Reduce - to a simple fraction. 
 
 all 
 
 b b a 
 
 2. Find the greatest common divisor of x* — 6 x 2 — 8 x — 3 
 and 4 x 3 — 12 x — 8. 
 
 3. Given V^ + x + Vl3 — a; = 6, to find £B. 
 
 4. Given x 4 — 21 a; 2 = 100, to find four values for x. 
 
 5. Find the value of a$ + aH^ + b^ when a — 8 and 5 = 64. 
 
 6. Given 1 2 2 _ 2 j- , to find x and y. 
 
 7. Given (a; 2 -as): y'a; : : ^/x : a:, to find values of x. 
 
 8. Expand r- into a series. 
 
 (2 a - 3)2 
 
 9. Compute the value of the continued fraction 
 
 1_ 
 
 12 + - 1 
 
 1 + 1 
 
 2 + 1 
 3 
 
 June, 1886. 
 !. Dlvlde __ _ __ 3 by — - b + c2 — fi2 . 
 
 2. Divide x*y-% -2 + x~ 2 y$ hy a^y^-aT^. 
 
34 YALE EXAMINATION PAPERS. 
 
 3. Multiply V— a + c $b by v"— a — c -^'*- 
 
 4. In — . make the denominator rational, and c( 
 
 A/3^1 
 
 pute the value of the expression to three places of decimals 
 
 5. Given a + x — Va 2 + x *JW- + x 1 , to find x. 
 
 r. c i ,i ,• S x + y = 12, 
 
 b. Solve the equations -j „ , ., _, 
 ^ ( a; 2 + jr = 74. 
 
 7. If A : B = C : Z>, prove by the principles of proportion 
 that A 2 - B 2 : B 2 = C 2 - D 2 : D 2 . 
 
 8. Find the sum of the infinite series \ + 5 V + tsif + e * c> 
 
 1 
 
 9. Expand to four terms by the Binomial Theorem 
 
 yi+*' 
 
 September, 1886. 
 
 1. Divide -j ^ , by y • 
 
 a: s — 2xy + y 2 x — y 
 
 2. Multiply a* — a 2 ^ + s^P- a&+«M— iHy at+bk 
 
 3. Free the fraction — „.- 8 „,-a J_ „-i ^ ronl ne £ at > ve 
 
 1 — x~ 8 y~ 2 + x 
 exponents. 
 
 ,_,.,. 7 a; + 9 / 2x~ 1\ . 
 
 4. Find * from —? \x — J = 7. 
 
 f a - y + *, 
 
 5. Find a;, y, ahd » from -< 6 = a; + s, 
 
 ( c ~ x + y. 
 
 6. Multiply ft — 5 + 2 V 17 ! by x - 5 - 2 v/^- 
 
ALGEBRA. 35 
 
 7. Make the denominator of the following fraction rational: 
 
 Vx — \/x + y 
 •\/x + V% + y 
 
 n r, , ■, • 1 2 4 
 
 8. Solve the equation -r + = - • 
 
 x — X x — -i o 
 
 9. If a : b = c : d, prove by the principles of proportion 
 
 that 
 
 a + b + c + d_a — b + e — d 
 
 a + b — c — d a — b — c + d 
 
 10. In a geometrical progression, having given first term, 
 ratio, and sum of series, write formula for last term. 
 
 11. Expand to 4 terms (a + x) *. 
 
 June, 1887. 
 
 1. Eesolve each of the following expressions into three 
 
 factors : 
 
 a*b + 8ac s bm% 4c 3 x 2 + 4c 2 se?/ + cy 2 . 
 
 __..., a b , b a, 
 
 2. Divide = r— ; by ■ r- -— • 
 
 a — b a + b a ~- b a + b 
 
 4. Solve V* + 40 = 10 — «Jx. 
 
 5. Solve mx 2 + mn — 2m \/n x + nx 2 . 
 
 6. Given — : — : : 3 : 7, and x 2 - y 2 = 9, to find x and y 
 
 a; y 
 
 7. Expand. by the Binomial Theorem 3 b (2 x — y)2. 
 
36 ALGEBRA. 
 
 September, 1887. 
 
 1. Free the following from brackets, and combine the terms 
 containing x and y. 
 
 a + b {x — 3a(y — 2x) + 3x[4a + 2(46 + 3)]}. 
 
 2. Factor a 8 — 27 5 s c 6 and m*x* — 2m*x !t y 2 + m 2 y*. 
 „ ,_. ,,. . x 4 — b* , x 2 -\-bx 
 
 3 - Mult1 ^ x*-2bx + b* hj -x~=V 
 
 4. Multiply a% — a? + 1 — a~b + a~% by a% + 1 + a~k 
 
 5. Solve - + | = 1. 
 
 a 6 
 
 * + * = l. 
 a e 
 
 o c 
 
 6. Solve the equation 4 a; 2 — «j3x + 16 = 2x\ 
 
 7. Make the denominator rational in the fraction 
 
 8. Expand by the Binomial Theorem 
 
 a* - bk 
 1 
 
 (a -b)%. 
 June, 1888. 
 
 1. Remove the parentheses from the following expression 
 and reduce it to its simplest form. 
 
 5x~(3x-i)-[7x + (2-9*)]. 
 
 2. Eesolve each of the following expressions into as many 
 factors as possible. 
 
 (a.) x* - 1. 
 
 (6.) (x 2 + y i -z 2 ) 2 -4:X ,i y\ 
 
 3. Divide - by . h 
 
 1 — x 1 + x J l — x\. + x' 
 
YALE EXAMINATION PAPERS. 37 
 
 4. Solve the equations 
 
 3 1_5 
 x y 4 
 
 a; y 
 
 5. Solve the equation v^ — 3 — V2cb + 8 = — 3. 
 
 6. Solve the equation x s — x^ = 256. 
 
 7. Multiply a; + 3 - 2 V 17 ! by a; + 3 + 2 V^l- 
 
 8. Expand (a; 3 + b)~^ to four terms. 
 
 g»* iyi3 «<i4 
 
 9. Given the series y = x — — + -r — -^ + etc., to find 
 
 ^ 4 o 
 
 the value of x in terms of y. 
 
 June, 1889. 
 
 1. Simplify 
 
 3a; _4j, + 5[_4a;- {3y - (2 + Ix - 2y) - 4}]. 
 
 2 . Solve §-^+2 = 0, *_?-2 = 0. 
 
 x y x y 
 
 3. Simplify ^-^ - ^-,) -r- {, 
 
 x + y _ x — y \ 
 x — y x + yj 
 
 4. Give the four factors of the first degree in x and y of 
 the expression 9 x* — 40 x 2 y* + 16 y k . 
 
 5. Solve 6a; 2 — 5a; = 6; solve also x (1 — x) = a x* + &• 
 
 3-5^^3 
 
 6. Eationalize the denominator of 
 
 5 + 3 
 
38 ALGEBRA. 
 
 7. Expand . in ascending powers of x. 
 
 V« — as 2 
 
 8. State and prove the formula expressing (s) the sum of 
 n terms of a geometric progression having the first term a 
 and the ratio r. 
 
 September, 1889. 
 
 1. Simplify 
 
 -[13 x -Uy -3 {2x-2(-y + 2x)-7ij\]. 
 
 2. Solve A 1= + 1 = 0, 4= ^=-1 = 0. 
 
 x + a x — a 
 
 „ „. ,., a; x x — a x + a 
 
 3. Simplify ■ ; , 
 
 r J x — a x + a x + a x — a 
 
 x — a x + a 
 
 4. Factor completely (x — y)" 1 — (a? — y 1 ). 
 
 5. Solve 12 a; 2 — 25 a; = - 12. 
 Solve also (2 + a;) (1 — a;) = a x*. 
 
 6. Express as a fraction with a rational denominator 
 
 4-3 -y/^5 
 3 + 2 V^5' 
 
 _2 
 
 7. Expand (a — 2x) s in ascending powers of x. 
 
 8. What is the number of permutations of the letters of 
 the word muriate taken three at a time ? Prove that the 
 number of permutations of any number of things taken t at 
 a time is 1 . 2 . 3 . . . £ times the number of combinations of 
 the things taken t at a time. 
 
EXAMINATION PAPEKS. 39 
 
 EXAMINATIONS FOR ADMISSION TO AMHERST 
 COLLEGE. 
 
 June, 1878. 
 
 1. Divide , a \~ m \ , by a * + am . 
 
 a* — 2 am -j- vr a — m 
 
 a? 1 
 
 2. Reduce — r to its lowest terms. 
 
 ab — b 
 
 1 4- af 
 
 3. Given b — ~ = : to find x. 
 
 1 — x 
 
 4. 2#+3y=23; 5x — 1y= 10; find a; and y. 
 
 5. Find the cube root of a 3 VT? 
 
 6. Divide 4 V24ac by 2 V^a. 
 
 7. A father's age is twice that of his son ; but 10 years ago 
 it was three times as great. What is the age of each i 
 
 8. If 1 be added to the numerator of a fraction, its value is 
 J ; and if 1 be added to the denominator, its value is J. What 
 is the fraction 1 
 
 September, 1878. 
 
 1. Resolve 1 — 36 y 2 into two factors. 
 
 2. Find the least common multiple of 9 a 8 , 12 a 2 X s , and 
 24a*V. 
 
 a — x 4- d 
 
 3. Find the sum ot x -j- - and 
 
 b m — y 
 
 _ . . , x , x 4- 1 
 4. Divide^— T by— - I - 
 
40 ELEMENTARY ALGEBRA. 
 
 I x 
 
 5. Given 8 a = ^ — ; — , to find x. 
 1 -\- x 
 
 6. Reduce 7 V^ a " — 27 a 2 6 to its simplest form. 
 
 7. Find the square root of 4a 4 — 12a 3 + 5a 2 -f- 6a -f" *• 
 
 8. What numbers are those whose difference is 20 and the 
 quotient of the greater divided by the less is 3 % 
 
 June, 1879. 
 
 1. Eeduce 3a — (2a — [a -\- 2]) to_ its simplest form. 
 
 2. Find the greatest common divisor oi2x s — 7x*-\-hx — 6 
 and 3a 3 — 7 x* — 7* + 3. 
 
 1 — x I -\- x 
 
 3. Eeduce — — to its simplest form. 
 
 — 1 — 
 
 1— x^ \+x 
 
 4. Eesolve a e — W into four factors. 
 
 \\x 3 
 
 5. Given 7x = 3x 4- 7 : find x. 
 
 4 ' 
 
 6. A crew can row 20 miles in 2 hours down stream, and 12 
 miles in 3 hours against the stream. Eequired the rate per hour 
 of the current, and the rate per hour of the crew in still water. 
 
 7. Extract the square root of 9 a; 4 — 12a^ + 16a; 2 — 8 a -f 4. 
 
 8. Express 2x 3 yi — 3« -1 y* — x~*y~% with positive exponents. 
 
 9. Divide # m_1 by x " and reduce the quotient to its sim- 
 plest form. 
 
 10. Multiply a + b y '(— 1) by a ~ b ^/(— 1). 
 
EXAMINATION PAPERS. 41 
 
 September, 1879. 
 
 1. Interpret a 2 ; a? ; a -2 ; a$. 
 
 2. Multiply a — 6 by e — d, and deduce the rule that "like 
 signs give -|- and unlike give — ." 
 
 3. Separate into prime factors 3m* x — 3n*x. 
 
 4. Explain the reason of the following equations : 
 
 a m a n = a m + n ; a m -=- a" = a m ~ n ; (a m ) n = a™. 
 
 _ 2x — 9 x—3 . x 25 — 3x 
 
 5- -27 r~ + 18=-^— ' finda: - 
 
 6. .4 left a certain town at the rate of a miles an hour, and 
 in n hours was followed by B at the rate of b miles an hour. 
 In how many hours did B overtake A 1 
 
 7. The sum of two numbers is * and the difference is d. What 
 are the numbers 1 Show from the result that if from the greater 
 of two numbers you subtract one half the sum, the remainder 
 will be one half the difference. 
 
 8. Find the square root of a; 4 — 4 n?y -(- 6 x 2 y 2 — ix y 3 -\- y 4 . 
 
 9. Simplify 7 ^/(a 2 "6 2m V). 
 10. Subtract 3 tya s from 6 tya*. 
 
 June, 1880. 
 
 1. Eeduce (a -\- b — c) V« + V — (a — |— i — j— c) (as -J- y)l to 
 
 its simplest form. 
 
 2. Eesolve a 6 — b e into its prime factors. 
 3- Divide CM -p* by jj- 
 
 6b*dd 2 J V'dd'e 
 
42 ELEMENTARY ALGEBRA. 
 
 3a; 4- 2a x — 5a _ „ , 
 
 4. ■ : = 5 a : find x. 
 
 2 3 ' 
 
 5. What number multiplied by m gives a product a less than 
 n times the number % 
 
 6. 5a;-|-3y=19, Ix — 2y = 8 ; find x and y. 
 
 7. Find the square root of a -f- 2a$#! -(- a;. 
 
 8. Find the square root of Si\a i x~ 2 yiz~^. 
 
 ar 2 -J- b~ 5 
 
 9. Free _ 4 _^ from negative exponents, and reduce the 
 
 result to its simplest form. 
 
 10. Multiply a + b V~l by a — b */^—\, 
 
 Septembeh, 1880. 
 
 1. Factor 81 a 8 — 1. 
 
 2. Find the greatest common divisor of 
 
 a 8 — a 2 x -f- Zax? — 3x 3 , and a 2 — 5 ax -|- 4a£. 
 
 3. Reduce a -4- b r- — to a fractional form. 
 
 a — b 
 
 find x and y. 
 
 5. Find the square root of 9x* — 12^ -f- 16a 8 — 8 a; -)- 4. 
 
 6. Find the sum of ^/Za?b and V3xH. 
 
 7. Multiply 5ai by 3ai 
 
 8. Find the cube of a — bxi. 
 
EXAMINATION PAPERS. 43' 
 
 9. Find the cube root of (x -\- y) yx -\- y. 
 a-\-b V^^l ,0 — 6 a/ 11 "! 
 
 10. Simplify 
 
 a — bV— 1 a + 6V— 1 
 
 June, 1881. 
 
 1. Remove the parentheses and reduce to its simplest form 
 a — \a — [a — (a — #)]$. 
 
 2. Find the greatest common divisor of 8 a; 2 -4- 2 x — 3 and 
 12^ + 10^ — 4. 
 
 3. Resolve as 12 — y 11 into its simplest factors. 
 
 4. Reduce —„ -1 — - — to its simplest form. 
 
 x l — l'a; + l 1 — x 
 
 5. Find the square root of 
 
 9x? — 12*V + Wa?y* — 24 a!*/ + 4/ + 16*/. 
 
 6. y — a = 3 (x — 6) ; y — 6 = 2 (x — a); find x and y. 
 
 7. Multiply fyia by y'Qx. 
 
 8. A wine-merchant has two kinds of wine which cost 90 
 cents and 36 cents a quart, respectively. How much of each 
 must he take to make a mixture of 100 quarts worth 50 cents a 
 quart % 
 
 9. Divide xi -\- xi — 6 by xi — 2. 
 
 K— a 2 b s Y T 
 I to its simplest form. 
 
 11. If a : b = c : d, prove a -\- b : a — b = c -f- d : c — d. 
 
 12. 3a 2 — 4a; = 7; finds. 
 
44 ELEMENTARY ALGEBRA. 
 
 13. Insert three geometrical means between 13 and 208. 
 
 14. Demonstrate the two fundamental formuhe of Arithmeti- 
 cal Progression. 
 
 15. What is the Binomial Theorem? Give the first four 
 terms of (a -j- ar) 10 . 
 
 September, 1881. 
 
 €L x 
 
 1. Keduce 1 5 5 to a single fraction. 
 
 a — x <r — x i 
 
 2. Find all the factors of x« — 64 a 6 . 
 
 x . x a 
 
 3. - -r 7 1 — ; • Find x. 
 
 a b — a a -\- b 
 
 \l 
 
 (x + y) + S(x-y) = 80) 
 
 5. Find the greatest common divisor of x s -f- 3 x 2 -\- 4 x -\- 12, 
 and X s -4- 4 x 2 -\- 4 a; -4- 3. 
 
 6. Find the square root of 4 a; 4 — 1 2 a: 8 -|- 5 a: 2 -|- 6 a: -|- 1. 
 
 7. Find two numbers whose sum is a and whose difference is I. 
 
 8. Subtract \fa™b 2 from 3 a tyb, 
 
 9. 2a; = 4 + -. Find a;. 
 
 1 x 
 
 10. Insert two geometrical means between 24 and 192. 
 
 11. Expand (2 x -j- yf by the binomial theorem. 
 
EXAMINATION PAPERS. 45 
 
 June, 1882. 
 
 gSn y2n 
 
 1. Eeduce ■ — -: to its lowest terms. 
 
 x n -{- y" 
 
 2. Find a number of two digits such that it shall he equal to 
 7 times the sum of its digits, and if 27 be subtracted from the 
 number, the digits will be inverted. 
 
 3. Eeduce 4^2—6^/8 + 10 V^i to its simplest form. 
 
 4. Extract the square root of 81 a*x~' i y^z~\. 
 
 5. Solve the equation 3 as 2 — ix = 119. 
 
 6. Form a quadratic equation whose roots shall be — 3 
 and — 2. 
 
 7. Solve the equations < •'„ T ^ 
 
 { ar + xy = 
 
 = 15 
 -\-xy = 10 
 
 8. What proportions may be derived from the proportion 
 
 a : b = c : d% 
 
 9. Find the sum of the first n odd numbers 
 
 1 + 3-f 5. . .2n— 1. 
 
 10. What is the sum of the infinite descending series 
 
 J + ^+etc* 
 
 11. Give the first four terms of (1 + 2 as 2 )". 
 
 September, 1882. 
 
 1. Divide m 8 — w 3 by m 2 -\- mn -f- » 2 - 
 
 2. Multiply a; 4 — x s y + x*y* — xy s + y l by x -f- y. 
 
 3. Eeduce a to a fraction. 
 
46 ELEMENTARY ALGEBRA. 
 
 4. Given — 5 — ^ — - = ^ (a: + a) — a, to find *. 
 
 5. Develop (3 a; j . 
 
 6. A courier starts from a place and travels a miles a day ; 
 n days after he is followed by another, who travels b miles a 
 day ; in what time will the second overtake the first ? 
 
 7. Given \ , » ? „„ r . to find a; and a. 
 
 8. Divide 15 (a s b s )i by 3(ai 2 )i 
 
 9. Find the w 01 term of the series 2, 2$, 2f , etc. 
 
 10. Find the sum of n terms of the series 1 + i + £ + ^f, etc. 
 
 June, 1883. 
 
 1. Find the value of 
 
 6a — [ib — {4 a — (6 a — 4 6)}]. 
 
 2. Divide a.- 3 " — b« n by ar n — 6 2 ". 
 
 3. Show that a" = 1 ; also that «- m = 1 -r- a m . 
 
 4. Resolve a im — b im into its prime factors. 
 
 5. Find the greatest common divisor of 
 
 a 4 — V and a 3 + a?b — aW — b s . 
 
 6. Given 3 a a; — 25a; — \c — \mx = %c -\- \mx — % 
 — b x -\- 2 a x, to find x. 
 
 7. Divide the number a into two parts, such that the second 
 part shall equal m times the first part plus n. 
 
EXAMINATION PAPEKS. 47 
 
 o i 3« — 2x= 9) „ . 
 
 9. Multiply V* + c by V ft + c - 
 
 10. 3a; 2 — 4a; = 15; find a;. 
 
 11. Expand (1 -\- a; 2 ) 7 by the Binomial Formula. 
 
 12. Find the (2 w)th term of the series 1, 3, 5, 7 . . . 
 
 September, 1883. 
 
 1. Multiply a m - x J + aT'b-* — a"- 2 ^ 1 by ab. 
 
 2. Find the greatest common divisor of 
 
 3 a; 2 — 4 x y -f- y% and 5 a; 8 — 6 x 2 y -f- y % . 
 
 „ f 2a; — y = 9 ) „ , 
 3 -i 2y -3a; = -lll' finda;alld y- * 
 
 4. A boatman can row 6 miles an hour with the current, 
 and 2 miles an hour against it ; how far down the stream can 
 he row so as to return to his place of starting in 12 hours? 
 
 5. Extract the square root of 
 
 1 -f 4 a; 2 + 9 a: 4 + 4 a; + 6 a; 2 + 12a; 8 . 
 
 6. Find the square root of 64 -\/36 a 4 b~ 2 . 
 
 7. x 2 -|- \ x = \ ; find x. 
 
 8. Give the first three and the last three terms of (x — yf® 
 with their proper signs. 
 
48 ELEMENTARY ALGEBKA. 
 
 9. Find the sum 
 
 n—ln—2 n—3 ... 
 K etc., to n terms. 
 
 10. Insert two geometrical means between *£- and Jj 5 -. 
 
 June, 1884. 
 
 1. Write the equivalent of 4 , 4 without negative expo- 
 nents, and reduce the result to its simplest form. 
 
 2. Find the greatest common divisor of 
 
 x s — 2 x*y + xy 2 — 2 f, and 2 x* — 4 x*y — xf -f 2 y*. 
 
 3. Find the sum of 
 
 V9a6 8 , — yaH, and <\/a* 6 — 6 a 2 fe 2 + 9 a b*. 
 
 4. "What fraction is that which becomes § when its numera- 
 tor is increased by 3, and equal to J when its denominator is 
 diminished by 1 1 
 
 5. If \a — x: \a-\-x=b — y. 6 -^j show that 2 a; : y=a: b. 
 
 6. Solve 2 a; = 4 4--. 
 
 ' x 
 
 7. Given d the common difference, n the number of terms, 
 and s the sum of an arithmetical progression, to find the last 
 term I. 
 
 8. Insert five geometrical means between J and 21£. 
 
 9. Expand {f — e~ x )* by the Binomial Formula. 
 10. Solve x 2 — xy = — 2; xy — y 2 = — 3. 
 
ALGEBRA. 49 
 
 September, 1884. 
 
 1. Find the greatest common divisor of a: 8 — 2 a; 2 — x + 2 
 and x* — 3 x s + 3 x 2 — 3 x + 2. 
 
 2. The sum of the digits of a number of two figures is 9 ; 
 and if 9 be subtracted from the number the digits are re- 
 versed. What is the number? 
 
 ax +by =c\ 
 o. . f ; rind x and y. 
 
 a' x + b' y = c' ) ^ 
 
 a' x + V y 
 
 4. The sum of the squares of two consecutive numbers is 
 545. What are the numbers ? 
 
 5. Solve | 
 
 x — y = 2. 
 a?-f = 20. 
 
 6. Raise 2 \Z3aFy to the 4th power and reduce the result 
 to its simplest form. 
 
 7. What number added to 2, 20, 9, 34, will make the result 
 proportional ? 
 
 8.- Given d, the difference, n, the number of terms, and s, 
 the sum of an arithmetical progression ; find the formula for 
 I, the last term. 
 
 9. Find the sum of the infinite series 1 — J+i — i + etc. 
 
 10. Find the 4th term of (x — 2y) 12 by the Binomial 
 Theorem. 
 
50 AMHEKST EXAMINATION PAl'EliS. 
 
 June, 1885. 
 
 1. Divide *"* , by X ~=^- 
 
 x 2 +2xy + y 2 x + y 
 
 2. Factor x i - 2 a; 2 y 2 + y\ 
 
 3. Eeduce the product (3 x—'ifz-*) (5a: 8 y- 4 2 8 ) to its 
 simplest form, freed from negative exponents. 
 
 4. Solve Vx + 13 = 1 + Vx- 
 
 5. Solve ar 8 + 4 a x = 6. 
 
 6. Given i „ „ ' find x and w. 
 
 (. ar — y 2 = 21 ; 
 
 „ „ . 3x x — 4 a; — 10 „ , „ , 
 
 7. Given — -| ^ s — = a; — 6, to find a;. 
 
 8. What is the sum of the first twenty odd numbers ? 
 
 9. The first term of a geometrical progression is \, the 
 ratio £, the number of terms 7. Find the sum. 
 
 10. Give the fourth term of (2x — 3 y)~\ 
 
 September, 1885. 
 
 . _ b — c , , a — b 
 
 1. From a ^ — take — = c. 
 
 Li o 
 
 2. Divide a 6 — 1 into its prime factors. 
 
 4x s — 5x 2 + x 
 
 3. Eeduce to its lowest terms 
 
 4. Solve 
 
 a; 2 — 6 x + 1 
 3x — iy = — 6, 
 
 10 x + 2 y = 26. 
 5. Solve 2a; 2 -5x + 2 = 0. 
 
ALGEBRA. 51 
 
 6. Solve *' + »' = 3 J 
 
 * + y = 8. 
 
 7. What two numbers whose difference is d are to each 
 other as a to b. 
 
 8. Insert 5 arithmetical means between 2 and — 3. 
 
 9. What is the sum of the first 10 terms of the series 1, 2, 
 4, 8, etc. ? 
 
 10. Expand (a; 2 - 2 b) B . 
 
 June, 1886. 
 1. From " + b r 2c take 2 "-/ + ' . 
 
 _ _ , S^w + Sxv 2 , .„, 
 
 2. Reduce 7; — ; — ^ ^r— = to its lowest terms. 
 
 3ar + Qxy + 3y 
 
 _ a; — 4 5z + 14 1 
 
 3. Given 3 a; 4 = ^ To > to fincl ^ 
 
 4. Solve x + V* + 3 = 4 a; — 1. 
 
 5 . Given 2*-^ = 8, 4 y - *^ = 24* - *^f*; 
 find a; and y. 
 
 6. Find a/24 + \/54 - \/6. 
 
 7. A person sets out from a certain place, and goes at the 
 rate of 11 miles in 5 hours ; and, 8 hours after, another person 
 sets out from the same place, and goes after him at the rate of 
 13 miles in 3 hours. How far must the latter travel to over- 
 take the former ? 
 
 8. The 1st and 9th terms of an arithmetical progression 
 are 5 and 22. Find the sum of 21 terms. 
 
52 AMHERST EXAMINATION PAPERS. 
 
 9. Find the 12th term of the geometrical progression 
 
 \/2, -2, +2V2, -4, etc. 
 
 10. Find the first four terms of (2 x — 3 yf by the Bino- 
 mial Formula. 
 
 September, 1886. 
 
 1. Reduce (a + b — c) 2 + (a — b + c) 2 to its simplest form. 
 
 2 x 8 16 a; 6 
 
 2. Eeduce ^r—z =-; to its lowest terms. 
 
 3 x 3 — 24 x — 9 
 
 _ _. 3a: + 4 Ix-Z x - 16 
 
 3. Given = s — = — 3 — ; find a;. 
 
 5 2 4 
 
 4. Find x and ?/ from the equations 
 
 x-2 10 — a; _ y — 10 
 
 5 3 ~~ 4 ' 
 
 2y + 4 2x + y _ x + 13 
 
 3 8 ~~ 4 
 
 5. Solve 2a;— */2x — 1 = 03 + 2. 
 
 fi ,, U 2 +y 2 =50, 
 
 6. Solve i „ „ __ 
 
 (9a: + 7v = 70. 
 
 7. Reduce V45c 8 — \/80c 3 + ■v/5a 2 c to its simplest form. 
 
 8. Find the sum of the first 90 odd numbers by arithmet- 
 ical progression. 
 
 9. Find the sum of the geometrical progression 20, 19, 
 18^, etc. 
 
 10. Find the first four terms of (1 — a;) 12 by the Binomial 
 Formula. 
 
ALGEBRA. 53 
 
 June, 1887. 
 
 1. Eeduce (a + b — c) >\/x + y — (a + b + c) (x -f y)% to 
 its simplest form. 
 
 2. Resolve a — 5° into its prime factors. 
 
 3. Divide — ^ by — - — 
 
 6& 8 c*rf 2 6 2 c*a 2 e 
 
 , 3x + 2a x — 5a „ „ , 
 
 4. ^ 5 = 5 a ; find a;. 
 
 5. What number multiplied by m gives a product a less 
 than n times the number? 
 
 „ 5 a; + 3 w = 19 „. . 
 
 6 - 7,-2 y = 18 ;finda;and2/ - 
 
 7. Find the square root of a + 2 a? x? + x. 
 
 8. Find the square root of 81 a*x y$z ». 
 
 9. Find the roots of a x 2 + b x + c — 0. 
 
 10. sc 2 + xy = 10, x?/ — y 2 = — 3 ; find x and y. 
 
 September, 1887. 
 
 1. Eeduce a. — [2 6 — (3 c + 2 5 — a)] to its simplest form. 
 
 2. Divide drb m ~ n by a n - m d~ n . 
 C 1 
 
 c-1 • • , ., 
 
 3. Eeduce to its simplest form. 
 
 1--?- 
 c + 1 
 
 4 Given j _ * + „ V ~ 1Q K to find x and y. 
 (.2a; + 32/ = lJ) 
 
54 AMHERST EXAMINATION PAPERS. 
 
 5. Reduce a a/48 a 8 «? and \/fH to simpler forms, 
 
 6. Multiply 3^by 2y/g. 
 
 £*j £) gC X 10 
 
 7. Given — = — + o — 12 = — , to find x. 
 
 8. Find two numbers whose sum equals s, and whose dif- 
 ference equals d. 
 
 9. Solve the equation 3 a; 2 — 4 x = 119. 
 10. Find the first four terms of (x — 2 y) 7 . 
 
 June, 1888. 
 
 1. Resolve 16 a* b* m? — 8 a" b 2 m + 1 into its factors. 
 
 2. Find the greatest common divisor of 6 a; 3 — 6 x 2 y + 2 zy 2 
 -2/ and 12 a: 2 - 15 a: y + 3y 2 . 
 
 „ a: + 3 x-2 3x-h 1 „ , 
 
 3. — g g- = - lr - + j; find x. 
 
 5. a;-2y + 3s = 2; 2x-3y + z = l; Sx-y + 2z = 9; 
 find a; and y. 
 
 io 
 
 6. Solve the equations \/x + 5 = 
 
 V* + 12 
 
 7. Solve the equations x -f y = a; a; 2 + y 2 = S 3 . 
 
 8. Find the sum £ \/| and f \/^. 
 
ALGEBRA. 55 
 
 9. Demonstrate the fundamental formulae used in Arith- 
 metical Progression. Find the sum of the first n terms of the 
 progression 1, 3, 5, 7, etc. 
 
 10. Find the sum of the first n terms of a geometrical pro- 
 gression whose first term is a, and third term c. 
 
 September, 1888. 
 
 1. Find the value of s — when x = 
 
 x — 2a x — 2b a + b 
 
 2. Resolve 1 — c 4 into its prime factors. 
 
 1 — x 2 1 — v 2 
 
 3. Multiply together -z , -- 2 , and 1 + 
 
 1 + y' x + x 2 ' ~*~ 1 — x' 
 
 4. Solve the equation = a. 
 
 x — 3 x + 3 
 
 5. = + ~ = 18 ; -= — j = 21 ; find x and y. 
 
 6. a; 2 -f y 2 = 34 ; x y = 15 ; find a; and y. 
 
 7. Reduce —= to an equivalent fraction having 
 
 a — y a 2 — x 2 
 
 a rational denominator. 
 
 8. Find the ratio of an infinite decreasing geometrical pro- 
 gression of which the first term is 1, and the sum of the 
 terms is f. 
 
 9. Find the sum of the terms of an arithmetical progression 
 formed by inserting 9 arithmetical means between 9 and 109. 
 
 10. Expand (a — b) 6 by the Binomial Formula. 
 
56 AMHEKST EXAMINATION PAPERS. 
 
 June, 1889. 
 
 1. Find the greatest common divisor of 
 x 3 -2x 2 -x + 2 and a: 4 - 3 a: 8 + 3a; 2 - 3a; + 2. 
 
 2. Multiply a m — a n by 2 a — <z". 
 
 3. aa! + ^ = c l ; find sandy, 
 a' a; + 5' y = c' ) 
 
 4. The sum of the squares of two consecutive numbers is 
 545. What are the numbers ? 
 
 5 - X 2 V z ™ \ > find x and y- 
 
 x 2 — y 2 = 20 ) 
 
 •sr 
 
 /3a 5 , 1 . / 3 
 2 a \2b 
 
 6. Multiply 4ay -^y by 
 
 7. What number added to 2, 20, 9, 34, will make the sums 
 proportional ? 
 
 8. Given d the difference, n the number of terms, and s 
 the sum of an arithmetical progression ; prove the formula 
 for I, the last term. 
 
 9. Find the sum of the infinite series 1 — £ + £ — \-\- etc. 
 
 10. Find the fourth term of (x — 3 y) 12 by the Binomial 
 Theorem. 
 
 Septembee, 1889. 
 
 1. Factor c*—(a?-2ab + b 2 ); also x> - 5 x + 6. 
 
 2. Free _ — —j— from negative exponents. 
 
 cb o t)u y 
 
 y-a + 2 
 
 3. Simplify — 
 
 a -r.- j r .t. i.. 7a; + 9 / 2a;-l\ _ 
 
 4. Find a; from the equation — Ix — 1=7. 
 
 5. What are the numbers whose difference is 3, and the 
 difference of whose squares is 51 ? 
 
ALGEBRA. 57 
 
 6. Solve the equations 2x— 5y=9, 3x + 2y = 23. 
 
 7. What is the cube root of — 8 a ~ a 6 6 x ~ 2 ? 
 
 8. Find the square root of 4 a; 2 — 12 x y + 9 y 2 . 
 * T 9. Solve the equation a x 2 + 2 b x =■ c. 
 
 10. Given a; 2 + y" = 130 ; x 2 — y 2 = 32 ; find a; and ?/. 
 
 11. Find the sum of the series, l + £ + i+£ + etc. 
 
58 EXAMINATION PAPERS. 
 
 ENTRANCE EXAMINATIONS TO DARTMOUTH 
 COLLEGE. 
 
 1878. 
 
 1. Define term, factor, coefficient, exponent, power, root, equa- 
 tion. What is the degree of a term 1 When is a polynomial 
 homogeneous 1 
 
 2. Write the following without using the radical sign : 
 
 \fa; \ / a 2 ; A/a 2 + 6 2 — 2 aft. 
 
 3. Write the following without using negative exponents : 
 
 -2 7.-1 a ~ 2 
 
 a 2 l ah 1 ; ~^- 
 
 4. Multiply a — 6 V— 1 by a -f- &V— ^- Also a — b*\/—i 
 by a -f- cV — 1. 
 
 5. Raise a — b\/ — 1 to the 3d power. Simplify the radi- 
 cal(a 3 — 2a 2 6 + a6 2 )*. 
 
 6 . Solve t^L _ ^f 2 = 6. AlBo4-, + 6^ + e = 0. 
 
 a -f- a; a — a; a; 
 
 . , a? — 1 a; — 2 x-4-1 . . a* — (a — a;) 1 1 
 
 Also — — = — £ — . Also ' ^ = -. 
 
 2 3 6 a 4 -j- (a — *)* a 
 
 1880. 
 
 1. Define Algehra, factor, coefficient, exponent, fraction, equa- 
 tion. 
 
 2. Write the following without using the radical sign : 
 
ELEMENTARY ALGEBKA. 59 
 
 3. Write the following without using negative exponents : 
 
 *-*; p; (c- 1 )- 1 - 
 
 4. Write, in the simplest form, the values of 
 
 VI; Vl; Vitf; 8°; 8 3 ; 8~* 
 
 5. Find the product of a/o6 X — <**&* X (— «***) X 2 aft ; 
 also of (2 + V^l) (2 — V^). 
 
 6. Solve 
 
 x , a; a;— 2 .301 , a: — 2 
 
 ■ 6 = ; also — 1 — = .001 x + .6 — 
 
 a + \ a — I' 5 ' .5 .05 
 
 1881. 
 
 1. Define term, factor, coefficient, exponent, power, root, 
 equation. 
 
 ' 2. What is the degree of a term 1 
 
 When is a polynomial homogeneous ? 
 
 3. Write the following without using the radical sign : 
 
 tyc; <fc?; V« 2 + 6 2 — 2ab. 
 
 4. Write the following without using negative exponents : 
 
 30 • ( '/ 30 , , 
 
 5. Multiply a — c V — 1 by a -(- b V — 1. 
 
 6. Raise 6 — a V — 1 to the 3d power. 
 Simplify the radical Vc 3 — 2 c 2 6 -\- cb\ 
 

 
 EXAMINATION 
 
 FAPEKS. 
 
 a 2 - 
 
 ■f 
 
 a? — f 
 a — y 
 
 a. 
 
 «4 
 
 ■y 
 
 a; — 
 
 l 
 
 x — 2 x 
 
 4-1 
 
 2 
 
 
 3 
 
 6 
 
 a 
 
 
 &a°+c = 0. 
 
 
 60 
 
 7. Solve 
 
 8. Solve 
 
 1882. 
 
 1. Define positive and negative quantities, coefficient, ex- 
 ponent, similar terms, elimination, radical quantity. 
 
 2. Eesolve x 4 — y*, a 3 -\- 1, and 1 — |— a; 6 into their prime 
 
 factors. 
 
 „ „ . x4- 1 . a . 1 1 2a; 
 
 3. Solve ^-1=-; also __ — = __,; 
 
 also \™ + h y = c \. 
 
 4. Write the following without using the radical sign or 
 negative exponents : \/a?, \/l 4" 2 a 4" a *' a & _1 c a_1 » V* -2 - 
 
 5. Divide a? -\- a (b -\- c) V = l — 6 c hy a + J V— 1. 
 
 6. Eationalize the denominators of -g^ > 
 
 •\/c -y/c — -\/o 1 — W3 
 
 1883. 
 
 1. Divide a 2 -4- -= — 2 hy a. 
 
 1 a 2 a 
 
 2. Eesolve a 12 — a: 12 into six factors. 
 
 3. Find the least common multiple of 
 
 x s — x, x 2 — x — 2, and aj 8 -f 1. 
 
 4. Reduce 
 
 f a! ~ 1 _ y* x -~ x *\ r y z i ^ to its s i m pie S t form. 
 
ELEMENTARY ALGEBRA. 61 
 
 5. Solve the equations : 
 
 (i)2«-^ 3-J= T ; 
 
 (2) i 2 ^ 4 (3) > + * 4 
 
 V 3 6 ' \x y 12' 
 
 6. Write the values of 8-3, 8°, 16*. and (8 «*#■)«. 
 
 7. Multiply together \/abc 2 , a^b -1 ^, and a 365c \ 
 
 8. Divide a 8 — 6 2 by a* — V& 
 
 1884. 
 
 1. Resolve the following expressions into factors : 
 4 a 2_9&4. a 8 +l; 9 a 4 — 24a 2 6 2 + 16J 4 ; ^-(-2*— 3. 
 
 (1 ^s i _i_ a;S\ i 
 -j -[— - J - to its simplest form. 
 
 3. Solve the equations 
 
 (a) 3*-(3*-^)= : 
 
 z — 2 
 
 5 
 
 (a; + 2y=6 ( a; + y = 5a;y 
 
 l*-s = 1; l*+y= 8 ' 
 
 (d) v ^ FT+ v^rr=^ ! - 
 
 4. Find the value of 16* X 5° X 2~ 2 X 8~S [(— 2) 2 ]4. 
 
 5. Simplify the following expressions : 
 
 WC + V^fi (c) v^v^; 
 
 6. Divide n? -f- y 4 by at + 3^- 
 
62 
 
 DAETMOUTH EXAMINATION PAPERS. 
 
 1885. 
 
 1. Factor x* — 16 a 4 , x B + a 6 , x 2 — 2ax - b* + a 2 , 
 x * _ x _ 72. 
 
 2. Find the greatest common divisor and least common 
 multiple of x* — 3 x — 2 and x s — 2x i — x + 2. 
 
 2 + 1 
 
 1- 
 
 3. Simplify 
 
 2 * o 
 
 * and — = (1 _ V^) 2 . 
 
 1 _ a: +^ V- 3-1 
 
 x— i 
 
 4. Solve x 
 
 -( 
 
 1 -a l + a; \ _ x + 3 
 2 / 2 ' 
 
 2 x — 3y =y — x + 4. 
 
 6 Solve Va: + 2 — V* — 2 = <\/2 a;. 
 
 7. Find the value of 9| X 8 - ^ X 7° X i~i + (8 -2 X 3) -1 . 
 
 8. Multiply x — x- 1 by a: — x~\ 2 + V^3 by 2 V3, 
 and a;i — y* by a;3 + a;^ + y%. 
 
 1886. 
 
 1. Factor x 2 - 9 a 4 , a: 9 + 2/ 9 , a; 2 + 4 a:y — 4 + 4 j/ 2 , 
 j- 2 + 3 a x + 2 a 2 . 
 
 2. Reduce " 
 
 x*+l 
 » . 1 f l> + x \ x ~ 1) J *° J t s simplest form. 
 
 . * + 1. 
 
ALGEBRA. 63 
 
 3. Multiply x + x- 1 — 1 by x — a;- 1 + 1, and x% + y§ 
 by a;* — x y* + x* y — y t # 
 
 4. Divide a; — y by x* + y*, and a: 8 + x _s by a; + x~ 1 . 
 
 5. Write the value of 8^ X 9~* X 2 _1 x 3° X l -2 X -^81. 
 'x — 4 a: — 6\ _ x 
 
 6. Solve x — 3 
 
 V 
 
 7. Solve 
 
 4-§ = a; - 4 
 
 g - y = y - 5£ 
 
 and 
 
 x y 
 
 ? + » = * 
 
 La: y 
 
 1887. (See Preface.) 
 
 1. Give all the theorems used in factoring binomials. 
 
 2. Find the prime factors of 1 + a 6 , a 6 — P, a 4 + 4 4* 
 + 4 a 2 b 2 - c 8 , a; 2 - x - 20, a; 3 + x 2 - 8 a; - 12. 
 
 3. Find the G. C. D. of a; 4 + 4a; 8 + 12 a; 2 + 16 x + 16 and 
 4 a- 8 + 12 a; 2 + 24 x + 16. 
 
 4. Solve - + - = 7, --- = 2. 
 
 y x y x 
 
 5. Write the values of 27 _ 3, 27$, 27°, [(27 "V*] . 
 
 6. Reduce to simplest form 
 
 X ab. 
 
 (11 + 4VS) 1 , 
 
 (- 1 - \^S)». 
 
 7. Keduce to equivalent fractions having rational denomi- 
 nators 
 
 a c 
 
 3aH?c - * Ji + bi 
 
 8. Solve <\/x + 5 + y/x - 8 = V3. 
 
64 DARTMOUTH EXAMINATION PAPERS. 
 
 1888. (See Preface.) 
 
 1. Eemove the parentheses from 
 
 3a — {3a - [3a - (3a - 3a - 3a) - 3a] - 3a} - 3a, 
 and simplify the result. 
 
 2. Give the three theorems used in factoring binomials. 
 
 3. Factor 4 a? x* - 9 b* c% 8 6 2 c 8 + 48 P c 2 + 72 J 2 c, 
 x* - 3 X s - 14 a; 2 + 48 x - 32. 
 
 4. Eesolve 1 — a 8 into six factors. 
 
 5. Give two methods of finding the G. C. D. of two 
 
 quantities. 
 
 a* + b* 
 
 . a 2 1 2 
 
 6. Eeduce — = - X -s r. to a simple fraction and 
 
 11 a 8 + 6' r 
 
 a 6 
 lowest terms. 
 
 „ , (2 a- 1 - y- 1 = 22x", 
 7 - ^Is*-- 3^ = 18,0.' 
 
 8. A and B can do \ of a piece of work in 2 days. B can 
 do J of it in 6 days. How long would it take A to do J of it ? 
 
 9. Eeduce to simplest form 
 
 \/^r~> y/e + V^isxy/e-V^m (35-12 V§)*. 
 
 10. Eeduce to equivalent fractions having rational denomi- 
 nators 
 
 ac 1 
 
 c % (J + $) ' afl + $ 
 
 11. Solve 1 + * = 12. 
 
 x + y/x* — 1 * — V 5 " 2 — 1 
 
ALGEBRA. 
 
 65 
 
 1889. 
 
 1. Find the prime factors of 
 
 d? — W, a s + b% a"+b & + 2a i b s -x 2 -4:y 2 + 4:xy, 
 and a; 8 -8a! 2 + 13a;-6. 
 
 2. Find the greatest common divisor of x 3 — x 2 — 8 x + 12 
 and 3 x 2 — 2 x - 8. 
 
 3. Simplify the following expressions : 
 
 A 
 
 (a) ^r, (b) 
 
 to =i 
 
 «* — y* 
 I' 
 
 ** — ?/ 5 
 
 (d) *Wa' + ft.», (e) 4 * X8 ? X2 ? , 
 
 2* x 9* X 4* 
 
 (f) Vl4 + 4y6. 
 
 4. Solve the following equations : 
 'x — 1 a; — 9 
 
 (l)f-f 
 
 J =x — ix°; 
 
 a; y 
 
 (2) < 
 
 (3) 2 Vx^l + 4 V4a; - 4 = 5 Vsb. 
 
 La; ?/ 
 
66 EXAMINATION PAPEES. 
 
 EXAMINATIONS FOR ADMISSION TO BROWN 
 UNIVERSITY. 
 
 1878. 
 c 
 
 — 1 
 
 c — 1 
 
 1. Reduce to a simple fraction. 
 
 1 e — 
 
 c + l 
 
 2. Divide a into two parts, such that m times one shall he n 
 times the other. 
 
 3. If 4 be subtracted from both terms of a fraction, the value 
 will be £ ; and if 5 be added to both terms, the value will be g . 
 What is the fraction t 
 
 4. Given V* — 9 + VaT+TT= 10, to find x. 
 
 , n . , 3x — 6 3*— 3 „ 
 
 5. Given 2a; -] = ox — , to find x. 
 
 1879. 
 
 i ajj 1 « + l , a? -f- a: -j- 1 
 
 1. Add — r— -, -* J j— 7T, and — ' , ' — 
 
 x -\- 3 a? — 3x -f 9 as 3 -|- 27 
 
 2. Multiply (a -j- 6)J by (a + ft)*, and 
 
 Divide (a + 6) V^^l by (a — 6) V* 2 + 2 a + 1, 
 giving answers in simplest forms. 
 
 3. Given "'(« + «> _ *(» + «»r = (* + «*»)» 
 
 (a — 6) (a: — a) (a — 6) (a; — b) a? — a a 
 
 to find x. 
 
 4. Given -y/2 -f- x -\- *Jx = , to find x. 
 
 v2+ a; 
 
ELEMENTARY ALGEBRA. 67 
 
 5. Divide the number s into two such parts, that if m 2 ho 
 divided by the second, and this quotient multiplied by the first, 
 the product is the same as if n 2 be divided by the first and the 
 quotient multiplied by the second. 
 
 1880. 
 
 1. Find the H. C. D. of 
 
 Gx 3 — 8yx 2 + 2fx and Ux 2 — I5xy + 3/. 
 
 2. Given ax -j- by = c and mx =. ny -\-d, to find x and y. 
 
 3. Extract the cube root of 
 
 — 99a; 3 — 9s 5 + X s + 64 — 144* + 150a; 2 + 39z 4 . 
 
 3Vi — 4 15 +y^ 
 
 4 - Glven T+v^ = lo+vT t0 find * 
 
 5. Two pipes, A and B, will fill a cistern in 70 minutes, A 
 and C in 84 minutes, and B and C in 140 minutes. How long 
 will it take each to fill it alone 1 
 
 6. Given -\/5 -(- x -j- \/x = , to find x. 
 
 V5 + x 
 
 7. A gentleman bought two pieces of silk which together 
 measured 36 yards. Each cost as many shillings per yard as 
 there were yards in the piece, and the cost of the pieces were 
 to each other as 4 to 1. Required the number of yards in each 
 piece. 
 
 8. Given x -\- ^5 x -f 10 = 8, to find x. 
 
 9. Given m? — xl = 56, to find x. 
 
 10. Given x 2 *\- xy = 15 and xy — y 2 = 2, to find x and y. 
 
68 EXAMINATION PAPERS. 
 
 1881. 
 
 a ' J ' a ' c o c 
 
 2. There is a number consisting of two digits ; the number is 
 equal to seven times the sum of its digits ; if 27 be subtracted 
 from the number the digits interchange their places ; find the 
 number. 
 
 3. Eeduce to their simplest forms f> r Vf and ^250 x s y'^. 
 
 4. V* — 3 — *Jx — 14 — aAz — 155 = 0. 
 
 7 3 _22 
 
 5 ' a; 2 — 4~ a; + 2 — ¥' 
 
 6. x + y=3; a; 2 + y 2 = 29. 
 
 7. Develop by binomial formula (2 a — 3 b) 5 . 
 
 8. In an arithmetical progression given the last term — 47, 
 the common difference — 1, and the sum of the terms — 1118 ; 
 find the first term and the number of terms. 
 
 1882. 
 
 1 . Rationalize denominator — ■==- • 
 
 ■2 a/5 — Vl8 
 
 2. At what time between 8 and 9 o'clock is the hour hand 
 of a watch 20 minute-spaces in advance of the minute hand 1 
 
 3. Extract the cube root of - a/ -, and express the result 
 in the simplest form. 
 
 4. Multiply — 6 V^3 by — 2 V^2. 
 
 5. 7 (x + 7) — 7 ( 3a: + 50 ) _ 0> Find the va i ue f x . 
 
 (6x 2 — 5 xy 4-2 y 2 =z 12) _,. , , 
 
 "■"ioo.^oo r • Find x and y. 
 
 (3a; 2 -f 2xy— 3y 2 = — 3) 
 
 7. Develop by Binomial Formula (£a — §# 2 ) 4 . 
 
 8. In a Geometrical Progression, given the number of terms 
 8, the ratio \, and the sum of the terms 7\%, to find the first 
 term and the last term. 
 
ELEMENTARY ALGEBRA. 69 
 
 1883. 
 
 1. Factor a; 4 — y* ; also factor 4 a 4 — 8 a 3 x -}- 4 a 2 x\ 
 
 „ 6x4-7 2x — 2 2a;+l ^. , , 
 
 2. — — = -i — . h ind the 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 received two dollars more than he did. 
 Find the number of persons and what each received. 
 
 4. Multiply 
 
 a i — ai -f- 1 — a~i -\- a~i by «5 -|- 1 — «~i 
 
 5. V* + x -\- Va — x = *\/b. Find the value of x. 
 
 6. x -f- y = 4 ; — | — = 1, Find the 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 cur- 
 rent of 2 miles per hour, find the rate at which the crew would 
 row in still water. 
 
 8. Find the sum of six terms of the Geometrical Progression 
 of which f is the first term and § the second term. 
 
 1884. 
 
 — 3 y — 1 
 
 f-l^+^H-- 
 
 j 2x — y 3;/ — 2 \ 
 Find the values of x and y. 
 
 12 
 3 — 2x 
 
70 ELEMENTAEY ALGEBRA. 
 
 2. A and B run a mile. First A gives B a start of 44 yards 
 and beats him by 51 seconds ; at the second heat A gives B a 
 start of 1 minute 15 seconds, and is beaten by 88 yards. Find 
 the times in which A and B can run a mile separately. 
 
 3. Extract the square root of 
 
 25 a; 4 — 30 a x* + 49 a 2 x 2 — 24 a 8 * + 16 a 4 . 
 
 4. Simplify \^ i7b y^ -\(b) }' 
 
 5. 3 x 2 — 4. x — 4 = 0. Find values of x. 
 
 1. What are eggs a dozen when two more in a shilling's 
 worth lowers the price one penny per dozen 1 
 
 7. Develop by Binomial Formula (J a 2 — § 6) 6 . 
 
 8. Sum to 20 terms 2, 6, 10, 14, . . . 
 
 9. Sum to 6 terms 3 + 2 + j + . 
 10. Find the sixth term of 3, G, 12, . . . 
 
 June, 1885. 
 
 1. Find highest common divisor of 15 a 2 x 3 — 20 a 2 x 2 
 -65 a 2 a; -30 a 2 and 12 bx 3 + 20 bx 2 - 16 4a; - 166. 
 
 9 h 4. _ -i_ k ( z ~ ^ _ s ( z + *) _ kz _ 2 
 
 a + A; z(s + &) k{s-k) k 2 -z 2 
 
 3. A number is compounded of three figures whose sum is 
 17. The figure of the hundreds is double that of the units. 
 When 396 is subtracted the order of the figures is reversed. 
 What is the number? 
 
 4. Multiply 2 V 11 ^ - 3 V^2 and 4 v^ + 6 V^2. 
 
 5. Reduce to an equivalent fraction wifli a rational denomi- 
 nator 
 
 <\/x — 4 <\/x — 2 
 
 2*/x + 3 V* — 2 ' 
 
ALGEBRA. 71 
 
 6. V2x-3 — V8 x + 1 + Vl8 a; - 92 = 0. Find value 
 of x. 
 
 7. 2x 2 -2xy - y 2 = 3; x 2 + 3 xy + if = 11. Find 
 values of x and y. 
 
 8. In an arithmetical progression, given the last term, — 47; 
 the common difference, — 1 ; and the sum of the terms, — 1118 ; 
 find the first term and the number of terms. 
 
 9. In a geometrical progression, given the first term, § ; the 
 ratio, — £ ; and the number of terms, 7 ; find the sum of the 
 terms. 
 
 10. Develop by Binomial Formula (a 2 b — \x a~ 2 ) 4 - 
 
 Septembeb, 1885. 
 
 1. Find the least common multiple of 2 x 3 — 3 x 2 — x -f 1 
 and 6 x 3 - x 2 + 3 x - 2. 
 
 „ x a — hex x ac — ibx _,. n . 
 
 2. . ttt = ~ s^ ■ Find value of x. 
 
 2 2bc bo ooo 
 
 3. A number is compounded of three figures whose sum is 
 17. The figure of the units is two thirds that of the hundreds. 
 When 297 is subtracted the order of the figures is reversed. 
 What is the number ? 
 
 4. Multiply 3 y/^1 - 2 V^2 and 4 V^2 - 2 V^i. 
 
 5. Eeduce to an equivalent fraction with a rational denom- 
 inator 
 
 -vV - 1 + V* 2 + i ' 
 
 91 
 
 6. V3« + V3aT+13 = • Find value of x. 
 
 y3 a; + 13 
 
72 BROWN EXAMINATION PAPERS. 
 
 7. (2x-5) 2 -(2x-l) 2 = 8x-5x 2 ~5. 
 
 8. In an arithmetical progression, given the first term, — J ; 
 the number of terms, 18; and the last term, 5; find the com- 
 mon difference and sum of terms. 
 
 9. In a geometrical progression, given last term, — 12; sum 
 of terms,— 23}§; and ratio, 2; find first term and number of 
 terms. 
 
 10. Develop by Binomial Formula (£ ab 2 — § a 2 5 -1 ) 5 . 
 
 June, 1886. 
 
 1. Multiply 5x p - s y r+a — 2x''- 1 y r+1 — x"- 2 i/ r + 2 by 
 3x p+i y r - 1 + 4a; ?+5 2/'— 2 — x"+ s y r . 
 
 2. Simplify 
 
 b [a b c + a + cj 
 
 6 + i * 
 
 c 
 
 3. Given J — P > Find the values of a; and w. 
 
 2sa;+iy = 2'. 
 
 4. The smaller of two numbers divided by the larger is 
 .21 with a remainder of .04162. The greater divided by 
 the smaller is 4 with .742 for a remainder. What are these 
 numbers ? 
 
 5. Given ^^ - £=2 = Q*--J* + (11 _ x) . 
 to find value of a;. 
 
 2 2 
 
 + ~ ■ = *. 
 
 x + V2 — x 2 x— \/2 — x 2 
 
 7. Expand (2 x% + 3 sb* y) 4 . 
 
 8. Find sum of terms in a geometrical progression. 
 
ALGEBRA. 73 
 
 September, 1886. 
 
 Find values of a: and y. 
 
 2. y/a, + x + y/a — x = \/6. Find values of x. 
 
 3. A number is compounded of three figures whose sum is 
 17. The figure of the hundreds is double that of the units. 
 When 396 is subtracted the order of the figures is reversed. 
 What is the number ? 
 
 4. 3 x 2 — 4 x — 4 = 0. Find values' of x. 
 
 5. Find sum of six terms of the geometrical progression of 
 which f is the first term and § the second term. 
 
 June, 1887. 
 
 -, , 3a;+7 ... 
 1- 5 y g - = 13 i> 
 
 ix-3 2x + Sy „ -r,. ■, -, 
 
 o . = — 5|. Iind values of x and y. 
 
 2. 2x-Sy = 8. 
 y-3z = -ll. 
 
 x — 2y + As — 17. Find values of x, y, and z. 
 
 3. A boy spent his money in oranges. If he had bought 5 
 more, each orange would have cost a half-cent less ; if 3 less, 
 a half-cent more. How much did he spend, and how many 
 did he buy ? 
 
74 CROWN EXAMINATION PAPERS. 
 
 4. Multiply Vi> + 1 + Vj° — q by \/p + 2 — Vi? — ?• 
 
 5. Multiply V— 5 + a by y/— b — V— «• 
 
 6. 7 a; — 3 x 2 + 14 = 0. Complete the square and find the 
 value of x. 
 
 7. \/x+ 3 + \/3 a; — 3 = 10. Find the value of sb. 
 
 8. In an arithmetical progression there are given the first 
 term, 4; the number of terms, 10; and the sum of the terms, 
 175. Find common difference and the last term. 
 
 9. Expand by the Binomial Formula (2 a% — 3 J*) 5 . 
 
 September, 1887. 
 
 ., _, . „ 4?/ — 6 . 2 x — 4 
 
 1. Given 3x ^ = 4 — , 
 
 to find values of x and y. 
 
 2. Add -^16 x'tf, \ r £aFy*, and 6xy v8*". 
 
 3. A man bought a certain number of eggs for 2 dollars. 
 If he had paid 5 cents more per dozen, he would have received 
 two dozens less for the same money. How many dozens did 
 he buy, and what did he pay per dozen ? 
 
 4. 2 a/2 x - 3 — V3a3 — 7 = V4a; — 11. Find values of x. 
 
 5. 3 x 2 — 4 x = 55. Find values of a;. 
 
 6. In an arithmetical progression, given the first term, 3; 
 the number of terms, 15; the sum of the terms, — 165; to find 
 the common difference and last term. 
 
 7. Expand (2 x — 3 y 2 ) 5 by the Binomial Formula. 
 
ALGEBRA. 75 
 
 
 
 
 
 June, 1888. 
 
 
 
 
 
 (Omit one from each set.) 
 
 
 
 
 
 I. 
 
 1. 
 
 Kesolve 64 
 
 x> 
 
 — 
 
 xy 6 into five factors, 
 
 2. 
 
 Simplify 
 
 X 
 
 X 2 
 
 y 
 
 + 
 
 + 
 t 
 
 X 
 
 2/ 2 
 
 -y K yl 
 
 a b a' b' 
 
 3. Given — | — = c, 1 — = c', to find values of x and y. 
 
 x y x y 
 
 II. 
 
 1. Add 3 x V« 8 — a 2 x, — A a -\/4 « a; 2 — 4 a; 3 , and 
 5 / \/aFx^~^~aFx~ 3 . 
 
 2. Multiply 2 \/a— \^x by 3 V 17 ^ + 2 v'S. 
 
 3. Given r ^ — • = ~ , to find values of x. 
 
 x — 1 Zx 6 
 
 III. 
 
 1. Given 2x* - 3xy + y 2 = 35, 2 x - 3 y = 13, to find 
 values of x and #. 
 
 2. Expand (2 a — 3 J) 4 by Binomial Formula. 
 
 3. In an arithmetical progression, given the first term, 14 ; 
 the number of terms, 7 ; the sum of the terms, 59£ ; find the 
 common difference and last term. 
 
 September, 1888. 
 1. Find value of 
 
 x (y + z ) + y O — (y + z )1 — * \jh - x ( s — «)] 
 
 when x — 3, y = 2, z = 1. 
 
76 BROWN EXAMINATION PAPERS. 
 
 2. 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 walk- 
 ing 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. 
 
 3. Solve the equation Va; + <\/£ + x = -—= ■ 
 
 yx 
 
 . _ . ., t . 5 2 a: - 3 3 
 
 4. Solve the equation — — -p-r = — - - ■ 
 
 4 x + 2 2 (a: — 2) 6 
 
 5. Find the sum of 10 terms of the geometrical progression 
 in which the fourth term is 1 and the ninth term is 5 J 3 . 
 
 June, 1889. 
 
 (Omit one from each set.) 
 I. 
 
 1. Find the lowest common multiple of 6 x s + 11 x 2 — 46 x 
 + 24, and 12 a; 3 + 37 x 2 - 42 x + 8. 
 
 2. Simplify 
 x — y y — z « — a; 
 
 (sb + «) (y + s) (« + y) (x + z) {x + y) (y + z) 
 
 3. Solve 
 
 3^_y x 2y 
 
 J 3 2^5 __ 7 
 
 1 13 - ~6" 
 
 2 4 
 
 2 y — 3 a; = 23. 
 
 II. 
 
 1. A and B run a race of 480 feet. The first heat, A gives 
 B a start of 48 feet, and beats him by 6 seconds ; . the second 
 heat, A gives B a start of 144 feet, and is beaten by 2 seconds. 
 How many feet can each run in a second ? 
 
ALGEBRA. 77 
 
 2. Solve V3 x + 10 — V3a; + 25 = -3. 
 2 a; 2 + 3 a: - 5 2 a: 2 - a; - 1 
 
 3. Solve 
 
 3a: 2 + 4a:-l 3a; 2 -2a; + 7 
 
 III. 
 
 1. In an arithmetical progression, given the first term, — 3 ; 
 the common difference, 2\ ; and the sum of the terms, 143 ; 
 to find the last term and the number of terms. 
 
 2. In a geometrical progression, prove the formula for the 
 sum of n terms. 
 
 3. Solve 2a; 2 - 3f = 60, and 3a; 2 - 4 xy + y 2 = 64. 
 
 September, 1889. 
 
 (Omit any one.) 
 
 1. Eeduce to its simplest form, 
 
 3c 2 +c(2a— [5 c — {3a + c — 4a}]). 
 
 2. Find highest common factor of 
 
 x B -x i -5x s + 2x* + 6x and x* + x* - a; 8 - 2a; 2 - 2x. 
 
 3. Given ax + by = c, p x + q y = r. Find values of x 
 and y in terms of the other quantities. 
 
 4. Given 3 , - j*JL+J - d 2 *~ 3 
 
 3a; — 5 2 
 
 5. Divide a; 8 — y 2 by V» + 'V'y. 
 
78 EXAMINATION PAPERS. 
 
 EXAMINATIONS IN ALGEBRA FOR ADMISSION TO 
 MASSACHUSETTS INSTITUTE OP TECHNOLOGY. 
 
 June, 1878. 
 
 1. Factor 6 a 2 x s — 15 a b x 2 and (2 x — a) 2 — b 2 - 
 
 2. Solve the equation (x — 3) (2a; + l) = 2(a:+l) 2 — 14. 
 
 3. Find the least common multiple of 12ab 2 c and \5a?d\ 
 
 a T7- , xi. .2a+2b ,, 2a— 26 
 
 4. Find the sum of , and : — — • 
 
 a — b a -\- o 
 
 ( x — y— s = 10 \ 
 
 5. Solve the equations < 2x-\-y — s = H /■ • 
 
 v k + 2/ + 2z = — 3J 
 
 6. A numher consisting of two figures is 4 times the sum of 
 its digits ; and if 27 be added to the number, the order of the 
 digits will be inverted. What is the number 1 
 
 4 5 12 
 
 7. Solve the equation _1_ — ' 
 
 ^ x+1^ x-{-2 x+3~ u - 
 
 8. Extract the square root of x~ s — 6 x~ 5 -\- 1 1 a; -2 — 6x-\-x*. 
 
 9. Solve the equation W x _]_ /J a \ x — . 
 
 September, 1878. 
 
 1. Factor 8 a s x 2 — 18 a 7 , also x 2 — 11 a: + 30. 
 
 2. Reduce 3a-| — 2a to a single 
 
 fraction. 
 
 3. Divide — — - — by -^— , , and reduce the result to its 
 
 ax-f-a* ar -\- ax 
 
 simplest form. 
 
ELEMENTAEY ALGEBRA. 79 
 
 a at 4.1, *• x—Zax x-\-\9a 
 
 4. Solve the equation — f- ^ = 20 a —. • 
 
 (f+3y=21 
 
 5. Solve the equations < 
 
 (J — 3a;=^25 
 
 6. Simplify the following expressions : 
 
 . / Sn 6\6 
 
 V729 a 6 * 2 , the cube root of 216 a Sn c e , Ixt yA . 
 
 7. Add V32 and \/l8- Multiply the cube root of I2a 2 x 2 
 by the cube root of 18 ax 4 , and express the result in the simplest 
 form. 
 
 x+l x — 2 1 
 
 8. Solve the equation 
 
 x -|- 2 x—1 6 
 
 9. Divide 10 into two such parts that their product shall be 
 12 times their difference. 
 
 June, 1879. 
 
 1. Divide 3x 3 +lla;— 2a; 4 + 12 + 4a: 2 by x + i + 2a?. 
 
 2. Factor 27 x 8 — 125. 
 
 x -\- 2 y . x 
 
 3. Simplify the fraction — ^±^ ^-— . 
 
 v J x x-\-2y 
 
 x-\-y y 
 
 4. A said to B, "Nine years ago my age was to yours as 
 3 to 5." B replied, " In twelve years my age will be to yours 
 as 4 to 3." Bequired their ages five years ago. 
 
 5. Simplify t^. 
 
 6. Extract the square root of 
 
 x*y-i — ixiy-%-\-§ — i x-% yi -\- x~ s y$. 
 
80 EXAMINATION PAPERS. 
 
 7. Subtract \/i8 from Vl62, multiply the remainder by 
 v4, and reduce the product to its simplest form. 
 
 8. Solve the equation V5+~l - V^H = g. 
 
 y^TT+va^r 2 
 
 9. Find two numbers whose difference multiplied by the 
 greater produces 35, and whose sum multiplied by the less 
 produces 18. 
 
 September, 1879. 
 
 CC ft* cc — 1 
 
 1. Reduce - — : 1 — to its lowest terms. 
 
 1 -f- x 1 — x x i 
 
 2. Solve the equation 
 
 (a — ma;) 2 — (n — ma;) 2 — (a 2 — an) = 0. 
 
 3. What principal, when put at simple interest for a years, 
 at b per cent, will amount to c dollars? 
 
 4. Divide 2 — 4artyi + 2 x-if by 2 + xiyi + aHyl. 
 
 5. Extract the seventh root of 8 ^/2, reducing the result to 
 its simplest form. 
 
 6. Reduce — == ~ l ~ to an equivalent fraction 
 
 Vl — x-\- ^\-\-x 
 
 with a rational denominator. 
 
 3 — a; 5 
 
 7. Solve the equation 
 
 3 — a; 2 6 
 
 8. Solve the equation x 2 — 2 V* 2 + 4 x — 5 = 13 — 4 a:. 
 
 9. The fore-wheel of a carriage makes 25 more revolutions 
 than the hind-wheel in going 300 yards ; but if the circumfer- 
 ence of each wheel were increased by one yard, the fore-wheel 
 would only make 15 revolutions more than the hind- wheel in 
 going the same distance. Required the circumference of each 
 wheel. 
 
ELEMENTARY ALGEBRA. 81 
 
 June, 1880. 
 
 1. Find the value of 
 x$y 
 
 , when x = 4, y = 9, a = y i , and b = a; 2 . 
 
 2. Substitute y — 3 for a; in x s -\- 2x 2 — 15 a; — 36, and 
 arrange the result. 
 
 3. Reduce ■ — r- 7 -4- ■ v -4 — ; 77, to its simplest form. 
 
 a-\-b ' a — b ' a 2 — b* 
 
 a 2 +2 + or* a 2 +l 
 
 4. Reduce T , = — to 
 
 a 2 — 1 
 
 5. Solve the equation — — :— - = 5. 
 
 x — 1 ' as — (— 2 
 
 6. Add -\/250 and yT6, multiply the sum by }V3, and find 
 the sixth power of the product. 
 
 7. Solve the equation yx-\- 13 — yx — 2 = 3. 
 
 2 ^ 1 
 
 8. Solve the equation — = 0. 
 
 ^ 3a? 7 — a; 
 
 September, 1880. 
 
 1. Find the value oi a -\- b (x -\- y)i — {a — b) (x — y)-i 
 when a = 5, 6 = 1, x = 12, and y = 4. 
 
 #2 J ^2 I g x _|_ 9 a;2 
 
 2. Multiply together -^-^ _^__, and -^ and 
 reduce the result to its simplest form. 
 
 „„,,, ,. x — 2 x-i-i 9 
 
 3. Solve the equation = — '-— — — - . 
 
 x a;-|-l 2as 
 
 4. Solve the equations 
 
 x + y_x-y = 8 and « + y,« = y ==1L 
 
 2 3 3 4 
 
82 EXAMINATION PAPERS. 
 
 5. Find the product of ($?)*, (ffi)* and (^)*. 
 
 6. Show that 
 
 (7 + 4 -s/H) (2 — V3) = (2 V2 + a/6 — V3 — 2) (a/2 + 1). 
 
 7. Solve the equation 
 
 Vx 4 — 3a; 2 + 5 — V»: 4 — 5 a: 2 + 8 = 1. 
 8. Solve the equation (2 « — 3) 2 = 8 x. 
 
 June, 1881. 
 
 1. Remove the parentheses and reduce 
 
 2a— [56+ {3c— (a+[26 — 3a + 4c])}]. 
 
 2. Factor the following expressions : 
 
 a; 2 -(-2a;y-j-?/ 2 — 4 and 9 — a; 4 — 4 ^ 2 -|- 4 a; 2 y. 
 
 3. Find the greatest common divisor of 
 
 a; 2 _6a; + 8 and 4a: 3 — 21 x 2 + 15 x-\- 20. 
 
 4. Find the least common multiple of 
 
 ax 2 -\- a 2 x, x 2 — a 2 , and x s — a 8 . 
 
 5. Simplify 
 
 x — 1 a; 2 — 1 x a — 1 
 4 a 2 
 
 x — 3a-\- 
 
 6. Eeduce 31— to its simplest form. 
 
 2 a 2 
 
 x : — 
 
 a-\-x 
 
 _ _ , 3a; — 1 2a; + l 4a; — 5 
 
 7. Solve -^ . — _ = 4. 
 
ELEMENTARY ALGEBRA. 83 
 
 3 15 2 3 
 
 8. Solve — 1 — = -, =1, and state method of 
 
 x y 4 as y 
 
 elimination. 
 
 a/x 4 \f x 2 
 
 9. Reduce = == to an equivalent fraction hav- 
 
 2^ + 3^-2 
 ing a rational denominator. 
 
 10. Solve the quadratic equations, *\/x -\- \/x -\- 5 = 5, 
 5_ 3a;-j-l 1 ^ _ _ g _ _ ^ 
 
 x x 2 4 
 
 September, 1881. 
 
 1. Eeduce 3x — (5x — [4x — (y — x)]) — ( — x — 3y) 
 to its simplest form. 
 
 2. Factor the following expressions : 
 
 a*—!, mi—Qz — y)*, a 2 — b 2 -\-2bc — c 2 . 
 
 3. Find the greatest common divisor of 
 
 a;* _ a; 8 + 2 a; 2 -{- a; + 3 and a: 4 + 2 x 3 — x — 2. 
 i. Simplify the following expressions : 
 
 a , 6 , 2«6 11 2a; 
 
 ^T+6 + ^Tj + ^3~^' and ^jT^ + ^TZ^ "~ ^+P 
 
 5. Divide 9 -J t— 1 — ; by 3 -J — and reduce to the 
 
 ar — y x — y 
 
 simplest form. 
 
 2 3 1 
 
 6. Solve the equation — - - = -= =-• 
 
 ^ x — 1 aj-j-1 a; 2 — 1 
 
 7. Solve the simultaneous equations : 
 
 xArV x — y~~ x + y os — y 
 
 8. Multiply 2a:3 — 3 a;* — 4-|-ari by 3a=S + x — 2a;S. 
 
 2 
 
 9. Solve the equation */^ — aJ x — 3 — —=■ 
 
 ■yx 
 
84 EXAMINATION PAPEKS. 
 
 10. Solve the quadratic equations 
 
 x 5 — x 15 
 
 5 — x x 4 
 
 x s — 6 a: 8 = 16. 
 (3a; + l) (4a; 2 — 25) = 0. 
 
 June, 1882. 
 
 1. Factor the following expressions : 
 
 4a: 2 — Mxy + 2y 2 , a; 2 -f 5 a; -f 6, a; 8 — 8y*. 
 
 2. Find the greatest common divisor of 
 
 2a; 8 — 4a; 2 — 13a; — 7 and 6a; 8 — llx 2 — 37 a; — 20. 
 
 3. Find the least common multiple of 
 
 4(1+ a;), 4(1— a;), and 2(1— x 2 ). 
 
 .,. ... 2 a -4-5 2 a — b 6 ab 
 
 4. Simplify '-= r-T s IT • 
 
 a — b a-\-b a 2 — 6 2 
 
 5. Multiply a" -\- a? and y'a together. 
 
 OiOC (JOT 
 
 6. Solve the equation - 1- a -I = 0. 
 
 b — ex c 
 
 7. Solve the simultaneous equations 
 
 x — 4 V -\-2 . , x , ii — 2 
 
 - 0, and s + — t — = 3. , 
 
 5 10 ' 6 ^ 4 
 
 8. Extract the square root of 
 
 x* — 2 x *y + 3 xY — 2 xtf -f- y\ 
 
 9. Solve the quadratic equation 
 
 _ 14a: — 9 _ a; 2 — 3 
 8a; — 3 ~~ a;+l " 
 
 10. Solve the simultaneous quadratic equations 
 
 - + - = 5, and - 2 4-- = 13. 
 x ' y x' ' y 2 
 
ELEMENTAKY ALGEBRA. 85 
 
 Septembee, 1882. 
 
 1. Factor the following : 
 
 9m 2 — 24m + 16; x 2 — 2xy + y 2 — z\ 
 
 2. Find the greatest common divisor of 
 
 12a; 3 — 9a? 2 4- 5a; 4- 2 and 24a; 2 -f 10a; -f- 1. 
 
 3. Find the least common multiple of 
 
 x 2 —l; x 2 -\-2x—3; 6x 2 — x — 2. 
 
 4. Simplify ^ ; — 
 
 x — a ■ (x — a) 1 (x — af 
 
 5. Show that 
 
 (a + b \/—l) (a — b */^-£) — (a + b -\- */2ab) (a + 6 — .y/2al) . 
 
 x 2 — a a — x 2x a 
 
 6. Solve the equation —7 7 — = -y . 
 
 o * 03c 
 
 7. Solve the simultaneous equations 
 
 # — a; 8 ' 7 
 
 8. Extract the cube root of 
 
 a? _j_ 3 a: 5 + 6a; 4 + 7 a* -f 6a; 2 + 3a; + 1. 
 
 9. Solve the quadratic equations 
 
 ? 4. ^ _ t- — 0, and 19a; 4 + 216a: 7 = a;. 
 3~ 4 3a r 
 
 10. Solve the simultaneous quadratic equations 
 
 x 1 V 1 1 ff 1 b a 
 - + y = 1 and - 4- - = 4. 
 a x ' y 
 
86 EXAMINATION PAPEES. 
 
 June, 1883. 
 
 1. Resolve into factors 
 
 1 — a; 2 — y 2 4-2x7/, and 9a; 2 — ^-. 
 
 25 
 
 2. Find the greatest common divisor of 
 
 x 2 — 1, X s — 1, and x 2 -j- x — 2. 
 
 3. Find the least common multiple of 
 
 x 2 — 1, x 2 + 2 a; — 3, and a 2 + 4 a; -f 3. 
 
 4. Reduce the following fractions to their simplest form : 
 
 . _ 2 a; — 15 x , , 1 
 
 s + 5 — — ; «-J T + 1 — 
 
 1' x + 1 
 
 ' x 
 
 5. A fraction which is equal to § is increased to -^ when a 
 certain number is added to both its terms, and is diminished 
 to g when the same number plus one is subtracted from both. 
 Find the fraction. 
 
 6. Solve the quadratic 
 
 a; 2 — 4 to n x 
 
 (m — nf. 
 
 (m -\- n) 2 
 
 7. Find the quadratic whose roots are — 1 and f . 
 
 8. Proportion. 
 
 T „ra c e q ,, ,a + c-r-e + ff a 
 
 9. Arithmetical Progression. 
 
 Find the sum of a -\- 4 a -j- 7 a -\- ■ • • to 10 terms. 
 
 10. Geometrical Progression. 
 
 Find the sum of 8 -f- 4 + 2 -j to 10 terms. 
 
 11. Binomial Theorem. 
 Expand (x — 3) 6 . 
 
ELEMENTARY ALGEBRA. 87 
 
 September, 1883. 
 1. Find the value of 
 
 ( x — a, y x — 2a-\-b _a-\-h 
 
 — a y 
 
 — b) 
 
 for x = 
 
 hi x-\-a — 2b 2 
 
 2. Find the greatest common divisor of 
 
 3 x s — 3x 2 y -j- xy 2 — y s and 4sb 2 ?/ — 5xy 2 -f- y' 
 
 3. Solve the equation 
 
 1 1 a — b 
 
 x — a x — b x 2 — ab 
 
 4. Solve the simultaneous equations 
 
 x-a y-b ^ x + y-b g -y- g=ft 
 
 b a a b 
 
 5. Solve the quadrate 4 a?x = (a. 2 — 5 2 -j- x) 2 - 
 
 6. Solve the simultaneous equations 
 
 x* -\- 3xy = 54 and set/ -j- 4«/ 2 = 115. 
 
 7. If T = - , show that - = ' • 
 
 b d c ^/ c 2 _|_ g* 
 
 8. Arithmetical Progression. 
 
 Find the sum of 15 terms of the series £, — f , — -y-, 
 
 9. Geometrical Progression. 
 
 Sum the series 6, — 2, §, to infinity. 
 
 10. Binomial Theorem. 
 Write the expansion of (3 — 2 a; 2 ) 5 . 
 
88 EXAMINATION PAPERS. 
 
 June, 1884. 
 
 1. Divide x*y % — x 2 — y 2 +l by #y — x — y -\- I. 
 
 2. Find the G. C. D. and L. C. M. of the three following 
 expressions : 
 
 3x*— Sx; ix 2 — 6x; (2 x — 2) (2 x — 3). 
 
 3. Simplify 
 
 («+»-i>£+M + (a+a)( V» 1)(> ~ 1) - 
 
 4. Solve each of the two following equations : 
 
 («) ^^-^ = 2; (6) V^+V^+^=^- 
 
 5. Solve the simultaneous equations 
 
 ax -\- by = 2 ; a 6 (a: 4" #) = a "I - &• 
 
 6- Simplify i (aJ y-\ 
 
 7. Solve the equation 
 
 7-^^H * = 0. 
 
 x — 3 ' (x — 1) (x — 3) 
 
 8. Solve the simultaneous equations, 
 
 (*+l) (y+2) = 10; xy = 3. 
 
 9. Find two numbers such that their sum, their difference, 
 and the sum of their squares may be to each other as 4, 1, 
 and 17. 
 
 10. Show that if 6 is a mean proportional between a and c,. 
 
 (a 2 + b*) (6 2 4- c 2 ) = (a 6 + 6 cf. ■ 
 
 11. Expand by the Binomial Theorem (a -{- 2 "/af. 
 
INST. OF TECHNOLOGY EXAMINATION PAPERS. 89 
 
 September, 1884. 
 
 2 y + 4 s 2 
 
 f 
 
 1. Simplify » a , after substituting 1 — a; 2 for y. 
 
 # 2 
 
 2. Resolve a 12 — J 12 into six factors. 
 Solve the following equations : 
 
 a o o a 
 
 , x* + 2x x 2 -x + 2 x-2 . 
 
 4. p- J - 1 5 — = 0. 
 
 5. V2as + 3 — Va + 1 = V3 x — 8. 
 
 6. Prove that the square of half the sum of any two unequal 
 numbers is less than half the sum of their squares. 
 
 7. Expand by the Binomial Theorem f a J • 
 
 8. Insert two arithmetical means between 24 and 81. Also 
 insert two geometrical means between the same numbers. 
 
 June, 1885. 
 1. Divide a 3 — a 2 by a* — a?. 
 
 2. Factor x 2 — x — 30, (x - y) s - y 3 , 
 
 _4>i + l . 
 
 X. 
 
 , , z 2 — V 2 t a + b i a ~ h 
 
 3. Find the value of -„-rhi wnen x = 1 and V = — n" 
 
 x* + y 2 a — o a-\- b 
 
90 ALGEBRA. 
 
 Solve the following equations : 
 4 ^ + 2 ^ = 3 . 
 
 X — X — c 
 
 5. J±*£i; = 3 -x. 
 1 + 2a;- 1 
 
 6. x - ^z 8 - 2 a; 2 = 2. 
 
 7. (x + 2) (y - 3) = 10, xy = 15. 
 
 8. Find the cube of 1 + \/— 3. 
 
 9. The sum of three terms in arithmetical progression 
 beginning with § is equal to the sum of three terms in geo- 
 metrical progression beginning with §, and the common 
 difference is equal to the ratio. What are the two series ? 
 
 September, 1885. 
 
 1. Factor x 2 — 6 x — 16 and 1 — 9 a + 8 a 2 . 
 
 2. Find highest common factor of x 2 + x — 6 and 
 2 a; 2 - 11 x + 14. 
 
 „ „■ ,-,- £ 3x 2*« 
 
 3. Simplify + 
 
 x — y x + y x' — y' 
 
 Solve the following equations : 
 
 , x 1 — 2ax 2x — 1 
 
 4- k + — o + r- = °- 
 
 2 2o a 2 
 
 7 3 22 
 
 5 
 
 x 2 - 4 a + 2 5 
 6. Vx-32 + \/x = 16. 
 
INST. OF TECHNOLOGY EXAMINATION TAPERS. 91 
 
 7. Find the sum of 16 terms of the arithmetical progres- 
 sion §, f, f, . . . 
 
 8. Find the sum to infinity of the geometrical series 
 1 a 2 a. 4 
 
 ar a; 4 
 
 9. Expand (3 x — 2 y) 6 by the Binomial Theorem. 
 
 June, 1886. 
 
 1. Find the value of — ; , when x = ■ 
 
 b a a— o 
 
 „ »ii, ,, x — « x + a 2 x (3 a — a;) 
 
 2. Add together / ^ _,_ , 2 > 
 
 (x + af {x - a) 2 ( x -a)(x+ a) 2 
 
 3. Solve „ * ., + 
 
 2* — 1 4a; — 3 a; — 1 
 
 4. Solve a; + \/x 2 — a 2 = b. 
 
 x n \/x 2n 4 
 
 5. Show that - ■ — is the reciprocal of 
 
 ( Vx n + 2 + \/x n -2 y 
 
 6. Show that (- 1 -f V^3) 8 + (- 1 - V^Y = 16. 
 
 _ „ . 6 x 5 (x — 1) 
 
 7. Solve - + - = -±- 1 . 
 
 x 6 4 
 
 8. Solve x 2 + x y — 15, xy — y 2 = 2. 
 
 9. Find the 4th term of (a — 2 6) 10 . 
 
 £ 10. How many terms of 16 + 24 + 32 + 40 + . . . amount 
 to 1840 ? 
 
92 ALGEBRA. 
 
 September, 1886. 
 
 x + 1 2a: — 1 3a: + 2 
 
 1. Simplify ^— ^ + a »_ x _ 2 - -^ ■ 
 
 2. Eesolve a 12 — b 12 into its prime factors. 
 
 „ „ , x — 5 5x — 7 Zx — 1 5 — x 
 
 3 . solve -g-+— g 4~ =_ 8— 
 
 4. Find the continued product of 
 
 VoT~b, 'tya^l, and \/(a 2 - b 2 )*. 
 
 5. Extract the square root of 41 + 12 V5- 
 Solve the following equations : 
 
 6. a - + U b - = 0. 
 x b a 
 
 7. *Jx + i = ■s/x + J, 
 
 g \ V2 a; — y=^/x — y + \, 
 1 x 2 + 4y = 17. 
 
 9. There are two Dumbers whose geometrical mean is § of 
 their arithmetical mean ; and if the two numbers be taken for 
 the first two terms of an arithmetical progression, the sum of 
 its first three terms is 36. What are the numbers ? 
 
 June, 1887. 
 
 2 
 
 + 3x^+4 
 
 v ujJia, Awl, 
 
 X* -\- 3 X 
 
 1. Eeduce to its lowest terms —7 jr— j- 
 
 a: 4 + 3a: 2 
 
 2. Solve yJL- = -L- . 
 
 — x a — x 
 
 3. simplify r("-y + "(«")" + i; 
 
 v J I (a")"-" J 
 
INST. OF TECHNOLOGY EXAMINATION PAPERS. 93 
 
 4. Solve V2 x + 1 — V* + 3 = V«- 
 
 5. Solve a; 2 + 2/ 2 = 20, * 2 — xy = 8. 
 
 6. Solve 2a: 3 + 8a;- 3 = 17. 
 
 7. Reduce to the form A + B V— 1. 
 
 i + V- 1 
 
 8. The 1st term of an arithmetical progression is 2, and 
 the difference between the 3d and 7th terms is 6. Find the 
 sum of the first 12 terms. 
 
 September, 1887. 
 
 1. Divide x" + x~ 2 by x$ + x~%. 
 
 2. Resolve into two factors a? + b 2 — <? — d 2 + 2 (a b + c d). 
 
 3. Solve x — a + \/x 2 — 2 ax = b. 
 
 n i 3x — ^/x* — 8 /—^ - 
 
 4. Solve = x + Vx 1 — 8. 
 
 x - <vV - 8 
 
 5. Solve (x — y) (x - 3 y) = 24, a; - 2 y = 5. 
 
 6. Form the quadratic equation whose roots are a + b — c 
 and a — b + c. 
 
 7. Insert three geometrical means between 3| and 18. 
 
 8. Give the first, third, and fifth terms of the expansion by 
 
 / ,- « s \ 10 
 the Binomial Theorem of ( x yy + — — I . 
 
 V 2 Vx/ 
 
94 ALGEBRA. 
 
 June, 1888. •''•>' ■' 
 
 PRELIMINARY. 
 
 1. Factor 8 ex — 12 cy + 2 ax — 3 ay, 
 
 and 2am — b 2 +m 2 + 2bn + a 2 — n*. 
 
 2. Find the G. C. D. & L. C. M. of 2 z 4 - 11 x* + 3x 2 + 10* 
 and 3 x* — Ux i - 6 x 2 + 5 x. 
 
 x + 2y x 
 x + y y 1 
 
 3 - Sim P lif y x~+2f — — and 1+ i • 
 
 y *+y 1+ i 
 
 X 
 
 4. Solve the equations 
 
 5 — 2x 3 — 2x 
 
 («■) 
 
 a; + 1 " x + 4 
 
 2 a ia s 3 a 2 v ' 
 
 5. At what time between 4 and 5 o'clock is the minute 
 hand of a watch exactly 5 minutes in advance of the hour 
 hand ? 
 
 6. Solve the simultaneous equations 
 
 5a: -3y + 2« = 41. 
 2a;+ y— z = ll. 
 5x + 4:y — 2z = 36. 
 
 7. Extract the square root of 
 
 x 2 + 4y 2 + 9z 2 -4:xy+ Gxz-I2yz. 
 
 8. Reduce to an equivalent fraction having a rational de- 
 nominator 
 
 \/x — 4 ^/x — 2 
 
 2 V* + 3 1/3^2 ' 
 
INST. OF TECHNOLOGY EXAMINATION PAPERS. 95 
 
 PINAL. 
 
 1. Solve the equation 2 x 2 + 3 x — 5 ^2x 2 + 3a; + 9 = — 3. 
 
 2. Solve the simultaneous equations 
 
 ( x 2 + x y + 4 y 2 = 6. 
 I 3x 2 + 8y 2 = U. 
 
 3. Factor x i — 7x 2 y 2 + y\ 
 
 4. A person saves $270 the first year, $210 the second, and 
 so on. In how many years will a person who saves every year 
 $ 180 have saved as much as he ? 
 
 5. Expand (m~§ + 2n)\ 
 
 Find 5th term of (cc -1 - 2y%) n . 
 
 6. Form the equation whose roots are — § and f . 
 
 7. Derive the formula for the sum of a series in geometrical 
 progression. 
 
 8. Find three numbers in geometrical progression such that 
 their sum shall be 14 and the sum of their squares 84. 
 
 COMPLETE. 
 
 1. Simplify ^-^ - ^-p - ^ {^p - V j rb )- 
 
 2. Solve — ^-r + — ^— =2, x + y = 2 a. 
 
 a -\- o a — o 
 
 3. Extract the square root of 
 
 4a 4 -12a 8 S + 29 a 2 S 2 - 30 ai 3 + 25 6 4 
 
 , si „ pliIy (^y x ( ^y 
 
96 ALGEBRA. 
 
 9a-l 55 , 
 5. Solve - 
 
 l ~ 6 
 
 x 
 
 X 
 
 6. Solve \/a + x + V a — x — 2 yfx. 
 
 „ „ , x — y .. x + 3 « 
 
 7 . Solves 2^ = 4, j, - —^ = 1. 
 
 8. Find the sum of 18 terms of the series ? t , — 1, — 2§, 
 
 September, 1888. 
 
 4 sc 2 -f 3 x — 10 
 
 1. Reduce to its lowest terms 
 
 4 t x i + 7x 2 -3x-15 
 
 i 
 
 '«*(^V : f-.) , -(^-^) 
 
 3. A fraction becomes | by the addition of 3 to the numer- 
 ator and 1 to the denominator. If 1 be subtracted from the 
 numerator and 3 from the denominator it becomes £. Find 
 the fraction. 
 
 4. Solve (x -ay=(x-2 a) (x 2 + 4 a 2 )k 
 
 5. Form the quadratic equation whose roots are 
 
 (a + b) 2 , . 
 
 ■*- and b — a. 
 
 a — 6 
 
 6. Expand by the Binomial Theorem lx + - j • 
 
 7. Divide 111 into three parts such that the products of 
 each pair may be in the ratios 4:5:6. 
 
 8. Find the sum to infinity of the geometrical progression 
 
INST. OP TECHNOLOGY EXAMINATION PAPERS. 97 
 
 Mat, 1889. 
 PRELIMINARY. 
 
 1. Find the greatest common divisor of 2 x 4 — 12x 3 + 19 x 2 
 -6* + 9 and ix s - 18 x 2 + 19 x - 3. 
 
 „ „. ,.„ a 3 -b s a + b (a 2 -ab + b 2 ) 2 
 
 2. Simplify ^ T ^ X— jX (fl , + o ft + &2)2 ■ 
 
 3. One tap will empty a vessel in 80 minutes, a second in 
 200 minutes, and a third in 5 hours. How long will it take 
 to empty the vessel if all the taps are opened ? 
 
 4. Solve — ■*-= + -JL- = 2 a, ^f = 1. 
 
 a+b a— b iab 
 
 x 2 4 
 
 5 Extract the square root of x 4 — x 3 + — + 4 x — 2 H — 2 • 
 
 6. Factor 2 ram — b 2 + m 2 + 2 bn + a 2 — n 2 , and 2 c 8 w 
 + 8c 2 m — 42 cm. 
 
 7. Which is the greater, \/10 or \/46, and why ? 
 
 8. Extract the square root of 75 — 12 \/21. 
 
 PINAL. 
 
 1. Write out the first four terms, the last four terms, and 
 the middle term of (x — 2 y) u . 
 
 2. Find the sum of the first n terms of the series 1, 2, 3, . . . 
 
 3. Find three geometrical means between 2 and 162. 
 
 4. Show that in the equation x 2 + p x + q = 0, the sum 
 of the roots is —p, and the product of the roots q. 
 
 5. Find the four roots of the equation x* — 3x 2 a 2 + a 4 = 0. 
 
98 ALGEBRA. 
 
 6. A number consists of two figures whose product is 21 ; 
 and if 22 is subtracted from the number and the sum of the 
 squares of its figures added to the remainder, the order of the 
 figures will be inverted. What is the number ? 
 
 7. Solve 3 x" + 15 x — 2 V^ 2 + 5 x + 1 = 2. 
 
 8. Form the equation whose roots are (a — |), (b -j- §). 
 
 COMPLETE. 
 
 1. Simplify 
 
 ( 
 
 x K y-y 1 \( x x f-r^ ^ ( 1 _' j, y-r 
 
 x + y J\ x 2 + y 2 
 
 2. Reduce to its lowest terms 
 
 15 x 2 + 24 x - 10 
 
 o «■ yt (a + b)t + (a - b)* , (a + ft)* - (a - b)* 
 
 3. Simplify ^ '- ^ '- H '- '— ■ 
 
 (a + J)* _ ( a - i)* (« + 6)*+ (a -6)* 
 
 4. A certain number when divided by a second gives a 
 quotient 3 and a remainder 2 ; if 9 times the second number 
 be divided by the first, the quotient is 2 and the remainder 11. 
 Find the two numbers. 
 
 5. Solve xb-at = (x- &)*. 
 
 6. Solve x* + 4 a b x 2 = (a 2 - Wf. 
 
 7. If A is the sum of the odd terms, and B of the even terms, 
 in the expansion of (x + a)*, show that A 2 — B 2 — (x 2 --a 2 )*. 
 
 8. If x— y is a mean proportional between y and y + z—2x, 
 show that x is a mean proportional between y and s. 
 
 9. The second term of a geometrical progression is 54, and 
 the fifth term 16. Find the series. 
 
INST. OF TECHNOLOGY EXAMINATION PAPEKS. 99 
 
 September, 1889. 
 FINAL. 
 
 1. Solve (a.) 3 x 2 - 53 x + 34 = 0. 
 
 (b.) \/a + x + V* — * = V^- 
 
 2. For what value of m will the equation 2x 2 + Sx + m = 
 have equal roots ? For what value, imaginary roots ? 
 
 3. Solve x 2 + 3 x y = 54, a; y + 4 y 2 = 115. 
 
 4. Find two numbers such that their sum, their difference, 
 and the sum of their squares may be to each other as 4, 1, 17. 
 
 5. In a geometrical progression, given the number of terms 
 8, the ratio \, and the sum of the terms 7}J ; find the last 
 term and the first term. 
 
 / 2 1\B 
 
 6. Expand {a? y — b *) ' 
 
 7. Four numbers are in arithmetical progression. The 
 product of the 1st and 3d is 27, and of the 2d and 4th, 72. 
 What are the numbers ? 
 
 COMPLETE. 
 
 1. Resolve into four factors 
 
 (a: 2 + y 2 — z 2 — u 2 ) 2 — 4 (x y — s u) 2 . 
 
 Q s+2a (x + a) 2 
 
 2. Solve or — 7 rn- 
 
 x — 2b (x — by 
 
 2 2 
 
 show that x^ — y* = 1. 
 4. Solve — ^> + 
 
 x + 3 x + 4 
 
 5. Solve x (y + 3) = 5, y (a; + 1) = 4. 
 
 6. Expand by the Binomial Theorem (a* — 2 b^) . 
 
 7. Solve x% + x~% = V5 — x~k 
 
 8. Find a series in arithmetical progression whose fourth 
 term is 4, and the sum of the first ten terms, 50.