A COMPLETE SET

OF

MALE PUPIL TEACHERS'

EXAMINATION

QUESTIONS IN ALGEBRA

TO SEPTEMBER 1879 (INCLUSIVE).

WITH ANSWERS.

Compiled, Classified, and Graduated by

W. J. DICKINSON,

EMERLY LECTURER ON Grammar and EUCLID AT THE BATTERSEA TRAINING COLLEGE⚫

AUTHOR OF

'FRACTICAL ENGLISH GRAMMAR AND ANALYSIS,' THE DIFFICULTIES OF GRAMMAR AND ANALYSIS SIMPLIFIED,' 'HOW TO TEACH GRAMMAR AND ANALYSIS,' AND THE DIFFICULTIES OF EUCLID SIMPLIFIED."

LONDON: JOSEPH HUGHES,

THREE TUNS PASSAGE, PATERNOSTER SQUARE. E.C.

SIMPLIFICATION (involving a Knowledge of the Whole of the Preceding Parts).

1. Simplify a(a-1) (a− 2) (a−3). What is its numerical value when a=4?

2. Simplify 4x-[(4x — 43) (4x + 43') 41) (4x + 41) — {4x + (4x + 41) (4x-4y)}+4], and find its numerical value when x=1 and

y= 0.

3. If x=100 and y = 50, find the numerical value of x(x+y) +x− {x(x-y)+(x+y) (x − y)-66}.

4. Explain the effect of a bracket or vinculum when used in algebraical expressions.

Simplify 3(x2+y2)− {x2+2xy + y2 - (2+y-x2-y2)}.

5. Simplify (a+b+c)2−a(b+c− a)−b(a+c—b) — c(a+b−c), and find its value when a = 1, b = 2, c= 3.

6. Reduce x3+3 — (3x2y + 3xy2) − {x3 — 3x2y — (3xy2 — y3)}. 7. Prove that axb-bxa; and simplify

-6225x-19y-[31-{4x-(51-62)}].

8. What are the rules for the removal of a bracket? Simplify {a-(b-c+d)} {a−b−(c–d)}, and find its value if a = 4, b=3, c= 2, d= 1.

9. Simplify 3(a - b) + 2(c− a) — 4{b+2(c−a)},

also (a+b-c)2+ (b + c−a2)+(c+a-b).

=

10. Explain the terms coefficient, factor, index, like quantities, binomial; and if x=1,y=-2, 3, find the value of [x-}{v−4(2-x-2y)}].

11. Simplify

the

(x+4)2+(x −4)2 +(x+6)2 + (x −6)2 − 3(x+8) (x − 8).

12. Simplify 1-2x − (1 − (1 − (1 − 2x)} − (1 − 2x)], and find square of the answer.

13. Simplify {(a+b)2 + (a−b)2 } × { (a+b)2 − (a−b)2}.

GREATEST COMMON MEASURE.

1. Find the G. C. M. of x2-a2 and x3-a3.

2. Find the G. C. M. of 9ax2y2 and 15a2xy, also of 3x5 – 20x2 +15x+2 and x1 — 4x+3.

3. Find the G. C. M. of x4 + 3x3 + 12x-16, and x3- 13x+12. What values of x will make both expressions vanish?

4. Find the G. C. M. of x3-x2+x+3, and x++x3 − 3x2

-x+2.

5. Find the G. C. M. of 20a++a2 - 1, and 25a*+5a3 — a − 1.

6. Give the rule for finding the greatest common measure of two compound algebraical quantities. Apply it to find the greatest common measure of 4x3-8x2y-xy2+2y3, and x2-4y2.

7. Find the G. C. M. of 12x3+4x2-3x-1, and 8x3- 4x3

2x + I.

8. Find the G. C. M. of 48x3 +8x2+31x+15, and 24x3 +22x2+17x+5.

9. Find the G. C. M. of 3a3-3a2b+ab2-b3, and 4a3-ab - 3ab2 by inspection or otherwise.

10. Find the G. C. M. of 4x3+x-5, and 3x3 — 7x2+4

11. Find the G. C. M. of 6a3 – 6a2 + 2a − 2, and 12a2 — 15a +3.

12. Find the G. C. M. of 8x3- 10x2+7x-2, and 6x3- 11x2 +8x-2.

13. Find the G. C. M. of x2+11x+30, and 9x3 +53x2 — 9x - 18.

14. Find the G. C. M. of x2+10x2+ 21x, and x3 – 9x.

15. Find the G. C. M. of 2x3-15x+14, and x1 — 15x2+28x

- 12.

16. Find the G. C. M. of 3x4 — 4x3 — x − 2, and 2x1 — 3x3 + 2.1o 2x-3.

17. Find the G. C. M. of 3x5 - 10x3+15x+8, and x3-214 -6x+4x2+13x+6.

18. Prove that if a quantity c be a common measure of a and b, it will also measure the sum or difference of any multiple of a and b, as ma±nb; and find the G. C. M. of 3a3 - 22α-15, and 5a454a2+18a.

19. Find the G. C. M. of xa+x2y2+ya, and x*+2x3y+3x°y2 +2x+y4.

20. Prove the rule for finding the G. C. M. of two quantities. Find the G. C. M. of 21(x2-xy+y2), 28(x6 −y6).

21. Find by inspection the G. C. M. of (a2b- ab2) and ab(a2-b2)2, also of 6(x2-1) and 8(x2-3x+2).

22. Find by inspection or otherwise the G. C. M. of 4(x3+a3), and 6(x2 - 2ax — 3a2).

LEAST COMMON MULTIPLE.

1. Find the L. C. M. of x2-a2, and x3-a3.

2. Find the L. C. M. of x+2, x2 − 1, x2+x−2.

-

3. Find the L. C. M. of 15a2+16ab – 15b2, and 9a2 – 2562.. 4. Find the L. C. M. of x3 + x2+x+1, and x3 − x2 + x − 1. 5. Find the L. C. M. of 12x3 + 4x2 - 3x − 1, and 8x-4x2

- 2x + 1.

6. Find the L. C. M. of x4+x2y2+y4, and x3-y3.

7. Prove the rule for finding the L. C. M. of two algebraical quantities.

8. Find the L. C. M. of x3+2ax2+a2x+2a3, and x3-2ax2 +a2x-2a3.

1. Simplify

} {x(x+1)(x+2)+x(x − 1) (x − 2)} +}(x − 1)a(x+1).