8. Extract the square root of

(1) 16x-3213 +24x2-8x+I.

(2) 9a2x2 - 12abxy+6axz+4b2y2 — 4byz+z2.

9. Solve the equations

(1) (x−3)+3(x−4)=}(x−5)+1(x+1).

10. Find a number such that one-third of it exceeds

of the remainder by 10.

11. A carriage, horse, and harness are worth together 1447.; the carriage is worth of the horse's value, and the harness of the difference between the values of the horse and carriage; what is the value of each?

12. In what time will a cistern which holds 670 gallons be filled by two taps running simultaneously, one of which lets in 14 gallons in five minutes and the other 5 gallons in three minutes?

1. If a=1, b=3, c=5, d=7, find the value of

a-2b-[3c-d-{3a—(5b—c—8d)} — 2b].

2. Simplify (x-3) — (3 — x)(1 − x) —(x-3)(5 2x).

3. From

(a+b)2—2bx take {(a+b)(a−x) — (a—b)(b−x)}.

13

x2 — 5x3y + 13x2y2 — 13xy3+y1 by x2—¿xy+}y2.

5. Find the G.C.M. of

2x5-112-9 and 4x+11x+81,

and the L.C.M. of

5a3-20ac2, a2 + ac-2c2, a2+3ac+2c3.

9. Find a number which is as much greater than 63

as its half is less than 93.

10. A and B have the same income; A puts by of his, but B, by spending every year 80%. more than A, finds himself at the end of three years 20/. in debt. What is the income of each ?

11. A person swimming in a stream which runs 1 mile per hour finds that it takes him four times as long to swim a mile up the stream as it does to swim the same distance down. At what rate does he swim?

12. The wages of 10 men and 8 boys for a day amount to 1l. 17s.; if four men together receive one shilling more than six boys, what are the wages of each man and boy?

XXIX.*

1. Define am, where m is a positive integer. Prove that (a1)3= (a3)1, a* =3⁄4ja.

2. Divide

a+b—c−3(b$—c$)(c3—a3)(aμ+b3) by a3+b3—c3,

and x+x+4(x2+x-2)+6 by x2+x2+2.

3. Reduce to their simplest forms

I

(1)

I

+

I

(2x−1)2+ (2x+1)(1 −2x)* 2x2(2x— 1)(1 − 4x2)*

5. Find (1) the G.C.M., (2) the L.C.M. of the three

quantities

12xy(x3y — 213μ3+xy3), 20(x3—2x3μ3+x21⁄4o), and

8y(x7 — xyб);

and (3) the L.C.M. of bic, cat, abt, and ab‡ct.

6. Express in their simplest forms

(1) ¿√3+1V192+81); (2) √36× 48.

(3)

; (4) {(at)}}}.

This and the papers which follow include Quadratic Equations and Proportion.

8. If a and B are the roots of x2-px+q=0, (1) shew that the product of x-a and x-ẞ produces the expression x2-px+q, and (2) shew that

9. If a b::cd::e: f, shew that

(1) ad=bc, and (2) a: b::a+c+e:b+d+f.

10. Two navvies A and B working together can dig out 20 cubic yards of earth in a day. They work together for 6 days, after which A leaves off work and B continues for 4 days more. It is then found that they have dug out 156 cubic yards. How much can each dig in a day?

2. Shew that if 2s=a+b+c, then

16s(s—a)(s—b)(s—c)=2b2c2+2c2a2 +2a2b2 — aa — ba —c1.

3. Multiply

a√ 1 — b2 + b √ I—a2 by a√1-b2-b √ ī —aa ;

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