6. Find the G.C.M. of

20x1+x2— 1 and 25x1+5x3 — x — I,

and the L.C.M. of

x2+5x+4, x2+2x-8, x2+7x+12.

7. Extract the square root of

aa—a2b+2a2b2 + 1b2 —b3 +ba.

9. Find two numbers whose difference is 1, and the sum of their squares amounts to 313.

10. A man agreed to work for 30 days, on condition. that for every day's work he should receive 3s. 4d., and that for every day's absence from work he should forfeit Is. 6d. ; at the end of the time he received 37. 11s.: how many days did he work, and how many was he absent?

(1) s2+(s—a)2 + (s—b)2 + (s—c)2 = a2 + b2 + c2 ;
(2) (s—a)3 + (s - b)3 + (s −c)3 + 3(s—a) (s—b)c
= {(s—a)+(s—b)}3.

XXV.

1. Remove the brackets from

x-[3ax-{2x-(a−2x)b—3a(b−x)}+2b(x−2a)].

4. Find the G.C.M. of

a3-2ab2-b3 and a3 +2a2b+2ab2+b3;

7. Prove that the difference of the squares of any unequal numbers is equal to the product of their sum and difference.

Resolve into rational factors.

8. Two persons start at noon from towns 60 miles apart. One walks at the rate of 4 miles an hour, but stops 21 hours on the way. The other walks at the rate of 3 miles an hour without stopping. When and where will they meet?

9. Divide 225 into 2 parts, so that three times the greater may exceed 7 times the less by 45.

three and a half times as old as B. Find their present

ages.

XXVI.

1. If a=2, b=3, x=6, y=5, what is the value of a+2x−{b+y−[a—x—(b−2y)]} ?

2. Add together

ax-by, x+y, and (a−1)x−(b + 1)y.

3. Multiply-by+

y

x

6. Find the G.C.M. of

3x3+x2+4x−4, and 6x3-7x2 + 5x−2,

and the L.C.M. of

x2-4x2, 4x3+4x2-24x, x2-x-6.

9. Divide 91 into two such parts that, the greater being divided by their difference, the quotient may be 7.

10. The ages of two persons are in the ratio of 3 to 4; but 10 years ago the ratio was that of 2 to 3: what are their ages?

11. When are the hands of a clock together between the hours of 6 and 7?

12. A person distributes 22s. in charity, giving 6d. to each child, Is. to each woman, and is. 6d. to each man ; there are twice as many women as men, and twice as many children as women; how many persons are relieved altogether?

13. Simplify

(1) a ̄b3 c¬1× a ̄3blc÷abc

(2) √56 × √63 × √ 50.

XXVII.

1. If a=1, b-1, c= 2, find the value of

20ab-17bc+16ac − 5 (a2 + b2 + c2).

2. From (a+1)2x— (b− 1)2y take (a + 1)ax−(b−1)by. 3. Multiply a2x2+abxy+b2y2 by ax-by; and divide a1x1+a2b2x2y2+bay1 by a2x2+abxy +b2y2.

4. Resolve into their simple factors

(1) 323-12a;

(2) a2-a-2;
(3) 2x2-7xy+6y2.

5. Find the G.C.M. of

20a++a2-1 and 25a+5a3-α — I.