ROYAL MILITARY ACADEMY,

WOOLWICH,

Papers in Competitive Examination for Admission to.

ALGEBRA.

WEDNESDAY, 29TH NOVEMBER, 1876. 2 P.M. TO 5 P.M:

1. "The product of two algebraical quantities having like signs is positive, and having unlike signs, negative." State briefly the steps by which this rule is obtained, and shew its arithmetical correctness when in the product obtained for (ab) x (c-d), a=12, b=9, c=10, d=4. Tod. Alg., Arts. 47, 48.

(a − b) × (c — d) = (12 − 9) × (10 − 4) = 3 × 6 = 18,

ac+bd-ad-bc120+36-48-90=18,

which proves rule in case given.

If x and y differ in value, explain why

(x − y)2 = (y — x)2.

Shew how the rule of signs stated above leads to the introduction of imaginary quantities in Algebra. Tod. Alg., Arts. 354, 355.

by

2. Multiply (1+x + 2x2)2 − (1 − x − 2x2)2

(1 + x − 2x2)2 − (1 − x + 2x2)2,

and find the product of x2-(a+b)x+ab by x2+(a−b)x-ab,

B

and examine what the product becomes if in it either

a or b be substituted for x.

(1 + x + 2x2)2 − (1 − x − 2x2)2 = 4x (1 + 2x),

(1 + x − 2x2)2 − (1 − x + 2x2)2 = 4x (1 − 2x);

therefore the first required product = 16×2 (1 − 4x3). {x2 + (a−b) x − ab} {x2 − (a + b) x + ab}

.

=

=

{(x2 - bx) + (ax−ab)} {(x2 — bx) — (ax — ab)} · (x2 — bx)2 — (ax — ab)”,

if xa orb the expression just obtained becomes equal

to zero.

3. Divide x 2bx3 — (a2 — b2) x2 + 2a bx - a2b2 by x2 - (a+b) x + ab.

Reduce to its simplest form

(x − y) (y − 2) + (x − y) (z − x) + (y − z) (≈ – x)

x (≈ −x) + y (x − y) + z (y − z)

Results x2+(a - b) x-ab, 1.

4. If m be a whole number, prove that (x+y) is

divisible by x+y" when m is odd.

Write down the three last terms of the quotient of

x+y divided by 2+y, and examine how many terms the quotient will contain.

If we divide x + y by x" + y" as in ordinary division,

after two operations we obtain a quotient x-xy"

and a remainder y (x + y); and, therefore, we see

that x+y will be divisible by x+y" without remainder,