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In the following pages are given, with complete solutions, the Mathematical problems set at the competitive examination for admission to the Royal Military College at Sandhurst, and for First Appointments in the Royal Marine Light Infantry, held under the direction of the Civil Service Commissioners in June and July 1883.

12, CARDIGAN ROAD, RICHMOND,

AND

42, THE COMMON, WOOLWICH,

August 1883.

MILITARY ENTRANCE EXAMINATION.

ROYAL MILITARY COLLEGE

AND

ROYAL MARINE LIGHT INFANTRY.

MATHEMATICS (1).

Saturday, 30th June, 1883. 10 A.M. to 1 P.M.

(Great importance will be attached to accuracy in results.)

1. Reduce to its simplest form

(ax + by)3 + (ax — by)3 + (bx — ay)3 + (bx + ay)3

by (a+b)2x2 - 3ab (x2 - y2).

(2) Find the Greatest Common Measure of

Sols. (1) Expanding and collecting—

(ax + by)3 + (ax − by)3 + (bx − ay)3 + (bx + ay)3

= 2 (a3 + b3) x3 + 6 (a + b) ab xy2

= 2 (a + b) x { (a2

Also (a + b)2x2 - 3ab (x2 —

ab + b2) x2 + 3ab y2}.

y2) = (a2 — ab + b2) x2 + 3ab y2.

Hence the required quotient is 2 (a + b) x. Ans.

B 2

Every measure of two quantities is a measure both of the sum and difference of any multiples of the quantities.

The Greatest Common Measure is not altered if either of the quantities be multiplied or divided by a factor which is not a measure of the other.

Subtracting 28 from xa and dividing the result by 3 gives y

Subtracting 10x (x − 3) from 8 gives 41x + 123.

Dividing 41x + 123 by 41 gives x 3.

Hence x 3 is the Greatest Common Measure of (a) and (B).

3.-(1) Prove that

(a2 + x2)++ (a2 — x2)+

a4

(a2 + x2)* − (a2 — x2)+ − {x - 1 } *

a2

=

Sols. (1) Multiplying both numerator and denominator by the

4.-(1) Prove that the cube of a number which has n digits cannot have less than 3 n 2 nor more than 3 n digits.

(2) Find the cube root of 40 353607; (3) and of 1 3x to four terms in ascending powers of x.

Sols. (1) 10" is represented by 1 and n zeros, and is therefore the least number containing n + 1 digits.

10"-1, in like manner, is the least number containing n digits. Hence a number which has n digits must be less than 10", and not less than 10"-1.

3n-3

Therefore the cube of such a number cannot be less than 103n-3, which has 3n 2 digits, and must be less than 103", the least number with 3n+ 1 digits, and therefore cannot have more than 3 n digits.

= a, (2) when x = a + √ a2 – 1;

Hence, substituting 2a for a in the above expressions,

1

202

1

ક

1

= 4a3—2, 23+3=-8a3-6a, and a* += 16a* — 16a2 + 2. When x = 64, 2-23 x§ + 2−3 3− x3 — 10 (27x) x}

1

16-10-83+33-33

2

5

=

= ↓ √ 3 × 32 + 1 × √3 × 16-10

8√3+