MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

NOVEMBER, 1880.

PRELIMINARY EXAMINATION.

I. EUCLID (Books I.-IV. and VI.). (Obligatory.)

1. Distinguish between a postulate and an axiom. In which propositions are the 2nd postulate and the 12th axiom of Euclid respectively used for the first time? What axioms are used in Prop. IV., Book I.?

2. The angles at the base of an isosceles triangle are equal to one another; and if the equal sides be produced, the angles on the other side of the base shall be equal to one another.

State fully the converse of this proposition.

3. If one side of a triangle be produced, the exterior angle shall be greater than either of the interior opposite angles.

Complete the figure which is necessary for the proof of the second part of this proposition.

How many of the exterior angles of any triangle must be obtuse?

4. Parallelograms upon the same base, and between the same parallels, are equal to one another.

Of all triangles that can be drawn upon a given base and between the same parallels, shew that an isosceles triangle has the least perimeter.

5. Describe a square that shall be equal to a given rectilineal figure. Divide a given straight line into two parts, such that the rectangle contained by them shall be equal to the square of their difference.

6. If two circles touch one another internally, the straight line which joins their centres, being produced, shall pass through the point of con

tact.

If two circles touch externally at E, and AB, CD be any two parallel diameters of the circles, shew that the straight lines AD, BC will pass through E.

7. If a straight line touch a circle, the straight line drawn from the centre to the point of contact shall be perpendicular to the line touching the circle.

8. The angles in the same segment of a circle are equal to one another.

If PQ, RS, two chords of a circle, intersect within the circle, shew that their inclination to one another is equal to one half of an angle at the centre of the circle standing upon an arc equal to the sum of the arcs PR and QS.

9. Inscribe an equilateral and equiangular pentagon in a given circle. Give a construction (without proof) for the inscription of a regular twenty-sided figure in a given circle.

IO. If the vertical angle of a triangle be bisected by a straight line which also cuts the base, the segments of the base shall have the same ratio which the other sides of the triangle have to one another.

Find a point C in ACB, a given segment of a circle, such that the straight line AC is double of the straight line CB.

11. Describe a rectilineal figure which shall be similar to one given rectilineal figure, and equal to another given rectilineal figure.

II. ARITHMETIC. (Obligatory.)

(Including the use of Common Logarithms.)

[N.B.-Great importance will be attached to accuracy in numerical results.]

I. Add together 2 of 5, 3 of %, 71 of 31%, and 4 of 51.

9. Express I furlong 4 poles as the decimal of a mile.

IO. What is the difference between 0057 of a lb. troy and of a dwt.? Express the answer as the fraction (vulgar) of a dwt.

II. Find, by Practice, the dividend on £3,245. 15s. at 13s. 9d. in the £.

12. Find the amount of £2,060 in 3 years at 4 per cent. simple interest, neglecting fractions of a penny.

13. Multiply by duodecimals 3 ft. 5 in. by 4 ft. 9 in., and the product by 8 ft. 7 in. What does the answer become when expressed in cubic feet, cubic inches, and the fraction of a cubic inch?

14. A, B, and C working together can do a piece of work in 6 days. A could do it alone in 24 days. After working together for 2 days A is taken ill. How long will B and C take to finish it?

16. Extract the square root of 107101801 and the cube root of 160103007.

17. Find the difference between the interest and the true discount on £264. 10s. for 3 years at 5 per cent. per annum simple interest.

18. A starts from London for Epsom, distant 14 miles, walking at the rate of 3 miles an hour. B starts from London on the same road 1 hours later, driving 8 miles an hour. At what distance from Epsom will B over

take A?

What

19. 40 lbs. troy of standard gold can be coined into 1,869 sovereigns, the proportion of pure gold to alloy in standard gold being 22 to 2. weight of pure gold is there in a sovereign?

20.

The average temperature for Monday, Tuesday, and Wednesday was 53°. The average for Tuesday, Wednesday, and Thursday was 56o, that for Thursday being 60o. What was the temperature on Monday?

21. Find, by logarithms, a fourth proportional to the 5th power of 11, the 4th power of 7, and the 5th power of 5.

of

22. Find log 001155, and calculate to 6 places of decimals the value

III. ALGEBRA. (Obligatory.)

[N.B.-Great importance will be attached to accuracy in numerical results.]

I. State and prove the rules for the removal of brackets from algebraical expressions.

Find the value of

when x=

2.

Divide 2a2x2- 2 (3b − 4c) (b − c) y2 + abxy by ax + 2 (b−c) y, and find the coefficients of x2, x3, and x4 in (x+a)3 × (x − a)5.

3. Shew that the product of the greatest common measure and the least common multiple of two quantities is equal to the product of the quantities themselves.

Find the greatest common measure and least common multiple of

7. Find the condition that the equation ax2+ bx+c=0 may have equal roots.

Also find the condition that the roots of a2x2+b2x+c2=o may be the squares of the roots of ax2 + bx+c=0.