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MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

NOVEMBER, 1893.

OBLIGATORY EXAMINATION.

I. EUCLID (Books I.-IV. AND VI.).

[Ordinary abbreviations may be employed; but the method of proof must be geometrical. Proofs other than Euclid's must not violate Euclid's sequence of propositions. Great importance will be attached to accuracy.]

I. Any two sides of a triangle are together greater than the third side.

ABC is any rectilineal angle less than the angle of an equilateral triangle, and D and E are two points within it; find the points F and G in AB and BC respectively, such that the sum of the lines DF, FG, GE has the least possible value.

2. Equal triangles on the same base and on the same side of it are between the same parallels.

Use this proposition to show that the straight line joining the middle points of two sides of a triangle is parallel to the third side.

3. Describe a parallelogram equal to a given rectilineal figure, and having an angle equal to a given rectilineal angle.

4. If a straight line be divided into two equal, and also into two unequal parts, the squares on the two unequal parts are together double of the square on half the line and of the square on the line between the points of section.

Express the proposition as an algebraical formula.

5. Draw a straight line from a given point, either without or in the circumference, which shall touch a given circle.

From a given point P without the circle ABC, draw a straight line PAB cutting the circle in A and B so that AB may be equal to a given straight line D.

6. Distinguish between the angle of a segment and the angle in a segment and define similar segments of circles.

ABCD... is a straight line, on parts of which AB, AC, AD, ... similar segments of circles AFB, AGC, AHD, ... are described; show that (1) all these circles have a common tangent at A, (2) any straight line AFGH... drawn from ▲ and cutting the circles in F, G, H, cuts off similar segments.

...

7. On a given straight line describe a segment of a circle containing an angle equal to a given rectilineal angle.

Given the altitude, the vertical angle, and the perimeter of a triangle, construct it.

8. Describe a circle about a given triangle.

Point out and prove any facts concerning the opposite angles or sides of a quadrilateral (1) inscribed in, (2) described about, a circle.

9. When is the first of four magnitudes said to have the same ratio to the second that the third has to the fourth?

Illustrate this definition by giving Euclid's proof of the proposition that triangles of the same altitude are to one another as their bases.

10. Parallelograms which are equiangular to one another have to one another the ratio which is compounded of the ratios of their sides.

II.

In any right-angled triangle, any rectilineal figure described on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle.

If BFC, CDA, AEB be equilateral triangles described externally on the sides of a triangle ABC right-angled at A; and if AG be drawn perpendicular to BC to meet BC in G; show that the triangles BFG, CFG are respectively equal to the triangles AEB, CDA.

II. ALGEBRA.

(Up to and including the Binomial Theorem, the theory and use of Logarithms.)

I.

2.

3.

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

Divide (x+y)+(x2 −12)2+(x − y) by 3x2+12.

Resolve each of the following into three real factors:
4x3- 23x2+28x, ja+1172 – 180, a6+27b6.

Find the Highest Common Factor of

5x4-16x3+20x+7 and 30x3-71x2+49.

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Divide I into two fractions such that the sum of their cubes is.

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8. Show that a ratio of greater inequality is diminished by adding any, the same, quantity to both terms of the ratio.

9. If b is half the Harmonic Mean between a and c, then

a3 − b3+c3+3abc = (a − b+c)3.

10. Prove by Mathematical Induction that, when n is a positive integer, (3n+1)7′′ – 1 is always divisible by 9.

II.

When x = 3, find the two greatest terms in the expansion of

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Show, with the aid of the logarithmic tables supplied, that approximately

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