MATHEMATICAL EXAMINATION PAPERS FOR ADMISSION INTO Royal Military Academy, JUNE, 1892. 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. The sum of any two sides of a triangle is greater than the third side, and their difference is less than the third side. 2. If two quadrilaterals ABCD, EFGH have the four angles A, B, C, D respectively equal to the four angles E, F, G, H; and likewise the sides AB, DC respectively equal to the sides EF, HG; show that if AD and BC meet when produced the quadrilaterals are equal in every respect. 3. Define parallel straight lines; and show that parallelograms on the same base and between the same parallels are equal to one another. 4. Enunciate the proposition which is represented in algebraical symbols by (a+b)2 + a2 = 2a(a+b)+b2; and give the construction by which the proposition is proved. If in a quadrilateral ABCD, the sides AD and BC are each perpendicular to the side AB, show that the square on DC is less than the sum of the squares on the three other sides by twice the rectangle contained by AD and BC. 5. Divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts may be equal to the square on the other part. 6. Define circle, tangent, angle in a segment. If a point be taken within a circle, from which there fall more than two equal straight lines to the circumference, that point is the centre of the circle. 7. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles which this line makes with the line touching the circle shall be equal to the angles which are in the alternate segments of the circle. If the bisectors of the adjacent angles A and B of a quadrilateral ABCD meet in E, and the bisectors of the adjacent angles B and C, C and D, D and A meet respectively in F, G, H; show that a circle can be described round the quadrilateral EFGH. 8. If from any point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it, and if the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, be equal to the square on the line which meets the circle, the line which meets the circle shall touch it. 9. About a given circle, describe a triangle equiangular to a given triangle. IO. Inscribe an equilateral and equiangular hexagon in a given circle. Show that if A ̧Ã‚Â ̧Â... be an equilateral and equiangular figure of 24 sides, the straight lines A110 and A5418 are at right angles. II. If two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals, the triangles shall be equiangular to one another, and shall have those angles equal which are opposite to the homologous sides. 12. Similar triangles are to one another in the duplicate ratio of their homologous sides. If C be the centre of a circle, OPCQ a straight line cutting the circle in P and Q, OT a tangent to the circle, and PN another tangent cutting OT in N; show that the triangles OPN and OTC bear the same ratio to one another that OP bears to OQ. II. ALGEBRA. (Up to and including the Binomial Theorem; the theory and use of Logarithms.) [N.B.--Great importance will be attached to accuracy.] 1. Multiply a3 +2a2b+3ab2+363 by a3-3a2b+3ab2. Also show that 2. (x − 4 y)3 − ( y − 4x)3 + (2y − 3x)3 − (2x − 31)3 = 30(x − y)(x+y)2. Divide x6 — (p1 -- 2pq)x1-p2q2x2-q1 by x3-p2x2+pqx – q2. 3. Find the highest common factor of 4x4-29x2+25 and 2x2+x3- 14x2 - 4x+15. Also resolve each of these expressions into the simplest possible factors. 5. A square of carpet is cut up into strips so as to cover a border (a+b) feet wide round the floor of a room, whose length is 10a feet and breadth 100 feet. What was the length of the side of the square? |