MATHEMATICAL EXAMINATION PAPERS FOR ADMISSION INTO Royal Military Academy, Woolwich, NOVEMBER, 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. Make a triangle of which the sides shall be equal to three given straight lines; and show that if ABC be a triangle with the vertical angle at A greater than either of the base angles, it is not possible to form a second triangle with sides equal to those of ABC and base equal to twice BC. 2. Give Euclid's Axiom on Parallels, and those on the addition and subtraction of equals to and from equals and unequals. Prove that if a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another; and the exterior angle equal to the interior and opposite angle on the same side; and also the two interior angles on the same side together equal to two right angles. 3. ABC being an isosceles triangle, and D any point in the base BC; show that the perpendiculars to BC through the middle points of BD and DC divide AB and AC at H and K respectively, so that BH= AK and AH=CK. 4. If a straight line be bisected, and produced to any point, the rectangle contained by the whole line thus produced, and the part of it produced, together with the square on half the line bisected, is equal to the square on the straight line which is made up of the half and the part produced. Prove this; and give its equivalent in algebraical symbols. 5. Prove that in every triangle, the square on the side subtending an acute angle, is less than the squares on the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall on it from the opposite angle, and the acute angle. If in a quadrilateral ABCD, the square on AB is greater than the squares on the three other sides BC, CD, DA by twice the rectangle CD, DA, show that the angle ACB is obtuse. 6. Prove that the straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and that no straight line can be drawn from the extremity, between that straight line and the circumference, so as not to cut the circle. Give the corollary to this proposition. 7. Prove that the angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same arc. If the diagonals of a quadrilateral inscribed in a circle cut at right angles, show that the angles which a pair of opposite sides of the quadrilateral subtend at the centre of the circle are supplementary. 8. If from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it; prove that the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, is equal to the square on the line which touches it. 9. Inscribe a circle in a given triangle; and indicate the construction when the circle is required to touch one side externally. IO. Inscribe a regular pentagon in a given circle. If a regular pentagon and a regular hexagon be on the same base and on the same side of it, prove that the pentagon is wholly within the hexagon. II. In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, prove that the triangles on each side of it are similar to the whole triangle, and to one another. If AB, BC be two sides of a regular figure, L and M their respective middle points, and O the centre of the inscribed circle, show that the triangle BLM has to the triangle OLM the duplicate ratio of that which a side of the figure has to the diameter of the circle. 12. Prove that in equal circles, angles, whether at the centres or circumferences, have the same ratio which the circumferences on which they stand have to one another. II. ALGEBRA. (Up to and including the Binomial Theorem, the theory and [N.B.-Great importance will be attached to accuracy.] a Define the meaning of the symbol and from your definition '3. show that a na bnb' where a, b, n are positive integers. Arrange in order of magnitude, a, b, n being positive integers, (iii.) 5x-14+ √3x-13=2√√x+√2x+1. 7. Prove that a quadratic equation cannot be satisfied by more than two values of the unknown quantity. In a certain quadratic the coefficients of x2 and x are 1 and 2 respectively, and the addition of 8 to each of the roots changes the sign but not the magnitude of the third term. Find the original quadratic, and the coefficient of x in the transformed equation. 8. Insert five arithmetic means between a 26 and 3a+b. The last term of an arithmetic progression is ten times the first, and the last but one is equal to the sum of the 4th and 5th. Find the number of the terms, and show that the common difference is equal to the first term. 9. Find the number of arrangements that can be made of the letters of the word infinite, (1) when they are taken all together, (2) when they are taken four together so that each arrangement has two vowels and two consonants. IO. Write down the first five terms in the expansion of (1 − 4x)₺ by the Binomial Theorem, and show that the coefficient of x" may be written in the form and, by means of the logarithm tables supplied, find the fifth root of 66901 × 337 correct to five places of decimals. |