9. From one of the angles of a rectangle a perpendicular is drawn to its diagonal, and from the point of their intersection straight lines are drawn perpendicular to the sides which contain the opposite angle; shew that if p and q be the lengths of the perpendiculars last drawn, and c the diagonal of the rectangle, 10. *Shew that the area of any triangle is equal to-- ANNUAL EXAMINATION. PURE MATHEMATICS. Examiner PAPER B.-ALGEBRA. PROFESSOR H. W. SEGAR, M.A. N.B. For a Pass only, questions marked with an asterisk need not be attempted. 1. (i.) Prove the ordinary process for finding the H.C.F. of two algebraical expressions. a (ii.) Give the original meanings of a÷b, and a : b, and b establish their practical identity. *(iii.) Prove generally (vi.) Prove that, if the terms of a fraction not equal to unity be positive, the addition of the same positive quantity to each makes the fraction more nearly equal to unity. (v.) Find the sum of a G. P. (vi.) Prove logam = log.mx log.b. (vii.) Prove the Binomial Theorem for the case when the index is a positive integer. (ii.) (x* + 4) (x2 + 14) (x2 + 1) (**· − 4) · 4. Two persons, A and B, run a race to go five times round a certain course. When A has gone three laps, B is 150 yards behind him. A then slackens speed and goes at B's rate, while B quickens his rate and goes at A's first rate. A wins by 30 yards. Find the length round the course, and compare the original speeds of A and B. 5. Shew that if one root of the equation x2 + px + q = = 0 be a 0, then its other root is (ii.) (a + b)2 + (a2 + b2) + (a b) + and p, q, r be in A.P., then x, y, z are in H.P. 9. A gentleman invites a party of m+n friends to dinner, and places m of them at one table and n at another, both tables being round. Find the number of ways in which he can arrange them among themselves. Apply the Binomial Theorem to find √17 to four places of decimals. 12. *If the two expressions + px2 + qx + r, x3 + p2x2 + q1x + y2, have the same quadratic factor, prove that ANNUAL EXAMINATION. APPLIED MATHEMATICS. Examiner PROFESSOR H. W. SEGAR, M.A. N.B. --For a Pass only, questions marked with an asterisk need not be attempted. (b) Shew that if two bodies in motion be subjected to equal accelerations their relative velocity is not altered. 2. If, in Attwood's machine, the string can bear a tension equal to only one-fourth the sum of the weights, shew that the 3. Find the range in vacuo of a rifle-bullet projected with a velocity of 1200 feet per second, the direction of projection making an angle sın -11 with the horizon. 4. * The measures of an acceleration and a velocity, when referred to (a+b) feet and (m + n) seconds, and (a - b) feet and (m − n) seconds respectively, are in the inverse ratio of their measures when referred to (a - b) feet and (mn) seconds, and (a+b) feet and (m+n) seconds. Their measures, when referred to a feet and m seconds, and b feet and n seconds, are as ma: nb ; shew that |