III. Insert n harmonic means between a and b. If the pth term of a series in H. P. is q and the qth term p show that the pqth term is 1. IV. Find the number of combinations of n things taken r at a time. In how many ways can a pack of 52 cards be distributed equally among four persons so that,-(a) each may have ace, king, queen and knave of the same suit,—(b) each may have ace, king, queen and knave all of different suits. V. Find the numerically greatest term in the expansion of (1+x) n taking n to be a positive fraction and a to lie between one and zero. VI. Show that the coefficient of x11 in the expansion of √(1 + x)3 (1-x)5 in ascending powers of x is 7.9.11.23 8 L 4. VII. Triangles on the same base and between the same parallels are equal to one another. ABCD is a quadrilateral figure whose diagonals intersect in E. AD BC are produced to meet at O. OP is cut off from OA equal to AD, and OQ from OB equal to BC. Shew that the triangle OPQ is equal to the difference of the triangles AEB, DEC. VIII. Angles in the same segment of a circle are equal to one another. If two triangles have one side of the one equal to one side of the other, the angles opposite to the equal sides equal and the areas equal, they shall be equal in all respects. IX. If a line be drawn parallel to the base of a triangle, it cuts the sides proportionally. P, Q are points in the sides CB, CA of a triangle such that BP is twice CP and AQ thrice CQ. AB, QP are produced to meet in R. Shew that RB is twice AB. X. Similar triangles are to one another in the duplicate ratio of their homologous sides. FRIDAY, 20TH DEC., 2 TO 5 P.M. I. Define the sine of angle, also its circular measure. The circular measure of an angle containing a given number of degrees can only be expressed approximately. Why is this? II. Find all the positive angles that have the same tangent as an angle a in the first quadrant; also all the negative angles that have the same cosine as a. If tan x cota, find the general value of x. III. Determine the proper sign in each of the equations, when A is an angle in the fourth quadrant. For what values of A is V. In any triangle find an expression for the cosine of an angle in terms of the sides. P and P' are any points in a line parallel to a given line AB, show that cot APB cot AP'B α CP2 CP'2 where C is the middle point of AB. VI. Find the least angle of the triangle whose sides are 7, 9, and 11 respectively. VII. Find an expression for the radius of the circle circumscribing a given triangle. PP' is the chord of a circle parallel to the diameter AB, and AP' and BP are drawn intersecting in Q. If R and R' be the radii of the circles circumscribing PQP' and AQB respectively, shew that R'2 R2 = a 2 where AB = 2a. VIII. Define a prism and a pyramid. Cubes, parallelopipeds and tetrahedrons are particular cases of other solids defined by Euclid. Shew this. IX. If two straight lines be at right angles to the same plane they shall be parallel to one another. X. If two straight lines meeting one another be parallel to two others that meet one another and are not in the same plane with the first two, the first two and the other two shall contain equal angles. A pyramid with a rectangular base is cut by a plane parallel to the base, shew that the section is a rectangle. XI. Prove that a straight line can be drawn at right angles to a plane from a given point in it by the following construction :through A drawn any line AB in the given plane and in the same plane drawn AC at right angles to AB; in any other plane passing through AB draw AD at right angles to AB, and in the plane CAD draw AE at right angles to AC; AE will be perpendicular required. L B. A. DEGREE EXAMINATION, 1879. MONDAY, 10TH FEB., 10 A.M. TO 1 P.M. ENGLSH POETRY. M. R. WELD, B.A. 6 I. Give a brief sketch of the plot of Chaucer's Prologue,' and render in modern English his descriptions of three of the principal personages who met at the Tabard. Mrs. Browning speaks of Chaucer's "'infantine "Familiar clasp of things divine," and of his love of conviviality "That stain upon his lips is wine." Quote a few lines illustrative of each point. II. a. Modernise the following, paying particular attention to the meanings of the words in italies. 5. In a gowne of faldyng to the kne. A daggere hangyng on a laas hadde he Aboute his nekke under his arm adown." "He waytede after no pompe aud reverence, But Cristes lore, and his apostles twelve. He taughte, but first he folwede it himselve." |