and determine the limits between which n must lie in order that the equation may have real roots. 10. The figures which express the pounds and the pence in a certain sum of money will change places if £2. 19s. 9d. be added to it, and those which express the shillings and the pence would be interchanged by subtracting 25. 9d. What alteration would be produced in the sum of money by interchanging the figures which express the pounds and shillings? II. Find the sum of a given arithmetical series. 300 trees are planted regularly in rows in the shape of an isosceles triangle, and the numbers in successive rows decrease by one. How many trees are there in the row which forms the base of the triangle? 12. Find the number of permutations of the letters forming the word treasurers taken all together. 13. Write down the middle terms in the expansions of Show also that only one term in the expansion of has a negative sign, if p is any positive number not less than unity. IV. PLANE TRIGONOMETRY. (Obligatory.) (Including the solution of triangles.) [N.B.-Great importance will be attached to accuracy.] I. Express in circular measure and in degrees to three places of decimals an angle which is subtended at the centre of a circle 3 square inches in area by an arc of 1 inch. [=] 2. Define the trigonometrical ratios of A involved in the equation— cot A +tan A= sec A cosec A, and establish its truth by a geometrical construction. 3. Show how the formula for tan (A+B) in terms of tan A, tan B, may be deduced from the formula for sin (A+B). (1) Tan2 60o – 2 tan2 45°=cot2 30° – 2 sin2 30° – & cosec2 45°. cot2 A-tan2 A' 6. Find all the values of A which satisfy the equation 7. cos A - cos A sin A - sin3 A = 1. If x be the least positive angle of which the sine is equal to sin A, find a general formula for A in terms of x. 8. Prove that in any triangle ABC, with the usual notation, If AD be drawn perpendicular to the plane of ABC, and DB, DC be joined, show that sin DCB. sin DBA = sin DCA . sin DBC. 9. If the sides of a triangle be 7·152 in., 8·263 in., 9°375 in., find its 10. Find the radii of the three circles escribed to a triangle in terms of the sides. Show that the sum of the products of these radii two and two is equal to the square on the semi-perimeter of the triangle. II. A straight flagstaff, leaning due east, is found to subtend an angle a at a point, in the plain upon which it stands, a yards west of the base. At a point b yards east of the base the flagstaff subtends an angle ß. Find at what angle it leans. FURTHER EXAMINATION. V. PURE MATHEMATICS (1). [Great importance will be attached to accuracy in results. Full marks may be obtained by doing about three-quarters of this paper.] I. If two straight lines be at right angles to the same plane they shall be parallel to one another. Only one line can be drawn from a given point perpendicular to a given plane, and it is the shortest line from the point to the plane. 2. If a solid angle be contained by three plane angles any two of them are together greater than the third. 3. Find the radius of the sphere inscribed in a given tetrahedron. 4. Define a right cone and prove that every section of it by a plane meeting it on opposite sides of the vertex is a hyperbola. Prove that the distance between the foci of the hyperbola is equal to the sum of the distances of its vertices from the vertex of the cone. 5. In the parabola prove the propositions (1) PN2=4AS. AN, (2) AT=AN, the letters having their ordinary signification. 6. PV is any chord of a parabola. Through O, O', two points on PV equidistant from its extremities, chords ROQ, R'O'Q' are drawn parallel to one another. Show that the rectangle contained by RO, OQ is equal to the rectangle contained by R'O', O'Q. 7. In the ellipse prove that the locus of Y, the foot of the perpendicular from S upon the tangent at P, is the auxiliary circle. 8. Find the polar equation of the straight line joining the points (1, 2) and (3, 4), the axis of y being the prime radius. 9. Find the equation of a circle referred to rectangular axes. Prove that the circles x2+y2 − 2ax+c2=0 and x2+y2 — 2by — c2 = 0 intersect at right angles. IO. Find the equation of the tangent at any point Q of an ellipse referred to conjugate diameters PCp and DCd as axes. Let this tangent meet the diagonals, or the diagonals produced, of the parallelogram circumscribing the ellipse at the extremities of Pp and Dd in R and S, and let RS be bisected in T. If X, Y are the co-ordinates of Q referred to Pp and Dd as axes, and if X', Y' are the co-ordinates of T, show that II. axes. 12. Find the equation of the hyperbola referred to the asymptotes as Prove that the equation of the axes of the central conic is x2-y2=0, ax2+2bxy+ay2=d 13. Find the polar equation to the tangent to an ellipse, the focus being the pole and the major axis the initial line. VI. PURE MATHEMATICS (2). [Great importance will be attached to accuracy in results.] I. In the expansion of (x-y-z)9 find the coefficient of x3y2+, and the number of terms. 2. Prove that the arithmetic mean of n positive quantities a, b, c, d... &c. is greater than their geometric mean. 3. Find the annuity, to continue for four years, which can be purchased for £750, supposing that a perpetual annuity is worth 25 years' purchase. The first payment is made at the expiration of a year from the purchase of the annuity. 5. Find the coefficient of x in the expansion of ascending powers of x. 6. Solve the equation— I-COS 20=2 (cos a cos - cos 2a). 8. Show that the perimeter of a triangle: perimeter of the inscribed |