2. Translate and comment on (a) Atrox certamen aderat, ni Fabius rem expedisset.. (b) Qui vicissitudinem imperitandi, quod unum exaequandae sit libertatis, sustulerint. (c) Sollers nunc hominem ponere, nunc deum. (d) Nodos et vincula linea rupit, Quis innexa pedem malo pendebat ab alto. (e) Neptuni sacro Danais de poste refixum. 3. Explain legem centuriatis comitiis tulere, ut quod tributim plebes iussisset, populum teneret-lex agraria-comitia consulis subrogandi-quid debeas, o Roma, Neronibus testis Metaurum flumen -Ceaeque et Alcaei minaces Stesichorique graves Camenae-Cydonius arcus. 3. Eliminate from the equations ax + bx + c = 0, a'xm + b'x2 + c′ = 0. 4. If p = a+b+c, q=bc + ca + ab — ƒ2 — g2 — h2, r = abc + 2fgh — aƒ2 — bg2 — ch2, prove that `q2 — 2pr = (bc — ƒ2)2 + (ca — g2)2 +(ab — h2)2: + 2(af — gh)2 + 2(bg — hƒ)2 + 2(ch—ƒg)2. 5. If 6. If Aal2 + 2hlm + bm2, Bal'2+2hl'm' + bm'2, H= all' + h(lm' + l'm) + bmm'; and if these equations are also true when a, b, h, A, B, H are replaced by a', b', h', A', B', H', prove that AB' A'B-2HH' = (lm' — l'm)2 (ab' + a'b — Qhh'). p = a + b + c, 7. Define a" when m is fractional or negative, and prove that 8. Find the number of combinations of n things r at a time, and find for what value of r the number of combinations is greatest. 9. Prove the binomial theorem for a positive integral exponent. If a, be the coefficient of x" in the expansion of (1+x)", prove that GEOMETRY AND TRIGONOMETRY. The Board of Examiners. 1. Prove that the sum of the squares on the four segments of two chords which intersect at right angles within a circle is independent of the position of the point of intersection. State and prove the corresponding proposition for the case in which the chords intersect without the circle. 2. If OR is the radius of a circle, and in it and in its continuation points P and Q are taken so that the rectangle OP, OQ is equal to the square on OR, the locus of P is called the inverse with respect to the circle of any locus which Q is caused to trace. Prove that in general the inverse of a circle with respect to a circle is a circle, and interpret the case in which the circumference of the circle to be inverted contains the centre of the circle with respect to which the inversion is to be made. 3. A diagonal of a polygon of an odd number of sides being defined as the longest line that can be drawn from any angular point in it to another, prove that the diagonals of such a polygon when it is regular form a polygon similar to it. 4. If two triangles have one angle in one equal to one angle in the other, the sides about another angle in each proportional, and the third angles either both acute or both not acute, the triangles are similar. State and establish briefly the various sets of relations between the sides and angles of two quadrilaterals necessary and sufficient to secure similarity between them. 5. Find the locus of a point which moves so that the angles subtended at it by the two fixed parts of a finite straight line are equal. 6. Prove by a geometrical construction directly from the definitions of the trigonometrical ratios that 8. With the usual notation for a triangle, prove that the sides of the triangle formed by joining the feet of the perpendiculars from the angular points on the opposite sides are a cos A, b cos B, and c cos C; and show that the ratio of its perimeter to that of the original triangle is (a2 + b2 + c2)2 — 2 (a + b4 +04) 2abc (a+b+c) 9. Obtain a trigonometrical statement of the relations that must hold amongst the sides and angles of a pentagon in order that it may be inscribable in a circle. |