10. If a and ß be the roots of the equation x2 (1 + a)x + 1)(1 + a + a2) = O, prove that a2 + B2 a, and form the equation whose roots are a2 and B2. = II. A sphere whose diameter is one foot, is cut out of a cubic foot of lead, and the remainder is melted down into the form of another sphere; find its diameter ( 3*1416). = 12. The diameter of the earth being 7900 miles and that of the moon 2160, compare the areas of their surfaces, and find the radius of a sphere, whose surface is equal to their sum. 13. Define the terms logarithm, mantissa, and characteristic, and prove (1) loga. logab (2) = I ; loga (my) = m logax + n logay. 14. Find log 98 and log (343), given log 2 = 30103, log 7 = '845098, and solve the equation (3)x+4 = (25)3x+2, given log10 5 = '6989700. II. EUCLID AND TRIGONOMETRY. 1. Show that if a straight line be divided into any two parts, the square on the whole line is equal to the squares on the two parts, together with twice the rectangle contained by the parts. If the two complements are together equal to the squares on the two parts, show, either by geometry or algebra, that the straight line is bisected. 2. In every triangle, the square on the side subtending either of the acute angles is less than the squares on the sides containing that angle by twice the rectangle contained by either of those sides, and the straight line intercepted between the acute angle and the perpendicular let fall on it from the opposite angle. Prove that in every triangle the squares on the two sides are together double of the squares on half the base and on the straight line joining its point of bisection to the vertex. 3. Show that the opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles. Also show that the sum of one pair of opposite sides of any quadrilateral described about a circle is equal to the sum of the other pair. 4. If from a point without a circle two straight lines be drawn, one of which cuts the circle and the other touches it, show 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. Find the locus of points from which the tangents drawn to two intersecting circles are equal. 5. Describe an isosceles triangle, having each of the angles at the base double of the third angle. Also show that this problem supplies a geometrical construction for determining sin 18°. 6. Inscribe an equilateral and equiangular hexagon in a given circle, and compare the area of this hexagon with that of a similar one described about the circle. 7. If two triangles have one angle of the one equal to one angle of the other, and the sides about those equal angles proportionals, the triangles shall be equiangular, and shall have those angles equal which are opposite to the homologous sides. If two chords AB, AC drawn from any point A in the circumference of the circle ABC, be produced to meet the tangent at the other extremity of the diameter through A in D, E; prove that the triangle AED is similar to ABC. 8. In right-angled triangles the rectilineal figure described upon the side opposite to the right angle is equal to the similar and similarly described figures upon the sides containing the right angle. 9. What is meant by the unit of circular measure? Find the length of that part of a circular railway curve which subtends an angle of 22° to a radius of a mile (π = 3*1416). 10. Prove from a figure that cos (A - B) = cos A cos B + sin A sin B, when A lies between 315° and 360°, and A B between 180° and 225°. Show, à priori, the reason for the four different values of sin A 2 (b+c) cos A+ (c + a) cos B + (a + b) cos C = a + b + c ; and if r, R, raro re are the radii of the circles inscribed in, circumscribed about, and escribed to the triangle ABC, 15. An observer is situated in a boat vertically beneath the centre of the roadway of a suspension bridge. He finds that its length subtends at his eye an angle a. At a measured distance q down stream, at a point immediately opposite the centre of the roadway, he finds it subtends an angle B. Supposing the surface of the river horizontal, find an expression for the length of the roadway and its height above the surface of the stream. July 1882. I. ALGEBRA AND MENSURATION. 1. Explain the law of indices; and show that 2. Find the value of (x2 + y2), when x = 4. Show that in the process of finding the greatest common measure of two expressions, the introduction of a new factor does not affect the result. Find the G.C.M. of a2x3 + a3 2abx3 + b2x2 + a3b2 – 2a1b and 2a2x2 - 5a2x2 + 3ao — 2b2x2 + 5a2b2x2 — 3a2b2. 5. Prove that, when n+1 digits of a square root have been obtained by the ordinary rule, n more may be obtained by division only. Apply this to find the square root of 3 to six places of decimals. Find the cube root of 8a(362 — a2) + 862 No862 – 3a2. 8. A, B, and C run a mile race at uniform speed; A wins by 160 yards, B comes in second, beating C by 7613 yards in distance and by minute in time. What is the pace of each? 9. Prove from first principles that Find to three places of decimals the values of x from the equation = 4771213. having given log 2 = *3010300, log 3 10. Explain what is meant by the characteristic of a logarithm. Given log 10969100 and log ·i = I 0457575, and the logs given in Q. 9, find the logarithms of 21, 21, and 2. = Also having given that log 7 *9923057, find the 540th root of '00007. 8450980 and log 9'824394 = 11. A right pyramid whose base is a square of 7 inches side and whose perpendicular height is 8 inches is cut into two parts by a plane parallel to the base and 6 inches from it. Find the volumes of the two parts and their total surfaces. (In the following questions consider π = 22.) 12. A piece of paper in the form of a circular sector, of which the radius is 7 inches and the curved side 11 inches, is formed into a conical cup. Find the area of the conical surface, and also of the base of the cone. |