90. Show that every periodic continued fraction is equal to one of the roots of a quadratic equation of which the coefficients are rational. can be transformed to an equation in which the term in xn-r is absent by solving an equation of the rth degree. (50) Find the condition that the roots, a, B, 7, of the equation x3 + 3 px + q = 0 may all be real, and show that (a2 + B2 + z3)2 = 2 (a1 + B* + y1). (50) 92. Let P be a point within the angle BAC, and let the angles PAB and PAC be denoted by a y and a respectively. Two circles are drawn to pass through P and to touch AB and AC; show that the ratio of their radii equals cos y + √(sina - sin2 7): 93. (a) If sin a = m sin B, find the value of sin (a-B) sin (a + B) when a becomes indefinitely small. (b) Taking cos 25° 50′ to equal 0·9, find the range of the values of 0, between 0° and 360°, for which √(1 + cos 26) + √(1 − cos 20) > 9 5 (45) 94. (a) If A, B, C are the angles of a plane triangle, show that and by expanding the determinant deduce that 1- cos2A - cos2B cos2 C2 cos A cos B cos C = 0. sin2 sin2 = cos2a + cos2 - 2 cos a cos B cos 7. (55) 95. (a) Write down the series commonly denoted by e, and show that it is convergent, and that its sum to infinity is an incommensurable number. (b) Show that eit is a periodical function. (c) A man reasoned thus :- 96. Expand log (1 - 2x cos + 2) in ascending powers of a Stages 5, 6 and 7, and Honours in Division II. INSTRUCTIONS. You may take Stage 5, or Stage 6, or Stage 7, or Honours, but you must confine yourself to one of them. The figures in descriptive geometry should not only be constructed with ruler and compasses, but the construction should in all cases be explained and its accuracy demonstrated. The examination in this subject lasts for three hours. Stage 5. You are not permitted to answer more than eight questions. 1. The vertical trace of a plane makes an angle of 25° with the ground line, and its horizontal trace an angle of 50°. A point P is distant from the plane on its upper side, 2" in front of the vertical plane and 3" above the horizontal plane. A sphere of diameter 3" has its centre at P. Construct the horizontal projection of the line of intersection of the plane and sphere. (24) 2. A right circular cylinder, whose diameter is 2", lies on the horizontal plane of projection, and its axis makes an angle of 50° with the vertical plane of projection. Construct the traces of a plane touching the cylinder and making an angle of 40° with the horizontal plane. Also construct and measure the angle which this tangent plane makes with the vertical plane of projection. 3. Establish the equation to a straight line in the form a cos ay sin a p = 0. (24) AB, AC, are two given lines which contain an angle 4 3 a, where tan α = Points P and Q are taken on AB and AC respectively, such that AP + 5 AQ= 16. Show that the centre of the circle which circumscribes the variable triangle APQ lies on a fixed straight line, whose distance from A is √2. (24) 4. Find the coordinates of the centre, and the radius of the Find also the coordinates of the point on the circle furthest from the origin. Determine the length intercepted by the circle on the line a parabola referred to 5. Having given the equation of rectangular coordinate axes, y2 = 8x, find the equation of the normal at a point (x, y1) on the curve, and show that the subnormal is constant. If the normal at a point, P, whose abscissa is 18, cut the parabola again at Q, show that 6. A rod AB of length moves with its extremities on two fixed lines which intersect each other at right angles. If P be the point which divides AB in the ratio 2 to 3, show that the locus of P is an ellipse, and state its eccentricity. Find the points on AB which describe ellipses whose 7. Find the position of the foci of the hyperbola x2-7y2 = 14, and show that it is confocal with the ellipse 9x225y2225. (25) If P be a point of intersection of these curves, and the ordinate at P, when produced, meet the auxiliary circle of the ellipse at Q, prove that Q lies on an asymptote of the hyperbola. (25) 8. Find what loci are represented by the following equations: (1) 3x24xy+6y2 = 5; 4x2 + 4xy + y2 — 5x + 1 = 0. Give the length of the latus rectum in each case. (26) In the ellipse given by the equation x2 + 3xy + 3y2+ y 12x 28=0 find the equation of the tangent at the point (1, 3). 10. Find the maximum and minimum values of x2 (2x+3) — 12 (x + 1), and distinguish between them. (24) Find the height of the right circular cone of maximum volume, the length of the slant edge being 6 inches. (24) 12. An ellipse of semi-axes 5, 3, is divided into two segments by a line parallel to the minor axis at a distance 1.5 from the centre. Find the area of the smaller segment, and the volume of the solid generated by the revolution of this area about the major axis. (25) Stage 6. You are not permitted to answer more than eight questions. 21. A plane passes through the points (2, 1, 4) (7, 1, 6) and the point of bisection of the line joining the points (3, 8, 5), (5, 2, 7). Show that this plane also passes through (14, 5, 10) and is inclined to the coordinate plane Oy, at 22. Obtain the equations of the line joining the points A (3, 7, 1), B (2, 1, 3) and give its direction cosines. (35) Find the coordinates of the point on AB which is nearest to the line of intersection of the two planes is the equation of the plane which touches the ellipsoid x2 y2 22 + + =1 a2 b2 c2 at the point (x', y', z′). (36) Find the equations of the two planes which contain the line given by |