ANALYTICAL GEOMETRY AND CALCULUS. PRELIMINARY HONOURS. 1. Find the equations of the lines which are a unit distance from the origin and which pass through the point of intersection of the lines 6x+5y=10 and 9x+8y=15. 2. Find the equation and draw the graph of the curve which is the inverse of the parabola y2=4x with respect to the circle x2+y2=9. 3. The polar of the point (1, 2) with respect to a circle. through the point (1, 3) is 4r-4y=11. Find the equation of the circle and draw the circle and the polar. - 4. Find the equation of the tangent to the parabola y24ar at the point (x, y), (a) using the derivative, and (b) without using the derivative. 6. (a) Find the maximum and minimum points and points of inflection of y=4x-4x2+2. (b) A barge whose deck is 12 ft. below the level of a dock is drawn up to it by means of a cable attached to a ring in the floor of the dock, the cable being hauled in by a windlass on deck at the rate of 8 ft. per minute. How fast is the barge moving towards the dock when 16 ft. away? Ꮎ 7. (a) Integrate (1) do. (2) Sx/a2 −x2 dx. (3) cos Ꭿ ' b) Find the area enclosed between the two parabolas y2=4x and x2-4y. PRELIMINARY HONOUR MATHEMATICS. MODERN SYNTHETIC GEOMETRY. 1. (a) A, B, C are three collinear points, and P is any point in the plane. Prove AB.CP+BC.AP2+CA.BP2-AB.BC.CA. (b) From (a) deduce a theorem of elementary geometry for the triangle PAC, when (i) PB is perpendicular to AC, (ii) B bisects AC. 2. Obtain a test for the concurrence of lines drawn perpendicular to the sides of a triangle; and apply it to prove the altitudes of a triangle concurrent. 3. (a) P and Q are the two limiting points of a limiting. point system of circles, and the common points of a common point system. Find the inverse of the doubly infinite system when (i) the centre of inversion is a point other than P or Q, (ii) P is the centre of inversion. (b) The polars of a given point with respect to all the circles of either system in (a) are concurrent. 4. X, Y, Z are collinear points on the sides of a triangle ABC. Show that the circles on AX, BY, CZ as diameters are coaxal. 5. Prove that the diameters and medians of a tetrahedron are concurrent, and find the ratio in which each is divided at the point of concurrence. 6. Find expressions for the lengths of the diagonals of a parallelepiped in terms of the edges and angles at a corner. 7. (a) Find a point P from which tangents to four spheres are equal. (b). How are the spheres of (a) situated when (i) the point P is at infinity, (ii) the point P may vary along a straight line? 8. (a) Apply the prismoidal formula to find the volume of (i) a pyramid, (ii) a zone of a sphere. (b) Assuming that the prismoidal formula holds for a sphere, show that it holds for a spheroid. TRIGONOMETRY. PRELIMINARY HONOURS. 1. In the following equation, assuming that the angles involved in the inverse functions lie between +90° and -90°, find the approximate value of 0 in degrees. 02 cot20(1+sin20)-cot-13+tan-11 =cosec1√5+(4+02) cos20. 2. (a) In what quadrant and between what limits does A lie in order that, 2 sin A=√1+sin 2 A-v1-sin 2 A. (b) Prove that, cos (y-2)+cos(≈—x)+cos(x−y)+1 3. A, B, C and D are four points in a straight line running east and west. B and C lie on the shores of a lake, the distance BC being its width. P is a point to the south so situated that APB=25°; [ BPC=17°, and ▲ CPD=18°. If AB=2 miles and CD=1.5 miles, find the width of the lake. 4. Develop an expression adapted to logarithms for finding a side of a ▲, the angle opposite and the sides including it being given. 5. The sides of a A are 48, 84 and 79. If the circumradius is R and the radius of the inscribed circle of its pedal A is p, find the ratio of R to p. (Use logarithms throughout). |