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ANALYTICAL GEOMETRY AND CALCULUS.
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=4r 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 y2-4ar at the point (x1, y1), (a) using the derivative, and (b) without using the derivative.
(b) Differentiate with respect to x,
tan n x
6. (a) Find the maximum and minimum points and points of inflection of
(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?
7. (a) Integrate (1)
(3) S4x3 + x2
(2) Sx√a2 −x2 dx.
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
(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.
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
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,
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 ▲ 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).