Find the two middle terms in the expansion of (x + y) 2m+1. 9. Define the sine, cosine, and tangent of an angle of any magnitude, and prove that sin (— A) = - sin A, 11. Shew how to solve a triangle, having given two sides and the included angle. If a = 65, b = 16, C′ = 60°, find the other angles, having given log 34771213, L tan 46° 20′ = 10·0202203, log 78450980, L tan 46° 21' 10.0204731. 12. Shew how to find the distance between two inaccessible points. From a point on a hill of uniform slope the angle of elevation of a tower on the summit is a, and from a point a feet nearer to the tower the angle of elevation is 6. If y be the inclination of the hill to the horizontal, shew that the height of the tower is a sin (ay) sin (6 − y) sec y cosec (6 — a). PURE MATHEMATICS.-PART II. The Board of Examiners. 1. Find the polar equation of a straight line. Shew that the polar equation of the straight line which passes through the points whose polar coordinates are a cos a, a; a cos ß, ß is 2. Find the equation of the tangent at any point of the circle x2 + y2+2gx + 2fy + c = 0. Hence or otherwise shew that the circle x2 + y2+2g'x + 2f'y + c = 0 will cut the above circle at right angles if 3. Find the locus of the middle points of a system of parallel chords of a parabola. If a chord of a parabola pass through a fixed point its middle point traces out a second parabola whose latus rectum is half of that of the first parabola. 4. If a chord of an ellipse be parallel to a fixed line, shew that the sum of the eccentric angles of the extremities of the chord is constant. Hence or otherwise shew that the locus of the middle points of a set of parallel chords is a line through the centre. 5. State and prove the rule for finding the differential coefficient of a function of a function. Find the differential coefficients of tan log z, log tan x, (log )tanz. 6. Enunciate and prove Leibnitz's theorem. Find the n differential coefficient of (ax+2bx + c) sin pr sin qx. 7. Assuming that f(x) can be expanded in a series of positive integral powers of x, shew that the expansion must be x2 x3 ƒ (x) =ƒ (0) + xƒ'(0) + =__ƒ′′ (0) + ««ƒ”(0) + ...••• Expand tan x in ascending powers of x as far as the term in x7. 8. Investigate a rule for finding maxima and minima values of a function of one independent variable. Find the maxima and minima values of 9. State and prove the rule for integration by parts. Integrate the expressions x1×a, x2 sin x, eax cos(bx + c). 10. Shew how to find the partial fraction corresponding to a single factor of the second degree in the decomposition of a rational fraction. Integrate x2 + a2 when f(x) has the values sina, cosa, sin3x, cos2x. 12. Find a formula for the volume of a solid of revolution. The curve am+n-4y=x(2a - x)" revolves about the axis of x; find the volume generated, m, n being odd positive integers. PURE MATHEMATICS.-PART III. The Board of Examiners. 1. If u be a function of x, y, z where x, y, z are functions of t, state and prove the rule for finding the differential coefficient of u with respect to t. If find (x, y, t) = 0, (x, y, t) = 0 d2y d2x 2. Determine the conditions for a maximum value of a function of two independent variables. Discuss the case in which the conditions are satisfied by a single relation between x, y. and apply it to find the radius of curvature at any point of an ellipse. 4. Shew how to find the envelop of a plane curve whose equation contains three parameters connected by two equations. Find the envelop of a circle whose centre lies on an ellipse, and whose radius is equal to the distance of the centre of the circle from an axis of the ellipse. |