If A = 60°, and b = 2c, find B, C. 4. Shew how to solve a triangle having given two sides and the angle opposite one of them. If a 1, c = √2, A = 30°, find b, B, C. UPPER MATHEMATICS. Professor Nanson. Candidates must answer satisfactorily in each of the three divisions of this paper. I.-1. Find the locus of the middle points of a system of parallel chords of a parabola. 2. Prove that in the ellipse PN2 CA2 - CN2 – CB2 : CA2. 3. Prove that the rectangle contained by the distances of any point on a hyperbola from its two asymptotes is of constant magnitude. 4. Prove that in any conic SG: SP SA: AX. II.-1. Prove that an equation of the nth degree cannot have more than n roots. 2. Prove that an infinite series is convergent if from and after some fixed term the ratio of each term to the preceding is numerically less than some quantity which is itself numerically less than unity. 3. Enunciate and prove the exponential theorem. 4. State and prove the rule for forming successive convergents to a given continued fraction. 5. Define a recurring series, and find a formula for the sum of n terms of a recurring series of the second order. III.-1. State and prove De Moivre's theorem. 2. Prove the formula for the expansion of sin 0 in ascending powers of 0. 3. Express cos" in terms of cosines of multiples of 0. 4. Find the sum of the cosines of a series of angles in arithmetical progression. 5. Prove the rule of proportional differences in the case of the natural sine. the result to find the differential coefficient of x”. 2. Differentiate log cot x, tan-1 1 + cos x , log.x. cos 3. State and prove Leibnitz's theorem. Find the nth differential coefficient of (ax2 + bx + c) cos mx. 4. Shew how to find the value of an expression which assumes the indeterminate form 0/0. Find the values when x is zero of 5. State and prove a rule for finding maxima and minima values of a function of one variable. Find when sin (a — x) sin x sin (a + x) is a maximum or minimum. 6. Shew how to find the polar coordinates of a point. whose rectangular coordinates are given. Assuming that the square of the distance between the points whose rectangular coordinates 7. Find the perpendicular distance of the point x, y from the line x cos a + y sin a = = p. If a straight line be such that the sum of the perpendiculars upon it from a number of fixed points is zero, shew that it will pass through a fixed point. 8. Find the equation of the tangent at any point of a parabola. The two tangents from a point P to a parabola make angles 01, 02 with its axis. If tan 0, tan 02 be constant, find the locus of P. 9. Find the equation of the normal at any point of an ellipse, and shew that the normal at any point bisects the angle between the focal distances of that point. 10. Prove that the sum of the squares of conjugate diameters of an ellipse is constant. Find when the difference of the squares on conjugate diameters is greatest. 11. Define a definite integral and an indefinite integral, and state the relation between them. a Prove that f(x) dx = ["4 (a−x)dx. 12. State and prove the formula for integration by parts. Find the values of S√ a2 = x2 dx, S ecos (bx + c) dx. 13. Shew how to find the partial fraction corresponding to a single factor of the first degree when a rational fraction is resolved into partial fractions. Integrate (x2 + a2) (x2 + b2) (x2 + c2)' 14. Find the area of a loop of the curve a1y2 = x2 (a2 — x2). 15. Find the volume generated by the revolution about the axis of x of the loop of the curve in question 14. MIXED MATHEMATICS.-I. Professor Nanson. 1. Find the shortest distance between two straight lines whose equations are given. 2. Find the equations of the principal planes of a conicoid. 3. Shew that there are two systems of generating lines on a hyperboloid of one sheet. 4. Find the measure of torsion at any point of a |