い 8. The equations of the sides of a triangle are Trace the curve and find the magnitude of its para meters. 10. Find the condition that the straight lines may intersect, whose equations are Determine the conditions necessary in order that the planes 1 x + n'y + m1z — o n1x + my + 11z = 0 m1x + l1y + n z = o may have a common line of intersection, and shew that the equations of that line are x (1 71 m1 n1) = y (m m1 l1 n1) = z (n n1 two straight lines may be drawn wholly coinciding with the surface. TRIGONOMETRY AND THEORY OF EQUATIONS. Examiner.-J. SUTCLIFFE, M. A. 1. Determine the relations which must exist be tween a, b, c, d that ax" + by" may be divisible by cr † dy without remainder. If from the product of n consecutive numbers beginning with a, we subtract the product of n consecutive beginning with b, prove that the difference is divisible by a - b. 2. Assuming the binomial theorem for positive integral values of the index, and denoting the series 1 + mx + m (m-1) If p be a prime, prove that 1 + (p~2) + (p-2)2 + (p −2)3 +......+ (p-2)p-2 is a multiple of p, except when p = 3. 5. State the relations between the roots of an equation and its co-efficients. If the roots are unknown and the co-efficients known, since by these relations we have as many equations as unknown quantities, why can we not always find the roots from these equations. 2 Solve the equation - 13 x2+39 x 270 whose roots are in geometrical progression. 6. Prove that impossible roots enter rational equations by pairs. If ea-1 be a root of the equation 7. Prove that an odd number of the real roots of the equation f (x) = 0 lies between every adjacent two of the real roots of f (x) = 0. 8. Define the sine and cosine of an angle, and prove by geometrical figures that for all values of A sin (90 + A) =cos A and cos (90 + A) Trace the changes in sign of the expression 2 A) sin A. 9. Find the radii r1. ̃1⁄2. r、 of the inscribed circles of a triangle, and prove that the square of the perimeter of the triangle = 4 (", ", + r2 r3 + 71 13). r2 r 1 10. Assuming De Moivre's Theorem for positive integral indices, prove it for fractional indices, and apply the theorem to find the cube root of 11. Find the sum of the series cos 2a + cos 4a + cos 6a + ...... + cos 2ra and deduce its value when r is a multiple of . 12. In any spherical triangle prove that 1. where R is the radius of the small circle circumscribing the triangle. DIFFERENTIAL AND INTEGRAL CALCULUS. Examiner.-J. SUTCLIFFE, M. A. 1. Find the differential co-efficient with regard to r of the following functions. x ex , log (Sin x), e + x cos (Sin x) cos (sin x) 2. State and prove Maclaurin's theorem. Expand to the function (e + e-x)n. 3. Shew how by means of differentiation to determine the true value of a function which for a particular 4. Find the cone of greatest volume which has a given slant side. 5. Eliminate by differentiation the functions from z = x ƒ (y) + y F (x). 6. If y = f (x) and x = (r, 0) y = F (r, 0) shew dy; how to transform an expression involving a y, x an expression involving r, 0, iuto dr do 7. Shew how to find the points of inflection of a curve y = = ƒ (x): and apply the method to the curve (ay — x2)3 = a3· (x — a). 3 5 8. Find the expression for the radius of curvature in terms of p and r. When the angle between the perpendicular and radius vector is a maximum or minimum then that the radius of curvature 9. Trace the curve whose equation is p 3 (a* — *)* and sin 3æ. 1 + x of the figure included between the parabolas whose equations are y* 4 ax and x2 4 ay. 12. The radii of the ends of a frustum of a sphere are rir, and its height is h: prove that its volume = Th 2 6 (3r,' + 3r," 2 2 + h2). OPTICS AND ASTRONOMY. Examiner.-K. S. MACDONALD, M. A. 1. Prove that rays diverging from a point and incident nearly at right angles on a concave spherical surface, converge after reflection nearly to another point such that the sum of the reciprocals of the distances of the two points from the reflector is double of the reciprocal of the radius. |
