. LOWER MATHEMATICS. Professor Nanson. (Candidates must answer satisfactorily in each of the three divisions of this paper.) I.-1. Find the locus of a point equidistant from two given points. In a given circle find the points. equidistant from two given points. 2. Prove that two tangents, and two only, can be drawn to a circle from an external point. Shew that the tangents make equal angles with the line joining the points of contact. 3. Inscribe in or describe about a given circle regular figures of 3, 6, 12, 24 ... sides. Describe a regular hexagon having each of its sides equal to a given straight line. 4. If two triangles have their angles respectively equal they are similar, and those sides' which are opposite to the equal angles are homologous. Through a given point draw a straight line so that the perpendiculars upon it from two given points may be in a given ratio. II.1. State and prove the rule for finding the highest common factor of two compound expressions. Find the highest common factor of x3. + 3x2 X 3 and 4 + 4x3 12x 9. - 2. Shew how to solve two. simultaneous equations of the form ax2 + by2 + 2hxy + 2gx + 2fy + c = 0 Solve the equations x.2 y2 a2 mn √2. 3. Shew that (am)” — aTMn for all values of m and n. = 4. If A vary as B when C is constant, and A vary as C when B is constant, prove that when B, C both vary A varies as BC. A varies as B directly and as C inversely; and Aa when B = b and C = c ; find the I value of A when Bc and C = b. III.-1. Define the trigonometrical ratios of an angle, and express all the other ratios in terms of the cotangent. a •If tan 0= , and be in the third quad rant, a and b being both positive, find the values of sin 0 and cos 0. 2. If A, B be positive angles whose sum is less than a right angle, prove that cos (A + B) : cos A cos B sin A sin B. 3. Prove that in any triangle a2 = b2 + c2 2bc cos A. The sides of a triangle are 3, 5, 7; find the greatest angle. 4. Shew how to solve a triangle having given two sides and the included angle. ? If b c (√3-1) and A 30°, find B and C. UPPER MATHEMATICS. Professor Nanson. (Candidates must answer satisfactorily in each of the three divisions of this paper.) I.-1. Prove that in the parabola PN2 4AS. AN. = 2. Prove that the tangent at any point of an ellipse makes equal angles with the focal distances of the point. 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 the semi-latus rectum is a harmonic mean between the segments of any focal chord. II.-1. State and prove the relations between the roots and the coefficients of an equation of the nt degree. 2. Prove that an infinite series is convergent if from and after some fixed term the ratio of each term to the preceding term is numerically less than some quantity which is itself numeriIcally 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. State and prove the rule for finding the sum of In terms of the series whose nth term is the reciprocal of (an+b) (an+1+b). . . . (an + m −1 + b). III.-1. Shew how to find all the values of the expression (cos 0 + i sin 0)". 2. Prove the formula for the expansion of cos 0 in ascending powers of 0. 3. Prove that tan -1xx 205 + 4. Sum to n terms the series cos a + c cos (a + ß) + c2 cos (a+2B)+ 5. Prove the rule of proportional differences in the case of the natural sine. ADVANCED MATHEMATICS. Professor Nanson. 1. Find the limit when x is zero of log (1 + x), and apply the result to find the differential coefficient of log a. 2. Find the differential coefficients of cos log tan x, (sin x)cos, log sin tan æ. 3. State the conditions under which the equation f(x + h) = f(x) + hf' (x + Oh) is true, and assuming these conditions to hold, prove the equation. 4. Shew how to find the value of an expression which assumes the indeterminate form 1°. Find the values when x is zero of 1 (x cot x), (cos x) cot. 5. Shew how to find the maxima and minima values of a function of one independent variable. Find when sin x cos"x is a maximum or minimum, m and n being positive integers. |