INTERMEDIATE HONOUR MATHEMATICS. ANALYTICAL GEOMETRY AND CALCULUS. (Omit one of the first four questions and one of the last two). 1. Graph (a) r2=a2 cos 20. (b) (x2+4x-12) y3-x2. 2. Show that two tangents and three normals, real or imaginary, can be drawn from any point to the parabola y=4ax. 3. What is meant by conjugate diameters of a conic? Find the condition that two diameters of an ellipse shall be conjugate. 4. If an ellipse and a hyperbola have common foci, they cut at right angles. 5. What conic does each of the following equations represent? Graph any one of them: (a) 9x2+24xy+16y2—52x+14y—6—0. 6. (a) Differentiate L tan x-x x=0 x - sin x (c) Find the point of inflection of the curve (y+1)3—r—2, and sketch the curve in the neighborhood of the point of inflection. 7. (a) Show that the subtangent of the curve y=ae/e is a constant. (b) Expand log(1+sin x) to four terms in powers of x. 9. Find the area between the curve y2 (4a2-—x2)=x2 and the line 2a. 10. Find the volume of the paraboloid of revolution generated by revolving the arc of the parabola y2-4r between the origin and 2 about its axis. FACULTY OF ARTS. MATHEMATICS. INTERMEDIATE HONOURS. Algebra II. 1. (a) Prove that if a>1; Σ is convergent, and hence prove the series Eu, is convergent if na Lt. n Sun >I. (b) Test the following for convergence, or divergence, (i) Σ n+ I I 2 3 4. (ii) (n+2)(n+3) 2 3 4 5 n is 2. (a) On what principle does the method of undetermined coefficients depend? Apply this method to find Σn3. (b) Find the general term and the sum of n terms of the series 4, 13, 35, 94, 262, 755. 3. (a) Prove the error of the convergent Pr < I In In In+1 and explain the application of this result. (b) Form the convergents intermediate to the 1st and. 3rd principle convergents of 207 779 4. (a) If p is prime and a is prime to p, show that ap-1 is a multiple of p. (b) If b is prime, shew that 2-3-1 is a multiple of p. 5. A die is thrown three times and the sum of the numbers thrown is 15. Find the chance that the first number thrown was 4. 6. (a) Explain briefly the relations existing between the frequency of occurrence of an error and its magnitude. To what class of error does this apply? (b) A quantity is measured 6 times and the results are 38.002, 38.013, 38.008, 38.006, 38.008, 38.005. Find the probable error of a single observation and of the mean. |