6. Solve the equation: cos 30+cos 40+cos 50+cos 60 = 0. 7. Sum to n terms the series, sin2a+sin2 2a+sin2 3a+sin2 4a++.... 8. Eliminate from the equations, (a+b) tan (0-6)=(a—b) tan (0+6). 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 y2=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+16y-52x+14y-6=0. (b) 3x2-2xy+3y2+10x-14y-3-0. (c) 4xy+6x-8y+1=0. 6. (a) Differentiate (c) Find the point of inflection of the curve (y+1)=x-2, and sketch the curve in the neighborhood of the point of inflection. 7. (a) Show that the subtangent of the curve y=ae/o is a constant. of x. (b) Expand log(1+sin x) to four terms in powers. 8. Evaluate (c) S9 x2 + 3x2-x-3 9x2+12x+8 x2 (b) Sx2 log x dx, dx, (d) S sin3 0 cos3 @ dẹ. 9. Find the area between the curve y2(4a2-x2)=x2 and the line r=2a. 10. Find the volume of the paraboloid of revolution generated by revolving the arc of the parabola y2-4r between the origin and r=2 about its axis. (b) Test the following for convergence, or divergence, 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 and explain the application of this result. (b) Form the convergents intermediate to the 1st and 3rd principle convergents of 779 207 4. (a) If p is prime and a is prime to p, show that ap-1 is a multiple of p. (b) If bis 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. |