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Hello, I am currently sitting in the British library. I have never seen it so quiet here, and there is a note on the desks from the Chief Executive explaining how the library is having to cope with massive budget cuts. Anyway, the reason I came was that many of the past papers that I am looking for are in fact available here. I have in front of me now bound sets of Oxford & Cambridge Higher Certificate Mathematical Papers for the years 1918-27, 1928-32, 1933-1937, 1938-42 and 1943-49. This is the first paper in the first book (I have tried my best to transcribe the mathematical expressions in a way that is understandable): ============================================ Mathematics, Group III (Paper 1) and Subsidiary Subject (15a) Arithmetic, Algebra and Trigonometry. 1918. 2 1/2 Hours. Not more than NINE questions should be attempted by any Candidate. The easier Questions A, B, C, D, E should be attempted only by Candidates who offer SUBSIDIARY SUBJECT (15a), and must not be attempted by those who offer GROUP III as their Principal Subject. 1. Using the tables of logarithms calculate the values of (1) 10^(-3/5) (2) (3/5)^(-10) Which of the two results is the greater? Can you give any general reason for anticipating which is the greater? 2. If q^2 is approximately equal to p, so that p-q^2 is small, verify that the percentage error in taking q to represent the square root of p is nearly equal to 50(p - q^2) / p Hence, or by direct calculation, estimate the percentage error in taking 63/19 as the square root of 11. 3. A boy thinks of an odd number: he multiplies the number by 3 and divides by 2, finding that the quotient is even. He again multiplies the quotient by 3 and divides by 2; and states that his result is 175. Prove that he is wrong: and, assuming that his only error is in taking the final figure to be 5, find what was the original number. Test your result. 4. Verify that if ps=qr, the value of the fraction (p+qx)/(r+sx) does not depend on the value of x. Show further that, if ad=bc, the value of the fraction (bcd+cda+dab+abc+x(a+b)(c+d))/((a+c)(b+d)+x(a+b+c+d)) is independent of the value of x 5. A motor-boat can travel at 12 miles an hour in still water. It makes a trip on a river of 15 miles down-stream and returns against the current; show that, whatever may be the speed of the current, the trip takes longer than the boat would take over the same distance in still water. Find the speed of the current, if the extra time taken is 10 minutes. 6. Prove that 1.2+2.3+3.4+...+n(n+1) = n(n+1)(n+2) / 3 and that 1.3+3.5+5.7+...+(2n-1)(2n+1) = n(4n^2 + 6n - 1) / 3 7. If w = (sqrt(13) - 1) / 2, prove that w^2 + w = 3, (2 - w)(3 + w) = 3; and that (5 - w) / (2 - w) = 4 + w 8. By means of the binomial theorem, right down the expanded form of (1+x)^5; and prove that its value for x=1/20 is slightly less than 40/31. Deduce or prove in any way that War Savings Certificates, which may be bought for 15s. 6d., and are worth £1 at the end of 5 years, give a better investment than compound interest at 5 per cent. per annum. 9. Write down the series for log_e (1 +x) in ascending powers of x, stating the limits of x in order that the expansion may be valid. Prove that, if n>2, 1/2 log_10 ((n + 2)/(n-2)) - log_10 ((n + 1)/(n - 1)) = m (1/z + (1/3).(1/z^3) + (1/5).(1/z^2) + ...), where z=1/2 (n^3 - 3n) and m=log_10 (e). Verify this result from the tables when n=6, taking m to be 0.4343. 10. If t = tan x, verify that sin 2x = 2t/(1=t^2), cos 2x = (1-t^2)/(1+t^2) Prove that if sec 2x + tan 2x = v, then t = (v - 1)/(v + 1), and find expressions for sin 2x and cos 2x in terms of v. 11. The relation between y, the angle of refraction, and x, the angle of incidence, of a ray of light on a block of glass, is given by sin y = (2 sin x) / 3 Plot a graph of y as x varies from 0 to 90 degrees, taking an inch to represent 10 degrees. Find from the graph the value of the angle x, at which x - y = 30 degrees, and show from the form of the graph that x - y is greatest when x=90 degrees A. A man's salary is fixed at the rate of £15 a month; after deduction of income-tax, he receives £13. 13s. 9d. At what rate in the pound does he pay income-tax? B. Find the factors of 42x^2 + xy - 30y^2 Verify that 2x - a is a factor of 4x^3 - 3a^2 x + a^3, and obtain all the factors. C. Solve the equations: (1) 2/x + y/2 = 1 = 3/x + y/3 (2) x (y - 1) = 8, y (x - 1) = 9 D. Plot graphs of (1) y = x^2 (2) y = 30x from x = -2.5 to x = +2.5, using the same lines of reference and taking an inch to represent unity. Deduce the approximate values of the roots of the quadratic x^2 + x = 3. E. A tower AB is observed by a man C standing on the roof of a house 400 feet away (horizontal distance). The height of C above the level of B is 40 feet and the angle ACB is observed to be 20 degrees. Calculate the height of the tower to the nearest foot. Verify by means of a diagram drawn to the scale of one inch to 100 feet. ================================ Best wishes, Sabbir.
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