 DM: Nice research Sabbir. I'm actually surprised at how straightforward those questions are. I would have expected them to be more challenging. The other thing that would cause me difficulty there is the pre-decimal money. Do they have any copies of the old Special papers there? I would imagine they would be considerably harder.
Just for reference, these were the papers offered in 1918: Paper 1: Arithmetic, Algebra and Trigonometry, 2 1/2 hours, [choose 9 questions from 11 (+5)] Paper 2: Pure Geometry and Trigonometry, 2 1/2 hours, [9 from 12 (+6)] Paper 3: Analytical Geometry, 1 1/2 hours [7 from 8 (+4)] Paper 4: Differential and Integral Calculus, 1 1/2 Hours [7 from 8 (+4)] Paper 5: Statics and Dynamics, 2 1/2 Hours [9 from 12 (+6)] Paper 6: Mathematical Distinction Paper, 2 Hours [9 questions] Paper 7: Mathematical Distinction Paper, 2 Hours [9 questions] Paper 8: Mathematical Distinction Paper, 2 Hours [9 questions] [The first distinction paper seems to be mainly algebra, the second geometry and the third calculus]. This is Paper 6,as per request: ====================== Group III (Paper 6) Mathematical Distinction Paper 1918. 2 Hours. 1. Determine the values of A, B, C so that 1 + Ax + Bx^3 + Cx^5 may be divisible by 1+x, the quotient being a perfect square. verify that then the equation 1 - (Ax + Bx^3 + Cx^5)^2 = (1 - x^2) (1 - 12x^2 + 16 x^4)^2 is identically true. [ It is supposed that the coefficients A, B, C are different from zero.] 2. Wriet down in terms of w the condition that the quadratic S + w S' = 0 should have equal roots in x, where S = px^2 + 2qx + r, S' = x^2 + c. Show that if the values of w which satisfy this condition are equal, and if all the coefficients are real, then (1) if c > 0, S = p S' ; (2) if c < 0, S and S' have a factor in common. What is trhe conclusion when c = 0? 3. If x, y are both positive and x^m + y^m = const., prove that x^n + y^n decreases as x increases if n > m, and x < y, but increases if n < m and x < y. Show that if x, y, z are positive and x^m + y^m + z^m = 3 c^m, then x^n + y^n + z^n > or < 3 c^n, according as n > or < m. [Both n and m are supposed positive]. 4. By considering the expansion of log {(1 - at)(1 - bt)(1 - ct)} in ascending powers of t, or otherwise, prove that if a + b + c = 0 and a^n + b^n + c^n = s_n, then s_4 = 1/2 (s_2)^2, s_5 = 5/6 s_2 s_3, s_7 = 7/6 s_3 s_4 5. From the vertices of a triangle ABC are drawn perpendiculars p, q, r to a straight line which is in the plane of ABC but passes entirely outside the triangle. Prove that a^2(p - q)(p - r) + b^2(q - p)(q - r) + c^2(r- p)(r-q) = 4 D^2 where D denotes the area of the triangle. 6. Prove that, if n is an integer, sin nx / sin x can be expressed as a polynomial of degree (n-1) in cos x, and that the coefficient of the term of highest degree is 2^(n-1). Prove that, if n is odd and equal to 2p+1, sin nx / sin x = n \prod_{r=1}^{r=p} {1 - (sin^2 x / sin^2 (r pi / n)}, and that \prod_{r=1}^{r=p} { 2 sin (r pi / n) } = sqrt(n) [Note. The remaining coefficients in the original polynomial need not be determined explicitly). 7. Two perpendicular lines CA, CB each of length L are drawn on level ground; borings are taken at each of the points A, B, C, and rock is found at depths a, b, c respectively. prove that, if the face of the rock is a plane, the inclination x of this plane to the horizontal is given by L^2 tan^2 x = (c - 1)^2 + (c - b)^2 Prove also that the volume of a triangular prism with vertical faces, bounded at the top and bottom by the plane ABC and the face of the rock, is 1/6 L^2 (a + b + c) 8. Explain how to solev an equation of the form ax^4 + bx^3 + cx^2 +bx + a = 0 by writing y = x + 1/x. Illustrate by means of the equation x^4 + x^3 +x^2 +x + 1 = 0. Hence or otherwise prove that if d is the side of the decagon (regular polygon of ten sides) and p that of the pentagon (regular polygon of five sides) inscribed in a circle of radius r, then d^2 + rd = r^2, p^2 = r^2 + d^2 9. Draw a rough graph of the curve y = x + 1 - 1/x + 1/(1-x) , and verify that it consists of three parts, along each of which y steadily increases with x. Verify that, for any real value of x, the equation in x, x + 1 - 1/x + 1/(1-x) = c, has three real roots, one negative, one between 0 and 1, and the thirds greater than 1. Prove also that if one root is p, the others are q = 1 - 1/p, r = 1/(1-p) =========================== I will see what I can do about making some of these papers available. I don't realistically think I will be able to spend the time to sit in the library Latex-ing them all (today is very much a one off as my car is currently being serviced!), so sorry for that. I hope that the above is sufficient to whet your appetite for the time being. Best wishes, Sabbir.
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