1999_Paper III

# 1999_Paper III - STEP III • Recent STEP papers together...

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Unformatted text preview: STEP III • Recent STEP papers, together with Hints and Solutions, can be obtained from OCR Publications, Mill Wharf, Mill Street, Birmingham, B6 4BU . • There is a useful booklet entitled Advanced Problems in Mathematics which is also available from OCR, at a modest price. • You can get individual help from the Mathematics Millenium Project on-line maths site http://www.nrich.maths.org.uk/. STEP III, 1999 2 Section A: Pure Mathematics 1 Consider the cubic equation x 3- px 2 + qx- r = 0 , where p = 0 and r = 0. (i) If the three roots can be written in the form ak- 1 , a and ak for some constants a and k , show that one root is q/p and that q 3- rp 3 = 0 . (ii) If r = q 3 /p 3 , show that q/p is a root and that the product of the other two roots is ( q/p ) 2 . Deduce that the roots are in geometric progression. (iii) Find a necessary and sufficient condition involving p , q and r for the roots to be in arithmetic progression. 2 (i) Let f ( x ) = (1 + x 2 ) e x . Show that f ( x ) > 0 and sketch the graph of f ( x ). Hence, or otherwise, show that the equation (1 + x 2 )e x = k, where k is a constant, has exactly one real root if k > 0 and no real roots if k 6 0. (ii) Determine the number of real roots of the equation (e x- 1)- k tan- 1 x = 0 in the cases (a) 0 < k 6 2 /π and (b) 2 /π < k < 1. 3 Justify, by means of a sketch, the formula lim n →∞ ( 1 n n X m =1 f (1 + m/n ) ) = Z 2 1 f ( x ) d x . Show that lim n →∞ ‰ 1 n + 1 + 1 n + 2 + ··· + 1 n + n = ln 2 . Evaluate lim n →∞ ‰ n n 2 + 1 + n n 2 + 4 + ··· + n n 2 + n 2 . STEP III, 1999 3 4 A polyhedron is a solid bounded by F plane faces, which meet in E edges and V vertices. Youvertices....
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1999_Paper III - STEP III • Recent STEP papers together...

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