2001_Paper II - STEP II 2001 2 Section A 1 Use the binomial...

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STEP II, 2001 2 Section A: Pure Mathematics 1 Use the binomial expansion to obtain a polynomial of degree 2 which is a good approximation to (1 - x ) when x is small. (i) By taking x = 1 / 100, show that (11) 79599 / 24000, and estimate, correct to 1 significant figure, the error in this approximation. (You may assume that the error is given approximately by the first neglected term in the binomial expansion.) (ii) Find a rational number which approximates (1111) with an error of about 2 × 10 - 12 . 2 Sketch the graph of the function [ x/N ], for 0 < x < 2 N , where the notation [ y ] means the integer part of y . (Thus [2 . 9] = 2, [4] = 4.) (i) Prove that 2 N k =1 ( - 1) [ k/N ] k = 2 N - N 2 . (ii) Let S N = 2 N k =1 ( - 1) [ k/N ] 2 - k . Find S N in terms of N and determine the limit of S N as N → ∞ . 3 The cuboid ABCDEFGH is such AE , BF , CG , DH are perpendicular to the opposite faces ABCD and EFGH , and AB = 2 , BC = 1 , AE = λ . Show that if α is the acute angle between the diagonals AG and BH then cos α = 3 - λ 2 5 + λ 2 Let R be the ratio of the volume of the cuboid to its surface area. Show that R < 1 / 3 for all possible values of λ . Prove that, if R 1 / 4, then α arccos(1 / 9).
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STEP II, 2001 3 4 Let f( x ) = P sin x + Q sin 2 x + R sin 3 x . Show that if Q 2 < 4 R ( P - R ), then the only values of x for which f( x ) = 0 are given by x = , where m is an integer. [You may assume that sin 3 x = sin x (4 cos 2 x - 1).] Now let g( x ) = sin 2 nx + sin 4 nx - sin 6 nx, where n is a positive integer and 0 < x < π/ 2. Find an expression for the largest root of the equation g( x ) = 0, distinguishing between the cases where n is even and n is odd. 5 The curve C 1 passes through the origin in the x y plane and its gradient is given by d y d x = x (1 - x 2 ) e - x 2 . Show that C 1 has a minimum point at the origin and a maximum point at ( 1 , 1 2 e - 1 ) . Find the coordinates of the other stationary point. Give a rough sketch of C 1 .
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