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Unformatted text preview: STEP I, 2000 2 Section A: Pure Mathematics 1 To nine decimal places, log 10 2 = 0 . 301029996 and log 10 3 = 0 . 477121255. (i) Calculate log 10 5 and log 10 6 to three decimal places. By taking logs, or otherwise, show that 5 × 10 47 < 3 100 < 6 × 10 47 . Hence write down the first digit of 3 100 . (ii) Find the first digit of each of the following numbers: 2 1000 ; 2 10 000 ; and 2 100 000 . 2 Show that the coefficient of x 12 in the expansion of x 4 1 x 2 ¶ 5 x 1 x ¶ 6 is 15, and calculate the coefficient of x 2 . Hence, or otherwise, calculate the coefficients of x 4 and x 38 in the expansion of ( x 2 1) 11 ( x 4 + x 2 + 1) 5 . 3 For any number x , the largest integer less than or equal to x is denoted by [ x ]. For example, [3 . 7] = 3 and [4] = 4. Sketch the graph of y = [ x ] for 0 6 x < 5 and evaluate Z 5 [ x ] d x. Sketch the graph of y = [ e x ] for 0 6 x < ln n , where n is an integer, and show that Z ln n [ e x ] d x = n ln n ln( n !) . STEP I, 2000 3 (i) 4 Show that, for 0 6 x 6 1, the largest value of x 6 ( x 2 + 1) 4 is 1 16 . (ii) Find constants A , B , C and D such that, for all x , 1 ( x 2 + 1) 4 = d d x Ax 5 + Bx 3 + Cx ( x 2 + 1) 3 ¶ + Dx 6 ( x 2 + 1) 4 . (iii) Hence, or otherwise, prove that 11 24 6 Z 1 1 ( x 2 + 1) 4 d x 6 11 24 + 1 16 . 5 Arthur and Bertha stand at a point O on an inclined plane. The steepest line in the plane through O makes an angle θ with the horizontal. Arthur walks uphill at a steady pace in a straight line which makes an angle...
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 Spring '12
 rotar
 Math, Probability, Euclidean geometry

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