2003_Paper III - STEP III, 2003 2 Section A: 1 Pure...

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STEP III, 2003 2 Section A: Pure Mathematics 1 Given that x + a > 0 and x + b > 0 , and that b > a , show that d d x arcsin ± x + a x + b ² = b - a ( x + b ) a + b + 2 x and find d d x arcosh ± x + b x + a ² . Hence, or otherwise, integrate, for x > - 1 , (i) Z 1 ( x + 1) x + 3 d x , (ii) Z 1 ( x + 3) x + 1 d x . [You may use the results d d x arcsin x = 1 1 - x 2 and d d x arcosh x = 1 x 2 - 1 . ] 2 Show that 2 r C r = 1 × 3 × ··· × (2 r - 1) r ! × 2 r , for r > 1 . (i) Give the first four terms of the binomial series for (1 - p ) - 1 2 . By choosing a suitable value for p in this series, or otherwise, show that X r =0 2 r C r 8 r = 2 . (ii) Show that X r =0 (2 r + 1) 2 r C r 5 r = ( 5 ) 3 . [ Note: n C r is an alternative notation for ± n r ² for r > 1 , and 0 C 0 = 1 .]
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STEP III, 2003 3 3 If m is a positive integer, show that (1 + x ) m + (1 - x ) m 6 = 0 for any real x . The function f is defined by f ( x ) = (1 + x ) m - (1 - x ) m (1 + x ) m + (1 - x ) m . Find and simplify an expression for f 0 ( x ). In the case m = 5 , sketch the curves y = f ( x ) and y = 1 f ( x ) . 4 A curve is defined parametrically by x = t 2 , y = t (1 + t 2 ) . The tangent at the point with parameter t , where t 6 = 0 , meets the curve again at the point with parameter T , where T 6 = t . Show that T = 1 - t 2 2 t and 3 t 2 6 = 1 . Given a point P 0 on the curve, with parameter t 0 , a sequence of points P 0 , P 1 , P 2 , . . . on the curve is constructed such that the tangent at P i meets the curve again at P i +1 . If t 0 = tan 7 π 18 , show that P 3 = P 0 but P 1 6 = P 0 . Find a second value of t 0 , with t 0 > 0 , for which P 3 = P 0 but P 1 6 = P 0 . 5 Find the coordinates of the turning point on the curve y = x 2 - 2 bx + c . Sketch the curve in the case that the equation x 2 - 2 bx + c = 0 has two distinct real roots. Use your sketch to determine necessary and sufficient conditions on b and c for the equation x 2 - 2 bx + c = 0 to have two distinct real roots. Determine necessary and sufficient conditions on b and c for this equation to have two distinct positive roots.
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2003_Paper III - STEP III, 2003 2 Section A: 1 Pure...

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