2002_Paper I - STEP I, 2002 Section A: 1 2 Pure Mathematics...

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STEP I, 2002 2 Section A: Pure Mathematics 1 Show that the equation of any circle passing through the points of intersection of the ellipse ( x + 2) 2 + 2 y 2 = 18 and the ellipse 9( x - 1) 2 + 16 y 2 = 25 can be written in the form x 2 - 2 ax + y 2 = 5 - 4 a . 2 Let f ( x ) = x m ( x - 1) n , where m and n are both integers greater than 1. Show that the curve y = f ( x ) has a stationary point with 0 < x < 1. By considering f 00 ( x ), show that this stationary point is a maximum if n is even and a minimum if n is odd. Sketch the graphs of f ( x ) in the four cases that arise according to the values of m and n . 3 Show that ( a + b ) 2 6 2 a 2 + 2 b 2 . Find the stationary points on the curve y = ( a 2 cos 2 θ + b 2 sin 2 θ ) 1 2 + ( a 2 sin 2 θ + b 2 cos 2 θ ) 1 2 , where a and b are constants. State, with brief reasons, which points are maxima and which are minima. Hence prove that | a | + | b | 6 ( a 2 cos 2 θ + b 2 sin 2 θ ) 1 2 + ( a 2 sin 2 θ + b 2 cos 2 θ ) 1 2 6 (2 a 2 + 2 b 2 ) 1 2 . 4 Give a sketch of the curve y = 1 1 + x 2 , for x > 0. Find the equation of the line that intersects the curve at x = 0 and is tangent to the curve at some point with x > 0 . Prove that there are no further intersections between the line and the curve. Draw the line on your sketch. By considering the area under the curve for 0 6 x 6 1, show that π > 3 . Show also, by considering the volume formed by rotating the curve about the y axis, that ln 2 > 2 / 3 . [ Note : Z 1 0 1 1 + x 2 d x = π 4 . ]
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STEP I, 2002 3 5 Let f ( x ) = x n + a 1 x n - 1 + ··· + a n , where a 1 , a 2 , . . . , a n are given numbers. It is given that f ( x ) can be written in the form f ( x ) = ( x + k 1 )( x + k 2 ) ··· ( x + k n ) . By considering f (0), or otherwise, show that k 1 k 2 . . . k n = a n . Show also that
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2002_Paper I - STEP I, 2002 Section A: 1 2 Pure Mathematics...

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