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2000_Paper III

# 2000_Paper III - STEP III 2000 2 Section A Pure Mathematics...

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Unformatted text preview: STEP III, 2000 2 Section A: Pure Mathematics 1 Sketch on the same axes the two curves C 1 and C 2 , given by C 1 : xy = 1 , C 2 : x 2- y 2 = 2 . The curves intersect at P and Q . Given that the coordinates of P are ( a, b ) (which you need not evaluate), write down the coordinates of Q in terms of a and b . The tangent to C 1 through P meets the tangent to C 2 through Q at the point M , and the tangent to C 2 through P meets the tangent to C 1 through Q at N . Show that the coordinates of M are (- b, a ) and write down the coordinates of N . Show that P MQN is a square. 2 Use the substitution x = 2- cos θ to evaluate the integral Z 2 3 / 2 x- 1 3- x ¶ 1 2 d x. Show that, for a < b , Z q p x- a b- x ¶ 1 2 d x = ( b- a )( π + 3 √ 3- 6) 12 , where p = (3 a + b ) / 4 and q = ( a + b ) / 2. 3 Given that α = e i π 3 , prove that 1 + α 2 = α . A triangle in the Argand plane has vertices A , B , and C represented by the complex numbers p , qα 2 and- rα respectively, where p , q and r are positive real numbers. Sketch the triangle ABC . Three equilateral triangles ABL , BCM and CAN (each lettered clockwise) are erected on sides AB , BC and CA respectively. Show that the complex number representing N is (1- α ) p- α 2 r and find similar expressions for the complex numbers representing L and M . Show that lines LC , MA and NB all meet at the origin, and that these three line segments have the common length p + q + r . STEP III, 2000 3 4 The function f ( x ) is defined by f ( x ) = x ( x- 2)( x- a ) x 2- 1 . Prove algebraically that the line y = x + c intersects the curve y = f ( x ) if | a | > 1, but there are values of c for which there are no points of intersection if | a | < 1....
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2000_Paper III - STEP III 2000 2 Section A Pure Mathematics...

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