# STEP II - Section A Pure Mathematics 1 Two curves have...

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Unformatted text preview: Section A: Pure Mathematics 1 Two curves have equations x 4 + y 4 = u and xy = v , where u and v are positive constants. State the equations of the lines of symmetry of each curve. The curves intersect at the distinct points A , B , C and D (taken anticlockwise from A ). The coordinates of A are ( α, β ), where α > β > 0. Write down, in terms of α and β , the coordinates of B , C and D . Show that the quadrilateral ABCD is a rectangle and find its area in terms of u and v only. Verify that, for the case u = 81 and v = 4, the area is 14. 2 The curve C has equation y = a sin( π e x ) , where a > 1. (i) Find the coordinates of the stationary points on C . (ii) Use the approximations e t ≈ 1 + t and sin t ≈ t (both valid for small values of t ) to show that y ≈ 1- πx ln a for small values of x . (iii) Sketch C . (iv) By approximating C by means of straight lines joining consecutive stationary points, show that the area between C and the x-axis between the k th and ( k + 1)th maxima is approximately ‡ a 2 + 1 2 a · ln ‡ 1 + ( k- 3 4 )- 1 · . 3 Prove that tan ( 1 4 π- 1 2 x ) ≡ sec x- tan x . ( * ) (i) Use ( * ) to find the value of tan 1 8 π . Hence show that tan 11 24 π = √ 3 + √ 2- 1 √ 3- √ 6 + 1 . (ii) Show that √ 3 + √ 2- 1 √ 3- √ 6 + 1 = 2 + √ 2 + √ 3 + √ 6 . (iii) Use ( * ) to show that tan 1 48 π = q 16 + 10 √ 2 + 8 √ 3 + 6 √ 6- 2- √ 2- √ 3- √ 6 . 4 The polynomial p( x ) is of degree 9 and p( x )- 1 is exactly divisible by ( x- 1) 5 ....
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STEP II - Section A Pure Mathematics 1 Two curves have...

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