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Wellington Girlsβ College Mathematics Department Achievement Standard 90638 (version 2) Manipulate real and complex numbers, and solve equations May 2010 QUESTION ONE (a) Solve πππ β ππ β π = π. Express your solution in simplified surd form. (b) Solve the equation π 3π₯β2 = 4 π 4 (c) Write (3πππ ( 6 )) as a complex number in the form a+bi. (d) Solve (e) π§ = 2π is a solution to x 3 ο« x2 ο« kx ο« 4 ο½ 0 , where k is a real number. Find the value of k. (f) π§ = π₯ + ππ¦ is any non zero complex number. If π§ + π§ = π, with k real, prove that either π¦ = 0 or π₯ 2 + π¦ 2 = 1. x ο« 3 ο½ 3x . 1 QUESTION TWO 4 3 2 1 u -3 -2 -1 0 1 2 3 4 Re -1 -2 v -3 (a) Write u in polar form, rcisο± . (b) Write uv in rectangular form, a ο« bi . (c) Solve log x ο 2 27 ο½ 3 . (d) Find all the solutions to π€ 3 = 8, where w is a complex number. (e) Hence or otherwise solve (π§ 2 β 2)3 = 8, where z is a complex number.