Throughout this question the use of a calculator is not permitted. The complex number with modulus 1 and argument 1/3π is denoted by w. The complex number 1 + 2i is denoted by u. The complex number v is such that |v| = 2|u| and arg v = arg u + 1/3π.

1. (i)Express w in the form x + iy, where x and y are real and exact.[1]
2. (ii)Sketch an Argand diagram showing the points representing u and v.[2]
3. (iii)Explain why v can be expressed as 2uw. Hence find v, giving your answer in the form a + ib, where a and b are real and exact.[4]

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