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Date November 2011 Marks available 6 Reference code 11N.1.hl.TZ0.2
Level HL only Paper 1 Time zone TZ0
Command term Find Question number 2 Adapted from N/A

Question

Find the cube roots of i in the form a+bi, where a, bR.

Markscheme

i=cosπ2+isinπ2     (A1)

z1=i13=(cosπ2+isinπ2)13=cosπ6+isinπ6(=32+12i)     M1A1

z2=cos5π6+isin5π6(=32+12i)     (M1)A1

z3=cos(π2)+isin(π2)=i     A1

Note: Accept exponential and cis forms for intermediate results, but not the final roots.

 

Note: Accept the method based on expanding (a+b)3. M1 for attempt, M1 for equating real and imaginary parts, A1 for finding a = 0 and b=12, then A1A1A1 for the roots.

 

[6 marks]

Examiners report

A varied response. Many knew how to solve this standard question in the most efficient way. A few candidates expanded (a+ib)3 and solved the resulting fairly simple equations. A disappointing minority of candidates did not know how to start.

Syllabus sections

Topic 1 - Core: Algebra » 1.7 » Powers of complex numbers: de Moivre’s theorem.
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