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, b∈R.
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.