| Date | May 2016 | Marks available | 6 | Reference code | 16M.2.hl.TZ1.9 |
| Level | HL only | Paper | 2 | Time zone | TZ1 |
| Command term | Find | Question number | 9 | Adapted from | N/A |
Question
Two distinct roots for the equation \({z^4} - 10{z^3} + a{z^2} + bz + 50 = 0\) are \(c + {\text{i}}\) and \(2 + {\text{i}}d\) where \(a,{\text{ }}b,{\text{ }}c,{\text{ }}d \in \mathbb{R},{\text{ }}d > 0\).
Write down the other two roots in terms of \(c\) and \(d\).
Find the value of \(c\) and the value of \(d\).
Markscheme
other two roots are \(c - {\text{i}}\) and \(2 - {\text{i}}d\) A1
[1 mark]
METHOD 1
use of sum of roots (M1)
\(2c + 4 = 10\)
\(c = 3\) A1
use of product of roots M1
product is \((c + {\text{i}})(c - {\text{i}})(2 + {\text{i}}d)(2 - {\text{i}}d)\) A1
\(({c^2} + 1)(4 + {d^2})\left[ { = 10(4 + {d^2})} \right] = 50\) A1
Note: The line above can be awarded if they have used their value of \(c\).
\(d = 1\) A1
METHOD 2
\({z^4} - 10{z^3} + a{z^2} + bz + 50 = ({z^2} - 2cz + {c^2} + 1)({z^2} - 4z + 4 + {d^2})\) M1A1
compare constant terms or coefficients of \({z^3}\) (M1)
\(4 + 2c = 10\)
\(({c^2} + 1)(4 + {d^2}) = 50\) A1
\(c = 3,{\text{ }}d = 1\) A1A1
[6 marks]
Examiners report
Most students using the sum and product of roots were able to work this problem through. There were many candidates who were attempting to multiply out, with varying degrees of success.
Most students using the sum and product of roots were able to work this problem through. There were many candidates who were attempting to multiply out, with varying degrees of success.
