A2 June 2020 Paper 1 Q4
4 It is given that \(1 - 3\mathrm{i}\) is one root of the quartic equation
\[z^4 - 2z^3 + pz^2 + rz + 80 = 0\]where \(p\) and \(r\) are real numbers.
(a) Express \(z^4 - 2z^3 + pz^2 + rz + 80\) as the product of two quadratic factors with real coefficients. [4 marks]
(b) Find the value of \(p\) and the value of \(r\). [2 marks]
| Scheme | Marks | AO |
|---|---|---|
| Identifies \(1 + 3i\) as a second root of the quartic equation. | B1 | 1.1b |
| Uses a pair of conjugate roots to find a quadratic factor. | M1 | 1.1a |
| Finds one correct quadratic factor. | A1 | 1.1b |
| Correctly expresses the quartic as the product of two quadratic factors. | A1 | 1.1b |
Typical solution
\(z = 1 + 3\mathrm{i}\) is another root.
\(z^2 - 2z + 10\) is a factor.
\[(z^2 - 2z + 10)(z^2 + bz + 8) \equiv z^4 - 2z^3 + pz^2 + rz + 80\]Comparing \(z^3\)-terms gives \(b = 0\)
\(\therefore\) the quartic is
\[(z^2 - 2z + 10)(z^2 + 8)\]| Scheme | Marks | AO |
|---|---|---|
| Substitutes \(1 + 3i\) or \(1 - 3i\) or one of their roots from their factorisation in (a) into the quartic equation and compares Re and Im parts or compares the coefficients of \(z^2\) and \(z\) from the (possibly partial) expansion of their product of quadratics with the given quartic. | M1 | 1.1a |
| Finds the correct values of \(p = 18\) and \(r = -16\) | A1 | 1.1b |
| (6 marks) |