AS October 2020 Paper 1 Q7
7.
\[\mathrm{f}(z) = z^4 + az^3 + bz^2 + cz + d\]where \(a\), \(b\), \(c\) and \(d\) are real constants.
The equation \(\mathrm{f}(z) = 0\) has complex roots \(z_1\), \(z_2\), \(z_3\) and \(z_4\)
When plotted on an Argand diagram, the points representing \(z_1\), \(z_2\), \(z_3\) and \(z_4\) form the vertices of a square, with one vertex in each quadrant.
Given that \(z_1 = 2 + 3\mathrm{i}\), determine the values of \(a\), \(b\), \(c\) and \(d\). (6)
| Scheme | Marks | AO |
|---|---|---|
| \(z_2 = 2 - 3\mathrm{i}\) | B1 | 1.1b |
| \((z_3 =)\ p - 3\mathrm{i}\) and \((z_4 =)\ p + 3\mathrm{i}\) May be seen in an Argand diagram | M1 | 3.1a |
| \((z_3 =)\ {-4} - 3\mathrm{i}\) and \((z_4 =)\ {-4} + 3\mathrm{i}\) May be seen in an Argand diagram, but the complex numbers used in their method takes precedence | A1 | 1.1b |
| \[\left(z^2 - 4z + 13\right)\left(z^2 + 8z + 25\right)\]or\[\left(z - (2 - 3\mathrm{i})\right)\left(z - (2 + 3\mathrm{i})\right)\left(z - (-4 - 3\mathrm{i})\right)\left(z - (-4 + 3\mathrm{i})\right)\]or\[a = -\left[(2 - 3\mathrm{i})+(2 + 3\mathrm{i})+(-4 - 3\mathrm{i})+(-4 + 3\mathrm{i})\right]\]and\[\begin{aligned}b = {} &(2 - 3\mathrm{i})(2 + 3\mathrm{i})+(2 - 3\mathrm{i})(-4 - 3\mathrm{i})+(2 - 3\mathrm{i})(-4 + 3\mathrm{i})\\ &+(2 + 3\mathrm{i})(-4 - 3\mathrm{i})+(2 + 3\mathrm{i})(-4 + 3\mathrm{i})+(-4 - 3\mathrm{i})(-4 + 3\mathrm{i})\end{aligned}\]and\[c = -\left[\begin{aligned}&(2 - 3\mathrm{i})(2 + 3\mathrm{i})(-4 - 3\mathrm{i})+(2 - 3\mathrm{i})(2 + 3\mathrm{i})(-4 + 3\mathrm{i})\\ &+(2 - 3\mathrm{i})(-4 - 3\mathrm{i})(-4 + 3\mathrm{i})+(2 + 3\mathrm{i})(-4 - 3\mathrm{i})(-4 + 3\mathrm{i})\end{aligned}\right]\]and\[d = (2 - 3\mathrm{i})(2 + 3\mathrm{i})(-4 - 3\mathrm{i})(-4 + 3\mathrm{i})\]or Substitutes in one root from each conjugate pair and equates real and imaginary parts and solves simultaneously\[\begin{aligned}&(2 \pm 3\mathrm{i})^4 + a(2 \pm 3\mathrm{i})^3 + b(2 \pm 3\mathrm{i})^2 + c(2 \pm 3\mathrm{i}) + d = 0\\ &(-4 \pm 3\mathrm{i})^4 + a(-4 \pm 3\mathrm{i})^3 + b(-4 \pm 3\mathrm{i})^2 + c(-4 \pm 3\mathrm{i}) + d = 0\end{aligned}\] | dM1 | 3.1a |
| \(a = 4, b = 6, c = 4, d = 325\) \(\mathrm{f}(z) = z^4 + 4z^3 + 6z^2 + 4z + 325\) | A1 A1 | 1.1b 1.1b |
| (6) | ||
| (6 marks) |
Notes
B1: Seen \(2 - 3\mathrm{i}\)
M1: Finds the third and fourth roots of the form \(p \pm 3\mathrm{i}\). May be seen in an Argand diagram
A1: Third and fourth roots are \(-4 \pm 3\mathrm{i}\). May be seen in an Argand diagram
dM1: Uses an appropriate method to find \(\mathrm{f}(z)\). If using roots of a polynomial at least 3 coefficients must be attempted.
A1: At least two of \(a\), \(b\), \(c\), \(d\) correct
A1: All \(a\), \(b\), \(c\) and \(d\) correct
Note: Using roots \(2 \pm 3\mathrm{i}\) and \(-2 \pm 3\mathrm{i}\) leads to \(z^4 + 10z^2 + 169\) Maximum score B1 M1 A0 M1 A0 A0