Challenging +1.2 This question requires understanding that complex roots come in conjugate pairs for real polynomials, and recognizing that a square's vertices relate by 90° rotations (multiplication by ±i). While it involves multiple concepts (conjugate roots, geometric transformations, Vieta's formulas), the structure is heavily scaffolded: given one vertex and told it's a square with one vertex per quadrant makes finding the other roots straightforward. The algebra to find coefficients is routine expansion. More challenging than basic complex arithmetic but less demanding than unscaffolded geometric problems.
7.
$$f ( z ) = z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$$
where \(a\), \(b\), \(c\) and \(d\) are real constants.
The equation \(\mathrm { f } ( \mathrm { z } ) = 0\) has complex roots \(\mathrm { z } _ { 1 } , \mathrm { z } _ { 2 } , \mathrm { z } _ { 3 }\) and \(\mathrm { z } _ { 4 }\) 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 \(z _ { 1 } = 2 + 3 i\), determine the values of \(a , b , c\) and \(d\).
\((z_3=)\ p-3\text{i}\) and \((z_4=)\ p+3\text{i}\); may be seen in Argand diagram
M1
3.1a — Finds third and fourth roots of form \(p\pm3\text{i}\)
\((z_3=)-4-3\text{i}\) and \((z_4=)-4+3\text{i}\); may be seen in Argand diagram
A1
1.1b — Third and fourth roots are \(-4\pm3\text{i}\)
Uses appropriate method to find \(f(z)\): e.g. \((z^2-4z+13)(z^2+8z+25)\), or expanding from all four linear factors, or using Vieta's formulas. If using roots, at least 3 coefficients must be attempted
dM1
3.1a
\(a=4,\ b=6,\ c=4,\ d=325\)
A1
1.1b — At least two of \(a,b,c,d\) correct
\(f(z)=z^4+4z^3+6z^2+4z+325\)
A1
1.1b — All of \(a,b,c,d\) correct
Note: Using roots \(2\pm3\text{i}\) and \(-2\pm3\text{i}\) leads to \(z^4+10z^2+169\); maximum score B1 M1 A0 M1 A0 A0
## Question 7:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $z_2=2-3\text{i}$ | B1 | 1.1b |
| $(z_3=)\ p-3\text{i}$ and $(z_4=)\ p+3\text{i}$; may be seen in Argand diagram | M1 | 3.1a — Finds third and fourth roots of form $p\pm3\text{i}$ |
| $(z_3=)-4-3\text{i}$ and $(z_4=)-4+3\text{i}$; may be seen in Argand diagram | A1 | 1.1b — Third and fourth roots are $-4\pm3\text{i}$ |
| Uses appropriate method to find $f(z)$: e.g. $(z^2-4z+13)(z^2+8z+25)$, or expanding from all four linear factors, or using Vieta's formulas. If using roots, at least 3 coefficients must be attempted | dM1 | 3.1a |
| $a=4,\ b=6,\ c=4,\ d=325$ | A1 | 1.1b — At least two of $a,b,c,d$ correct |
| $f(z)=z^4+4z^3+6z^2+4z+325$ | A1 | 1.1b — All of $a,b,c,d$ correct |
**Note:** Using roots $2\pm3\text{i}$ and $-2\pm3\text{i}$ leads to $z^4+10z^2+169$; maximum score **B1 M1 A0 M1 A0 A0**
7.
$$f ( z ) = z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$$
where $a$, $b$, $c$ and $d$ are real constants.\\
The equation $\mathrm { f } ( \mathrm { z } ) = 0$ has complex roots $\mathrm { z } _ { 1 } , \mathrm { z } _ { 2 } , \mathrm { z } _ { 3 }$ and $\mathrm { z } _ { 4 }$ 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 $z _ { 1 } = 2 + 3 i$, determine the values of $a , b , c$ and $d$.
\hfill \mbox{\textit{Edexcel CP AS 2020 Q7 [6]}}