| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.8 This is a substantial multi-part question requiring: (a) substitution of a complex number into a cubic to find k, (b) using the conjugate root theorem and polynomial division to find remaining roots, (c) sketching a circle in the Argand diagram, and (d) finding maximum argument using geometric reasoning with tangent lines from the origin. While the individual techniques are standard for Further Maths Pure, the combination of algebraic manipulation with complex numbers, geometric interpretation, and optimization of argument makes this moderately challenging. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute \(x=-1+\sqrt{7}\,\mathbf{i}\) and attempt expansions of \(x^2\) and \(x^3\) | \*M1 | |
| Use \(\mathbf{i}^2=-1\) correctly at least once and solve for \(k\) | DM1 | \(2(20-4\sqrt{7}\,\mathbf{i})+3(-6-2\sqrt{7}\,\mathbf{i})+14(-1+\sqrt{7}\,\mathbf{i})+k=0\) |
| Obtain answer \(k=-8\) | A1 | SC B1 only for those showing no working for cube and square but obtaining \(k=-8\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempt division by \((x+1-\sqrt{7}\,\mathbf{i})\) as far as \(2x^2+z_1x+\ldots\) | \*M1 | See division on next page |
| Use \(\mathbf{i}^2=-1\) correctly at least once and obtain \(2x^2+z_1x+z_2+\) remainder | DM1 | |
| Obtain answer \(k=-8\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State answer \(-1 - \sqrt{7}\)i | B1 | Can be seen simply stated on its own, or in a list of roots. Allow if stated clearly in part 10(a). |
| Carry out a method for finding a quadratic factor with zeros \(-1 + \sqrt{7}\)i and \(-1 - \sqrt{7}\)i | M1 | Or state \(\left(x-\left(-1+\sqrt{7}\text{i}\right)\right)\left(x-\left(-1-\sqrt{7}\text{i}\right)\right)(2x-p)\) |
| Obtain \(x^2 + 2x + 8\) | A1 | Or obtain \(\left(-1+\sqrt{7}\text{i}\right)\left(-1-\sqrt{7}\text{i}\right)p = -8\); Or obtain \(\left(-1+\sqrt{7}\text{i}\right)+\left(-1-\sqrt{7}\text{i}\right)+\frac{p}{2} = -\frac{3}{2}\) |
| Obtain root \(x = \frac{1}{2}\), or equivalent, via division or inspection | A1 | Needs to follow from the working. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show a circle with centre \(-1 + \sqrt{7}\)i | B1 | (See diagram in mark scheme) |
| Show circle with radius 2 and centre not at the origin; there needs to be some evidence of scale e.g. radius marked or a scale on the axes | B1 | If the scales are very different from each other then B1 for centre in correct position and B1 for an ellipse. If there is more than one circle the max score is B1. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out a complete method for calculating the maximum value of arg \(z\) for correct circle | M1 | e.g. \(\frac{\pi}{2} + \tan^{-1}\frac{1}{\sqrt{7}} + \frac{\pi}{4}\). Can be implied by \(155.7°\). |
| Obtain answer \(2.72\) radians | A1 | CAO. The question requires radians. |
## Question 10(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $x=-1+\sqrt{7}\,\mathbf{i}$ and attempt expansions of $x^2$ and $x^3$ | \*M1 | |
| Use $\mathbf{i}^2=-1$ correctly at least once and solve for $k$ | DM1 | $2(20-4\sqrt{7}\,\mathbf{i})+3(-6-2\sqrt{7}\,\mathbf{i})+14(-1+\sqrt{7}\,\mathbf{i})+k=0$ |
| Obtain answer $k=-8$ | A1 | SC B1 only for those showing no working for cube and square but obtaining $k=-8$ |
**Alternative method for 10(a):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt division by $(x+1-\sqrt{7}\,\mathbf{i})$ as far as $2x^2+z_1x+\ldots$ | \*M1 | See division on next page |
| Use $\mathbf{i}^2=-1$ correctly at least once and obtain $2x^2+z_1x+z_2+$ remainder | DM1 | |
| Obtain answer $k=-8$ | A1 | |
**Total: 3 marks**
## Question 10(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State answer $-1 - \sqrt{7}$i | **B1** | Can be seen simply stated on its own, or in a list of roots. Allow if stated clearly in part 10(a). |
| Carry out a method for finding a quadratic factor with zeros $-1 + \sqrt{7}$i and $-1 - \sqrt{7}$i | **M1** | Or state $\left(x-\left(-1+\sqrt{7}\text{i}\right)\right)\left(x-\left(-1-\sqrt{7}\text{i}\right)\right)(2x-p)$ |
| Obtain $x^2 + 2x + 8$ | **A1** | Or obtain $\left(-1+\sqrt{7}\text{i}\right)\left(-1-\sqrt{7}\text{i}\right)p = -8$; Or obtain $\left(-1+\sqrt{7}\text{i}\right)+\left(-1-\sqrt{7}\text{i}\right)+\frac{p}{2} = -\frac{3}{2}$ |
| Obtain root $x = \frac{1}{2}$, or equivalent, via division or inspection | **A1** | Needs to follow from the working. |
---
## Question 10(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show a circle with centre $-1 + \sqrt{7}$i | **B1** | (See diagram in mark scheme) |
| Show circle with radius 2 and centre not at the origin; there needs to be some evidence of scale e.g. radius marked or a scale on the axes | **B1** | If the scales are very different from each other then B1 for centre in correct position and B1 for an ellipse. If there is more than one circle the max score is B1. |
---
## Question 10(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out a complete method for calculating the maximum value of arg $z$ for correct circle | **M1** | e.g. $\frac{\pi}{2} + \tan^{-1}\frac{1}{\sqrt{7}} + \frac{\pi}{4}$. Can be implied by $155.7°$. |
| Obtain answer $2.72$ radians | **A1** | CAO. The question requires radians. |10 The complex number $- 1 + \sqrt { 7 } \mathrm { i }$ is denoted by $u$. It is given that $u$ is a root of the equation
$$2 x ^ { 3 } + 3 x ^ { 2 } + 14 x + k = 0$$
where $k$ is a real constant.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $k$.
\item Find the other two roots of the equation.
\item On an Argand diagram, sketch the locus of points representing complex numbers $z$ satisfying the equation $| z - u | = 2$.
\item Determine the greatest value of $\arg z$ for points on this locus, giving your answer in radians.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q10 [11]}}