| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Given one complex root of cubic or quartic, find all roots |
| Difficulty | Standard +0.3 This is a straightforward application of the complex conjugate root theorem and Vieta's formulas. Given one complex root, students immediately write down its conjugate, then use sum/product of roots to find k and the real root. The arithmetic involves complex numbers but follows standard procedures with no novel insight required—slightly easier than average due to its routine nature. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02j Cubic/quartic equations: conjugate pairs and factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State answer \(-1-\sqrt{3}i\) | B1 | If \(-\frac{1}{2}\) given as well at this point, still just B1 |
| Total | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Substitute \(x = -1+\sqrt{3}i\) in the equation and attempt expansions of \(x^2\) and \(x^3\) | M1 | Need to see sufficient working to be convinced a calculator has not been used |
| Use \(i^2 = -1\) correctly at least once | M1 | Allow for relevant use at any point in the solution |
| Obtain \(k = 2\) | A1 | |
| Carry out a complete method for finding a quadratic factor with zeros \(-1+\sqrt{3}i\) and \(-1-\sqrt{3}i\) | M1 | Could use factor theorem. Need to see working. M1 for correct testing of correct root or allow M1 for three unsuccessful valid attempts |
| Obtain \(x^2 + 2x + 4\) | A1 | Using factor theorem, obtain \(f\!\left(-\frac{1}{2}\right) = 0\) |
| Obtain root \(x = -\frac{1}{2}\), or equivalent, via division or inspection | A1 | Final answer |
| Total | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Carry out a complete method for finding a quadratic factor with zeros \(-1+\sqrt{3}i\) and \(-1-\sqrt{3}i\) (multiplying two linear factors or using sum and product of roots) | M1 | Need to see sufficient working to be convinced a calculator has not been used |
| Use \(i^2 = -1\) correctly at least once | M1 | Allow for relevant use at any point in the solution |
| Obtain \(x^2 + 2x + 4\) | A1 | Allow M1A0 for \(x^2 + 2x + 3\) |
| Obtain linear factor \(kx+1\) and compare coefficients of \(x\) or \(x^2\) and solve for \(k\) | M1 | Can find the factor by inspection or by long division. Must get to zero remainder |
| Obtain \(k = 2\) | A1 | |
| Obtain root \(x = -\frac{1}{2}\) | A1 | Final answer |
| Note: Verification that \(x = -\frac{1}{2}\) is a root is worth no marks without a clear demonstration of how the root was obtained |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use equation for sum of roots of cubic and use equation for product of roots of cubic | M1 | |
| Use \(i^2 = -1\) correctly at least once | M1 | Allow for relevant use at any point in the solution |
| Obtain \(-\frac{5}{k} = -2 + \gamma\), \(\;-\frac{4}{k} = 4\gamma\) | A1 | |
| Solve simultaneous equations for \(k\) and \(\gamma\) | M1 | |
| Obtain \(k = 2\) | A1 | |
| Obtain root \(\gamma = -\frac{1}{2}\) | A1 | Final answer |
| Total | 6 |
## Question 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State answer $-1-\sqrt{3}i$ | B1 | If $-\frac{1}{2}$ given as well at this point, still just B1 |
| **Total** | **1** | |
---
## Question 5(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Substitute $x = -1+\sqrt{3}i$ in the equation and attempt expansions of $x^2$ and $x^3$ | M1 | Need to see sufficient working to be convinced a calculator has not been used |
| Use $i^2 = -1$ correctly at least once | M1 | Allow for relevant use at any point in the solution |
| Obtain $k = 2$ | A1 | |
| Carry out a complete method for finding a quadratic factor with zeros $-1+\sqrt{3}i$ and $-1-\sqrt{3}i$ | M1 | Could use factor theorem. Need to see working. M1 for correct testing of correct root or allow M1 for three unsuccessful valid attempts |
| Obtain $x^2 + 2x + 4$ | A1 | Using factor theorem, obtain $f\!\left(-\frac{1}{2}\right) = 0$ |
| Obtain root $x = -\frac{1}{2}$, or equivalent, via division or inspection | A1 | Final answer |
| **Total** | **6** | |
---
## Question 5(ii) Alternative Method 1:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Carry out a complete method for finding a quadratic factor with zeros $-1+\sqrt{3}i$ and $-1-\sqrt{3}i$ (multiplying two linear factors or using sum and product of roots) | M1 | Need to see sufficient working to be convinced a calculator has not been used |
| Use $i^2 = -1$ correctly at least once | M1 | Allow for relevant use at any point in the solution |
| Obtain $x^2 + 2x + 4$ | A1 | Allow M1A0 for $x^2 + 2x + 3$ |
| Obtain linear factor $kx+1$ and compare coefficients of $x$ or $x^2$ and solve for $k$ | M1 | Can find the factor by inspection or by long division. Must get to zero remainder |
| Obtain $k = 2$ | A1 | |
| Obtain root $x = -\frac{1}{2}$ | A1 | Final answer |
| | | Note: Verification that $x = -\frac{1}{2}$ is a root is worth no marks without a clear demonstration of how the root was obtained |
---
## Question 5(ii) Alternative Method 2:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use equation for sum of roots of cubic and use equation for product of roots of cubic | M1 | |
| Use $i^2 = -1$ correctly at least once | M1 | Allow for relevant use at any point in the solution |
| Obtain $-\frac{5}{k} = -2 + \gamma$, $\;-\frac{4}{k} = 4\gamma$ | A1 | |
| Solve simultaneous equations for $k$ and $\gamma$ | M1 | |
| Obtain $k = 2$ | A1 | |
| Obtain root $\gamma = -\frac{1}{2}$ | A1 | Final answer |
| **Total** | **6** | |
---5 Throughout this question the use of a calculator is not permitted.
It is given that the complex number $- 1 + ( \sqrt { } 3 ) \mathrm { i }$ is a root of the equation
$$k x ^ { 3 } + 5 x ^ { 2 } + 10 x + 4 = 0$$
where $k$ is a real constant.\\
(i) Write down another root of the equation.\\
(ii) Find the value of $k$ and the third root of the equation.\\
\hfill \mbox{\textit{CAIE P3 2019 Q5 [7]}}