| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.3 This is a standard Further Maths question on complex roots with real coefficients. It requires knowing that complex roots come in conjugate pairs and applying Vieta's formulas, but follows a predictable structure with clear signposting through multiple parts. The techniques are routine for FP1 students, though slightly above average A-level difficulty due to being Further Maths content. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02j Cubic/quartic equations: conjugate pairs and factor theorem4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\delta = 1-j\) | B1 | |
| There must be a second real root because complex roots occur in conjugate pairs | E1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\alpha+\beta+\gamma+\delta=1\) | B1 | |
| \(1+(1+j)+\gamma+(1-j)=1 \Rightarrow \gamma=-2\) | M1, A1 [3] | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((z-1)(z+2)(z-(1+j))(z-(1-j))\) | B1 | Correct factors from their roots |
| \(=(z^2+z-2)(z^2-2z+2)\) | M1 | Attempt to expand using all 4 factors |
| \(= z^4 - z^3 - 2z^2 + 6z - 4\) | ||
| \(\Rightarrow a=-2\), \(b=6\), \(c=-4\) | A3 [5] | One mark for each of \(a\), \(b\) and \(c\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\alpha\beta+\alpha\gamma+\alpha\delta+\beta\gamma+\beta\delta+\gamma\delta = a = -2\) | M2, A1 | Use of root relationships; \(a=-2\) cao |
| \(\alpha\beta\gamma+\alpha\beta\delta+\alpha\gamma\delta+\beta\gamma\delta = -b = -6 \Rightarrow b=6\) | A1 | \(b=6\) cao |
| \(\alpha\beta\gamma\delta = c = -4\) | B1 [5] | \(c=-4\) (SC ft on their 2nd real root) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(-z) = z^4+z^3-2z^2-6z-4\) | B1 | ft on their \(a\), \(b\), \(c\), simplified |
| Roots of \(f(-z)=0\) are \(-1\), \(2\), \(-1+j\) and \(-1-j\) | B1 [2] | For all four roots, cao |
# Question 8:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\delta = 1-j$ | B1 | |
| There must be a second real root because complex roots occur in conjugate pairs | E1 [2] | |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\alpha+\beta+\gamma+\delta=1$ | B1 | |
| $1+(1+j)+\gamma+(1-j)=1 \Rightarrow \gamma=-2$ | M1, A1 [3] | cao |
## Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(z-1)(z+2)(z-(1+j))(z-(1-j))$ | B1 | Correct factors from their roots |
| $=(z^2+z-2)(z^2-2z+2)$ | M1 | Attempt to expand using all 4 factors |
| $= z^4 - z^3 - 2z^2 + 6z - 4$ | | |
| $\Rightarrow a=-2$, $b=6$, $c=-4$ | A3 [5] | One mark for each of $a$, $b$ and $c$ |
**OR:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\alpha\beta+\alpha\gamma+\alpha\delta+\beta\gamma+\beta\delta+\gamma\delta = a = -2$ | M2, A1 | Use of root relationships; $a=-2$ cao |
| $\alpha\beta\gamma+\alpha\beta\delta+\alpha\gamma\delta+\beta\gamma\delta = -b = -6 \Rightarrow b=6$ | A1 | $b=6$ cao |
| $\alpha\beta\gamma\delta = c = -4$ | B1 [5] | $c=-4$ (SC ft on their 2nd real root) |
## Part (iv):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(-z) = z^4+z^3-2z^2-6z-4$ | B1 | ft on their $a$, $b$, $c$, simplified |
| Roots of $f(-z)=0$ are $-1$, $2$, $-1+j$ and $-1-j$ | B1 [2] | For all four roots, cao |8 The function $\mathrm { f } ( z ) = z ^ { 4 } - z ^ { 3 } + a z ^ { 2 } + b z + c$ has real coefficients. The equation $\mathrm { f } ( z ) = 0$ has roots $\alpha , \beta$, $\gamma$ and $\delta$ where $\alpha = 1$ and $\beta = 1 + \mathrm { j }$.\\
(i) Write down the other complex root and explain why the equation must have a second real root.\\
(ii) Write down the value of $\alpha + \beta + \gamma + \delta$ and find the second real root.\\
(iii) Find the values of $a , b$ and $c$.\\
(iv) Write down $\mathrm { f } ( - z )$ and the roots of $\mathrm { f } ( - z ) = 0$.\\
\hfill \mbox{\textit{OCR MEI FP1 2011 Q8 [12]}}