| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Moderate -0.3 This is a straightforward application of the complex conjugate root theorem for polynomials with real coefficients, followed by routine expansion of factors. Part (i) requires immediate recall of theory, part (ii) is basic plotting, and part (iii) involves algebraic expansion that, while lengthy, follows a standard procedure with no novel problem-solving required. Slightly easier than average due to its mechanical nature. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| \(2+2\text{j}\) and \(-1-\text{j}\) | B2 [2] | 1 mark for each |
| Answer | Marks | Guidance |
|---|---|---|
| Argand diagram with all four roots plotted correctly in conjugate pairs | B2 [2] | 1 mark for each correct pair |
| Answer | Marks | Guidance |
|---|---|---|
| \((x-2-2\text{j})(x-2+2\text{j})(x+1+\text{j})(x+1-\text{j})\) | M1, B2 | Attempt to use factor theorem; Correct factors minus 1 each error; B1 if only errors are sign errors |
| \(= (x^2-4x+8)(x^2+2x+2)\) | A1 | One correct quadratic with real coefficients (may be implied) |
| \(= x^4 - 2x^3 + 2x^2 + 8x + 16\) | M1 | Expanding |
| \(\Rightarrow A=-2,\ B=2,\ C=8,\ D=16\) | A2 [7] | Minus 1 each error; A1 if only errors are sign errors |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sum\alpha = 2\); \(\alpha\beta\gamma\delta = 16\) | B1, B1 | |
| \(\sum\alpha\beta = \alpha\alpha^* + \alpha\beta + \alpha\beta^* + \beta\beta^* + \beta\alpha^* + \beta^*\alpha^*\) | M1 | |
| \(\sum\alpha\beta\gamma = \alpha\alpha^*\beta + \alpha\alpha^*\beta^* + \alpha\beta\beta^* + \alpha^*\beta\beta^*\) | M1 | |
| \(\sum\alpha\beta = 2\), \(\sum\alpha\beta\gamma = -8\) | A1 | Both correct |
| \(A=-2,\ B=2,\ C=8,\ D=16\) | A2 [7] | Minus 1 each error; A1 if only errors are sign errors |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt to substitute in one root; Attempt to substitute in a second root | M1, M1, A1 | Both correct |
| Equating real and imaginary parts to 0; Attempt to solve simultaneous equations | M1, M1, A2 [7] | Minus 1 each error; A1 if only errors are sign errors |
# Question 9:
## Question 9(i):
| $2+2\text{j}$ and $-1-\text{j}$ | B2 [2] | 1 mark for each |
## Question 9(ii):
| Argand diagram with all four roots plotted correctly in conjugate pairs | B2 [2] | 1 mark for each correct pair |
## Question 9(iii):
| $(x-2-2\text{j})(x-2+2\text{j})(x+1+\text{j})(x+1-\text{j})$ | M1, B2 | Attempt to use factor theorem; Correct factors minus 1 each error; B1 if only errors are sign errors |
| $= (x^2-4x+8)(x^2+2x+2)$ | A1 | One correct quadratic with real coefficients (may be implied) |
| $= x^4 - 2x^3 + 2x^2 + 8x + 16$ | M1 | Expanding |
| $\Rightarrow A=-2,\ B=2,\ C=8,\ D=16$ | A2 [7] | Minus 1 each error; A1 if only errors are sign errors |
### OR (via symmetric functions):
| $\sum\alpha = 2$; $\alpha\beta\gamma\delta = 16$ | B1, B1 | |
| $\sum\alpha\beta = \alpha\alpha^* + \alpha\beta + \alpha\beta^* + \beta\beta^* + \beta\alpha^* + \beta^*\alpha^*$ | M1 | |
| $\sum\alpha\beta\gamma = \alpha\alpha^*\beta + \alpha\alpha^*\beta^* + \alpha\beta\beta^* + \alpha^*\beta\beta^*$ | M1 | |
| $\sum\alpha\beta = 2$, $\sum\alpha\beta\gamma = -8$ | A1 | Both correct |
| $A=-2,\ B=2,\ C=8,\ D=16$ | A2 [7] | Minus 1 each error; A1 if only errors are sign errors |
### OR (via substitution):
| Attempt to substitute in one root; Attempt to substitute in a second root | M1, M1, A1 | Both correct |
| Equating real and imaginary parts to 0; Attempt to solve simultaneous equations | M1, M1, A2 [7] | Minus 1 each error; A1 if only errors are sign errors |
---9 Two complex numbers, $\alpha$ and $\beta$, are given by $\alpha = 2 - 2 \mathrm { j }$ and $\beta = - 1 + \mathrm { j }$.\\
$\alpha$ and $\beta$ are both roots of a quartic equation $x ^ { 4 } + A x ^ { 3 } + B x ^ { 2 } + C x + D = 0$, where $A , B , C$ and $D$ are real numbers.\\
(i) Write down the other two roots.\\
(ii) Represent these four roots on an Argand diagram.\\
(iii) Find the values of $A , B , C$ and $D$.
\hfill \mbox{\textit{OCR MEI FP1 2008 Q9 [11]}}