| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.3 This is a straightforward application of the complex conjugate root theorem for polynomials with real coefficients. Part (i) requires recalling that complex roots come in conjugate pairs (2-j and 2j), while part (ii) involves forming quadratic factors and expanding—routine algebraic manipulation with no novel insight required. Slightly above average difficulty due to being Further Maths content and requiring careful algebra, but still a standard textbook exercise. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| \(2 - j\) | B1 | — |
| \(2j\) | B1 [2] | — |
| Answer | Marks | Guidance |
|---|---|---|
| \((x-2-j)(x-2+j)(x+2j)(x-2j)\) | M1, M1 | M1 for each attempted factor pair |
| \(= (x^2-4x+5)(x^2+4)\) | A1, A1 | A1 for each quadratic – follow through sign errors |
| \(= x^4 - 4x^3 + 9x^2 - 16x + 20\) | A4 [8] | Minus 1 each error – follow through sign errors only |
| Answer | Marks | Guidance |
|---|---|---|
| \(-A = \sum\alpha = 4 \Rightarrow A = -4\) | M1, A1 | M1s for reasonable attempt to find sums |
| \(B = \sum\alpha\beta = 9 \Rightarrow B = 9\) | M1, A1 | S.C. If one sign incorrect, give total of A3 for A, B, C, D values |
| \(-C = \sum\alpha\beta\gamma = 16 \Rightarrow C = -16\) | M1, A1 | If more than one sign incorrect, give total of A2 for A, B, C, D values |
| \(D = \sum\alpha\beta\gamma\delta = 20 \Rightarrow D = 20\) | M1, A1 [8] | — |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt to substitute two correct roots into \(x^4 + Ax^3 + Bx^2 + Cx + D = 0\) | M1, M1 | One for each root |
| Produce 2 correct equations in two unknowns | A2 | One for each equation |
| \(A = -4,\ B = 9,\ C = -16,\ D = 20\) | A4 [8] | One mark for each correct. S.C. If one sign incorrect, give total of A3; if more than one sign incorrect, give total of A2 |
## Question 9:
### Part (i)
$2 - j$ | B1 | —
$2j$ | B1 **[2]** | —
### Part (iii)
$(x-2-j)(x-2+j)(x+2j)(x-2j)$ | M1, M1 | M1 for each attempted factor pair
$= (x^2-4x+5)(x^2+4)$ | A1, A1 | A1 for each quadratic – follow through sign errors
$= x^4 - 4x^3 + 9x^2 - 16x + 20$ | A4 **[8]** | Minus 1 each error – follow through sign errors only
So $A = -4,\ B = 9,\ C = -16,\ D = 20$
**OR (using symmetric functions):**
$-A = \sum\alpha = 4 \Rightarrow A = -4$ | M1, A1 | M1s for reasonable attempt to find sums
$B = \sum\alpha\beta = 9 \Rightarrow B = 9$ | M1, A1 | S.C. If one sign incorrect, give total of A3 for A, B, C, D values
$-C = \sum\alpha\beta\gamma = 16 \Rightarrow C = -16$ | M1, A1 | If more than one sign incorrect, give total of A2 for A, B, C, D values
$D = \sum\alpha\beta\gamma\delta = 20 \Rightarrow D = 20$ | M1, A1 **[8]** | —
**OR (substitution method):**
Attempt to substitute two correct roots into $x^4 + Ax^3 + Bx^2 + Cx + D = 0$ | M1, M1 | One for each root
Produce 2 correct equations in two unknowns | A2 | One for each equation
$A = -4,\ B = 9,\ C = -16,\ D = 20$ | A4 **[8]** | One mark for each correct. S.C. If one sign incorrect, give total of A3; if more than one sign incorrect, give total of A29 The quartic equation $x ^ { 4 } + A x ^ { 3 } + B x ^ { 2 } + C x + D = 0$, where $A , B , C$ and $D$ are real numbers, has roots $2 + \mathrm { j }$ and - 2 j .\\
(i) Write down the other roots of the equation.\\
(ii) Find the values of $A , B , C$ and $D$.
\hfill \mbox{\textit{OCR MEI FP1 2005 Q9 [10]}}