| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| 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 straightforward Further Maths question testing standard results about complex conjugate roots and substitution to solve a quartic. Part (i) is immediate recall, part (ii) uses Vieta's formulas with simple arithmetic, and part (iii) is a routine substitution y²=x that reduces to the original quadratic. While it's Further Maths content, it requires no novel insight—just applying well-practiced techniques in sequence. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(16 + 30i\) | B1 [1] | State correct value |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a = -32\) | M1, A1 | Use \(a = -(\text{sum of roots})\); obtain correct answer |
| \(b = 1156\) | M1, A1 [4] | Use \(b = \text{product of roots}\); obtain correct answer |
| *Alternative:* | M1 | Substitute, expand and equate imaginary parts |
| A1 | Obtain \(a = -32\) | |
| M1 | Equate real parts | |
| A1 | Obtain \(b = 1156\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(p^2 - q^2 = 16\) and \(pq = -15\) | M1 | Attempt to equate real and imaginary parts of \((p+iq)^2\) & \(16-30i\) or root from (ii) |
| A1 | Obtain both results c.a.o. | |
| M1 | Obtain quadratic in \(p^2\) or \(q^2\) | |
| M1 | Solve to obtain \(p = (\pm)5\) or \(q = (\pm)3\) | |
| A1 | Obtain 2 correct answers as complex nos | |
| \(\pm(5 \pm 3i)\) | M1 | Attempt at all 4 roots |
| A1 [7] | State other two roots as complex nos |
## Question 9:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $16 + 30i$ | B1 **[1]** | State correct value |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a = -32$ | M1, A1 | Use $a = -(\text{sum of roots})$; obtain correct answer |
| $b = 1156$ | M1, A1 **[4]** | Use $b = \text{product of roots}$; obtain correct answer |
| *Alternative:* | M1 | Substitute, expand and equate imaginary parts |
| | A1 | Obtain $a = -32$ |
| | M1 | Equate real parts |
| | A1 | Obtain $b = 1156$ |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $p^2 - q^2 = 16$ and $pq = -15$ | M1 | Attempt to equate real and imaginary parts of $(p+iq)^2$ & $16-30i$ or root from (ii) |
| | A1 | Obtain both results c.a.o. |
| | M1 | Obtain quadratic in $p^2$ or $q^2$ |
| | M1 | Solve to obtain $p = (\pm)5$ or $q = (\pm)3$ |
| | A1 | Obtain 2 correct answers as complex nos |
| $\pm(5 \pm 3i)$ | M1 | Attempt at all 4 roots |
| | A1 **[7]** | State other two roots as complex nos |
---9 One root of the quadratic equation $x ^ { 2 } + a x + b = 0$, where $a$ and $b$ are real, is $16 - 30 \mathrm { i }$.\\
(i) Write down the other root of the quadratic equation.\\
(ii) Find the values of $a$ and $b$.\\
(iii) Use an algebraic method to solve the quartic equation $y ^ { 4 } + a y ^ { 2 } + b = 0$.
\hfill \mbox{\textit{OCR FP1 2011 Q9 [12]}}