| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2020 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Challenging +1.2 This is a structured Further Maths question on complex roots with real coefficients. Parts (a)-(c) are routine applications of conjugate root theorem and Vieta's formulas requiring straightforward calculation. Part (d) requires converting to modulus-argument form and using De Moivre's theorem, which is standard Further Pure content but involves multiple steps and careful algebraic manipulation, elevating it slightly above average difficulty. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02q De Moivre's theorem: multiple angle formulae4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| 9 | (a) | β=1−i 2oe |
| Answer | Marks |
|---|---|
| (b) | 3 ( )( ) |
| Answer | Marks |
|---|---|
| 2 | M1 |
| A1 | 2.1 |
| 1.1 | 3 |
| Answer | Marks |
|---|---|
| (c) | ( ( ))( ( )) |
| Answer | Marks |
|---|---|
| ⇒2x3−5x2 +8x−3=0 i.e. p =−5, q =8 | M1 |
| A1 | 3.1a |
| 1.1 | Multiply out (can be seen in (b)) |
| Alternate Method | M1 |
| A1 | Use of symmetry forms for roots |
| Answer | Marks |
|---|---|
| (d) | 1 2 1 |
| Answer | Marks |
|---|---|
| 2n | M1 |
| Answer | Marks |
|---|---|
| A1 | 2.1 |
| Answer | Marks |
|---|---|
| 2.1 | Either α or β seen in mod/arg form |
Question 9:
9 | (a) | β=1−i 2oe | B1 | 2.2a
[1]
(b) | 3 ( )( )
αβγ= ,αβ= 1+i 2 1−i 2 =3
2
1
⇒γ=
2 | M1
A1 | 2.1
1.1 | 3
Use of αβγ= to find the 3rd root.
2
Alternatively, find x2 −2x+3and divide
[2]
(c) | ( ( ))( ( ))
x− 1+i 2 x− 1−i 2 ( 2x−1 )=0
⇒ ( x2 −2x+3 )( 2x−1 )=0
⇒2x3−4x2 +6x−x2 +2x−3=0
⇒2x3−5x2 +8x−3=0 i.e. p =−5, q =8 | M1
A1 | 3.1a
1.1 | Multiply out (can be seen in (b))
Alternate Method | M1
A1 | Use of symmetry forms for roots
a 5
α+β+γ=− = ⇒ p =−5
2 2
q
αβ+βγ+γα=
2
1( ) ( )( )
= 1+i 2+1−i 2 + 1+i 2 1−i 2
2
=1+3=4⇒q=8
[2]
M1
A1
Use of symmetry forms for roots
(d) | 1 2 1
α=1+i 2 = 3 +i =32 ( cosθ+isinθ)
3 3
1 2
where cosθ= , sinθ= ⇒tanθ= 2
3 3
1 2 1
β=1−i 2 = 3 −i =32 ( cosθ−isinθ) oe
3 3
n n
⇒αn =32 ( cosnθ+isinnθ) , βn =32 ( cosnθ−isinnθ)
n
⇒αn +βn =2×32 ×cosnθ AG
n 1
⇒αn +βn +γn =2×32 ×cosnθ+
2n | M1
A1
M1
A1 | 2.1
1.1
2.1
2.1 | Either α or β seen in mod/arg form
For both of them – accept exponentials
Derivation of αn or βn
[4]9 You are given that the cubic equation $2 x ^ { 3 } + p x ^ { 2 } + q x - 3 = 0$, where $p$ and $q$ are real numbers, has a complex root $\alpha = 1 + \mathrm { i } \sqrt { 2 }$.
\begin{enumerate}[label=(\alph*)]
\item Write down a second complex root, $\beta$.
\item Determine the third root, $\gamma$.
\item Find the value of $p$ and the value of $q$.
\item Show that if $n$ is an integer then $\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } = 2 \times 3 ^ { \frac { 1 } { 2 } n } \times \cos n \theta + \frac { 1 } { 2 ^ { n } }$ where $\tan \theta = \sqrt { 2 }$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2020 Q9 [9]}}