| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Parametric polynomials with root conditions |
| Difficulty | Challenging +1.8 This is a challenging Further Maths question requiring theoretical understanding of complex roots (conjugate pairs) and multi-step problem-solving. Part (i) tests conceptual knowledge about continuity and the intermediate value theorem. Part (ii) requires finding complex roots (likely by factoring or using the cubic formula), identifying the conjugate pair, then calculating triangle area on an Argand diagram—demanding both algebraic manipulation and geometric insight beyond standard A-level content. |
| Spec | 4.01b Complex proofs: conjecture and demanding proofs4.02g Conjugate pairs: real coefficient polynomials4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| 8 | (i) | DR |
| Answer | Marks |
|---|---|
| real. | E1 |
| E1 | 2.1 |
| Answer | Marks | Guidance |
|---|---|---|
| Or | N | |
| Based on graph of y(cid:32)f(x). If coefficient of x3 is | E1 | E1 |
| Answer | Marks |
|---|---|
| If coefficient of x3 is negative then y is positive for | I |
| Answer | Marks | Guidance |
|---|---|---|
| 8 | (ii) | DR |
| Answer | Marks |
|---|---|
| = 3 3 (square units) | M1 |
| Answer | Marks |
|---|---|
| [7] | 3.1a |
| Answer | Marks |
|---|---|
| 1.1 | f(±1) or f(±5) or f(±7) |
Question 8:
8 | (i) | DR
Either
If it has a complex root, the complex conjugate is also
a root. Hence complex roots occur in pairs and,
as the equation has three roots, at least one must be
real. | E1
E1 | 2.1
2.4
Or | N
Based on graph of y(cid:32)f(x). If coefficient of x3 is | E1 | E1 | M | E
positive then y is negative for large negative x and
positive for large positive x, so graph (which is
continuous) cuts x-axis at least once – giving a real
root of f(x)(cid:32)0.
If coefficient of x3 is negative then y is positive for | I
CE1
large negative x and negative for large positive x, so
graph (which is continuous) again cuts x-axis at least
once – giving a real root of f(x)(cid:32)0.
[2]
8 | (ii) | DR
f(cid:11)(cid:16)5(cid:12)(cid:32)0
(cid:159) z + 5 is a factor
z3(cid:14)9z2 (cid:14)27z(cid:14)35(cid:32)(cid:11)z(cid:14)5(cid:12)(cid:11) z2 (cid:14)4z(cid:14)7 (cid:12)
(cid:11)z(cid:14)2(cid:12)2
z2 (cid:14)4z(cid:14)7(cid:32)0 (cid:159) (cid:14)3(cid:32)0
(cid:159) roots are −5, (cid:16)2(cid:114) 3i
1
Roots form triangle, area (cid:117)2 3(cid:117)3
2
= 3 3 (square units) | M1
A1
B1
M1
A1
M1
I
C
A1
[7] | 3.1a
1.1
1.1
1.1
M
2.1
3.2a
1.1 | f(±1) or f(±5) or f(±7)
correctly must be seen
N
Finding z + 5 is a factor
Correct factorising
E
Attempt at solving quadratic
must be seen
Any valid method attempted
on any 3 distinct non-collinear
roots, however obtained, must
be seen
FT their roots as above
A0 for decimal answerIn this question you must show detailed reasoning.
\begin{enumerate}[label=(\roman*)]
\item Explain why all cubic equations with real coefficients have at least one real root. [2]
\item Points representing the three roots of the equation $z^3 + 9z^2 + 27z + 35 = 0$ are plotted on an Argand diagram.
Find the exact area of the triangle which has these three points as its vertices. [7]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core AS Q8 [9]}}