| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Session | Specimen |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.3 This is a standard Further Maths question on complex roots with real coefficients. Students must recognize that 2-3i is also a root, form the quadratic factor (x-(2+3i))(x-(2-3i)), then divide to find the other quadratic factor. While it requires multiple steps and is from Further Maths content, it follows a well-established procedure with no novel insight needed, making it slightly easier than average overall. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02j Cubic/quartic equations: conjugate pairs and factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (i) | DR |
| Answer | Marks |
|---|---|
| (cid:159)f(x)(cid:32) (cid:11) x2 (cid:16)4x(cid:14)13 (cid:12)(cid:11) x2 (cid:14)6x(cid:14)13 (cid:12) | B1 |
| Answer | Marks |
|---|---|
| [4] | 2.2a |
| Answer | Marks |
|---|---|
| 1.1 | n |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt to derive second quadratic | Must come from (cid:11)x(cid:16)(cid:68)(cid:12)(cid:11)x(cid:16)(cid:69)(cid:12) | |
| 2 | (ii) | DR |
| (cid:159)Roots are 2(cid:114)3i,(cid:16)3(cid:114)2i | A1 | |
| [1] | 1.1 | m |
Question 2:
2 | (i) | DR
(cid:68)(cid:32)2(cid:14)3i(cid:159)(cid:69)(cid:32)2(cid:16)3i
(cid:159)x2 (cid:16)4x(cid:14)13(cid:32)0 is a quadratic factor
(cid:159)f(x)(cid:32) (cid:11) x2 (cid:16)4x(cid:14)13 (cid:12)(cid:11) x2 (cid:14)6x(cid:14)13 (cid:12) | B1
M1
M1
A1
[4] | 2.2a
1.1
1.1
1.1 | n
Attempt to create first quadratic
e
Attempt to derive second quadratic | Must come from (cid:11)x(cid:16)(cid:68)(cid:12)(cid:11)x(cid:16)(cid:69)(cid:12)
2 | (ii) | DR
(cid:159)Roots are 2(cid:114)3i,(cid:16)3(cid:114)2i | A1
[1] | 1.1 | m
Depends on M2 awarded in part (i)
--- 2(i) ---
2(i)
n
e
m
i
c
e
p
S
--- 2(ii) ---
2(ii)2 In this question you must show detailed reasoning.
The equation $\mathrm { f } ( x ) = 0$, where $\mathrm { f } ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } + 26 x + 169$, has a root $x = 2 + 3 \mathrm { i }$.\\
(i) Express $\mathrm { f } ( x )$ as a product of two quadratic factors.\\
(ii) Hence write down all the roots of the equation $\mathrm { f } ( x ) = 0$.
\hfill \mbox{\textit{OCR Further Pure Core 1 Q2 [5]}}