| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2020 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Direct nth roots: purely real or imaginary RHS |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths question requiring standard technique: convert 25i to modulus-argument form (r=25, θ=π/2), apply the nth root formula to get two roots with half the argument and square root of modulus, then plot on an Argand diagram. While Further Maths content, it's a direct application of a well-practiced algorithm with no problem-solving or novel insight required, making it slightly easier than average overall. |
| Spec | 4.02h Square roots: of complex numbers4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (a) | π |
| Answer | Marks |
|---|---|
| So the square roots are 5e4 and 5e 4 | B1 |
| Answer | Marks |
|---|---|
| A1 | 3.3 |
| Answer | Marks |
|---|---|
| 3.2a | Conversion soi |
| Answer | Marks |
|---|---|
| Alternate Method | B1 |
| Answer | Marks |
|---|---|
| A1 | Conversion and attempt to square |
| Answer | Marks | Guidance |
|---|---|---|
| (b) | B1 | 1.1 |
Question 4:
4 | (a) | π
i
25i=25e2 oe
π
i
25i=25e2
π 5π
i i
⇒ 25i =5e4 and 5e 4
π 5π
i i
So the square roots are 5e4 and 5e 4 | B1
M1
A1 | 3.3
3.1a
3.2a | Conversion soi
Attempt to find at least one root by rooting the
modulus and halving the argument soi
Both
Alternate Method | B1
M1
A1 | Conversion and attempt to square
a and b. Ignore ±
Both
25i =a+bi
5
⇒a =b=± 2
2
5 1 1
⇒ 25i =± 2 ( 1+i )=±5cos π+isin π
2 4 4
1 5
π π
⇒ 25i =5e4 and 5e4
[3]
(b) | B1 | 1.1 | All three but no extras. Scales etc are not required but
if no scale then the lines representing the roots should
be at 45° to axis.
Accept points.
No extras
Line representing 25i must be at least two times as
long
[1]
B1
M1
A1
Conversion and attempt to square
a and b. Ignore ±
Both4 In this question you must show detailed reasoning.
\begin{enumerate}[label=(\alph*)]
\item Determine the square roots of 25 i in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $0 \leqslant \theta < 2 \pi$.
\item Illustrate the number 25i and its square roots on an Argand diagram.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2020 Q4 [4]}}