OCR MEI Further Pure Core 2020 November — Question 10 7 marks

Exam BoardOCR MEI
ModuleFurther Pure Core (Further Pure Core)
Year2020
SessionNovember
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex numbers 2
TypeDirect nth roots: roots with geometric or algebraic follow-up
DifficultyStandard +0.3 This is a structured multi-part question on complex numbers requiring reading coordinates from a diagram, converting to exponential form, finding a transformation, and determining a value from roots. While it involves several steps, each part follows standard procedures (reading diagrams, applying multiplication in exponential form, using the relationship between roots and equations). The conceptual demand is moderate for Further Maths students, slightly above average difficulty due to the multi-step nature but not requiring novel insight.
Spec4.02b Express complex numbers: cartesian and modulus-argument forms4.02d Exponential form: re^(i*theta)4.02k Argand diagrams: geometric interpretation4.02r nth roots: of complex numbers
  1. Write down, in exponential ( \(r \mathrm { e } ^ { \mathrm { i } \theta }\) ) form, the complex numbers represented by the points \(\mathrm { A } , \mathrm { B }\), \(\mathrm { C } , \mathrm { D } , \mathrm { E }\) and F .
  2. When these complex numbers are multiplied by the complex number \(w\), the resulting complex numbers are represented by the points G, H, I, J, K and L. Find \(w\) in exponential form.
  3. You are given that \(\mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { J } , \mathrm { K }\) and L represent roots of the equation \(z ^ { 6 } = p\). Find \(p\).
Question 10:
AnswerMarks
10DR
let u = 2πarcosh y, u’ = 2π/√(y2 ​– 1)
​ ​ ​ ​ ​​ ​ ​ ​ ​ ​​
v’ = 1, v = y
​ ​​ ​
= 2π(2ln(2+√3) − √3)
AnswerMarks
​ ​M1
M1
A1
M1
A1
M1
A1cao
AnswerMarks
[7]1.1
3.1a
2.1
1.1
1.1
1.1
AnswerMarks
3.2aintegration by parts
condone missing 2π and
​ ​
incorrect limits
subst u = y2 ​– 1 or inspection
​ ​ ​​
use of arcosh x =
​​
ln[x+√(x2−​ 1)]
AnswerMarks
​​ ​​A1 for √y2−1 
Question 10:
10 | DR
let u = 2πarcosh y, u’ = 2π/√(y2 ​– 1)
​ ​ ​ ​ ​​ ​ ​ ​ ​ ​​
v’ = 1, v = y
​ ​​ ​
= 2π(2ln(2+√3) − √3)
​ ​ | M1
M1
A1
M1
A1
M1
A1cao
[7] | 1.1
3.1a
2.1
1.1
1.1
1.1
3.2a | integration by parts
condone missing 2π and
​ ​
incorrect limits
subst u = y2 ​– 1 or inspection
​ ​ ​​
use of arcosh x =
​​
ln[x+√(x2−​ 1)]
​​ ​​ | A1 for √y2−1
\begin{enumerate}[label=(\alph*)]
\item Write down, in exponential ( $r \mathrm { e } ^ { \mathrm { i } \theta }$ ) form, the complex numbers represented by the points $\mathrm { A } , \mathrm { B }$, $\mathrm { C } , \mathrm { D } , \mathrm { E }$ and F .
\item When these complex numbers are multiplied by the complex number $w$, the resulting complex numbers are represented by the points G, H, I, J, K and L.

Find $w$ in exponential form.
\item You are given that $\mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { J } , \mathrm { K }$ and L represent roots of the equation $z ^ { 6 } = p$.

Find $p$.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Pure Core 2020 Q10 [7]}}
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