OCR MEI Further Pure Core AS 2018 June — Question 9 9 marks

Exam BoardOCR MEI
ModuleFurther Pure Core AS (Further Pure Core AS)
Year2018
SessionJune
Marks9
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TopicComplex Numbers Argand & Loci
TypeRegion shading with multiple inequalities
DifficultyStandard +0.3 This is a straightforward Further Maths question testing basic complex number representations and region membership. Parts (i) and (ii) require simple conversions between forms using standard formulas. Part (iii) involves routine checks of modulus and argument conditions against given bounds. While it's Further Maths content, the techniques are mechanical with no problem-solving insight required, making it slightly easier than average overall.
Spec4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines
Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\left\{z : |z| \leq 4\sqrt{2}\right\} \cap \left\{z : -\frac{1}{4}\pi \leq \arg z \leq \frac{1}{4}\pi\right\}.$$ \includegraphics{figure_9}
  1. Find, in modulus-argument form, the complex number represented by the point P. [2]
  2. Find, in the form \(a + ib\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q. [3]
  3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
    lie within this region. [4]
Question 9:
AnswerMarks Guidance
9(i) argz1,
4
so z4 2  cos1isin1 
AnswerMarks
4 4B1
B1
AnswerMarks Guidance
[2]1.1
2.5allow 45 for both marks
9(ii) argz1,
3
so z4 2  cos1isin1 
3 3
AnswerMarks
2 22 6iM1
A1
A1
AnswerMarks
[3]1.1a
1.1
AnswerMarks Guidance
1.1o.e., must be exact
9(iii) DR
35i= 925 34
34 32 so not in the region
5.5(cos0.8isin0.8) has mod 5.5 and arg 0.8
5.5 32 5.656K
AnswerMarks
0.785 < 0.8 < 1.047 so is in the regionM1
A1
M1
A1
AnswerMarks
[4]3.1a
2.1
3.1a
AnswerMarks
2.1finding modulus (correct
method)
comparing mod or argument
checking both conditions
Question 9:
9 | (i) | argz1, |z|4 2
4
so z4 2  cos1isin1 
4 4 | B1
B1
[2] | 1.1
2.5 | allow 45 for both marks
9 | (ii) | argz1, |z|4 2
3
so z4 2  cos1isin1 
3 3
2 22 6i | M1
A1
A1
[3] | 1.1a
1.1
1.1 | o.e., must be exact
9 | (iii) | DR
|35i|= 925 34
34 32 so not in the region
5.5(cos0.8isin0.8) has mod 5.5 and arg 0.8
5.5 32 5.656K
0.785 < 0.8 < 1.047 so is in the region | M1
A1
M1
A1
[4] | 3.1a
2.1
3.1a
2.1 | finding modulus (correct
method)
comparing mod or argument
checking both conditions
Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by
$$\left\{z : |z| \leq 4\sqrt{2}\right\} \cap \left\{z : -\frac{1}{4}\pi \leq \arg z \leq \frac{1}{4}\pi\right\}.$$

\includegraphics{figure_9}

\begin{enumerate}[label=(\roman*)]
\item Find, in modulus-argument form, the complex number represented by the point P. [2]
\item Find, in the form $a + ib$, where $a$ and $b$ are exact real numbers, the complex number represented by the point Q. [3]
\item In this question you must show detailed reasoning.

Determine whether the points representing the complex numbers
\begin{itemize}
\item $3 + 5i$
\item $5.5(\cos 0.8 + i \sin 0.8)$
\end{itemize}
lie within this region. [4]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2018 Q9 [9]}}
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