OCR Further Pure Core AS 2020 November — Question 3 12 marks

Exam BoardOCR
ModuleFurther Pure Core AS (Further Pure Core AS)
Year2020
SessionNovember
Marks12
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex numbers 2
TypeModulus and argument calculations
DifficultyModerate -0.3 This is a standard Further Pure AS complex numbers question testing routine manipulations (conjugate, arithmetic, division), modulus-argument form conversion, and basic properties of multiplication. All parts follow textbook procedures with no novel insight required, though it's slightly easier than average due to straightforward calculations and the given structure guiding students through each step.
Spec4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide
In this question you must show detailed reasoning. The complex number \(7 - 4\text{i}\) is denoted by \(z\).
  1. Giving your answers in the form \(a + b\text{i}\), where \(a\) and \(b\) are rational numbers, find the following.
    1. \(3z - 4z^*\) [2]
    2. \((z + 1 - 3\text{i})^2\) [2]
    3. \(\frac{z + 1}{z - 1}\) [2]
  2. Express \(z\) in modulus-argument form giving the modulus exactly and the argument correct to 3 significant figures. [3]
  3. The complex number \(\omega\) is such that \(z\omega = \sqrt{585}(\cos(0.5) + \text{i}\sin(0.5))\). Find the following.
    [3]
Question 3:
AnswerMarks Guidance
3(a) (i)
3(7 – 4i) – 4(7 + 4i) = 21 – 12i – 28 – 16i
AnswerMarks
= –7 – 28iM1
A1
AnswerMarks
[2]1.1
1.1z* correct and brackets opened. Allow
sign mistake with -16i, but not z/z*
mix-up
AnswerMarks Guidance
(a)(ii) DR
(7 – 4i + 1 – 3i )2 = (8 – 7i)2 = 64 – 112i +49i2
AnswerMarks
15 – 112iM1
A1
AnswerMarks
[2]1.1
1.1Simplifying bracket and three term
expansion. Allow z z* mix-up here
AnswerMarks Guidance
(a)(iii) DR
z+1 7−4i+1 8−4i 6+4i
= = ×
z−1 7−4i−1 6−4i 6+4i
48+32i−24i+16 64 8
= = + i
AnswerMarks
36+16 52 52M1
A1
AnswerMarks
[2]1.1
1.1Simplifying and correct process for
“realising” the denominator
i
Allow =
64+8 16+2𝑖𝑖
AnswerMarks
52 13z*+1
Allow M1 if used correctly
z*−1
AnswerMarks
(b)DR
√(72 + (–4)2) = √65
 4
tan−1 ± 
 7
AnswerMarks
√65 (cos(–0.519) + isin(–0.519))B1
M1
A1
AnswerMarks
[3]1.1
1.1
AnswerMarks
2.57
Allow tan−1   if it is clear that this
4
being used correctly (eg from diagram)
to find the argument
or √65 cis(–0.519) or [√65, –0.519] or
AnswerMarks
√65 cis(5.76) etcMust be in the correct form ie c + is
not c – is. If using [r, ] then square
brackets must be seen.
𝜃𝜃
AnswerMarks
(c)DR
3
0.5 – –0.519
AnswerMarks
awrt 1.02B1ft
M1
A1ft
AnswerMarks
[3]1.1
1.1
AnswerMarks
1.1√585÷their z
Use of arg(z z ) = argz + argz
1 2 1 2
Must be seen
0.5 – their arg(z) in [0, π/2]
Question 3:
3 | (a) | (i) | DR
3(7 – 4i) – 4(7 + 4i) = 21 – 12i – 28 – 16i
= –7 – 28i | M1
A1
[2] | 1.1
1.1 | z* correct and brackets opened. Allow
sign mistake with -16i, but not z/z*
mix-up
(a) | (ii) | DR
(7 – 4i + 1 – 3i )2 = (8 – 7i)2 = 64 – 112i +49i2
15 – 112i | M1
A1
[2] | 1.1
1.1 | Simplifying bracket and three term
expansion. Allow z z* mix-up here
(a) | (iii) | DR
z+1 7−4i+1 8−4i 6+4i
= = ×
z−1 7−4i−1 6−4i 6+4i
48+32i−24i+16 64 8
= = + i
36+16 52 52 | M1
A1
[2] | 1.1
1.1 | Simplifying and correct process for
“realising” the denominator
i
Allow =
64+8 16+2𝑖𝑖
52 13 | z*+1
Allow M1 if used correctly
z*−1
(b) | DR
√(72 + (–4)2) = √65
 4
tan−1 ± 
 7
√65 (cos(–0.519) + isin(–0.519)) | B1
M1
A1
[3] | 1.1
1.1
2.5 | 7
Allow tan−1   if it is clear that this
4
being used correctly (eg from diagram)
to find the argument
or √65 cis(–0.519) or [√65, –0.519] or
√65 cis(5.76) etc | Must be in the correct form ie c + is
not c – is. If using [r, ] then square
brackets must be seen.
𝜃𝜃
(c) | DR
3
0.5 – –0.519
awrt 1.02 | B1ft
M1
A1ft
[3] | 1.1
1.1
1.1 | √585÷their z
Use of arg(z z ) = argz + argz
1 2 1 2
Must be seen
0.5 – their arg(z) in [0, π/2]
In this question you must show detailed reasoning.

The complex number $7 - 4\text{i}$ is denoted by $z$.

\begin{enumerate}[label=(\alph*)]
\item Giving your answers in the form $a + b\text{i}$, where $a$ and $b$ are rational numbers, find the following.
\begin{enumerate}[label=(\roman*)]
\item $3z - 4z^*$ [2]
\item $(z + 1 - 3\text{i})^2$ [2]
\item $\frac{z + 1}{z - 1}$ [2]
\end{enumerate}

\item Express $z$ in modulus-argument form giving the modulus exactly and the argument correct to 3 significant figures. [3]

\item The complex number $\omega$ is such that $z\omega = \sqrt{585}(\cos(0.5) + \text{i}\sin(0.5))$.

Find the following.
\begin{itemize}
\item $|\omega|$
\item $\arg(\omega)$, giving your answer correct to 3 significant figures
\end{itemize} [3]
\end{enumerate}

\hfill \mbox{\textit{OCR Further Pure Core AS 2020 Q3 [12]}}
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