WJEC Further Unit 1 2022 June — Question 1 12 marks

Exam BoardWJEC
ModuleFurther Unit 1 (Further Unit 1)
Year2022
SessionJune
Marks12
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex Numbers Arithmetic
TypeModulus and argument with operations
DifficultyStandard +0.3 This is a straightforward Further Maths question testing standard complex number operations: finding modulus/argument of a product, converting to polar form, manipulating complex fractions, and verifying that z·z̄ is real. All parts use routine techniques with no novel insight required, though the algebraic manipulation in part (b) requires care. Slightly above average difficulty due to being Further Maths content and multi-step nature.
Spec4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02d Exponential form: re^(i*theta)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide
  1. The complex numbers \(z , w\) are given by \(z = 3 - 4 \mathrm { i } , w = 2 - \mathrm { i }\).
    1. (i) Find the modulus and argument of \(z w\).
      (ii) Express \(z w\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).
    2. The complex numbers \(v , w , z\) satisfy the equation \(\frac { 1 } { v } = \frac { 1 } { w } - \frac { 1 } { z }\). Find \(v\) in the form \(a + \mathrm { i } b\), where \(a , b\) are real.
    3. The complex conjugate of \(v\) is denoted by \(\bar { v }\).
    Show that \(v \bar { v } = k\), where \(k\) is a real number whose value is to be determined.
Question 1:
Part a(i):
Method 1:
AnswerMarks Guidance
WorkingMark Guidance
\(zw = (3-4i)(2-i) = 6-3i-8i+4i^2\), \(zw = 2-11i\)B2 B1 for unsimplified expansion with 3 correct terms
\(\zw\ = \sqrt{2^2+(-11)^2} = 5\sqrt{5}\)
\(\arg zw = \tan^{-1}\left(-\frac{11}{2}\right) = -1.39\) or \(-79.7°\)B1 oe FT their \(zw\) if not in 1st quadrant
Method 2:
AnswerMarks Guidance
WorkingMark Guidance
\(\z\ = \sqrt{3^2+(-4)^2} = 5\), \(\
\(\arg z = \tan^{-1}\left(-\frac{4}{3}\right) = -0.927\) or \(-53.13°\), \(\arg w = \tan^{-1}\left(-\frac{1}{2}\right) = -0.464\) or \(-26.57°\)(B1) Both args oe
\(\zw\ = 5 \times \sqrt{5} = 5\sqrt{5}\)
\(\arg zw = -0.927 + -0.464 = -1.39\) or \(-79.7°\)(B1) oe FT args and mods (mods and args must be seen)
[4]
Part a(ii):
AnswerMarks Guidance
WorkingMark Guidance
\(5\sqrt{5}(\cos(-1.39) + i\sin(-1.39))\) OR \(5\sqrt{5}(\cos(-79.7°) + i\sin(-79.7°))\)B1 oe FT their mod and arg
[1]
Part b):
Method 1:
AnswerMarks Guidance
WorkingMark Guidance
\(\frac{1}{v} = \frac{1}{2-i} - \frac{1}{3-4i}\)
\(\frac{1}{v} = \frac{3-4i-2+i}{(3-4i)(2-i)}\)M1 Attempt to combine
\(\frac{1}{v} = \frac{1-3i}{2-11i}\)A1
\(v = \frac{2-11i}{1-3i}\)A1
\(v = \frac{2-11i}{1-3i} \times \frac{1+3i}{1+3i}\)M1 FT their \(v\), M0 for no working
\(v = \frac{35-5i}{10}\left(= \frac{7-i}{2}\right)\)
\(v = 3.5 - 0.5i\)A1 oe cao
Method 2:
AnswerMarks Guidance
WorkingMark Guidance
\(\frac{1}{v} = \frac{z-w}{zw}\)(M1) Attempt to combine
\(v = \frac{zw}{z-w}\) or \(\frac{1}{v} = \frac{1-3i}{2-11i}\)(A1)
\(v = \frac{2-11i}{1-3i}\)(A1)
\(v = \frac{2-11i}{1-3i} \times \frac{1+3i}{1+3i}\)(M1) FT their \(v\), M0 no working
\(v = \frac{35-5i}{10}\left(= \frac{7-i}{2}\right)\), \(v = 3.5-0.5i\)(A1) oe cao
Method 3:
AnswerMarks Guidance
WorkingMark Guidance
Attempt to realise at least one fraction e.g. \(\frac{1}{2-i} \times \frac{2+i}{2+i}\) OR \(\frac{1}{3-4i} \times \frac{3+4i}{3+4i}\)(M1) M0 no working
\(\frac{1}{v} = \frac{2+i}{5} - \frac{3+4i}{25}\)
\(\frac{1}{v} = \frac{7+i}{25}\)(A1)
\(v = \frac{25}{7+i}\)(A1)
\(v = \frac{25}{7+i} \times \frac{7-i}{7-i}\)(M1) FT their \(v\), M0 no working
\(v = \frac{35-5i}{10}\left(= \frac{7-i}{2}\right)\), \(v = 3.5-0.5i\)(A1) oe cao
[5]
Part c):
AnswerMarks Guidance
WorkingMark Guidance
\(\bar{v} = \frac{7+i}{2}\)B1 FT their \(v\) provided complex
\(v\bar{v} = \frac{7-i}{2} \times \frac{7+i}{2} = \frac{25}{2}\)B1 oe
[2]
[12] 
# Question 1:

## Part a(i):

**Method 1:**

| Working | Mark | Guidance |
|---------|------|----------|
| $zw = (3-4i)(2-i) = 6-3i-8i+4i^2$, $zw = 2-11i$ | B2 | B1 for unsimplified expansion with 3 correct terms |
| $\|zw\| = \sqrt{2^2+(-11)^2} = 5\sqrt{5}$ | B1 | FT their $zw$ ($zw$ must be seen) |
| $\arg zw = \tan^{-1}\left(-\frac{11}{2}\right) = -1.39$ or $-79.7°$ | B1 | oe FT their $zw$ if not in 1st quadrant |

**Method 2:**

| Working | Mark | Guidance |
|---------|------|----------|
| $\|z\| = \sqrt{3^2+(-4)^2} = 5$, $\|w\| = \sqrt{2^2+(-1)^2} = \sqrt{5}$ | (B1) | Both mods |
| $\arg z = \tan^{-1}\left(-\frac{4}{3}\right) = -0.927$ or $-53.13°$, $\arg w = \tan^{-1}\left(-\frac{1}{2}\right) = -0.464$ or $-26.57°$ | (B1) | Both args oe |
| $\|zw\| = 5 \times \sqrt{5} = 5\sqrt{5}$ | (B1) | FT args and mods |
| $\arg zw = -0.927 + -0.464 = -1.39$ or $-79.7°$ | (B1) | oe FT args and mods (mods and args must be seen) |
| | [4] | |

## Part a(ii):

| Working | Mark | Guidance |
|---------|------|----------|
| $5\sqrt{5}(\cos(-1.39) + i\sin(-1.39))$ OR $5\sqrt{5}(\cos(-79.7°) + i\sin(-79.7°))$ | B1 | oe FT their mod and arg |
| | [1] | |

## Part b):

**Method 1:**

| Working | Mark | Guidance |
|---------|------|----------|
| $\frac{1}{v} = \frac{1}{2-i} - \frac{1}{3-4i}$ | | |
| $\frac{1}{v} = \frac{3-4i-2+i}{(3-4i)(2-i)}$ | M1 | Attempt to combine |
| $\frac{1}{v} = \frac{1-3i}{2-11i}$ | A1 | |
| $v = \frac{2-11i}{1-3i}$ | A1 | |
| $v = \frac{2-11i}{1-3i} \times \frac{1+3i}{1+3i}$ | M1 | FT their $v$, M0 for no working |
| $v = \frac{35-5i}{10}\left(= \frac{7-i}{2}\right)$ | | |
| $v = 3.5 - 0.5i$ | A1 | oe cao |

**Method 2:**

| Working | Mark | Guidance |
|---------|------|----------|
| $\frac{1}{v} = \frac{z-w}{zw}$ | (M1) | Attempt to combine |
| $v = \frac{zw}{z-w}$ or $\frac{1}{v} = \frac{1-3i}{2-11i}$ | (A1) | |
| $v = \frac{2-11i}{1-3i}$ | (A1) | |
| $v = \frac{2-11i}{1-3i} \times \frac{1+3i}{1+3i}$ | (M1) | FT their $v$, M0 no working |
| $v = \frac{35-5i}{10}\left(= \frac{7-i}{2}\right)$, $v = 3.5-0.5i$ | (A1) | oe cao |

**Method 3:**

| Working | Mark | Guidance |
|---------|------|----------|
| Attempt to realise at least one fraction e.g. $\frac{1}{2-i} \times \frac{2+i}{2+i}$ OR $\frac{1}{3-4i} \times \frac{3+4i}{3+4i}$ | (M1) | M0 no working |
| $\frac{1}{v} = \frac{2+i}{5} - \frac{3+4i}{25}$ | | |
| $\frac{1}{v} = \frac{7+i}{25}$ | (A1) | |
| $v = \frac{25}{7+i}$ | (A1) | |
| $v = \frac{25}{7+i} \times \frac{7-i}{7-i}$ | (M1) | FT their $v$, M0 no working |
| $v = \frac{35-5i}{10}\left(= \frac{7-i}{2}\right)$, $v = 3.5-0.5i$ | (A1) | oe cao |
| | [5] | |

## Part c):

| Working | Mark | Guidance |
|---------|------|----------|
| $\bar{v} = \frac{7+i}{2}$ | B1 | FT their $v$ provided complex |
| $v\bar{v} = \frac{7-i}{2} \times \frac{7+i}{2} = \frac{25}{2}$ | B1 | oe |
| | [2] | |
| | **[12]** | |

---
\begin{enumerate}
  \item The complex numbers $z , w$ are given by $z = 3 - 4 \mathrm { i } , w = 2 - \mathrm { i }$.\\
(a) (i) Find the modulus and argument of $z w$.\\
(ii) Express $z w$ in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$.\\
(b) The complex numbers $v , w , z$ satisfy the equation $\frac { 1 } { v } = \frac { 1 } { w } - \frac { 1 } { z }$. Find $v$ in the form $a + \mathrm { i } b$, where $a , b$ are real.\\
(c) The complex conjugate of $v$ is denoted by $\bar { v }$.
\end{enumerate}

Show that $v \bar { v } = k$, where $k$ is a real number whose value is to be determined.\\

\hfill \mbox{\textit{WJEC Further Unit 1 2022 Q1 [12]}}
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