Pre-U Pre-U 9794/1 2015 June — Question 8 11 marks

Exam BoardPre-U
ModulePre-U 9794/1 (Pre-U Mathematics Paper 1)
Year2015
SessionJune
Marks11
TopicComplex Numbers Arithmetic
TypeDivision plus other arithmetic operations
DifficultyModerate -0.3 This is a straightforward multi-part question testing standard complex number operations: division by multiplying by conjugate, plotting on Argand diagram, finding argument using arctan, and algebraic manipulation. All parts are routine textbook exercises requiring only direct application of standard techniques with no problem-solving insight needed. Slightly easier than average due to simple arithmetic throughout.
Spec4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation
8 The complex numbers \(w\) and \(z\) are given by \(w = 3 - \mathrm { i }\) and \(z = 1 + \mathrm { i }\).
  1. Express \(\frac { z } { w }\) in the form \(p + \mathrm { i } q\) where \(p\) and \(q\) are real numbers.
  2. On the same Argand diagram, mark the points representing \(z , w\) and \(\frac { z } { w }\).
  3. Find the value in radians of \(\arg w\).
  4. Show that \(z + \frac { 2 } { z }\) is a real number.
(i)
Show or imply multiplication of denominator and numerator by \((3 + \mathrm{i})\) [M1]
Obtain either \(10\) for the denominator OR \(2 + 4\mathrm{i}\) for the numerator [A1]
Obtain \(\dfrac{2+4\mathrm{i}}{10}\) or simplified equivalent, e.g. \(0.2 + 0.4\mathrm{i}\), \(\dfrac{1+2\mathrm{i}}{5}\) [A1]
Subtotal: [3]
(ii)
Show relative position of:
\(z = (1, 1)\) [B1]
\(w = (3, -1)\) [B1]
\(z/w = (0.2, 0.4)\) [B1]
Subtotal: [3]
(iii)
Use \(\tan^{-1}\left(\pm\dfrac{1}{3}\right)\) or equivalent, e.g. \(\sin^{-1}\left(\pm\dfrac{1}{\sqrt{10}}\right)\) [M1]
Obtain \(-0.322\) or \(5.96\) [A1]
Subtotal: [2]
(iv)
State or imply \((1+\mathrm{i}) + \dfrac{2}{(1+\mathrm{i})}\) [B1]
Form LCM or multiply fraction by conjugate \(\left(= \dfrac{(1+\mathrm{i})^2 + 2}{1+\mathrm{i}}\right)\) [M1]
Obtain \(2\) AND state "real". CWO with all steps shown [A1]
Subtotal: [3]
Total: [11]
**(i)**
Show or imply multiplication of denominator and numerator by $(3 + \mathrm{i})$ [M1]
Obtain either $10$ for the denominator OR $2 + 4\mathrm{i}$ for the numerator [A1]
Obtain $\dfrac{2+4\mathrm{i}}{10}$ or simplified equivalent, e.g. $0.2 + 0.4\mathrm{i}$, $\dfrac{1+2\mathrm{i}}{5}$ [A1]
**Subtotal: [3]**

**(ii)**
Show relative position of:
$z = (1, 1)$ [B1]
$w = (3, -1)$ [B1]
$z/w = (0.2, 0.4)$ [B1]
**Subtotal: [3]**

**(iii)**
Use $\tan^{-1}\left(\pm\dfrac{1}{3}\right)$ or equivalent, e.g. $\sin^{-1}\left(\pm\dfrac{1}{\sqrt{10}}\right)$ [M1]
Obtain $-0.322$ or $5.96$ [A1]
**Subtotal: [2]**

**(iv)**
State or imply $(1+\mathrm{i}) + \dfrac{2}{(1+\mathrm{i})}$ [B1]
Form LCM or multiply fraction by conjugate $\left(= \dfrac{(1+\mathrm{i})^2 + 2}{1+\mathrm{i}}\right)$ [M1]
Obtain $2$ **AND** state "real". CWO with all steps shown [A1]
**Subtotal: [3]**
**Total: [11]**
8 The complex numbers $w$ and $z$ are given by $w = 3 - \mathrm { i }$ and $z = 1 + \mathrm { i }$.\\
(i) Express $\frac { z } { w }$ in the form $p + \mathrm { i } q$ where $p$ and $q$ are real numbers.\\
(ii) On the same Argand diagram, mark the points representing $z , w$ and $\frac { z } { w }$.\\
(iii) Find the value in radians of $\arg w$.\\
(iv) Show that $z + \frac { 2 } { z }$ is a real number.

\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2015 Q8 [11]}}
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