Plot the position of $z$ on an Argand Diagram.

Express $z$ in the form $z=a{\text{e}}^{\text{i}b}$, where $a, b\in \mathrm{ℝ}$, giving the exact value of $a$ and the exact value of $b$.

Find $w_1+w_2$ in the form ${\text{e}}^{\text{i}x}\left(c+\text{i}d\right)$.

Hence find $\text{Re}\left(w_1+w_2\right)$ in the form $A \cos \left(x-a\right)$, where $A>0$ and $0<a\le \frac{\mathrm{\pi }}{2}$.

Find the maximum value of $I_{\text{T}}$.

Find the phase shift of $I_{\text{T}}$.

         A1

[1 mark]

$z=\sqrt{2}{\text{e}}^{\frac{\text{i}\mathrm{\pi }}{4}}$          A1A1

Note: Accept an argument of $\frac{7\mathrm{\pi }}{4}$. Do NOT accept answers that are not exact.

[2 marks]

$w_1+w_2={\text{e}}^{\text{i}x}+{\text{e}}^{\text{i}\left(x-\frac{\mathrm{\pi }}{2}\right)}$

                $={\text{e}}^{\text{i}x}\left(1+{\text{e}}^{-\frac{\text{i}\mathrm{\pi }}{2}}\right)$          (M1)

                $={\text{e}}^{\text{i}x}\left(1-\text{i}\right)$          A1

[2 marks]

$w_1+w_2={\text{e}}^{\text{i}x}\times \sqrt{2}{\text{e}}^{-\frac{\text{i}\mathrm{\pi }}{4}}$           M1

$=\sqrt{2}{\text{e}}^{\text{i}\left(x-\frac{\mathrm{\pi }}{4}\right)}$           (A1)

attempt extract real part using cis form           (M1)

$\text{Re}\left(w_1+w_2\right)=\sqrt{2}\cos \left(x-\frac{\mathrm{\pi }}{4}\right)$  OR  $1.4142\dots \cos \left(x-0.785398\dots \right)$           A1

[4 marks]

$I_t=12 \cos \left(bt\right)+12 \cos \left(bt-\frac{\mathrm{\pi }}{2}\right)$           (M1)

$I_t=12\text{\hspace{0.05em}Re}\left({\text{e}}^{\text{i}bt}+{\text{e}}^{\text{i}\left(bt-\frac{\mathrm{\pi }}{2}\right)}\right)$           (M1)

$I_t=12\sqrt{2} \cos \left(bt-\frac{\mathrm{\pi }}{4}\right)$

max $=12\sqrt{2} \left(=17.0\right)$           A1

[3 marks]

phase shift $=\frac{\mathrm{\pi }}{4} \left(=0.785\right)$           A1

[1 mark]

This question produced the weakest set of responses on the paper. There seemed a general lack of confidence when tackling a problem involving complex numbers. Whilst most candidates could represent a complex number on the complex plane, far fewer had the ability to move between the different forms of complex numbers. This is clearly an area of the course that needs more attention when being taught. Part (c) is challenging but it should be noted that a candidate who has answered parts (a) and (b) with confidence should find this both straightforward, and also an example of a type of problem that is mentioned in the syllabus guidance.

This question produced the weakest set of responses on the paper. There seemed a general lack of confidence when tackling a problem involving complex numbers. Whilst most candidates could represent a complex number on the complex plane, far fewer had the ability to move between the different forms of complex numbers. This is clearly an area of the course that needs more attention when being taught. Part (c) is challenging but it should be noted that a candidate who has answered parts (a) and (b) with confidence should find this both straightforward, and also an example of a type of problem that is mentioned in the syllabus guidance.

This question produced the weakest set of responses on the paper. There seemed a general lack of confidence when tackling a problem involving complex numbers. Whilst most candidates could represent a complex number on the complex plane, far fewer had the ability to move between the different forms of complex numbers. This is clearly an area of the course that needs more attention when being taught. Part (c) is challenging but it should be noted that a candidate who has answered parts (a) and (b) with confidence should find this both straightforward, and also an example of a type of problem that is mentioned in the syllabus guidance.

This question produced the weakest set of responses on the paper. There seemed a general lack of confidence when tackling a problem involving complex numbers. Whilst most candidates could represent a complex number on the complex plane, far fewer had the ability to move between the different forms of complex numbers. This is clearly an area of the course that needs more attention when being taught. Part (c) is challenging but it should be noted that a candidate who has answered parts (a) and (b) with confidence should find this both straightforward, and also an example of a type of problem that is mentioned in the syllabus guidance.

This question produced the weakest set of responses on the paper. There seemed a general lack of confidence when tackling a problem involving complex numbers. Whilst most candidates could represent a complex number on the complex plane, far fewer had the ability to move between the different forms of complex numbers. This is clearly an area of the course that needs more attention when being taught. Part (c) is challenging but it should be noted that a candidate who has answered parts (a) and (b) with confidence should find this both straightforward, and also an example of a type of problem that is mentioned in the syllabus guidance.

This question produced the weakest set of responses on the paper. There seemed a general lack of confidence when tackling a problem involving complex numbers. Whilst most candidates could represent a complex number on the complex plane, far fewer had the ability to move between the different forms of complex numbers. This is clearly an area of the course that needs more attention when being taught. Part (c) is challenging but it should be noted that a candidate who has answered parts (a) and (b) with confidence should find this both straightforward, and also an example of a type of problem that is mentioned in the syllabus guidance.