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Question

HLPaper 2

Let z = 1 - i. Plot the position of z on an Argand Diagram.

[1]

Verified

Solution

Express z in the form $z = ae^{ib}$ , where a, b ∈ ℝ, giving the exact value of a and the exact value of b.

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Solution

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

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

Let $w₁ = e^{ix}$ and $w₂ = e^{i(x-π/2)}$ , where x ∈ ℝ. Find w₁ + w₂ in the form $e^{(ix)}(c + id)$ .

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Solution

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

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

Hence find $Re(w₁ + w₂)$ in the form $A \cos(x - α)$ , where $A > 0$ and $0 < α ≤ π/2$ .

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Solution

$w_{1}+w_{2}=\mathrm{e}^{\mathrm{i} x} \times \sqrt{2} \mathrm{e}^{-\frac{\mathrm{i} \pi}{4}}$ M1 $=\sqrt{2} \mathrm{e}^{\mathrm{i}\left(x-\frac{\pi}{4}\right)}$ (A1) attempt extract real part using cis form $\operatorname{Re}\left(w_{1}+w_{2}\right)=\sqrt{2} \cos \left(x-\frac{\pi}{4}\right)$ OR $1.4142 \ldots \cos (x-0.785398 \ldots)$

The current, I, in an AC circuit can be modelled by the equation $I = a \cos(bt - c)$ where b is the frequency and c is the phase shift.

Two AC voltage sources of the same frequency are independently connected to the same circuit. If connected to the circuit alone they generate currents $I_A$ and $I_B$ . The maximum value and the phase shift of each current is shown in the following table:

Current	Maximum value	Phase shift
$I_A$	12 amps	0
$I_B$	12 amps	π/2

When the two voltage sources are connected to the circuit at the same time, the total current $I_T$ can be expressed as $I_A + I_B$ .

Find the maximum value of $I_T$ .

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Solution

$I_{t}=12 \cos (b t)+12 \cos \left(b t-\frac{\pi}{2}\right)$ (M1) $I_{t}=12 \operatorname{Re}\left(\mathrm{e}^{\mathrm{i} b t}+\mathrm{e}^{\mathrm{i}\left(b t-\frac{\pi}{2}\right)}\right)$ (M1)

$I_{t}=12 \sqrt{2} \cos \left(b t-\frac{\pi}{4}\right)$ $\max =12 \sqrt{2}(=17.0)$ A1

Find the phase shift of $I_T$ .

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Verified

Solution

phase shift $=\frac{\pi}{4}(=0.785)$

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