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    Question
    HLPaper 2
    1.

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

    [1]
    Verified
    Solution

    2.

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

    [1]
    Verified
    Solution

    z=2e−iπ4z=\sqrt{2} \mathrm{e}^{-\frac{i \pi}{4}}z=2​e−4iπ​ A1A1

    Note: Accept an argument of 7π4\frac{7 \pi}{4}47π​. Do NOT accept answers that are not exact.

    3.

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

    [1]
    Verified
    Solution

    w1+w2=eix+ei(x−π2)w_{1}+w_{2}=\mathrm{e}^{\mathrm{i} x}+\mathrm{e}^{\mathrm{i}\left(x-\frac{\pi}{2}\right)}w1​+w2​=eix+ei(x−2π​)

    =eix(1+e−i2)=e^{i x}\left(1+e^{-\frac{i}{2}}\right)=eix(1+e−2i​) =eix(1−i)=e^{i x}(1-i)=eix(1−i)

    4.

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

    [1]
    Verified
    Solution

    w1+w2=eix×2e−iπ4w_{1}+w_{2}=\mathrm{e}^{\mathrm{i} x} \times \sqrt{2} \mathrm{e}^{-\frac{\mathrm{i} \pi}{4}}w1​+w2​=eix×2​e−4iπ​ M1 =2ei(x−π4)=\sqrt{2} \mathrm{e}^{\mathrm{i}\left(x-\frac{\pi}{4}\right)}=2​ei(x−4π​) (A1) attempt extract real part using cis form Re⁡(w1+w2)=2cos⁡(x−π4)\operatorname{Re}\left(w_{1}+w_{2}\right)=\sqrt{2} \cos \left(x-\frac{\pi}{4}\right)Re(w1​+w2​)=2​cos(x−4π​) OR 1.4142…cos⁡(x−0.785398…)1.4142 \ldots \cos (x-0.785398 \ldots)1.4142…cos(x−0.785398…)

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    5.

    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.

    [1]
    Verified
    Solution

    It=12cos⁡(bt)+12cos⁡(bt−π2)I_{t}=12 \cos (b t)+12 \cos \left(b t-\frac{\pi}{2}\right)It​=12cos(bt)+12cos(bt−2π​) (M1) It=12Re⁡(eibt+ei(bt−π2))I_{t}=12 \operatorname{Re}\left(\mathrm{e}^{\mathrm{i} b t}+\mathrm{e}^{\mathrm{i}\left(b t-\frac{\pi}{2}\right)}\right)It​=12Re(eibt+ei(bt−2π​)) (M1)

    It=122cos⁡(bt−π4)I_{t}=12 \sqrt{2} \cos \left(b t-\frac{\pi}{4}\right)It​=122​cos(bt−4π​) max⁡=122(=17.0)\max =12 \sqrt{2}(=17.0)max=122​(=17.0) A1

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    6.

    Find the phase shift of I_T.

    [1]
    Verified
    Solution

    phase shift =π4(=0.785)=\frac{\pi}{4}(=0.785)=4π​(=0.785)

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