Modulus-Argument Form and Loci (AQA A-Level Further Maths): Quizzes

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Modulus-Argument Form and Loci
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What is ∣z∣|z| for z=3+4iz = 3 + 4i?

55

What range must the principal argument θ\theta satisfy?

−π<θ≤π-\pi < \theta \leq \pi

If ∣z1∣=3|z_1| = 3 and ∣z2∣=4|z_2| = 4, what is ∣z1z2∣|z_1z_2|?

1212

If arg⁡(z1)=π3\arg(z_1) = \frac{\pi}{3} and arg⁡(z2)=π6\arg(z_2) = \frac{\pi}{6}, what is arg⁡(z1z2)\arg(z_1z_2)?

π2\frac{\pi}{2}

What locus does ∣z−2+3i∣=5|z - 2 + 3i| = 5 represent?

Circle, centre (2,−3)(2,-3), radius 55

What locus does arg⁡(z−4i)=π4\arg(z - 4i) = \frac{\pi}{4} represent?

Half-line from (0,4)(0,4) at angle π4\frac{\pi}{4}

What locus does ∣z−1∣=∣z−5∣|z - 1| = |z - 5| represent?

Perpendicular bisector of segment from 11 to 55

What region does ∣z−3∣<2|z - 3| < 2 represent?

Interior of circle, centre (3,0)(3,0), radius 22

If arg⁡(w)=π6\arg(w) = \frac{\pi}{6}, what is arg⁡(−2w)\arg(-2w)?

7π6\frac{7\pi}{6}

For z=r(cos⁡θ+isin⁡θ)z = r(\cos\theta + i\sin\theta), what is the real part aa?

rcos⁡θr\cos\theta

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