Complex transformations and mappings

A question is this type if and only if it involves a transformation w = f(z) mapping loci from the z-plane to the w-plane, requiring finding the image locus equation.

28 questions · Standard +1.0

4.02k Argand diagrams: geometric interpretation

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Edexcel FP2 Q22

10 marks Standard +0.8

1. On the same Argand diagram sketch the loci given by the following equations. $$|z - 1| = 1,$$ $$\arg(z + 1) = \frac{\pi}{12},$$ $$\arg(z + 1) = \frac{\pi}{2}.$$ [4]
2. Shade on your diagram the region for which $$|z - 1| \leq 1 \quad \text{and} \quad \frac{\pi}{12} \leq \arg(z + 1) \leq \frac{\pi}{2}.$$ [1]
1. Show that the transformation $$w = \frac{z - 1}{z}, \quad z \neq 0,$$ maps $|z - 1| = 1$ in the $z$-plane onto $|w| = |w - 1|$ in the $w$-plane. [3] The region $|z - 1| \leq 1$ in the $z$-plane is mapped onto the region $T$ in the $w$-plane.
2. Shade the region $T$ on an Argand diagram. [2]

Edexcel FP2 Q46

11 marks Standard +0.3

A complex number $z$ is represented by the point $P$ in the Argand diagram. Given that $$|z - 3i| = 3,$$

sketch the locus of $P$. [2]
Find the complex number $z$ which satisfies both $|z - 3i| = 3$ and $\arg (z - 3i) = \frac{3}{4}\pi$. [4] The transformation $T$ from the $z$-plane to the $w$-plane is given by $$w = \frac{2i}{w}.$$
Show that $T$ maps $|z - 3i| = 3$ to a line in the $w$-plane, and give the cartesian equation of this line. [5]

WJEC Further Unit 1 Specimen Q7

9 marks Standard +0.3

The complex numbers $z$ and $w$ are represented, respectively, by points $P(x, y)$ and $Q(u,v)$ in Argand diagrams and $$w = z(1 + z)$$

Show that $$v = y(1 + 2x)$$ and find an expression for $u$ in terms of $x$ and $y$. [4]
The point $P$ moves along the line $y = x + 1$. Find the Cartesian equation of the locus of $Q$, giving your answer in the form $v = au^2 + bu$, where $a$ and $b$ are constants whose values are to be determined. [5]