Transformations from the z-plane to the w-planeEdexcel International A Level Further Maths: Subtopic test

1. 1
   The transformation $T$ from the $z$-plane to the $w$-plane is given by $w=z^2$, where $z=x+iy$ and $w=u+iv$.

   (a)

   Find the image of the point $z=3-2i$.

   [1 mark]

   • A$5+12i$
   • B$5-12i$
   • C$13-12i$
   • D$9+4i$

   (b)

   The circle $|z|=3$ in the $z$-plane is mapped by $T$ to a circle in the $w$-plane. Which equation gives this circle?

   [1 mark]

   • A$|w|=6$
   • B$|w|=3$
   • C$|w|=9$
   • D$|w|=81$

   (c)

   Find a Cartesian equation for the image of the line $x=2$ under $T$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The transformation $T$ from the $z$-plane to the $w$-plane is given by $w=(1+i)z-2$.

   (a)

   Find the image of the point $z=1+i$.

   [1 mark]

   • A$-2+2i$
   • B$2i$
   • C$-2$
   • D$2+2i$

   (b)

   Which of the following describes $T$ as a combination of elementary transformations?

   [1 mark]

   • AAn enlargement of scale factor $2$ and an anticlockwise rotation of $\frac{\pi}{4}$ about the origin, followed by a translation of $-2$
   • BAn enlargement of scale factor $\sqrt2$ and a clockwise rotation of $\frac{\pi}{4}$ about the origin, followed by a translation of $-2$
   • CAn enlargement of scale factor $\sqrt2$ and an anticlockwise rotation of $\frac{\pi}{2}$ about the origin, followed by a translation of $-2$
   • DAn enlargement of scale factor $\sqrt2$ and an anticlockwise rotation of $\frac{\pi}{4}$ about the origin, followed by a translation of $-2$

   (c)

   The circle $|z|=1$ is mapped by $T$ to a circle in the $w$-plane. Find its centre and radius.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The transformation $T$ from the $z$-plane to the $w$-plane is given by $w=\frac{z+i}{z-i}$, $z\neq i$, where $z=x+iy$ and $w=u+iv$.

   (a)

   Show that the image of the real axis ($y=0$) lies on the circle $|w|=1$.

   [3 marks]

   (b)

   Find the image of the line $y=1$ and describe it geometrically.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The transformation $T$ maps the $z$-plane to the $w$-plane, where $w=\frac{1}{z-2}$, with $z=x+iy$ and $w=u+iv$. The circle $C$ in the $z$-plane has equation $|z-1|=1$.

   (a)

   Find the image of $C$ under $T$, giving its Cartesian equation, and explain why it is a straight line.

   [6 marks]

   (b)

   Find the image under $T$ of the region $R=\{z:\ |z-1|<1\ \text{and}\ \mathrm{Im}\,z>0\}$, stating the inequalities satisfied by $u$ and $v$.

   [6 marks]

   Total for question 4: 12 marks

End of questions