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

1. 1
   The transformation $T$ from the $z$-plane to the $w$-plane is given by $w=3z+2i$, where $z=x+iy$ and $w=u+iv$.

   (a)

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

   [1 mark]

   • A$3+3i$
   • B$1+3i$
   • C$3+5i$
   • D$5+3i$

   (b)

   Which of the following describes the image of the circle $|z|=2$?

   [1 mark]

   • Aa circle with centre $2i$ and radius $6$
   • Ba circle with centre $2i$ and radius $2$
   • Ca circle with centre $6i$ and radius $6$
   • Da circle with centre $0$ and radius $6$

   (c)

   Find the Cartesian equation of the image of the line $y=x$ in the $z$-plane.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

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

   [1 mark]

   • A$10-6i$
   • B$8-6i$
   • C$8+6i$
   • D$10$

   (b)

   Which of the following describes the image of the circle $|z|=3$?

   [1 mark]

   • Aa circle with centre $0$ and radius $3$
   • Ba circle with centre $0$ and radius $6$
   • Ca circle with centre $0$ and radius $\sqrt3$
   • Da circle with centre $0$ and radius $9$

   (c)

   Find the image of the line $y=x$ in the $w$-plane.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The transformation $T$ from the $z$-plane to the $w$-plane is given by $w=\frac{1}{z}$, where $z=x+iy$ and $w=u+iv$.

   (a)

   Show that the image of the line $x=1$ in the $z$-plane is a circle in the $w$-plane, and find its centre and radius.

   [3 marks]

   (b)

   The circle $|z-1|=1$ passes through the origin. Find the image of this circle in the $w$-plane.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   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)

   (i) Show that $z=\frac{i(w+1)}{w-1}$.
   (ii) Hence, or otherwise, show that the image of the real axis in the $z$-plane lies on the circle $|w|=1$.
   (iii) Find the image of the imaginary axis in the $z$-plane.

   [6 marks]

   (b)

   Find the image of the circle $|z|=3$ in the $w$-plane, giving its centre and radius.

   [6 marks]

   Total for question 4: 12 marks

End of questions