Further Pure 2: Further complex numbersEdexcel A-Level Further Maths: Topic test

1. 1
   The locus of points $z$ in the Argand diagram satisfying $|z-3|=2|z|$ is a circle $C$.

   (a)

   Which of these complex numbers lies on $C$?

   [1 mark]

   • A$3$
   • B$2\mathrm{i}$
   • C$1$
   • D$-1$

   (b)

   What is the greatest value of $|z|$ for points $z$ on $C$?

   [1 mark]

   • A$2$
   • B$3$
   • C$4$
   • D$1$

   (c)

   By writing $z=x+\mathrm{i}y$, show that $C$ has Cartesian equation $(x+1)^2+y^2=4$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The transformation $T$ from the $z$-plane to the $w$-plane is given by $w=\frac{z+1}{z-2}$, $z\neq2$.

   (a)

   What is the image of $z=0$ under $T$?

   [1 mark]

   • A$\frac12$
   • B$-2$
   • C$2$
   • D$-\frac12$

   (b)

   Which value of $z$ is mapped to $w=3$?

   [1 mark]

   • A$\frac72$
   • B$\frac52$
   • C$\frac{11}{2}$
   • D$-\frac72$

   (c)

   Express $z$ in terms of $w$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The locus $L$ in the Argand diagram is given by $\arg\left(\frac{z-3}{z+1}\right)=\frac{\pi}{4}$.

   (a)

   By writing $z=x+\mathrm{i}y$, show that $L$ lies on the circle with equation $(x-1)^2+(y-2)^2=8$.

   [3 marks]

   (b)

   Find the exact greatest value of $|z|$ for points $z$ on $L$, justifying that the point concerned lies on $L$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The transformation $T$ from the $z$-plane to the $w$-plane is given by $w=\frac{z}{z-4}$, $z\neq4$.

   (a)

   The line $l$ in the $z$-plane has equation $\mathrm{Re}(z)=2$. Show that the image of $l$ under $T$ is the circle $|w|=1$, excluding $w=1$.

   [6 marks]

   (b)

   Using the result $z=\frac{4w}{w-1}$, find the image under $T$ of the circle $|z-2|=2$, excluding $z=4$, and verify your answer using the point $z=2+2\mathrm{i}$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The transformation $T$ from the $z$-plane to the $w$-plane is given by $w=z^2$. The region $R$ in the $z$-plane is defined by $0\leq\arg z\leq\frac{\pi}{6}$ and $|z|\leq3$.

   (a)

   The point $z=\sqrt3+\mathrm{i}$ lies on the boundary of $R$. What is its image under $T$?

   [1 mark]

   • A$3+\mathrm{i}$
   • B$2+2\mathrm{i}$
   • C$4\mathrm{i}$
   • D$2+2\sqrt3\,\mathrm{i}$

   (b)

   Under $T$, the ray $\arg z=\frac{\pi}{6}$ is mapped onto which ray?

   [1 mark]

   • A$\arg w=\frac{\pi}{12}$
   • B$\arg w=\frac{\pi}{3}$
   • C$\arg w=\frac{\pi}{6}$
   • D$\arg w=\frac{\pi}{2}$

   (c)

   Describe the image of $R$ in the $w$-plane.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The region $R$ in the Argand diagram is defined by $|z-3+2\mathrm{i}|\leq2$ and $\mathrm{Re}(z)\geq3$.

   (a)

   Which of these complex numbers lies in $R$?

   [1 mark]

   • A$4-\mathrm{i}$
   • B$1-2\mathrm{i}$
   • C$3+\mathrm{i}$
   • D$2-2\mathrm{i}$

   (b)

   What is the greatest value of $\mathrm{Im}(z)$ for points in $R$?

   [1 mark]

   • A$-4$
   • B$-2$
   • C$0$
   • D$2$

   (c)

   Find the exact area of $R$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The transformation $T$ from the $z$-plane to the $w$-plane is given by $w=\frac{3}{z-2}$, $z\neq2$, where $w=u+\mathrm{i}v$.

   (a)

   The line $\mathrm{Re}(z)=3$ is mapped by $T$ to a curve in the $w$-plane. Show that the curve has equation $u^2+v^2=3u$.

   [3 marks]

   (b)

   Hence find the centre and radius of the circle, and verify that the image of $z=3+\mathrm{i}$ lies on it.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The locus of points $z$ satisfying $|z-6|=2|z-3|$ is a circle $C$. The region $R$ is defined by $|z-6|\geq2|z-3|$ and $0\leq\arg z\leq\frac{\pi}{4}$.

   (a)

   Show that $C$ has Cartesian equation $(x-2)^2+y^2=4$, and show that the inequality $|z-6|\geq2|z-3|$ describes the points inside or on $C$.

   [6 marks]

   (b)

   Find the exact area of $R$.

   [6 marks]

   Total for question 8: 12 marks

End of questions