Further loci and regions in the Argand diagramEdexcel A-Level Further Maths: Subtopic test

1. 1
   The locus of points $z$ in the Argand diagram satisfying $|z-1|=2|z-4|$ is a circle $C$.

   (a)

   Which gives the centre and radius of $C$?

   [1 mark]

   • Acentre $5$, radius $4$
   • Bcentre $5$, radius $2$
   • Ccentre $3$, radius $2$
   • Dcentre $7$, radius $2$

   (b)

   Which complex number lies on $C$?

   [1 mark]

   • A$5+4i$
   • B$3+2i$
   • C$5+2i$
   • D$7+2i$

   (c)

   Find the greatest value of $|z|$ for points $z$ on $C$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The region $R$ of the Argand diagram is defined by $\frac{\pi}{6}\leq\arg(z-1)\leq\frac{\pi}{3}$ and $\mathrm{Re}(z)\leq3$.

   (a)

   Which describes the locus $\arg(z-1)=\frac{\pi}{3}$?

   [1 mark]

   • AA circle with centre $1$
   • BA full straight line through $1$ at $\frac{\pi}{3}$ to the real axis
   • CA half-line from $1$ at $\frac{\pi}{3}$ to the negative real direction
   • DA half-line starting at $1$ (not including $1$) at $\frac{\pi}{3}$ to the positive real direction

   (b)

   Which complex number lies in $R$?

   [1 mark]

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

   (c)

   Find the exact area of $R$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The locus $L$ is given by $\arg\left(\frac{z-2}{z+2}\right)=\frac{\pi}{2}$.

   (a)

   Show that $L$ is part of the circle $x^2+y^2=4$, stating which part, where $z=x+iy$.

   [3 marks]

   (b)

   Find the least value of $|z-5i|$ for $z$ on $L$, and the value of $z$ at which it occurs.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The region $R$ of the Argand diagram is defined by $|z-2|\leq|z-2i|$ and $|z|\leq4$.

   (a)

   Show that $|z-2|\leq|z-2i|$ is equivalent to $y\leq x$, where $z=x+iy$. Hence describe $R$ fully and find its area.

   [6 marks]

   (b)

   For $z\neq0$ in $R$, find the range of values of $\arg z$ and the greatest value of $\mathrm{Im}(z)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions