FP2: Further complex numbersEdexcel International A Level Further Maths: Topic test

1. 1
   The complex number $z=\sqrt2\,\mathrm{e}^{-\mathrm{i}\pi/4}$.

   (a)

   Write $z$ in the form $x+\mathrm{i}y$.

   [1 mark]

   • A$1+\mathrm{i}$
   • B$\sqrt2-\sqrt2\,\mathrm{i}$
   • C$-1-\mathrm{i}$
   • D$1-\mathrm{i}$

   (b)

   Find $z^4$.

   [1 mark]

   • A$4$
   • B$-4$
   • C$-4\sqrt2$
   • D$-4\mathrm{i}$

   (c)

   Write $\frac{z^*}{z}$ in the form $\mathrm{e}^{\mathrm{i}\theta}$, where $z^*$ is the complex conjugate of $z$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the equation $z^4=-16$.

   (a)

   Find the modulus of each root of the equation.

   [1 mark]

   • A$2$
   • B$4$
   • C$16$
   • D$8$

   (b)

   Which of the following is a root of the equation?

   [1 mark]

   • A$2\mathrm{i}$
   • B$-2$
   • C$-\sqrt2+\sqrt2\,\mathrm{i}$
   • D$1+\mathrm{i}$

   (c)

   Show that the sum of the four roots of the equation is zero.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $z=\mathrm{e}^{\mathrm{i}\theta}$, where $\theta$ is real. For any integer $n$, $z^n+z^{-n}=2\cos n\theta$.

   (a)

   Expand $\left(z+\frac1z\right)^4$ and hence show that $16\cos^4\theta=2\cos4\theta+8\cos2\theta+6$.

   [3 marks]

   (b)

   Hence find the exact value of $\int_0^{\pi/2}\cos^4\theta\,\mathrm{d}\theta$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The locus $C$ in the $z$-plane has equation $|z-2|=2$. The transformation $T$ maps the $z$-plane to the $w$-plane by $w=\frac{z}{z-4}$, where $z\neq4$, $w=u+\mathrm{i}v$ and $u$, $v$ are real.

   (a)

   Show that $T$ maps $C$ onto a straight line in the $w$-plane, and find the equation of this line.

   [6 marks]

   (b)

   Find the image of the point $z=2+2\mathrm{i}$, and determine the region of the $w$-plane onto which $T$ maps the interior $|z-2|<2$ of $C$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The complex number $z=x+\mathrm{i}y$ satisfies $\arg\left(\frac{z-1}{z+1}\right)=\frac\pi2$.

   (a)

   Which of the following describes the locus of $z$?

   [1 mark]

   • AThe whole circle $|z|=1$
   • BThe line segment from $-1$ to $1$
   • CThe part of the circle $|z|=1$ above the real axis, excluding $z=\pm1$
   • DThe part of the circle $|z|=1$ below the real axis, excluding $z=\pm1$

   (b)

   Which of the following points lies on the locus?

   [1 mark]

   • A$z=\mathrm{i}$
   • B$z=-\mathrm{i}$
   • C$z=2$
   • D$z=1+\mathrm{i}$

   (c)

   Show that the locus lies on the circle $x^2+y^2=1$, and state the condition on $y$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The complex number $z=-2\sqrt3-2\mathrm{i}$.

   (a)

   Find the principal argument of $z$.

   [1 mark]

   • A$-\frac{\pi}{6}$
   • B$-\frac{5\pi}{6}$
   • C$\frac{5\pi}{6}$
   • D$\frac{7\pi}{6}$

   (b)

   Find $z^6$.

   [1 mark]

   • A$4096$
   • B$24$
   • C$4096\,\mathrm{i}$
   • D$-4096$

   (c)

   Express $z^3$ in the form $x+\mathrm{i}y$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Consider the equation $z^6=1$ over the complex numbers.

   (a)

   Find all six roots, giving each in the form $x+\mathrm{i}y$.

   [3 marks]

   (b)

   Hence factorise $z^6-1$ into linear and quadratic factors with real coefficients.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Let $w=1+\mathrm{e}^{\mathrm{i}\theta}$, where $-\pi<\theta<\pi$.

   (a)

   Show that $w=2\cos\frac\theta2\,\mathrm{e}^{\mathrm{i}\theta/2}$, and hence write down $|w|$ and $\arg w$.

   [6 marks]

   (b)

   Hence show that $1+5\cos\theta+10\cos2\theta+10\cos3\theta+5\cos4\theta+\cos5\theta=32\cos^5\frac\theta2\cos\frac{5\theta}2$.

   [6 marks]

   Total for question 8: 12 marks

End of questions