1.12 Complex numbers: Cartesian formIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   Let $z=3+4i$ and $w=1-2i$.

   (a)

   Find $z+w$.

   [1 mark]

   • A$4+2i$
   • B$4+6i$
   • C$2+6i$
   • D$6$

   (b)

   Find the modulus of $z$.

   [1 mark]

   • A$7$
   • B$25$
   • C$\sqrt7$
   • D$5$

   (c)

   Find $\frac{z}{w}$, giving your answer in the form $a+bi$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $u=2-3i$ and $v=4+i$.

   (a)

   Find $uv$.

   [1 mark]

   • A$8-3i$
   • B$5-10i$
   • C$11-10i$
   • D$5+14i$

   (b)

   Find the complex conjugate of $u-v$.

   [1 mark]

   • A$-2-4i$
   • B$-2+4i$
   • C$2+4i$
   • D$6-2i$

   (c)

   Use your GDC to find $u^{4}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the quadratic equation $x^{2}-6x+25=0$.

   (a)

   Show that the equation has no real solutions.

   [3 marks]

   (b)

   Solve the equation, giving your answers in the form $a+bi$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $f(x)=x^{2}-4x+13$ is defined for real $x$, and the graph of $y=f(x)$ is a parabola.

   (a)

   (i) Find the discriminant of $x^{2}-4x+13$ and explain what it tells you about the graph of $y=f(x)$.
   (ii) Solve $f(x)=0$, giving your answers in the form $a+bi$.
   (iii) Write $f(x)$ in the form $(x-h)^{2}+k$, and hence write down the coordinates of the vertex of the graph.

   [6 marks]

   (b)

   The roots of $f(x)=0$ are $z_1=2+3i$ and $z_2=2-3i$.
   (i) Write down the real part and imaginary part of $z_1$, and find its modulus and its argument in radians to 3 significant figures.
   (ii) Find $z_1+z_2$ and $z_1z_2$, and compare them with $-\frac{b}{a}$ and $\frac{c}{a}$ for $f$.

   [6 marks]

   Total for question 4: 12 marks

End of questions