Complex roots of quadratic equationsEdexcel International A Level Further Maths: Subtopic test

1. 1
   The quadratic equation $z^2-6z+13=0$.

   (a)

   Find the value of the discriminant $b^2-4ac$.

   [1 mark]

   • A$16$
   • B$-16$
   • C$88$
   • D$4$

   (b)

   Find the roots of the equation.

   [1 mark]

   • A$3\pm4\mathrm{i}$
   • B$-3\pm2\mathrm{i}$
   • C$6\pm4\mathrm{i}$
   • D$3\pm2\mathrm{i}$

   (c)

   The equation $z^2+kz+13=0$, where $k$ is a real constant, has no real roots. Find the range of possible values of $k$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The quadratic equation $z^2+4z+29=0$.

   (a)

   Find the roots of the equation.

   [1 mark]

   • A$-2\pm5\mathrm{i}$
   • B$2\pm5\mathrm{i}$
   • C$-4\pm10\mathrm{i}$
   • D$-2\pm25\mathrm{i}$

   (b)

   Find the modulus of each root.

   [1 mark]

   • A$29$
   • B$7$
   • C$\sqrt{29}$
   • D$3$

   (c)

   Show that $z=-2+5\mathrm{i}$ satisfies the equation.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The quadratic equation $2z^2-6z+5=0$.

   (a)

   Solve the equation, giving the roots in the form $p+q\mathrm{i}$.

   [3 marks]

   (b)

   For the root with positive imaginary part, find the modulus and the principal argument in radians to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The quadratic function $\text{f}(z)=z^2-4z+k$, where $k$ is a real constant.

   (a)

   (i) Show that the equation $\text{f}(z)=0$ has no real roots when $k>4$.
   (ii) Given that $k=13$, solve $\text{f}(z)=0$.

   [6 marks]

   (b)

   For $k>4$, the roots of $\text{f}(z)=0$ are represented by the points $P$ and $Q$ on an Argand diagram. Given that $PQ=8$, find the value of $k$ and the modulus of each root.

   [6 marks]

   Total for question 4: 12 marks

End of questions