Question 4 · A spiral on the Argand diagram
Rotation and enlargement as multiplication by $0.9\,e^{\frac{\pi}{6}i}$; expression for $z_n$; inequality with logarithms; positive imaginary axis
Maximum mark · 10 marksOn an Argand diagram, the point $P_0$ represents the complex number $z_0 = 6$. Every second the point rotates through an angle of $\dfrac{\pi}{6}$ radians about the origin, anticlockwise, and its distance from the origin is multiplied by $0.9$. Let $z_n$ be the complex number representing the position of the point after $n$ seconds.
- Write $z_1$ in Euler form and in Cartesian form. 3 p
- Write down an expression for $z_n$ in terms of $n$. 3 p
- Find the first time at which the point is less than $1$ unit from the origin. 2 p
- Find all the times at which the point lies on the positive imaginary axis. 2 p
Step-by-step solution
a) The first step
Euler form 2 marks, Cartesian form 1 mark.
b) Step $n$
Every second the modulus is multiplied by $0.9$ and the argument increases by $\frac{\pi}{6}$:
Modulus 1 mark, argument 1 mark, full expression 1 mark.
c) Less than one unit away
Dividing by $\ln 0.9 < 0$ reverses the inequality. Check: $6\,(0.9)^{17} \approx 1.0006$ is not yet below $1$, while $6\,(0.9)^{18} \approx 0.90$ is.
Setting up 1 mark, integer answer 1 mark.
d) On the positive imaginary axis
The argument must be $\frac{\pi}{2}$ plus a whole number of turns:
Equation 1 mark, general solution 1 mark.