Question 3(c) Merit

Five separate parts: a fraction of surds simplified, a complex number squared and plotted, an equation in a number and its conjugate, the fourth roots of a negative number, and a cube of x + 1/x.

Find the value(s) of the real constant d, given that z = 3 + di and z̄ = 10dz⁻¹.

The 2024 paper, page 10 ↗ Marking schedule, page 5 ↗

Work it out

Try it on paper first. Stuck? Take one hint at a time.

1. 1 Multiply both sides by z to clear the z⁻¹.

   zz̄ = 10d

2. 2 What is zz̄ for z = 3 + di?

   (3 + di)(3 − di) = 9 − d²i² = 9 + d². It is always |z|², a real number.
   9 + d² = 10d

3. 3 Solve the quadratic.

   d² − 10d + 9 = 0, so (d − 1)(d − 9) = 0 and d = 1 or d = 9.

4. 4 Check one.

   d = 1: z = 3 + i, and 10/(3 + i) = (10(3 − i))/10 = 3 − i = z̄ ✓

What the marker wanted

• Achieved 9 + d² = 10d, or the same equation still with i² in it.
• Merit Both values of d, with the working that gets them.

In our words, from NZQA’s 2024 marking schedule.

How did it go?

Got it moves it further out. Review brings it back tomorrow.