← Complex numbers: past papers

AS 91577 · 2023 paper · dividing in polar form, a modulus in k, an argument, cube roots, and a tangent to a circle

Question 2(c) Merit

Practises Argand diagram & polar form

Five separate parts: a quotient of two numbers in polar form, the modulus of a difference, an argument found from an equation, the cube roots of a negative number, and a line that touches a circular locus.

Find Arg(z) if 13z/(z + 1) = 11 − 3i.

The 2023 paper, page 5 ↗ Marking schedule, page 3 ↗

Work it out

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

  1. 1 Find z first. Clear the fraction.

    13z = (11 − 3i)(z + 1) = 11z − 3iz + 11 − 3i

  2. 2 Collect the z terms and factorise.

    2z + 3iz = 11 − 3i, so z(2 + 3i) = 11 − 3i and z = (11 − 3i)/(2 + 3i).

  3. 3 Divide with the conjugate.

    z = ((11 − 3i)(2 − 3i))/13 = (22 − 33i − 6i + 9i2)/13 = (13 − 39i)/13 = 1 − 3i

  4. 4 Where is 1 − 3i? Sketch it before you use a calculator.

    Real part positive, imaginary part negative: the fourth quadrant, so the argument is negative.

  5. 5 Find the angle.

    tan−1(3/1) = 1.249 rad below the real axis, so Arg(z) = −1.249 rad.
Answer Arg(z) = −1.249 rad (−71.6°), with z = 1 − 3i.

What the marker wanted

  • Achieved z on its own: z = (11 − 3i)/(2 + 3i).
  • Merit The value of Arg z. The schedule accepts −1.25 rad, −71.6°, or 288.4°.

In our words, from NZQA’s 2023 marking schedule.

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Common mistake

Putting the point in the wrong quadrant. NZQA’s 2023 report, in its general comments on the paper, says students struggled to find the size and angle of a number lying outside the first quadrant, and recommends a sketch of the Argand diagram rather than trusting the calculator.

How did it go?

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