Question 6
Regions of the complex plane are now taught in Unit 2, so this won’t be directly examined in your Units 3 and 4 external exam. It is still good practice with modulus and the Argand diagram.
The shaded region defined as $\{z:|z+2-i|\le5\}\cap\{z:\operatorname{Re}(z)<1\},\ z\in\mathbb{C}$ is best represented by
Not quite. That shades the part of the disc to the right of the line, where $\operatorname{Re}(z)>1$. In 2023, 6.9% of Queensland students chose (A).
That’s it. Step through the hints below to see why. In 2023, 71.5% of Queensland students chose (B).
Not quite. Check the centre. $|z+2-i|\le5$ is centred on $-2+i$, left of the imaginary axis and above the real axis. This disc is centred on $2-i$. In 2023, 6.0% of Queensland students chose (C).
Not quite. The shading is on the correct side of the line, but this disc is centred on $2-i$. The centre should be $-2+i$. In 2023, 15.3% of Queensland students chose (D).
Drag the dot to test a point. The shaded region is inside the circle and to the left of the dashed line.
This point satisfies both conditions, so it is in the shaded region.
This point is inside the disc but has $\operatorname{Re}(z)\ge1$, so it is not in the region.
This point has $\operatorname{Re}(z)<1$ but is outside the disc, so it is not in the region.
This point fails both conditions.
Write $|z+2-i|$ as $|z-(-2+i)|$. It is the distance from $z$ to $-2+i$, so $|z+2-i|\le5$ is a disc with centre $-2+i$ and radius $5$, including its edge.
$\operatorname{Re}(z)<1$ is everything to the left of the vertical line $\operatorname{Re}(z)=1$. The line is dashed because $<$ leaves it out.
The intersection is the part of the disc to the left of the line. The centre $-2+i$ is left of the imaginary axis and above the real axis, which rules out (C) and (D).
$|z+2-i|=|z-(-2+i)|\le5$ is a disc with centre $-2+i$ and radius $5$. $\operatorname{Re}(z)<1$ is the half-plane to the left of the dashed line $\operatorname{Re}(z)=1$.
The intersection is the part of the disc to the left of the line. Only (B) shows this, with the centre left of the imaginary axis and above the real axis.
How Queensland students answered in 2023
From the QCAA Specialist Mathematics subject report for 2023. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2023 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2023, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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