VCE Specialist Mathematics Past Exam Questions
Browse VCE Specialist Mathematics exam questions from 2015–2026, including NHT and VCE papers. Filter by topic, year and paper; open the original source.
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Section B, Question 2
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Estimated difficulty · Medium
Question 2 (10 marks) Let w cis 2 7 .
Search excerpt only. Open the original question for exact equations and diagrams.Source: 2023 VCE Specialist Mathematics examination 2Section B, Question 2(a)Question part (a)
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Report-based difficulty · MediumAverage marks: 60% of available · official reportFull marks: 63% of students · official report
a. Verify that w is a root of z7 − 1 = 0. 1 mark
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a. Verify by substitution that z = −5 is a real axis intercept of the graph of z i 2 3 2 3 and find the other real axis intercept. 2 marks
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Report-based difficulty · MediumAverage marks: 60% of available · official reportFull marks: 61% of students · official report
b. List the other roots of z7 − 1 = 0 in polar form. 1 mark
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b. Sketch the circle given by z i 2 3 2 3 on the Argand diagram below. Label the axes intercepts with their values. 3 marks Im(z) Re(z) 6 4 2 −2 −4 −6 −6 −4 −2 2 6 4 O 15 2023 SPEC
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Report-based difficulty · MediumAverage marks: 60% of available · official reportFull marks: 53% of students · official report
c. On the Argand diagram below, plot and label the points that represent all the roots of z7 − 1 = 0. 2 marks 1 −1 1 −1 O Im(z) Re(z) 13 2023 SPECMATH EXAM 2 d o n o t w r i t e i
Search excerpt only. Open the original question for exact equations and diagrams.Source: 2023 VCE Specialist Mathematics examination 2 · Report: 2023 VCE Specialist Mathematics 2 external assessment reportSection B, Question 2(c)Question part (c)
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c. The line given by z i z z 2 3 0 , where z, z0 ∈ C, is tangent to the circle given by z i 2 3 2 3 at z = −5. Find z0 in the form a + ib, where a, b ∈ R. 1 mark
Search excerpt only. Open the original question for exact equations and diagrams.Source: 2023 VCE (NHT) Specialist Mathematics examination 2Section B, Question 2(d)Question part (d)
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d. i. On the Argand diagram below, sketch the ray that originates at the real root of z7 − 1 = 0 and passes through the point represented by cis 2 7 . 1 mark 1 −1 1 −1 O Im(z) Re(z
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d. Find the area of the part of the circle given by z i 2 3 2 3, which lies below the real axis. Give your answer in the form p π + q, where p, q ∈ R. 3 marks
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ii. Find the equation of this ray in the form Arg(z − z0) = θ, where z0 ∈ C, and θ is measured in radians in terms of π. 1 mark
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e. A second circle is given by z i 2 3 2 3 , where z ∈ C. Sketch this circle on the axes provided in part b. You do not need to label the axes intercepts. 2 marks 2023 SPECMATH EXA
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Report-based difficulty · MediumAverage marks: 50% of available · official reportFull marks: 54% of students · official report
e. Verify that the equation z7 − 1 = 0 can be expressed in the form (z − 1)(z6 + z5 + z4 + z3 + z2 + z + 1) = 0. 1 mark
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