A-LevelMathematicsSeries9709/12 · Oct/Nov 2023 · Q56 Marks

The first, second and third terms of a geometric progression are sin θ, cos θ and 2 – sin θ respectively, where θ radians is an acute angle.

(a)Find the value of θ.[3]
(b)Using this value of θ, find the sum of the first 10 terms of the progression. Give the answer in the form b/(√c - 1), where b and c are integers to be found.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

[*M1] OE. Forming a correct equation in θ only using the terms of the GP and an attempt to clear fractions.
[DM1] Correct use of cos² θ + sin² θ = 1 and attempt to solve for sin θ.
[A1] AWRT A0 for θ = 30°. Condone inclusion of 5π/6 and/or
[B1] OE SOI Trigonometric values need to have been evaluated but allow decimal equivalents (0.5 and 1.73 AWRT)
[M1] Use of a correct formula for S₁₀, with their value of θ. Their √3 needs to come from (cos(their θ)/sin(their θ)) or (2-sin(their θ))/(cos(their θ)) OE
[A1] -121/(1-√3) or 165 AWRT scores B1M1A0.

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Topic: Series, A-Level Mathematics (syllabus 9709). Structured question, 6 marks.
Source: A-Level Mathematics, Oct/Nov 2023 Paper 1 Variant 2, question 5.

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