O-LevelAdditional MathematicsSeries4037/12 · Oct/Nov 2024 · Q1011 Marks

The first 3 terms of an arithmetic progression are 2 tan 2x, 5 tan 2x, 8 tan2x. Find the values of x, where -180° ≤ x ≤ 180°, for which the sum to 30 terms is 455√3.

(a)The first 3 terms of an arithmetic progression are 2 tan 2x, 5 tan 2x, 8 tan2x. Find the values of x, where -180° ≤ x ≤ 180°, for which the sum to 30 terms is 455√3.[5]
(b)The first 3 terms of a geometric progression are 5 cos²(θ - π/2), 20 cos⁴(θ - π/2), 80 cos⁶(θ - π/2), where -π/6 ≤ θ ≤ 7π/6. Find the values of θ for which this geometric progression has a sum to infinity.[6]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

[M1] 30/2 (4 tan2x + (29 × 3 tan 2x)) = 455√3
[A1] tan 2x = √3/3
[M1] for 1 correct solution (allow if in radians or from use of tan 2x = 0.577 or tan 2x = 0.58 e.g. 14.99°, 15.06°.)
[A1] for a second correct solution
[A1] for 2 further correct solutions and no extras in the range
[B1] r = 4cos²(θ - π/2)
[M1] 4cos²(θ - π/2) < 1
[M1] dep for one correct critical value
[A1] for all critical values and no extras in the range -π/6 < θ < 7π/6
[A1] -π/6 < θ < π/6 (excluding 0)
[A1] 5π/6 < θ < 7π/6 (excluding π)

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