A-LevelMathematicsTrigonometry9709/12 · Oct/Nov 2019 · Q67 Marks

Given that x > 0, find the two smallest values of x, in radians, for which 3 tan(2x + 1) = 1. Show all necessary working.

(a)Given that x > 0, find the two smallest values of x, in radians, for which 3 tan(2x + 1) = 1. Show all necessary working.[4]
(b)(i)Express f(x) in the form a cos2x + b, where a and b are constants.[1]
(b)(ii)Find the range of f.[2]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

[*M1] (2x + 1) = tan⁻¹(1/3) (= 0.322 or 18.4 OR -0.339 rad or 8.7°) Correct order of operations. Allow degrees.
[DM1] Either their 0.322 + π or 2π Or their -0.339 + π/2 or π Must be in radians
[A1] x = 1.23 or x = 2.80 AWRT for either correct answer, accept 0.39π or 0.89π
[A1] For the second answer with no other answers between 0 and 2.8 SC1 For both 1.2 and 2.8
[B1] 5 cos²x - 2 Allow a = 5, b = -2
[B1FT] -2 FT for sight of their b
[B1FT] 3 FT for sight of their a + b

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Topic: Trigonometry, A-Level Mathematics (syllabus 9709). Structured question, 7 marks.
Source: A-Level Mathematics, Oct/Nov 2019 Paper 1 Variant 2, question 6.

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