A-LevelMathematicsTrigonometry9709/12 · Oct/Nov 2023 · Q79 Marks

Verify the identity (2x – 1)(4x² + 2x – 1) = 8x³ – 4x + 1.

(a)Verify the identity (2x – 1)(4x² + 2x – 1) = 8x³ – 4x + 1.[1]
(b)Prove the identity (tan² θ + 1)/(tan² θ - 1) = 1/(1 - 2cos² θ).[3]
(c)Using the results of (a) and (b), solve the equation (tan² θ + 1)/(tan² θ - 1) = 4cos θ, for 0° ≤ θ ≤ 180°.[5]

Mark scheme

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

[B1] AG Six correct terms leading to the correct answer.
[*M1] For use of tan θ = sin θ/cos θ in the numerator and denominator.
[DM1] For use of sin² θ + cos² θ = 1 twice, in a correct expression, resulting in an expression in cos² θ.
[A1] AG
[B1] Replace LHS with RHS from (b) and clear fractions.
[*B1] Use of the expression from (a) with x = cos θ.
[DB1] OE For all three values.
[B2,1,0] B2 for three correct answers only, B1 for two correct answers and no others (but allow 36° instead of 144°) in the given range or 3 correct answers plus other values in the given range. Ignore answers outside of the given range. Accept AWRT 72.0, 144.0 . SC B1 for all 3 correct answers in radians and no others: π/3, 2π/5, 4π/5

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