(i) By first expanding cos(2x + x), show that cos 3x = 4 cos³ x − 3 cos x. 3 [4] (ii) Hence solve the equation cos 3x + 3 cos x + 1 = 0, for 0 ≤ x ≤ π. [2] (iii) Find the exact value of π cos²xdx. π [4]

Mark scheme

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

1. [M1] Use cos(A + B) formula to express cos3x in terms of trig functions of 2x and x
2. [M1] Use double angle formulae and Pythagoras to obtain an expression in terms of cos x only
3. [A1] Obtain a correct expression in terms of cos x in any form
4. [A1] Obtain cos3x = 4 cos³ x − 3 cos x AG
5. [M1] Use identity and solve cubic 4cos³x = −1 for x
6. [A1] Obtain answer 2.25 and no other in the interval Accept 0.717π M1A0 for 129.0°
7. [B1] Obtain indefinite integral (1/12)sin 3x + (3/4)sin x
8. [B1] Substitute limits in an indefinite integral of the form a sin 3x + b sin x, where ab ≠ 0
9. [M1] Substitute limits in an indefinite integral of the form a sin 3x + b sin x, where ab ≠ 0
10. [A1] Obtain answer (1/24)(9√3-11), or exact equivalent
11. [B1] cos x(1-sin²x)dx = sin x - (1/3)sin³ x(+C)
12. [B1] Substitute limits in an indefinite integral of the form a sin x + b sin³ x where ab ≠ 0
13. [M1] Substitute limits in an indefinite integral of the form a sin x + b sin³ x where ab ≠ 0
14. [A1] Obtain answer (1/24)(9√3-11), or exact equivalent