The function f is defined by f(x) = 3 sin²x-2cosx for 2≤x≤4, where x is in radians.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. [B2] 3(2sinxcosx) - (-2sinx) [=0] or better B1 for the correct derivative for either term; may be unsimplified
2. [B1] Factors out sinx and equates to 0: [2](sin x)(3cosx + 1) = 0 oe
3. [M1] sinx = 0 [theira cos x = -theirb] FT an expression of the form asinxcosx +bsinx for non-zero constants a and b; dep on previous B1
4. [A1] x = π as only solution dep on all previous marks awarded
5. If B2 B0 M0 then SC1 for dividing by sinx and using cos x = -1/3 to find x = 1.91 and 4.37 and clearly reject them.
6. [M1] For use of sin²x = 1- cos²x to write in terms of cosx only e.g. 3(1-cos²x) – 2cosx = 1 – 3cosx
7. [A1] Collects terms e.g. 3cos²x – cosx -2 = 0
8. [M1] Factorises the left-hand side or solves: (3cosx + 2)(cosx - 1) = 0
9. [A2] x = 2.3[0], x = 3.98 and no extras A1 for either; not from wrong working