The diagram shows the curve y = 2sinx√2+cosx, for 0 ≤ x ≤ 2π, and its minimum point M, where x = a. [Figure 11.1]

Mark scheme

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

1. [M1] (a) Use of correct product rule and correct chain rule.
2. [A1] (a) Obtain dy/dx = 2cosx√(2+cosx) - sin²x/√(2+cosx) or equivalent.
3. [*M1] (a) Equate the derivative to zero and obtain a 3-term quadratic equation in cos x, e.g., 3cos²x + 4cosx - 1 = 0.
4. [DM1] (a) Solve for cos x. cos x = (-2+√7)/3 or approx 0.215.
5. [A1] (a) Obtain α = 4.93. (2 decimal places)
6. [B1] (b) State or imply du = -sin x dx.
7. [M1] (b) Substitute throughout for u and du to express the integral in terms of u.
8. [A1] (b) Obtain -∫2√u du. OE. (Ignore limits at this stage).
9. [M1] (b) Integrate to obtain ku^(3/2) (+C).
10. [M1] (b) Use correct limits for u (1 and 3, corresponding to x=π and x=0 respectively), or for x (0 and π), correctly in the integrated expression.
11. [A1] (b) Obtain (4/3)(3√3-1) or 4√3 - 4/3. OE.