The diagram shows the curve y = √x cos x, for 0 < x < ¾π, and its minimum point M, where x = a. The shaded region between the curve and the x-axis is denoted by R. [Figure X.X]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.

1. [M1] Use correct product rule
2. [A1] Obtain correct derivative in any form e.g. dy/dx = (1/(2√x)) cosx - √x sinx
3. [A1] Equate derivative to zero and obtain tan a = 1/(2a)
4. [M1] Use the iterative process correctly at least once (get one value and go on to use it in a second use of the formula)
5. [A1] Show sufficient iterations to at least 4 d.p. to justify 3.29, or show there is a sign change in the interval (3.285, 3.295)
6. [A1] Obtain final answer 3.29
7. [B1] State or imply the indefinite integral for the volume is π ∫(√x cosx)² dx [If π omitted, or 2π or π/2 used, give B0 and follow through. 4/6 available]
8. [*M1] Use correct cos 2A formula, commence integration by parts and reach x(ax+b sin 2x) ± ∫ ax+b sin 2x dx
9. [A1] Obtain x(x/2 + (1/4)sin 2x) - ∫(x/2 + (1/4)sin 2x) dx, or equivalent
10. [A1] Complete integration and obtain (1/4)x² + (1/4)x sin 2x + (1/8)cos 2x
11. [DM1] Substitute limits x = 0 and x = π/2, having integrated twice
12. [A1] Obtain answer π(π²/16 - 1/4), or exact equivalent