The diagram shows the curve with equation y = 2x^(3/2) - 3x^(1/2) + 1 for x > 0. The curve crosses the x-axis at points A and B and has a minimum point M. [Figure 11]

Mark scheme

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

1. [B1] Differentiate to obtain 3x^1/2 - 3/2x^-1/2
2. [M1] Equate first derivative to zero and reach a solution for x^1/3 or x^-1/3 with no error in use of indices or complete square to find minimum point 2(a-3/4)² -1/8 where a = x^1/3 Substitution SOI if dealt with correctly later
3. [A1] Obtain x = 64/27 Or exact equivalent. SC B1 if no working shown. Ignore extra solution x = 0 .
4. [B1] y = -1/8 seen Or exact equivalent. Allow -0.125.
5. [M1] Recognise equation as quadratic in x-1/3 or equivalent and attempt solution 2a²-3a+1[=0] where a = x^1/3.
6. [A1] Obtain x^1/3 =1 and x^1/3 = ½ OE SC B1 if no M mark awarded.
7. [A1] Obtain 1 and 8 SC B1 if no M mark awarded.
8. [*M1] Integrate to obtain form k₁x⁵⁄² + k₂x³⁄² + x or 2 out of 3 correct terms Expect 4/5x⁵⁄² - 2x³⁄² + x.
9. [A1] Obtain correct 4/5x⁵⁄² - 2x³⁄² + x No other terms from a second integral.
10. [DM1] Apply their limits correctly Their limits must be from their working.
11. [A1] [Obtain -0.5 and conclude area is] 0.5