O-LevelAdditional MathematicsDifferentiation and integration4037/21 · May/June 2022 · Q117 Marks
The diagram shows part of the curves y = eˣᐟ² and y = cos 5x and part of the line x = π/4. The curves intersect at A. The curve y = cos 5x cuts the x-axis at B. The line x = π/4 cuts the x-axis at C and the curve y = eˣᐟ² at D. Find the exact area of the shaded region, ABCD. [Figure 11.1]
- (a)The diagram shows part of the curves y = eˣᐟ² and y = cos 5x and part of the line x = π/4. The curves intersect at A. The curve y = cos 5x cuts the x-axis at B. The line x = π/4 cuts the x-axis at C and the curve y = eˣᐟ² at D. Find the exact area of the shaded region, ABCD.[7]
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Mark scheme
Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.
- [B1] B (π/10, 0) soi
- [M2] ∫(π/4)_0 e^(x/2)dx - ∫(π/4)_0 cos 5x dx (M1 Integrates e^(x/2) to ke^(x/2), k≠0; M1 Integrates cos5x to k sin 5x, k>0 or k = -1/5)
- [A2] [2e^(x/2)]^(π/4)_0 - [1/5 sin 5x]^(π/4)_0 (A1 for each part correct)
- [M1] [2e^(π/8) - 2e⁰] - [1/5 sin(5π/4) - 1/5 sin0] M1 for correct attempt at subtraction and for one correct use of limits dependent on at least M1for integration and B1
- [A1] 2e^(π/8) - 11/5 isw
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