O-LevelAdditional MathematicsDifferentiation and integration4037/11 · May/June 2017 · Q97 Marks
Show that 5 + 4 tan²(x/3) = 4 sec²(x/3) + 1.
- (i)Show that 5 + 4 tan²(x/3) = 4 sec²(x/3) + 1.[1]
- (ii)Given that d/dx (tan(x/3)) = 1/3 sec²(x/3), find ∫ sec²(x/3) dx.[1]
- (iii)The diagram shows part of the curve y = 5 + 4 tan²(x/3). Using the results from parts (i) and (ii), find the exact area of the shaded region enclosed by the curve, the x-axis and the lines x = π/2 and x = π.[5]
This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.
Mark scheme
Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.
- [B1] 5+4 sec²(x/3) -1 leading to given answer use of correct identity
- [B1] 3tan(x/3)+c
- [M1] attempt to integrate using (i) and/or (ii)
- [A1] Area = [12tan(x/3) + x] all correct
- [DM1] correct method for evaluation using limits in correct order
- [A1] =(12tan(π/3)+π) - (12tan(π/6)+π/2)
- [A1] = 8√3+π/2
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