A-LevelMathematicsDifferentiation9709/33 · Oct/Nov 2025 · Q1112 Marks
The diagram shows the graph of y = sec²x√(3+2 tanx) for -¼π ≤ x ≤ ¼π, and its minimum point M. [Figure 11.1]
- (a)Find the x-coordinate of M.[6]
- (b)Using the substitution u = 3+2 tanx, find the exact value of the area of the region bounded by the curve, the x-axis and the lines x =−¼π and x = ¼π.[6]
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Mark scheme
Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.
- Differentiate √(3+2tanx) to obtain (1/2)sec²x(3+2tanx)^(-1/2)
- Differentiate sec²x to obtain 2secx secx tanx
- Use correct product (or quotient) rule
- [Obtain derivative, if correct 2sec²x tanx(3+2tanx)^½ + sec²x(3+2tanx)^(-1/2)] and equate derivative to zero and obtain an equation in one trig function
- Obtain 5tan²x + 6 tan x+1=0
- Obtain AWRT x = –0.197 only
- du/dx = 2 sec² x
- Reach an integral of the form ∫ A√u du
- Obtain (1/2) ∫ u^(1/2) du
- Obtain (1/3) u^(3/2)
- Substitute correct limits correctly in an expression of the form B u^(3/2) or B(3+2tanx)^(3/2) and obtain c[5^(3/2)-1^(3/2)]
- Obtain answer (5√5)/3 - 1/3
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