A-LevelMathematicsNumerical methods9709/33 · Oct/Nov 2019 · Q810 Marks
The diagram shows the graph of y = sec x for 0 ≤ x < 1/2π.
- (i)Use the trapezium rule with 2 intervals to estimate the value of ∫₀¹·² sec x dx, giving your answer correct to 2 decimal places.[3]
- (ii)Explain, with reference to the diagram, whether the trapezium rule gives an overestimate or an underestimate of the true value of the integral in part (i).[1]
- (iii)P is the point on the part of the curve y = sec x for 0 ≤ x < 1/2π at which the gradient is 2. By first differentiating 1/cos x, find the x-coordinate of P, giving your answer correct to 3 decimal places.[6]
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 (Oct/Nov 2019). The mark or mark type is shown in brackets.
- [B1] State or imply ordinates 1, 1.2116..., 2.7597...
- [M1] Use correct formula, or equivalent, with h = 0.6
- [A1] Obtain answer 1.85
- [B1] Explain why the rule gives an overestimate
- [M1] Differentiate using quotient or chain rule
- [A1] Obtain correct derivative in terms of sin x and cos x
- [M1] Equate derivative to 2, use Pythagoras and obtain an equation in sin x
- [A1] Obtain 2sin²x+sinx-2=0
- [M1] Solve a 3-term quadratic for x
- [A1] Obtain answer x = 0.896 only
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