A-LevelMathematicsNumerical methods9709/23 · May/June 2020 · Q79 Marks
It is given that ∫₀ᵃ ( 4/(2x + 1) + 8x) dx = 10, where a is a positive constant.
- (a)Show that a = √2.5 – 0.5 ln(2a + 1).[4]
- (b)Using the equation in part (a), show by calculation that 1 < a < 2.[2]
- (c)Use an iterative formula, based on the equation in part (a), to find the value of a correct to 4 significant figures. Give the result of each iteration to 6 significant figures.[3]
Mark scheme
Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.
- Integrate to obtain the form k₁ ln(2x + 1) + k₂x²
- Obtain correct 2ln(2x + 1) + 4x²
- Use limits correctly and attempt rearrangement
- Confirm a = √2.5 - 0.5 ln(2a + 1) AG
- Consider sign of a – √[2.5 – 0.5ln(2a + 1)] or equivalent for 1 and 2
- Obtain -0.3... and 0.6... or equivalents and justify conclusion
- Use iteration process correctly at least once
- Obtain final answer 1.358
- Show sufficient iterations to 6 sf to justify answer or show a sign change in the interval [1.3575, 1.3585]
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