A-LevelMathematicsNumerical methods9709/21 · May/June 2019 · Q69 Marks
The diagram shows the curve with equation y = (8 + x³)/(2 - 5x). The maximum point is denoted by M.
- (i)Find an expression for dy/dx and determine the gradient of the curve at the point where the curve crosses the x-axis.[4]
- (ii)Show that the x-coordinate of the point M satisfies the equation x = √(0.6x + 4x⁻¹).[2]
- (iii)Use an iterative formula, based on the equation in part (ii), to find the x-coordinate of M correct to 3 significant figures. Give the result of each iteration to 5 significant figures.[3]
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 2019). The mark or mark type is shown in brackets.
- [M1] Use quotient rule (or product rule) to differentiate
- [A1] Obtain dy/dx = (3x²(2-5x) – (-5)(8+ x³)) / (2-5x)² or equivalent
- [B1] State or imply curve crosses x-axis when x = -2
- [A1] Substitute -2 to obtain 1
- [M1] Equate numerator of first derivative to zero and rearrange as far as kx³ = ... or equivalent
- [A1] Confirm given result x = √(0.6x + 4x⁻¹)
- [M1] Use iterative process correctly at least once
- [A1] Obtain final answer 1.81
- [A1] Show sufficient iterations to 5 sf to justify answer or show a sign change in the interval [1.805, 1.815]
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