A-LevelMathematicsAlgebra9709/22 · Oct/Nov 2020 · Q710 Marks

A curve has equation y = f(x) where f(x) = x⁴ – 5x³ + 6x² + 5x – 15. As shown in the diagram, the curve crosses the x-axis at the points A and B with coordinates (a, 0) and (b, 0) respectively. [Figure 7.1]

(a)Use the factor theorem to show that (x - 3) is a factor of f(x).[2]
(b)By first finding the quotient when f(x) is divided by (x – 3), show that a= - 5 2-a[5]
(c)Use an iterative formula, based on the equation in part (b), to find the value of a 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 (Oct/Nov 2020). The mark or mark type is shown in brackets.

[1] Substitute x = 3 and attempt evaluation
[1] Obtain 0 and confirm factor x-3
[1] Divide quartic expression by x-3 at least as far as x³ + kx²
[1] Obtain x³ - 2x²
[1] Obtain x³-2x²+5 With no errors seen
[1] Attempt rearrangement of their cubic expression to x = ... Or a = ...
[1] Confirm a = -√(5/(2-a))
[1] Use iteration process correctly at least once Need to see 3 values including their starting value.
[1] Obtain final answer -1.24 Answer required to exactly 3 significant figures.
[1] Show sufficient iterations to 5 sf to justify answer or show a sign change in the interval [−1.245, -1.235]

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