A-LevelMathematicsNumerical methods9709/33 · May/June 2019 · Q68 Marks

The diagram shows the curve y = x⁴ – 2x³ – 7x − 6. The curve intersects the x-axis at the points (a, 0) and (b, 0), where a < b. It is given that b is an integer. [Figure X.X]

(i)Find the value of b.[1]
(ii)Hence show that a satisfies the equation a = −(1/3)(2 + a² + a³).[4]
(iii)Use an iterative formula based on the equation in part (ii) to determine a correct to 3 decimal places. Give the result of each iteration to 5 decimal places.[3]

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Mark scheme

Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.

[B1] State b = 3
[M1] Commence division by x – b and reach partial quotient x³ + kx²
[A1] Obtain quotient x³ + x² + 3x + 2 (There being no remainder)
[M1] Equate quotient to zero and rearrange to make the subject a
[A1] Obtain the given equation
[M1] Use the iterative formula a_(n+1) = −(1/3)(2 + a_n² + a_n³) correctly at least once
[A1] Obtain final answer -0.715
[A1] Show sufficient iterations to 5 d.p. to justify -0.715 to 3 d.p., or show there is a sign change in the interval (-0.7145, -0.7155)

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