A-LevelMathematicsNumerical methods9709/21 · Oct/Nov 2017 · Q79 Marks

The diagram shows the curve [Figure X.X] y = x² + 3x + 1 + 5 cos (x/2). The curve crosses the y-axis at the point P and the gradient of the curve at P is m. The point Q on the curve has x-coordinate q and the gradient of the curve at Q is -m.

(i)Find the value of m and hence show that q satisfies the equation x = a sin (x/2) + b, where the values of the constants a and b are to be determined.[4]
(ii)Show by calculation that –4.5 < q < -4.0.[2]
(iii)Use an iterative formula based on the equation in part (i) to find the value of q correct to 3 significant figures. Give the result of each iteration to 5 significant figures.[3]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.

[*M1] Differentiate to obtain form k₁x+k₂ + k₃ sin(x/2)
[A1] Obtain correct 2x+3-2.5sin(x/2) and deduce or imply gradient at P is 3
[DM1] Equate first derivative to their -3 and rearrange
[A1] Obtain x = 5/2 sin(x/2) - 3
[M1] Consider sign of their 2x+6-2.5sin(x/2) at -4.5 and -4.0 or equivalent
[A1] Complete argument correctly for correct expression with appropriate calculations
[M1] Use iteration formula correctly at least once
[A1] Obtain final answer -4.11
[A1] Show sufficient iterations to justify accuracy to 3 sf or show sign change in interval (-4.115,-4.105)

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