A-LevelMathematicsIntegration9709/12 · Oct/Nov 2021 · Q1111 Marks

The diagram shows the line x = 5/2, part of the curve y = (1/2)x + 7/10 - 1/(x-2)³ and the normal to the curve at the point A (3, 5/2). [Figure X.X]

(a)Find the x-coordinate of the point where the normal to the curve meets the x-axis.[5]
(b)Find the area of the shaded region, giving your answer correct to 2 decimal places.[6]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

[B1] dy/dx = 1/2 + 3(x-2)⁻⁴ OE. Allow unsimplified.
[*M1] Substituting x = 3 into their differentiated expression – defined by one of 3 original terms with correct power of x.
[*DM1] Negative reciprocal of their evaluated dy/dx.
[DM1] Using their normal gradient and A in the equation of a straight line. Dependent on *M1 and *DM1.
[A1] x = 4 or (4, 0)
[M1] For intention to integrate the curve (no need for limits). Condone inclusion of π for this mark.
[A1] 1/4 x² + 7/10 x + 1/(2(x-2)²) OE. Allow unsimplified. Condone inclusion of π for this mark.
[M1] Clear substitution of 3 and 2.5 into their integrated expression (with at least one correct term) and subtracting.
[A1] 0.48[24] If M1A1M0 scored then SC B1 can be awarded for correct answer.
[B1] [Area of triangle =] 0.6 OE
[A1] [Total area =] 1.08 Dependent on the first M1 and WWW.

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