A-LevelMathematicsIntegration9709/32 · May/June 2023 · Q1010 Marks

The diagram shows the curve y = (x + 5)√3 – 2x and its maximum point M. [Figure 10.1]

(a)Find the exact coordinates of M.[5]
(b)Using the substitution u = 3 – 2x, find by integration the area of the shaded region bounded by the curve and the x-axis. Give your answer in the form a√13, where a is a rational number.[5]

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Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

M1 Use the product rule correctly to obtain p(x+5)(3-2x)^n + q(3-2x)^m [Allow with incorrect chain rule. BOD over sign errors unless an incorrect rule is quoted.]
A1 Obtain correct derivative in any form [e.g. -(x + 5) / (√(3-2x)) + √(3-2x).]
DM1 Equate derivative to zero and obtain a linear equation [Allow with surd factor e.g. (1/√(3-2x))(-(x+5)+(3-2x)) = 0.]
A1 Obtain a correct linear equation. [e.g. (x + 5) + 3 – 2x = 0.]
A1 Obtain answer x = 8/3, y = (13/3)√(1/3) or (2√39)/9 [Or exact equivalent e.g. (8/3, 2√13/√9) or (8/3, 2√2197/3√27). Accept with x, y stated separately. ISW]
Alternative Method for Question 10(a):
M1 Obtain y² and differentiate [Ignore their left hand side i.e. their dy²/dx.]
A1 Obtain correct derivative in any form [e.g. -6x² - 34x – 20 .]
DM1 Equate derivative to zero and solve for x
A1 Ignore -5 if seen.
A1 Obtain answer x = 8/3, y = (2√39)/9 only [Or exact equivalent e.g. (8/3, 2√13/√3) or (8/3, 2√2197/3√27) ISW]
M1 Use the given substitution and reach a∫((13-u)/2)u^(1/2)du [OE Need to see -2 or -½ used. Condone if du missing or the integral sign is missing. Allow M1A0 for complete substitution into ∫x√(3-2x) dx to obtain first term of the line below.]
A1 Obtain correct integral -½∫(3-u)/2 √(u) du [OE e.g. -½∫( (3-u)/2 )√u du + ∫5√u du. Ignore limits at this stage. Condone if du missing.]
B1 SOI x = -5 and x = 3/2 [e.g. by u = 13 and 0. In any order and at any stage.]
DM1 Use correct limits the right way round in an integral of the form a( (2/3)u^(3/2) - (2/5)u^(5/2) ) [a( (26/3)√(13) - (5/3)√(13) )]
A1 Obtain answer (169/15)√13 or a = 169/15 [or exact equivalents.]