O-LevelAdditional MathematicsDifferentiation and integration4037/22 · Oct/Nov 2016 · Q79 Marks

3 cm P x cm R 8 S 14 cm [Figure 7]

(i)Show that the area, A cm², of the trapezium PQRS is given by A = (7 + x)√9-x².[2]
(ii)Given that x can vary, find the stationary value of A.[7]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2016). The mark or mark type is shown in brackets.

[B2/1/0] (i) h=√9-x² Must be clear that √9-x² is the height of the trapezium. 14+2x oe must be seen AG
A= (√9-x²)/2 (14+x+x) = √9-x² (7+x)
[M1] (ii) dA/dx = (√9-x²) + (7+x)½(9-x²)^-0.5 x−2x product rule on correct function
[A2/1/0] minus 1 each error, allow unsimplified.
[M1] dA/dx = 0 → 9-x² = 7x+x² equate to 0 and simplify to a linear or quadratic equation.
[A1] 2x²+7x-9=0 correct three term quadratic obtained
[A1] x=1
[A1] A=16√2 or 8√8 or √512 or 22.6 Extra positive answer loses penultimate A1. ignore negative solution.

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About this question

Topic

Differentiation and integration, O-Level Additional Mathematics (syllabus 4037). Structured question, 9 marks.

Source

O-Level Additional Mathematics, Oct/Nov 2016 Paper 2 Variant 2, question 7.

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