A-LevelMathematicsDifferentiation9709/12 · May/June 2021 · Q34 Marks

The equation of a curve is y = (x − 3)√x + 1 + 3. The following points lie on the curve. Non-exact values are rounded to 4 decimal places. A (2, k) В (2.9, 2.8025) C (2.99, 2.9800) D (2.999, 2.9980) E (3, 3)

(a)Find k, giving your answer correct to 4 decimal places.[1]
(b)Find the gradient of AE, giving your answer correct to 4 decimal places.[1]
(c)The gradients of BE, CE and DE, rounded to 4 decimal places, are 1.9748, 1.9975 and 1.9997 respectively. State, giving a reason for your answer, what the values of the four gradients suggest about the gradient of the curve at the point E.[2]

Mark scheme

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

[B1] 1.2679
[B1] 1.7321
[*B1] Sight of 2 or 2.0000 or two in reference to the gradient
[DB1] This is because the gradient at E is the limit of the gradients of the chords as the x-value tends to 3 or dx tends to 0.

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Topic: Differentiation, A-Level Mathematics (syllabus 9709). Structured question, 4 marks.
Source: A-Level Mathematics, May/June 2021 Paper 1 Variant 2, question 3.

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