A-LevelMathematicsDifferentiation9709/12 · Feb/Mar 2021 · Q1112 Marks

The diagram shows the curve with equation y = 9(x^(½) - 4x^(-½)). The curve crosses the x-axis at the point A. [Figure 11.1]

(a)Find the x-coordinate of A.[2]
(b)Find the equation of the tangent to the curve at A.[4]
(c)Find the x-coordinate of the maximum point of the curve.[2]
(d)Find the area of the region bounded by the curve, the x-axis and the line x = 9.[4]

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Mark scheme

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

[1] 9(x^(½) - 4x^(-½)) = 0 leading to 9x^(-½)(x-4) = 0 (OE. Set y to zero and attempt to solve.)
[1] x = 4 only (From use of a correct method.)
[2] B2; all 3 terms correct: 9, 1/2 x^(-1/2) and 6x^(-3/2) B1; 2 of the 3 terms correct
[1] At x = 4 gradient = 9(-1/16 + 6/32) = 9(1/8)
[1] Using their x = 4 in their differentiated expression and attempt to find equation of the tangent.
[1] Equation is y = 9/8(x-4) (or y = 9x/8 - 9/2 OE)
[1] Set their dy/dx to zero and an attempt to solve.
[1] x = 12 (Condone (±)12 from use of a correct method.)
[2] B2; all 3 terms correct: 9, x^(3/2)/(3/2), -4x^(1/2)/(1/2) B1; 2 of the 3 terms correct
[1] Apply limits their 4 → 9 to an integrated expression with no consideration of other areas.
[1] 6 (Use of π scores A0)

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