A-LevelMathematicsDifferentiation9709/12 · May/June 2017 · Q57 Marks

A curve has equation y = 3 + 12 / (2 - x).

(i)Find the equation of the tangent to the curve at the point where the curve crosses the x-axis.[5]
(ii)A point moves along the curve in such a way that the x-coordinate is increasing at a constant rate of 0.04 units per second. Find the rate of change of the y-coordinate when x = 4.[2]

Mark scheme

Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.

Crosses x-axis at (6, 0)
[B1] x = 6 is sufficient.
dy/dx = (0+) −12 (2−x)^-2 × (-1)
[B2,1,0] -1 for each incorrect term of the three or addition of + C.
Tangent y = ¾(x-6) or 4y = 3x-18
[M1 A1] Must use dy/dx, x= their 6 but not x = 0 (which gives m = 3), and correct form of line equation. Using y = mx + c gets A1 as soon as c is evaluated.
If x = 4, dy/dx = 3
dy/dt = 3×0.04 = 0.12
[M1 A1FT] M1 for ("their m" from dy/dx and x = 4) × 0.04. Be aware: use of x = 0 gives the correct answer but gets M0.

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Topic: Differentiation, A-Level Mathematics (syllabus 9709). Structured question, 7 marks.
Source: A-Level Mathematics, May/June 2017 Paper 1 Variant 2, question 5.

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