The diagram shows part of the curve y = e^(x/3). The tangent to the curve at P (9, e³) meets the x-axis at Q. [Figure showing a curve y=e^(x/3) and a tangent at P(9, e³) meeting x-axis at Q, shaded region between curve, axes and tangent]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2013). The mark or mark type is shown in brackets.

1. [B1] dy/dx = 1/3 e^(x/3)
2. [M1] m = 1/3 e³ For insertion of x = 9 into their dy/dx. 6.7 or better if correct.
3. [DM1] y-e³ = 1/3 e³ (x – 9) Using their evaluated m to find eqn y = 6.7x-40.2 or better if correct.
4. [A1] At Q y = 0, x = 6 Accept value that rounds to 6.0 to 2sf
5. [B1] Area triangle 1.5e³ or 30.1
6. [B1] ∫e^(x/3) dx = 3e^(x/3) oe
7. [M1] Uses limits of 0 and 9 in integrated function. + must see both values inserted if incorrect answer
8. [A1] 3e³ - 3 or 57.3
9. [M1] Area under curve subtract area of triangle
10. [A1] 1.5e³ - 3 or 27.1 Condone 27.2 if obtained from 57.3 - 30.1.