16N.2.SL.TZ0.S_4

pestleMathematicsAASLPaper 216N· sl-5-5-integration-introduction-areas-between-curve-and-x-axissource ↗

Let $f(x) = x{{\text{e}}^{ - x}}$ and $g(x) = - 3f(x) + 1$ .

The graphs of $f$ and $g$ intersect at $x = p$ and $x = q$ , where $p < q$ .

Find the value of $p$ and of $q$ .

[3]

Hence, find the area of the region enclosed by the graphs of $f$ and $g$ .

[3]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

valid attempt to find the intersection (M1)

eg $\,\,\,\,\,$ $f = g$ , sketch, one correct answer

$p = 0.357402,{\text{ }}q = 2.15329$

$p = 0.357,{\text{ }}q = 2.15$ A1A1 N3

[3 marks]

attempt to set up an integral involving subtraction (in any order) (M1)

eg $\,\,\,\,\,$ $\int_p^q {\left[ {f(x) - g(x)} \right]{\text{d}}x,{\text{ }}} \int_p^q {f(x){\text{d}}x - } \int_p^q {g(x){\text{d}}x}$

0.537667

${\text{area}} = 0.538$ A2 N3

[3 marks]

Examiners’ report

[N/A]