EXN.2.SL.TZ0.6

pestleMathematicsAASLPaper 2EXN· sl-2-4-key-features-of-graphs-intersections-using-technology, sl-5-11-definite-integrals-areas-under-curve-onto-x-axis-and-areas-between-curvessource ↗

Consider the curves $y = x^{2} \sin x$ and $y = - 1 - \sqrt{1 + 4 {(x + 2)}^{2}}$ for $- π \leq x \leq 0$ .

Find the $x$ -coordinates of the points of intersection of the two curves.

[3]

Find the area, $A$ , of the region enclosed by the two curves.

[4]

Markscheme / solution

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

attempts to solve $x^{2} \sin x = - 1 - \sqrt{1 + 4 {(x + 2)}^{2}}$ (M1)

$x = - 2.76, - 1.54$ A1A1

Note: Award A1A0 if additional solutions outside the domain are given.

[3 marks]

$A = \int_{- 2.762 \dots}^{- 1.537 \dots} (- 1 - \sqrt{1 + 4 {(x + 2)}^{2}} - x^{2} \sin x) d x$ (or equivalent) (M1)(A1)

Note: Award M1 for attempting to form an integrand involving “top curve” − “bottom curve”.

so $A = 1.47$ A2

[4 marks]

Examiners’ report

[N/A]