The curve \(y = {{\text{e}}^{ - x}} - x + 1\) intersects the x-axis at P.

(a)     Find the x-coordinate of P.

(b)     Find the area of the region completely enclosed by the curve and the coordinate axes.

(a)     Either solving \({{\text{e}}^{ - x}} - x + 1 = 0\) for x, stating \({{\text{e}}^{ - x}} - x + 1 = 0\), stating P(x, 0) or using an appropriate sketch graph.     M1

x = 1.28     A1     N1

Note: Accept P(1.28, 0) .

(b)     Area \( = \int_0^{1.278...} {({{\text{e}}^{ - x}} - x + 1){\text{d}}x} \)     M1A1

= 1.18     A1     N1

Note: Award M1A0A1 if the dx is absent.

[5 marks]

This was generally well done. In part (a), most candidates were able to find x = 1.28 successfully. A significant number of candidates were awarded an accuracy penalty for expressing answers to an incorrect number of significant figures.

Part (b) was generally well done. A number of candidates unfortunately omitted the dx in the integral while some candidates omitted to write down the definite integral and instead offered detailed instructions on how they obtained the answer using their GDC.