| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2009 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Area under curve using integration |
| Difficulty | Moderate -0.8 This is a straightforward two-part question requiring routine calculus techniques: (i) differentiate a polynomial, set equal to zero, and use second derivative test; (ii) integrate a simple polynomial between given limits. Both parts are standard textbook exercises with no problem-solving insight required, making this easier than average for A-level. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.08e Area between curve and x-axis: using definite integrals |
B1: $y = x^4 + 4x + 9$
(i) M1: Differential = $4x^3 + 4$
M1: Sets to 0 + solution $\to (-1, 6)$
B1: $2$nd differential = $12x^2$
A1: Positive, $\to$ Minimum
[4]
(ii) B1: $A = \frac{x^5}{5} +2x^2 +9x$
M1: Limits from 0 to 1 $\to$ Value at "1" − value at "0" in integral of $y$
A1: $11.2$
[3]4 The equation of a curve is $y = x ^ { 4 } + 4 x + 9$.\\
(i) Find the coordinates of the stationary point on the curve and determine its nature.\\
(ii) Find the area of the region enclosed by the curve, the $x$-axis and the lines $x = 0$ and $x = 1$.
\hfill \mbox{\textit{CAIE P1 2009 Q4 [7]}}