| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find curve equation from derivative (find unknown constant in derivative first) |
| Difficulty | Moderate -0.8 This is a straightforward integration question requiring basic power rule integration (x^{-3} → x^{-2}/-2), finding the constant of integration using given points, and then computing a definite integral for area. All steps are routine A-level techniques with no problem-solving insight required, making it easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.08b Integrate x^n: where n != -1 and sums1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Integrating \(\quad y = -k\frac{x^{-2}}{-2} (+ c)\) | B1 | ok unsimplified |
| Sub \((1,18)\) \(\quad 18 = \frac{k}{2} + c\) | M1 | Substitutes once (even if without + c) |
| Sub \((4,3)\) \(\quad 3 = \frac{k}{32} + c\) | M1 | 2nd substitution and solution of simultaneous equations for \(k\) and \(c\) |
| \(\rightarrow k = 32, c = 2\) | A1 | co [4] |
| (ii) Area \(= \left[-\frac{16}{x} + 2x\right]\) from 1 to 1.6 | B1 B1 | co |
| \(\rightarrow [-10 + 3.2] - [-16 + 2] = 7.2\) | M1 A1 | Use of limits in an integral. co. [4] |
$\frac{dy}{dx} = \frac{k}{x^3}$
**(i)** Integrating $\quad y = -k\frac{x^{-2}}{-2} (+ c)$ | B1 | ok unsimplified
Sub $(1,18)$ $\quad 18 = \frac{k}{2} + c$ | M1 | Substitutes once (even if without + c)
Sub $(4,3)$ $\quad 3 = \frac{k}{32} + c$ | M1 | 2nd substitution and solution of simultaneous equations for $k$ and $c$
$\rightarrow k = 32, c = 2$ | A1 | co [4]
**(ii)** Area $= \left[-\frac{16}{x} + 2x\right]$ from 1 to 1.6 | B1 B1 | co
$\rightarrow [-10 + 3.2] - [-16 + 2] = 7.2$ | M1 A1 | Use of limits in an integral. co. [4]9\\
\includegraphics[max width=\textwidth, alt={}, center]{d71002bb-b6f0-42a3-89fb-f2769d5c3779-3_791_885_1281_630}
The diagram shows a curve for which $\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { k } { x ^ { 3 } }$, where $k$ is a constant. The curve passes through the points $( 1,18 )$ and $( 4,3 )$.\\
(i) Show, by integration, that the equation of the curve is $y = \frac { 16 } { x ^ { 2 } } + 2$.
The point $P$ lies on the curve and has $x$-coordinate 1.6.\\
(ii) Find the area of the shaded region.
\hfill \mbox{\textit{CAIE P1 2008 Q9 [8]}}