| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find derivative of quotient |
| Difficulty | Easy -1.2 Part (a) is a straightforward quotient differentiation that can be simplified to 2x² + 5x⁻¹ before differentiating, avoiding the quotient rule entirely. Part (b) is a standard chain rule integration (reverse) followed by definite integral evaluation. Both parts require only routine application of basic calculus techniques with no problem-solving or insight needed. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(y = \frac{2x^3 + 5}{x} = 2x^2 + \frac{5}{x}\) | M1 | Knows to divide numerator by \(x\) |
| \(\frac{dy}{dx} = 4x - \frac{5}{x^2}\) or \(4x - 5x^{-2}\) | A1 + A1 | co |
| [3] | ||
| (b) \(\int(3x-2)^5 dx = \frac{(3x-2)^6}{6} \cdot 3 \, (+c)\) | B1 B1 | B1 without "\(\cdot 3\)". B1 for "\(\cdot 3\)". (ignore (+c)) |
| \(\int_0^{} (3x-2)^5 dx = \left[\frac{(3x-2)^6}{18}\right]\) | M1 | Uses limits after integration. |
| Limits used correctly \(\rightarrow -3\frac{1}{2}\) | A1 | co |
| [4] |
(a) $y = \frac{2x^3 + 5}{x} = 2x^2 + \frac{5}{x}$ | M1 | Knows to divide numerator by $x$
$\frac{dy}{dx} = 4x - \frac{5}{x^2}$ or $4x - 5x^{-2}$ | A1 + A1 | co
| [3] |
(b) $\int(3x-2)^5 dx = \frac{(3x-2)^6}{6} \cdot 3 \, (+c)$ | B1 B1 | B1 without "$\cdot 3$". B1 for "$\cdot 3$". (ignore (+c))
$\int_0^{} (3x-2)^5 dx = \left[\frac{(3x-2)^6}{18}\right]$ | M1 | Uses limits after integration.
Limits used correctly $\rightarrow -3\frac{1}{2}$ | A1 | co
| [4] |4
\begin{enumerate}[label=(\alph*)]
\item Differentiate $\frac { 2 x ^ { 3 } + 5 } { x }$ with respect to $x$.
\item Find $\int ( 3 x - 2 ) ^ { 5 } \mathrm {~d} x$ and hence find the value of $\int _ { 0 } ^ { 1 } ( 3 x - 2 ) ^ { 5 } \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2011 Q4 [7]}}