| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Finding maximum/minimum on curve |
| Difficulty | Standard +0.3 This is a standard C3 question covering quotient rule differentiation, finding turning points, and integration by substitution. Part (ii) is routine quotient rule application, part (iii) involves standard calculus techniques to locate and verify a minimum, and part (iv) uses a given substitution to simplify integration. All techniques are textbook exercises with clear guidance provided. Slightly above average difficulty due to the algebraic manipulation required in the substitution and the multi-step nature, but no novel insight needed. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08d Evaluate definite integrals: between limits1.08h Integration by substitution |
Fig. 9 shows the curve $y = \frac{x^2}{3x - 1}$.
P is a turning point, and the curve has a vertical asymptote $x = a$.
\includegraphics{figure_1}
\begin{enumerate}[label=(\roman*)]
\item Write down the value of $a$. [1]
\item Show that $\frac{dy}{dx} = \frac{x(3x - 2)}{(3x - 1)^2}$ [3]
\item Find the exact coordinates of the turning point P.
Calculate the gradient of the curve when $x = 0.6$ and $x = 0.8$, and hence verify that P is a minimum point. [7]
\item Using the substitution $u = 3x - 1$, show that $\int \frac{x^2}{3x - 1} dx = \frac{1}{27} \int \left( u + 2 + \frac{1}{u} \right) du$.
Hence find the exact area of the region enclosed by the curve, the $x$-axis and the lines $x = \frac{2}{3}$ and $x = 1$. [7]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 Q1 [18]}}