| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Reverse chain rule with linear composite |
| Difficulty | Moderate -0.3 Part (a) is a straightforward reverse chain rule application with a linear composite function—a standard C3 exercise requiring recognition that the derivative of (4x-1) gives the factor 4. Part (b) requires setting up an integral for area under y=1/x and evaluating ln(2a)-ln(a), then subtracting a rectangle area, but follows a predictable pattern for this type of question. Both parts are routine applications of C3 techniques with no novel problem-solving required, making this slightly easier than average. |
| Spec | 1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Obtain integral of form \(k(4x-1)^{-1}\) | M1 | any non-zero constant \(k\) |
| Obtain \(-\frac{1}{4}(4x-1)^{-1}\) | A1 | or equiv; allow \(+c\) |
| Substitute limits and attempt evaluation | M1 | for any expression of form \(k'(4x-1)^n\) |
| Obtain \(\frac{2}{21}\) | A1 4 | or exact equiv |
| (b) Integrate to obtain \(\ln x\) | B1 | |
| Substitute limits to obtain \(\ln 2a - \ln a\) | B1 | |
| Subtract integral attempt from attempt at area of appropriate rectangle | M1 | or equiv |
| Obtain \(1 - (\ln 2a - \ln a)\) | A1 | or equiv |
| Show at least one logarithm property | M1 | at any stage of solution |
| Obtain \(1 - \ln 2\) and hence \(\ln(\frac{e}{2})\) | A1 6 | AG; full detail required |
**(a)** Obtain integral of form $k(4x-1)^{-1}$ | M1 | any non-zero constant $k$
Obtain $-\frac{1}{4}(4x-1)^{-1}$ | A1 | or equiv; allow $+c$
Substitute limits and attempt evaluation | M1 | for any expression of form $k'(4x-1)^n$
Obtain $\frac{2}{21}$ | A1 4 | or exact equiv
**(b)** Integrate to obtain $\ln x$ | B1 |
Substitute limits to obtain $\ln 2a - \ln a$ | B1 |
Subtract integral attempt from attempt at area of appropriate rectangle | M1 | or equiv
Obtain $1 - (\ln 2a - \ln a)$ | A1 | or equiv
Show at least one logarithm property | M1 | at any stage of solution
Obtain $1 - \ln 2$ and hence $\ln(\frac{e}{2})$ | A1 6 | AG; full detail required7
\begin{enumerate}[label=(\alph*)]
\item Find the exact value of $\int _ { 1 } ^ { 2 } \frac { 2 } { ( 4 x - 1 ) ^ { 2 } } \mathrm {~d} x$.
\item \\
\includegraphics[max width=\textwidth, alt={}, center]{ebfdf170-99c6-4785-b9d7-201c3425b4c9-3_563_753_1681_735}
The diagram shows part of the curve $y = \frac { 1 } { x }$. The point $P$ has coordinates $\left( a , \frac { 1 } { a } \right)$ and the point $Q$ has coordinates $\left( 2 a , \frac { 1 } { 2 a } \right)$, where $a$ is a positive constant. The point $R$ is such that $P R$ is parallel to the $x$-axis and $Q R$ is parallel to the $y$-axis. The region shaded in the diagram is bounded by the curve and by the lines $P R$ and $Q R$. Show that the area of this shaded region is $\ln \left( \frac { 1 } { 2 } \mathrm { e } \right)$.
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2006 Q7 [10]}}