| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Simpson's rule estimation |
| Difficulty | Standard +0.3 This is a multi-part question requiring standard techniques: finding a tangent line (differentiation and point-slope form), applying Simpson's Rule (routine numerical integration), and geometric reasoning to relate two areas. Part (iii) requires recognizing that area B = triangle area - area A, which is straightforward once parts (i) and (ii) are complete. All techniques are standard C3 material with no novel insight required, making it slightly easier than average. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07m Tangents and normals: gradient and equations1.08e Area between curve and x-axis: using definite integrals1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Differentiate to obtain \(kx(5-x^2)^{-1}\) | M1 | Any non-zero constant |
| Obtain correct \(-2x(5-x^2)^{-1}\) | A1 | Or equiv |
| Obtain \(-4\) for value of derivative | A1 | |
| Attempt equation of straight line through \((2,0)\) with numerical value of gradient obtained from attempt at derivative | M1 | Not for attempt at eqn of normal |
| Obtain \(y = -4x + 8\) | A1 | 5 Or equiv |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply \(h = \frac{1}{2}\) | B1 | |
| Attempt calculation involving attempts at \(y\) values | M1 | Addition with each of coefficients 1, 2, 4 occurring at least once |
| Obtain \(k(\ln 5 + 4\ln 4.75 + 2\ln 4 + 4\ln 2.75 + \ln 1)\) | A1 | Or equiv perhaps with decimals; any constant \(k\) |
| Obtain 2.44 | A1 | 4 Allow \(\pm 0.01\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt difference of two areas | M1 | Allow if area of their triangle \(<\) area \(A\) |
| Obtain \(8 - 2.44\) and hence 5.56 | A1\(\checkmark\) | 2 Following their tangent and area of \(A\) providing answer positive |
## Question 8(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Differentiate to obtain $kx(5-x^2)^{-1}$ | M1 | Any non-zero constant |
| Obtain correct $-2x(5-x^2)^{-1}$ | A1 | Or equiv |
| Obtain $-4$ for value of derivative | A1 | |
| Attempt equation of straight line through $(2,0)$ with numerical value of gradient obtained from attempt at derivative | M1 | Not for attempt at eqn of normal |
| Obtain $y = -4x + 8$ | A1 | **5** Or equiv |
## Question 8(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply $h = \frac{1}{2}$ | B1 | |
| Attempt calculation involving attempts at $y$ values | M1 | Addition with each of coefficients 1, 2, 4 occurring at least once |
| Obtain $k(\ln 5 + 4\ln 4.75 + 2\ln 4 + 4\ln 2.75 + \ln 1)$ | A1 | Or equiv perhaps with decimals; any constant $k$ |
| Obtain 2.44 | A1 | **4** Allow $\pm 0.01$ |
## Question 8(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt difference of two areas | M1 | Allow if area of their triangle $<$ area $A$ |
| Obtain $8 - 2.44$ and hence 5.56 | A1$\checkmark$ | **2** Following their tangent and area of $A$ providing answer positive |
---8\\
\includegraphics[max width=\textwidth, alt={}, center]{d858728a-3371-4755-880c-54f96c5e5156-4_787_742_276_719}
The diagram shows part of the curve $y = \ln \left( 5 - x ^ { 2 } \right)$ which meets the $x$-axis at the point $P$ with coordinates $( 2,0 )$. The tangent to the curve at $P$ meets the $y$-axis at the point $Q$. The region $A$ is bounded by the curve and the lines $x = 0$ and $y = 0$. The region $B$ is bounded by the curve and the lines $P Q$ and $x = 0$.\\
(i) Find the equation of the tangent to the curve at $P$.\\
(ii) Use Simpson's Rule with four strips to find an approximation to the area of the region $A$, giving your answer correct to 3 significant figures.\\
(iii) Deduce an approximation to the area of the region $B$.
\hfill \mbox{\textit{OCR C3 2006 Q8 [11]}}