| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Simpson's rule estimation |
| Difficulty | Standard +0.8 This question requires finding a maximum by differentiation (product rule with exponential), applying Simpson's rule with 4 strips, then making a geometric deduction about how the two regions relate. The deduction in part (iii) requires insight into symmetry properties and careful geometric reasoning beyond routine integration, elevating it above standard C3 questions. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Obtain \(-2xe^{-x^2}\) as derivative of \(e^{-x^2}\) | B1 | |
| Attempt product rule | *M1 | allow if sign errors or no chain rule |
| Obtain \(8x^7e^{-x^2} - 2x^9e^{-x^2}\) | A1 | or (unsimplified) equiv |
| Either: Equate first derivative to zero and attempt solution | M1 | dep *M; taking at least one step of solution |
| Confirm 2 | A1 | 5 AG |
| Or: Substitute 2 into derivative and show attempt at evaluation | M1 | |
| Obtain 0 | A1 | (5) AG; necessary correct detail required |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt calculation involving attempts at \(y\) values | M1 | with each of 1, 4, 2 present at least once as coefficients |
| Attempt \(k(y_0 + 4y_1 + 2y_2 + 4y_3 + y_4)\) | M1 | with attempts at five \(y\) values corresponding to correct \(x\) values |
| Obtain \(\frac{1}{6}(0 + 4\times0.00304 + 2\times0.36788 + 4\times2.70127 + 4.68880)\) | A1 | or equiv with at least 3 d.p. or exact values |
| Obtain 2.707 | A1 | 4 or greater accuracy; allow \(\pm 0.001\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt \(4(y\text{ value}) - 2(\text{part (ii)})\) | M1 | or equiv |
| Obtain 13.3 | A1 | 2 or greater accuracy; allow \(\pm 0.1\) |
# Question 8:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain $-2xe^{-x^2}$ as derivative of $e^{-x^2}$ | B1 | |
| Attempt product rule | *M1 | allow if sign errors or no chain rule |
| Obtain $8x^7e^{-x^2} - 2x^9e^{-x^2}$ | A1 | or (unsimplified) equiv |
| Either: Equate first derivative to zero and attempt solution | M1 | dep *M; taking at least one step of solution |
| Confirm 2 | A1 | **5** AG |
| Or: Substitute 2 into derivative and show attempt at evaluation | M1 | |
| Obtain 0 | A1 | **(5)** AG; necessary correct detail required |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt calculation involving attempts at $y$ values | M1 | with each of 1, 4, 2 present at least once as coefficients |
| Attempt $k(y_0 + 4y_1 + 2y_2 + 4y_3 + y_4)$ | M1 | with attempts at five $y$ values corresponding to correct $x$ values |
| Obtain $\frac{1}{6}(0 + 4\times0.00304 + 2\times0.36788 + 4\times2.70127 + 4.68880)$ | A1 | or equiv with at least 3 d.p. or exact values |
| Obtain 2.707 | A1 | **4** or greater accuracy; allow $\pm 0.001$ |
## Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt $4(y\text{ value}) - 2(\text{part (ii)})$ | M1 | or equiv |
| Obtain 13.3 | A1 | **2** or greater accuracy; allow $\pm 0.1$ |
---8\\
\includegraphics[max width=\textwidth, alt={}, center]{1216a06e-7e14-48d7-a7ca-7acd8d71af5f-4_538_1443_262_351}
The diagram shows the curve with equation $y = x ^ { 8 } \mathrm { e } ^ { - x ^ { 2 } }$. The curve has maximum points at $P$ and $Q$. The shaded region $A$ is bounded by the curve, the line $y = 0$ and the line through $Q$ parallel to the $y$-axis. The shaded region $B$ is bounded by the curve and the line $P Q$.\\
(i) Show by differentiation that the $x$-coordinate of $Q$ is 2 .\\
(ii) Use Simpson's rule with 4 strips to find an approximation to the area of region $A$. Give your answer correct to 3 decimal places.\\
(iii) Deduce an approximation to the area of region $B$.
\hfill \mbox{\textit{OCR C3 2007 Q8 [11]}}