| Answer | Marks | Guidance |
|---|---|---|
| (i) Use quotient or product rule | M1 | |
| Obtain derivative in any correct form, e.g. \(e^{2x}\left(\frac{2}{x} - \frac{1}{x^2}\right)\) | A1 | |
| Equate derivative to zero and solve for \(x\) | M1 | |
| Obtain \(x = \frac{1}{2}\) | A1 | |
| Obtain \(y = 2e\) (or \(5.44\)) | A1* | 5 marks |
| (* allow \(y = 2e\) if \(x = \frac{1}{2}\)) | ||
| (ii) Show or imply correct ordinates \(7.389..., 13.390..., 27.299...\) | B1 | |
| Use correct formula, or equivalent, with \(k = 0.5\) and three ordinates | M1 | |
| Obtain answer \(15.4\) with no errors seen | A1 | 3 marks |
| (iii) Justify the statement that the rule gives an over-estimate | B1 | 1 mark |
(i) Use quotient or product rule | M1 |
Obtain derivative in any correct form, e.g. $e^{2x}\left(\frac{2}{x} - \frac{1}{x^2}\right)$ | A1 |
Equate derivative to zero and solve for $x$ | M1 |
Obtain $x = \frac{1}{2}$ | A1 |
Obtain $y = 2e$ (or $5.44$) | A1* | 5 marks |
(* allow $y = 2e$ if $x = \frac{1}{2}$) |
(ii) Show or imply correct ordinates $7.389..., 13.390..., 27.299...$ | B1 |
Use correct formula, or equivalent, with $k = 0.5$ and three ordinates | M1 |
Obtain answer $15.4$ with no errors seen | A1 | 3 marks |
(iii) Justify the statement that the rule gives an over-estimate | B1 | 1 mark |6\\
\includegraphics[max width=\textwidth, alt={}, center]{4029c46c-50a1-4d23-bc29-589417a6b7f5-3_501_497_269_826}
The diagram shows the part of the curve $y = \frac { \mathrm { e } ^ { 2 x } } { x }$ for $x > 0$, and its minimum point $M$.\\
(i) Find the coordinates of $M$.\\
(ii) Use the trapezium rule with 2 intervals to estimate the value of
$$\int _ { 1 } ^ { 2 } \frac { \mathrm { e } ^ { 2 x } } { x } \mathrm {~d} x$$
giving your answer correct to 1 decimal place.\\
(iii) State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (ii).\\
(i) Given that $y = \tan 2 x$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.\\
(ii) Hence, or otherwise, show that
$$\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \sec ^ { 2 } 2 x \mathrm {~d} x = \frac { 1 } { 2 } \sqrt { } 3$$
and, by using an appropriate trigonometrical identity, find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \tan ^ { 2 } 2 x \mathrm {~d} x$.\\
(iii) Use the identity $\cos 4 x \equiv 2 \cos ^ { 2 } 2 x - 1$ to find the exact value of
$$\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \frac { 1 } { 1 + \cos 4 x } \mathrm {~d} x$$
\hfill \mbox{\textit{CAIE P2 2006 Q6 [9]}}