| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2017 |
| Session | October |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Complete table then apply trapezium rule |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard C3/C4 techniques: substituting into a function, applying the trapezium rule formula, product rule differentiation, and solving dy/dx=0. All parts are routine applications of well-practiced methods with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07l Derivative of ln(x): and related functions1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.09f Trapezium rule: numerical integration |
| \(x\) | 0 | \(\frac { \pi } { 4 }\) | \(\frac { \pi } { 2 }\) | \(\frac { 3 \pi } { 4 }\) | \(\pi\) |
| \(y\) | 0 | 0.76679 | 0.15940 | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(0.41576\) | B1 | awrt \(0.41576\). Degrees gives \(0.068835...\) scores B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Strip width \(= \frac{\pi}{4}\) | B1 | awrt \(0.785\), may be implied by \(\frac{1}{2}\times\frac{\pi}{4}\times\{...\}\) or \(\frac{\pi}{8}\times\{...\}\) |
| Area \(\approx \frac{1}{2}\times\frac{\pi}{4}\times\{0+2(0.76679+0.41576+0.15940)+0\}\) | M1 | Correct structure for trapezium formula; do not condone missing brackets unless implied |
| \(1.0540\) | A1 | awrt \(1.0540\) (not \(1.054\)). Note degrees gives \(0.78149...\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Uses \(vu'+uv'\): \(\frac{dy}{dx} = 2e^{-x}\times\frac{1}{2}(\sin x)^{-\frac{1}{2}}(\cos x) - 2e^{-x}(\sin x)^{\frac{1}{2}}\) | M1A1A1 | M1: uses product/quotient rule with \(u/v=2e^{-x}\), \(u/v=(\sin x)^{0.5}\). A1: either term correct. A1: completely correct derivative |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{dy}{dx}=0 \Rightarrow 2e^{-x}\times\frac{1}{2}(\sin x)^{-\frac{1}{2}}(\cos x) - 2e^{-x}(\sin x)^{\frac{1}{2}} = 0\) | ||
| \(\cos x = 2\sin x\) | M1 | Sets \(\frac{dy}{dx}=0\), correct algebra to \(A\cos x = B\sin x\) |
| \(\tan x = \frac{1}{2} \Rightarrow x = 0.464\) | dM1A1 | dM1: divides by \(\cos x\) to reach \(\tan x = \alpha\), \(\alpha\neq\pm1\). A1: cso awrt \(0.464\) (do not allow \(0.148\pi\)) |
# Question 6:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $0.41576$ | B1 | awrt $0.41576$. Degrees gives $0.068835...$ scores B0 |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Strip width $= \frac{\pi}{4}$ | B1 | awrt $0.785$, may be implied by $\frac{1}{2}\times\frac{\pi}{4}\times\{...\}$ or $\frac{\pi}{8}\times\{...\}$ |
| Area $\approx \frac{1}{2}\times\frac{\pi}{4}\times\{0+2(0.76679+0.41576+0.15940)+0\}$ | M1 | Correct structure for trapezium formula; do not condone missing brackets unless implied |
| $1.0540$ | A1 | awrt $1.0540$ (not $1.054$). Note degrees gives $0.78149...$ |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Uses $vu'+uv'$: $\frac{dy}{dx} = 2e^{-x}\times\frac{1}{2}(\sin x)^{-\frac{1}{2}}(\cos x) - 2e^{-x}(\sin x)^{\frac{1}{2}}$ | M1A1A1 | M1: uses product/quotient rule with $u/v=2e^{-x}$, $u/v=(\sin x)^{0.5}$. A1: either term correct. A1: completely correct derivative |
## Part (d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{dy}{dx}=0 \Rightarrow 2e^{-x}\times\frac{1}{2}(\sin x)^{-\frac{1}{2}}(\cos x) - 2e^{-x}(\sin x)^{\frac{1}{2}} = 0$ | | |
| $\cos x = 2\sin x$ | M1 | Sets $\frac{dy}{dx}=0$, correct algebra to $A\cos x = B\sin x$ |
| $\tan x = \frac{1}{2} \Rightarrow x = 0.464$ | dM1A1 | dM1: divides by $\cos x$ to reach $\tan x = \alpha$, $\alpha\neq\pm1$. A1: cso awrt $0.464$ (do not allow $0.148\pi$) |
---6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{2a6d0dba-d948-4124-9740-a88c17b0be65-16_618_1018_228_456}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of the curve $C$ with equation $y = 2 \mathrm { e } ^ { - x } \sqrt { \sin x } , 0 \leqslant x \leqslant \pi$. The finite region $R$, shown shaded in Figure 1, is bounded by the curve and the $x$-axis.
\begin{enumerate}[label=(\alph*)]
\item Complete the table below with the value of $y$ corresponding to $x = \frac { \pi } { 2 }$, giving your answer to 5 decimal places.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 0 & $\frac { \pi } { 4 }$ & $\frac { \pi } { 2 }$ & $\frac { 3 \pi } { 4 }$ & $\pi$ \\
\hline
$y$ & 0 & 0.76679 & & 0.15940 & 0 \\
\hline
\end{tabular}
\end{center}
\item Use the trapezium rule, with all the values of $y$ in the completed table, to obtain an estimate for the area of the region $R$. Give your answer to 4 decimal places.
\item Given $y = 2 \mathrm { e } ^ { - x } \sqrt { \sin x }$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ for $0 < x < \pi$.
The curve $C$ has a maximum turning point when $x = a$.
\item Use your answer to part (c) to find the value of $a$, giving your answer to 3 decimal places.
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C34 2017 Q6 [10]}}