| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2015 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find stationary points |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard differentiation (including fractional powers), finding stationary points, and basic integration. Part (a) is routine differentiation, part (b) requires substituting x=4 into dy/dx=0 (simple algebra), and part (c) is definite integration of a polynomial. All techniques are standard Core 1/2 material with no problem-solving insight required, making it slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{dy}{dx} = 3x^2 + 10 \times \frac{3}{2}x^{\frac{1}{2}} + k\) | M1A1 | M1: fractional power dealt with correctly, becomes \(\frac{3}{2}x^{\frac{1}{2}}\); A1: all terms correct [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Substitutes \(x=4\) and \(\frac{dy}{dx}=0\): \(3(4)^2+15(4)^{\frac{1}{2}}+k=0 \Rightarrow k=-78\) | M1 A1* | M1: must see \(3(4)^2+15(4)^{\frac{1}{2}}+k=0\) or \(48+30+k=0\); printed answer so all working must be correct [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| When \(x=4\), \(y=-168\) | B1 | Substitute into \(y\) to find \(y\) |
| \(\int x^3+10x^{\frac{3}{2}}-78x\,(+168)\,dx = \frac{1}{4}x^4+\frac{10}{\frac{5}{2}}x^{\frac{5}{2}}-\frac{78}{2}x^2(+168x+c)\) | M1A1 | M1: attempt to integrate, at least one power increases; A1: correct unsimplified answer |
| Use limits \(0\) and \(4\) to give \(\pm432\) (or \(\pm240\) if \(168x\) included) | dB1 | Use limit \(4\) to give \(432\); depends on M1A1 |
| Rectangle area is \(4 \times 168 = 672\) | M1 | Calculates rectangle area; must be rectangle not triangle |
| \(R = 672 - 432 = 240\) | M1 A1 | M1: subtracts numerical areas; A1: \(240\) only [7] |
# Question 15:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{dy}{dx} = 3x^2 + 10 \times \frac{3}{2}x^{\frac{1}{2}} + k$ | M1A1 | M1: fractional power dealt with correctly, becomes $\frac{3}{2}x^{\frac{1}{2}}$; A1: all terms correct **[2]** |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Substitutes $x=4$ and $\frac{dy}{dx}=0$: $3(4)^2+15(4)^{\frac{1}{2}}+k=0 \Rightarrow k=-78$ | M1 A1* | M1: must see $3(4)^2+15(4)^{\frac{1}{2}}+k=0$ or $48+30+k=0$; printed answer so all working must be correct **[2]** |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| When $x=4$, $y=-168$ | B1 | Substitute into $y$ to find $y$ |
| $\int x^3+10x^{\frac{3}{2}}-78x\,(+168)\,dx = \frac{1}{4}x^4+\frac{10}{\frac{5}{2}}x^{\frac{5}{2}}-\frac{78}{2}x^2(+168x+c)$ | M1A1 | M1: attempt to integrate, at least one power increases; A1: correct unsimplified answer |
| Use limits $0$ and $4$ to give $\pm432$ (or $\pm240$ if $168x$ included) | dB1 | Use limit $4$ to give $432$; depends on M1A1 |
| Rectangle area is $4 \times 168 = 672$ | M1 | Calculates rectangle area; must be rectangle not triangle |
| $R = 672 - 432 = 240$ | M1 A1 | M1: subtracts numerical areas; A1: $240$ only **[7]** |
The images you've shared appear to be blank pages (aside from the header showing "PhysicsAndMathsTutor.com" and "January 2015 (IAL)"). There is no mark scheme content visible on these pages to extract.
Could you please share the actual pages containing the mark scheme questions and answers? These appear to be blank/spare pages from the document.15.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{3b99072a-cd16-4c1d-9e44-085926a3ba24-24_591_570_255_678}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{center}
\end{figure}
Figure 5 shows a sketch of part of the curve $C$ with equation
$$y = x ^ { 3 } + 10 x ^ { \frac { 3 } { 2 } } + k x , \quad x \geqslant 0$$
where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$
The point $P$ on the curve $C$ is a minimum turning point.\\
Given that the $x$ coordinate of $P$ is 4
\item show that $k = - 78$
The line through $P$ parallel to the $x$-axis cuts the $y$-axis at the point $N$.\\
The finite region $R$, shown shaded in Figure 5, is bounded by $C$, the $y$-axis and $P N$.
\item Use integration to find the area of $R$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2015 Q15 [11]}}