| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2018 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find indefinite integral of polynomial/power |
| Difficulty | Easy -1.2 This is a straightforward Core 1/2 question testing basic differentiation and integration of power functions. Part (a) requires applying the power rule to x^(2/3), and part (b) involves integrating each term separately using standard power rule for integration. Both are routine procedures with fractional powers, requiring no problem-solving or insight beyond direct application of learned rules. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x^{\frac{2}{3}} \rightarrow x^{-\frac{1}{3}}\) | M1 | Reducing the power of \(x^{\frac{2}{3}}\) by 1; may be implied by \(x^{\frac{2}{3}} \rightarrow x^{\frac{2}{3}-1}\) and no other powers of \(x\) |
| \(\left(\frac{dy}{dx}\right) = \frac{2}{9}x^{-\frac{1}{3}}\) | A1 | Correct expression. Allow equivalent exact simplified forms e.g. \(\frac{2x^{-\frac{1}{3}}}{9}\), \(\frac{2}{9x^{\frac{1}{3}}}\), \(\frac{2}{9\sqrt[3]{x}}\). Allow 0.222... or 0.2 with dot over 2 for \(\frac{2}{9}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x^{\frac{2}{3}} \rightarrow x^{\frac{5}{3}}\) or \(k \rightarrow kx\) | M1 | Increases power by 1 for one term. Must come from correct work integrating numerator and denominator separately is M0 |
| \(\frac{3}{5} \times \frac{2}{6}x^{\frac{5}{3}}\) or \(\frac{3}{6}x\) | A1 | One correct term, may be un-simplified including the power |
| \(\frac{1}{5}x^{\frac{5}{3}} + \frac{1}{2}x + c\) | A1 | All correct and simplified including \(+c\) all on one line. Allow \(\sqrt[3]{x^5}\) for \(x^{\frac{5}{3}}\); allow 0.2 for \(\frac{1}{5}\) and 0.5 for \(\frac{1}{2}\) |
## Question 1:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^{\frac{2}{3}} \rightarrow x^{-\frac{1}{3}}$ | M1 | Reducing the power of $x^{\frac{2}{3}}$ by 1; may be implied by $x^{\frac{2}{3}} \rightarrow x^{\frac{2}{3}-1}$ and no other powers of $x$ |
| $\left(\frac{dy}{dx}\right) = \frac{2}{9}x^{-\frac{1}{3}}$ | A1 | Correct expression. Allow equivalent exact simplified forms e.g. $\frac{2x^{-\frac{1}{3}}}{9}$, $\frac{2}{9x^{\frac{1}{3}}}$, $\frac{2}{9\sqrt[3]{x}}$. Allow 0.222... or 0.2 with dot over 2 for $\frac{2}{9}$ |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^{\frac{2}{3}} \rightarrow x^{\frac{5}{3}}$ or $k \rightarrow kx$ | M1 | Increases power by 1 for one term. Must come from correct work integrating numerator and denominator separately is M0 |
| $\frac{3}{5} \times \frac{2}{6}x^{\frac{5}{3}}$ or $\frac{3}{6}x$ | A1 | One correct term, may be un-simplified including the power |
| $\frac{1}{5}x^{\frac{5}{3}} + \frac{1}{2}x + c$ | A1 | All correct and simplified including $+c$ all on one line. Allow $\sqrt[3]{x^5}$ for $x^{\frac{5}{3}}$; allow 0.2 for $\frac{1}{5}$ and 0.5 for $\frac{1}{2}$ |
---\begin{enumerate}
\item Given that
\end{enumerate}
$$y = \frac { 2 x ^ { \frac { 2 } { 3 } } + 3 } { 6 } , \quad x > 0$$
find, in the simplest form,\\
(a) $\frac { \mathrm { d } y } { \mathrm {~d} x }$\\
(b) $\int y \mathrm {~d} x$\\
\hfill \mbox{\textit{Edexcel C12 2018 Q1 [5]}}