| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Finding curve equation from derivative |
| Difficulty | Moderate -0.3 This is a straightforward integration question requiring simplification of the integrand into powers of x (x^{1/2} + 9x^{-1/2}), basic integration using the power rule, finding the constant using the given point, then solving a simple quadratic equation in part (b). Slightly easier than average as it's a standard C1 technique with no conceptual challenges. |
| 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 | Marks | Guidance |
| \(f'(x) = \frac{x+9}{\sqrt{x}} = \frac{x}{\sqrt{x}} + \frac{9}{\sqrt{x}} = x^{\frac{1}{2}} + 9x^{-\frac{1}{2}}\) | M1A1 | M1: Correct attempt to split into 2 separate terms. Divides by \(x^{\frac{1}{2}}\) or multiplies by \(x^{-\frac{1}{2}}\). A1: \(x^{\frac{1}{2}} + 9x^{-\frac{1}{2}}\) or equivalent |
| \(f(x) = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + \frac{9x^{\frac{1}{2}}}{\frac{1}{2}} (+c)\) | M1A1 | M1: Independent method mark for \(x^n \rightarrow x^{n+1}\) on separate terms. A1: Allow un-simplified answers, no requirement for \(+c\) |
| \(\frac{(9)^{\frac{3}{2}}}{\frac{3}{2}} + \frac{9(9)^{\frac{1}{2}}}{\frac{1}{2}} + c = 0 \Rightarrow c = ...\) | M1 | Substitutes \(x = 9\) and \(y = 0\). If no \(c\) at this stage M0A0 follows |
| \(f(x) = \frac{2}{3}x^{\frac{3}{2}} + 18x^{\frac{1}{2}} - 72\) | A1 | No requirement to simplify; accept any correct un-simplified form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f'(x) = \frac{x+9}{\sqrt{x}} = 10 \Rightarrow x + 9 = 10\sqrt{x}\) | M1 | Sets \(f'(x) = \frac{x+9}{\sqrt{x}} = 10\) and multiplies by \(\sqrt{x}\). Must be setting original \(f'(x) = 10\) or equivalent correct expression \(= 10\) |
| \((\sqrt{x} - 9)(\sqrt{x} - 1) = 0 \Rightarrow \sqrt{x} = ...\) | dM1 | Correct attempt to solve relevant 3TQ in \(\sqrt{x}\) leading to solution for \(\sqrt{x}\). Dependent on previous M1 |
| \(x = 81,\ x = 1\) | A1, B1 | Note \(x = 1\) solution could be just written down and is B1 but must come from a correct equation |
## Question 10:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f'(x) = \frac{x+9}{\sqrt{x}} = \frac{x}{\sqrt{x}} + \frac{9}{\sqrt{x}} = x^{\frac{1}{2}} + 9x^{-\frac{1}{2}}$ | M1A1 | M1: Correct attempt to split into 2 separate terms. Divides by $x^{\frac{1}{2}}$ or multiplies by $x^{-\frac{1}{2}}$. A1: $x^{\frac{1}{2}} + 9x^{-\frac{1}{2}}$ or equivalent |
| $f(x) = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + \frac{9x^{\frac{1}{2}}}{\frac{1}{2}} (+c)$ | M1A1 | M1: Independent method mark for $x^n \rightarrow x^{n+1}$ on separate terms. A1: Allow un-simplified answers, no requirement for $+c$ |
| $\frac{(9)^{\frac{3}{2}}}{\frac{3}{2}} + \frac{9(9)^{\frac{1}{2}}}{\frac{1}{2}} + c = 0 \Rightarrow c = ...$ | M1 | Substitutes $x = 9$ and $y = 0$. If no $c$ at this stage M0A0 follows |
| $f(x) = \frac{2}{3}x^{\frac{3}{2}} + 18x^{\frac{1}{2}} - 72$ | A1 | No requirement to simplify; accept any correct un-simplified form |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f'(x) = \frac{x+9}{\sqrt{x}} = 10 \Rightarrow x + 9 = 10\sqrt{x}$ | M1 | Sets $f'(x) = \frac{x+9}{\sqrt{x}} = 10$ and multiplies by $\sqrt{x}$. Must be setting original $f'(x) = 10$ or equivalent correct expression $= 10$ |
| $(\sqrt{x} - 9)(\sqrt{x} - 1) = 0 \Rightarrow \sqrt{x} = ...$ | dM1 | Correct attempt to solve relevant 3TQ in $\sqrt{x}$ leading to solution for $\sqrt{x}$. Dependent on previous M1 |
| $x = 81,\ x = 1$ | A1, B1 | Note $x = 1$ solution could be just written down and is B1 but must come from a correct equation |10. A curve has equation $y = \mathrm { f } ( x )$. The point $P$ with coordinates $( 9,0 )$ lies on the curve.
Given that
$$\mathrm { f } ^ { \prime } ( x ) = \frac { x + 9 } { \sqrt { } x } , \quad x > 0$$
\begin{enumerate}[label=(\alph*)]
\item find $\mathrm { f } ( x )$.
\item Find the $x$-coordinates of the two points on $y = \mathrm { f } ( x )$ where the gradient of the curve is equal to 10
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2013 Q10 [10]}}