| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find curve equation from derivative (straightforward integration + point) |
| Difficulty | Moderate -0.8 This is a straightforward C1 integration question requiring only standard power rule integration (rewriting surds as fractional powers) and finding a constant using a given point. Part (a) is simple substitution and simplification. The techniques are routine with no problem-solving insight needed, making it easier than average but not trivial since it requires correct manipulation of fractional indices. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits |
| Answer | Marks |
|---|---|
| \(\sqrt{8} = 2\sqrt{2}\) seen or used somewhere (possibly implied) | B1 |
| \(\frac{12}{\sqrt{8}} = \frac{12\sqrt{8}}{8}\) or \(\frac{12}{2\sqrt{2}} = \frac{12\sqrt{2}}{4}\) | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| At \(x = 8\), \(\frac{dy}{dx} = 3\sqrt{8} + \frac{12}{\sqrt{8}} = 6\sqrt{2} + 3\sqrt{2} = 9\sqrt{2}\) | A1 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Integrating: \(\frac{3x^{3/2}}{\left(\frac{3}{2}\right)} + \frac{12x^{1/2}}{\left(\frac{1}{2}\right)} (+C)\) | M1 A1 A1 | \(C\) not required |
| At \((4, 30)\): \(\frac{3 \times 4^{3/2}}{\left(\frac{3}{2}\right)} + \frac{12 \times 4^{1/2}}{\left(\frac{1}{2}\right)} + C = 30\) | M1 | \(C\) required |
| \(f(x) = 2x^{3/2} + 24x^{1/2} - 34\) | A1, A1 | (6 marks) (9 marks) |
## Question 5:
### Part (a)
$\sqrt{8} = 2\sqrt{2}$ seen or used somewhere (possibly implied) | B1 |
$\frac{12}{\sqrt{8}} = \frac{12\sqrt{8}}{8}$ or $\frac{12}{2\sqrt{2}} = \frac{12\sqrt{2}}{4}$ | M1 |
Direct statement e.g. $\frac{6}{\sqrt{2}} = 3\sqrt{2}$ (no indication of method) is M0
At $x = 8$, $\frac{dy}{dx} = 3\sqrt{8} + \frac{12}{\sqrt{8}} = 6\sqrt{2} + 3\sqrt{2} = 9\sqrt{2}$ | A1 | (3 marks)
### Part (b)
Integrating: $\frac{3x^{3/2}}{\left(\frac{3}{2}\right)} + \frac{12x^{1/2}}{\left(\frac{1}{2}\right)} (+C)$ | M1 A1 A1 | $C$ not required
At $(4, 30)$: $\frac{3 \times 4^{3/2}}{\left(\frac{3}{2}\right)} + \frac{12 \times 4^{1/2}}{\left(\frac{1}{2}\right)} + C = 30$ | M1 | $C$ required
$f(x) = 2x^{3/2} + 24x^{1/2} - 34$ | A1, A1 | (6 marks) **(9 marks)**
---5. The curve $C$ with equation $y = \mathrm { f } ( x )$ is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { } x + \frac { 12 } { \sqrt { } x } , x > 0$.
\begin{enumerate}[label=(\alph*)]
\item Show that, when $x = 8$, the exact value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ is $9 \sqrt { } 2$.
The curve $C$ passes through the point $( 4,30 )$.
\item Using integration, find $\mathrm { f } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q5 [9]}}