| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Differentiate after index conversion |
| Difficulty | Moderate -0.8 This is a straightforward C1 differentiation question requiring expansion of brackets, simplification using index laws, then routine application of the power rule. All steps are mechanical with no problem-solving insight needed, making it easier than average but not trivial due to the algebraic manipulation required. |
| Spec | 1.02a Indices: laws of indices for rational exponents1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\left[(3-4\sqrt{x})^2 =\right] 9-12\sqrt{x}-12\sqrt{x}+(-4)^2 x\) | M1 | Attempt to expand with at least 3 terms correct |
| \(9x^{-\frac{1}{2}}+16x^{\frac{1}{2}}-24\) | A1, A1 | 1st A1: coefficient of \(\sqrt{x}=16\); 2nd A1: constant term \(=-24\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f'(x) = -\frac{9}{2}x^{-\frac{3}{2}} + \frac{16}{2}x^{-\frac{1}{2}}\) | M1 A1, A1ft | M1 for \(x^n \to x^{n-1}\); 1st A1 for \(-\frac{9}{2}x^{-\frac{3}{2}}\) and constant \(B\) differentiated to zero; 2nd A1ft follows through their \(Ax^{\frac{1}{2}}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f'(9) = -\frac{9}{2}\times\frac{1}{27}+\frac{16}{2}\times\frac{1}{3} = -\frac{1}{6}+\frac{16}{6}=\frac{5}{2}\) | M1 A1 | M1 for correct substitution of \(x=9\); accept \(\frac{15}{6}\), \(\frac{45}{18}\), \(\frac{135}{54}\), \(\frac{67.5}{27}\) or \(2.5\) |
# Question 9:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\left[(3-4\sqrt{x})^2 =\right] 9-12\sqrt{x}-12\sqrt{x}+(-4)^2 x$ | M1 | Attempt to expand with at least 3 terms correct |
| $9x^{-\frac{1}{2}}+16x^{\frac{1}{2}}-24$ | A1, A1 | 1st A1: coefficient of $\sqrt{x}=16$; 2nd A1: constant term $=-24$ |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f'(x) = -\frac{9}{2}x^{-\frac{3}{2}} + \frac{16}{2}x^{-\frac{1}{2}}$ | M1 A1, A1ft | M1 for $x^n \to x^{n-1}$; 1st A1 for $-\frac{9}{2}x^{-\frac{3}{2}}$ and constant $B$ differentiated to zero; 2nd A1ft follows through their $Ax^{\frac{1}{2}}$ |
## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f'(9) = -\frac{9}{2}\times\frac{1}{27}+\frac{16}{2}\times\frac{1}{3} = -\frac{1}{6}+\frac{16}{6}=\frac{5}{2}$ | M1 A1 | M1 for correct substitution of $x=9$; accept $\frac{15}{6}$, $\frac{45}{18}$, $\frac{135}{54}$, $\frac{67.5}{27}$ or $2.5$ |
---9.
$$f ( x ) = \frac { ( 3 - 4 \sqrt { } x ) ^ { 2 } } { \sqrt { } x } , \quad x > 0$$
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { f } ( x ) = 9 x ^ { - \frac { 1 } { 2 } } + A x ^ { \frac { 1 } { 2 } } + B$, where $A$ and $B$ are constants to be found.
\item Find $\mathrm { f } ^ { \prime } ( x )$.
\item Evaluate $\mathrm { f } ^ { \prime } ( 9 )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2009 Q9 [8]}}