| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Differentiate after index conversion |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring expansion of brackets with fractional indices, solving a simple equation, and routine integration. Part (a) is algebraic manipulation, part (b) is expanding and simplifying (standard technique), and part (c) is direct integration of power functions. While it involves multiple steps, each step uses standard C2 techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.02a Indices: laws of indices for rational exponents1.07i Differentiate x^n: for rational n and sums1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2x^{\frac{3}{2}} - 3x^{-\frac{3}{2}} = 0\) | M1 | |
| \(x^3 = \frac{3}{2}\) | ||
| \(x = \sqrt[3]{\frac{3}{2}}\) | M1 | |
| \(= 1.1447\ldots = 1.14\) (3 s.f.) | A1 cao | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f(x) = 4x^3 + 9x^{-3} - 12 + 5\), \(A = 4\) | B1 | |
| \(= 4x^3 + \dfrac{9}{x^3} - 7\), \(B = 9,\ C = -7\) | B1, B1 | (3) |
| \(\displaystyle\int_1^2 f(x)\,dx = \left[x^4 - \tfrac{9}{2}x^{-2} - 7x\right]_1^2\) — \(x^n \to x^{n+1}\) | M1 | |
| A2 ft | candidate's \(A,B,C\); \(-1\) each error | |
| \(= \left(2^4 - \tfrac{9}{2}\times 2^{-2} - 14\right) - \left(1 - \tfrac{9}{2} - 7\right)\) | M1 | use of limits |
| \(= 11\tfrac{3}{8}\) or \(11.375\) | A1 | (5) |
## Question 6:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2x^{\frac{3}{2}} - 3x^{-\frac{3}{2}} = 0$ | M1 | |
| $x^3 = \frac{3}{2}$ | | |
| $x = \sqrt[3]{\frac{3}{2}}$ | M1 | |
| $= 1.1447\ldots = 1.14$ (3 s.f.) | A1 cao | (3) |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(x) = 4x^3 + 9x^{-3} - 12 + 5$, $A = 4$ | B1 | |
| $= 4x^3 + \dfrac{9}{x^3} - 7$, $B = 9,\ C = -7$ | B1, B1 | (3) |
| $\displaystyle\int_1^2 f(x)\,dx = \left[x^4 - \tfrac{9}{2}x^{-2} - 7x\right]_1^2$ — $x^n \to x^{n+1}$ | M1 | |
| | A2 ft | candidate's $A,B,C$; $-1$ each error |
| $= \left(2^4 - \tfrac{9}{2}\times 2^{-2} - 14\right) - \left(1 - \tfrac{9}{2} - 7\right)$ | M1 | use of limits |
| $= 11\tfrac{3}{8}$ or $11.375$ | A1 | (5) |
---6. Given that $\mathrm { f } ( x ) = \left( 2 x ^ { \frac { 3 } { 2 } } - 3 x ^ { - \frac { 3 } { 2 } } \right) ^ { 2 } + 5 , x > 0$,
\begin{enumerate}[label=(\alph*)]
\item find, to 3 significant figures, the value of $x$ for which $\mathrm { f } ( x ) = 5$.
\item Show that $\mathrm { f } ( x )$ may be written in the form $A x ^ { 3 } + \frac { B } { x ^ { 3 } } + C$, where $A , B$ and $C$ are constants to be found.
\item Hence evaluate $\int _ { 1 } ^ { 2 } f ( x ) d x$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q6 [11]}}