| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | January |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Rewrite with fractional indices |
| Difficulty | Easy -1.2 This is a straightforward application of index laws requiring students to convert between radical and fractional index notation. Part (a) is direct recall (∜(x³) = x^(3/4)), and part (b) requires simple algebraic manipulation by subtracting exponents. No problem-solving or conceptual insight needed—purely mechanical application of rules taught early in C2. |
| Spec | 1.02a Indices: laws of indices for rational exponents |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\sqrt[4]{x^3} = x^{\frac{3}{4}}\) | B1 (Total: 1) | Accept \(k = \frac{3}{4}\) OE |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\frac{1-x^2}{\sqrt[4]{x^3}} = \frac{1}{\sqrt[4]{x^3}} - \frac{x^2}{\sqrt[4]{x^3}} = x^{-k} - \frac{x^2}{\sqrt[4]{x^3}}\) | M1 | Split followed by at least one correct index law used to remove denominator |
| \(= x^{-\frac{3}{4}} - x^{\frac{5}{4}}\) | A1F (Total: 2) | If incorrect, ft on c's non-integer \(k\) value from part (a), provided M1 awarded. Accept answer given in form of values for \(p\) and \(q\) |
## Question 3:
**Part (a):**
| Working | Mark | Guidance |
|---------|------|----------|
| $\sqrt[4]{x^3} = x^{\frac{3}{4}}$ | B1 (Total: 1) | Accept $k = \frac{3}{4}$ OE |
**Part (b):**
| Working | Mark | Guidance |
|---------|------|----------|
| $\frac{1-x^2}{\sqrt[4]{x^3}} = \frac{1}{\sqrt[4]{x^3}} - \frac{x^2}{\sqrt[4]{x^3}} = x^{-k} - \frac{x^2}{\sqrt[4]{x^3}}$ | M1 | Split followed by at least one correct index law used to remove denominator |
| $= x^{-\frac{3}{4}} - x^{\frac{5}{4}}$ | A1F (Total: 2) | If incorrect, ft on c's non-integer $k$ value from part (a), provided M1 awarded. Accept answer given in form of values for $p$ and $q$ |
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\begin{enumerate}[label=(\alph*)]
\item Write $\sqrt [ 4 ] { x ^ { 3 } }$ in the form $x ^ { k }$.
\item Write $\frac { 1 - x ^ { 2 } } { \sqrt [ 4 ] { x ^ { 3 } } }$ in the form $x ^ { p } - x ^ { q }$.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2012 Q3 [3]}}