| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Rewrite with fractional indices |
| Difficulty | Easy -1.2 This is a straightforward C1 question testing basic index law manipulation (subtracting powers when dividing) and routine integration of power functions. Part (a) requires only mechanical application of x^a / x^b = x^(a-b), and part (b) is standard integration with a boundary condition. No problem-solving or insight needed. |
| 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 |
|---|---|---|
| (a) \(p = \frac{1}{2}, q = 2\) or \(6x^{1/2}, 3x^{3}\) | B1, B1 | Accept any equivalent answers, e.g. \(p = 0.5, q = 4/2\) |
| (b) \(\frac{6x^{3}}{(3x^{1/2})^2} + \frac{3x^3}{3} \left(= 4x^{3/2} + x^3\right)\) | M1 A1ft | |
| \(x = 4, y = 90: 32 + 64 + C = 90 \Rightarrow C = -6\) | M1 A1 | |
| \(y = 4x^{3/2} + x^3 + \text{"their } -6\text{"} \) | A1 | |
| Notes | (a) Accept any equivalent answers, e.g. \(p = 0.5, q = 4/2\). (b) 1st M: Attempt to integrate \(x^n \to x^{n+1}\) (for either term). 1st A: ft their p and q, but terms need not be simplified (+C not required for this mark). 2nd M: Using \(x = 4\) and \(y = 90\) to form an equation in C. 2nd A: cao. 3rd A: answer as shown with simplified correct coefficients and powers – but follow through their value for C. If there is a 'restart' in part (b) it can be marked independently of part (a), but marks for part (a) cannot be scored for work seen in (b). Numerator and denominator integrated separately: First M mark cannot be awarded so only mark available is second M mark. So 1 out of 5 marks. |
(a) $p = \frac{1}{2}, q = 2$ or $6x^{1/2}, 3x^{3}$ | B1, B1 | Accept any equivalent answers, e.g. $p = 0.5, q = 4/2$
(b) $\frac{6x^{3}}{(3x^{1/2})^2} + \frac{3x^3}{3} \left(= 4x^{3/2} + x^3\right)$ | M1 A1ft |
$x = 4, y = 90: 32 + 64 + C = 90 \Rightarrow C = -6$ | M1 A1 |
$y = 4x^{3/2} + x^3 + \text{"their } -6\text{"} $ | A1 |
Notes | | (a) Accept any equivalent answers, e.g. $p = 0.5, q = 4/2$. (b) 1st M: Attempt to integrate $x^n \to x^{n+1}$ (for either term). 1st A: ft their p and q, but terms need not be simplified (+C not required for this mark). 2nd M: Using $x = 4$ and $y = 90$ to form an equation in C. 2nd A: cao. 3rd A: answer as shown with simplified correct coefficients and powers – but follow through their value for C. If there is a 'restart' in part (b) it can be marked independently of part (a), but marks for part (a) cannot be scored for work seen in (b). Numerator and denominator integrated separately: First M mark cannot be awarded so only mark available is second M mark. So 1 out of 5 marks.
---6. Given that $\frac { 6 x + 3 x ^ { \frac { 5 } { 2 } } } { \sqrt { } x }$ can be written in the form $6 x ^ { p } + 3 x ^ { q }$,
\begin{enumerate}[label=(\alph*)]
\item write down the value of $p$ and the value of $q$.
Given that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 x + 3 x ^ { \frac { 5 } { 2 } } } { \sqrt { } x }$, and that $y = 90$ when $x = 4$,
\item find $y$ in terms of $x$, simplifying the coefficient of each term.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2011 Q6 [7]}}