| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Rewrite with fractional indices |
| Difficulty | Moderate -0.8 This is a straightforward algebraic manipulation question requiring conversion of roots to fractional indices, simplification by subtracting exponents, followed by routine differentiation using the power rule. All steps are standard techniques with no problem-solving insight required, making it easier than average but not trivial due to the multi-step nature. |
| Spec | 1.02a Indices: laws of indices for rational exponents1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y = \frac{5x^2 + \sqrt{x^3}}{\sqrt[3]{8x}} = \frac{5x^2 + x^{\frac{3}{2}}}{...x^{\frac{1}{3}}}\) | M1 | Converts radicals to powers of \(x\), at least one power correct: \(\sqrt{x^3} \to x^{\frac{3}{2}}\) or \(\sqrt[3]{8x} \to ...x^{\frac{1}{3}}\) |
| \(= \frac{5x^{2-\frac{1}{3}}}{2} + \frac{x^{\frac{3}{2}-\frac{1}{3}}}{2}\) | M1 | Correct subtraction index law used at least once (may be implied) |
| \(= \frac{5}{2}x^{\frac{5}{3}} + \frac{1}{2}x^{\frac{7}{6}}\) | A1 A1 | First A1: one term correct (coefficient and index) or both indices correct. Second A1: fully correct expression |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{dy}{dx} = ...x^{\frac{2}{3}} + ...x^{\frac{1}{6}}\) | M1 | \(x^n \to ...x^{n-1}\) for at least one term where \(n \neq 1\) |
| \(\frac{dy}{dx} = \frac{25}{6}x^{\frac{2}{3}} + \frac{7}{12}x^{\frac{1}{6}}\) | A1ft A1 | A1ft: correct differentiation of one term with \(n \neq 1\), follow through from (a). A1: fully correct derivative in simplest form. Score A0 if "\(+c\)" included |
# Question 4:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y = \frac{5x^2 + \sqrt{x^3}}{\sqrt[3]{8x}} = \frac{5x^2 + x^{\frac{3}{2}}}{...x^{\frac{1}{3}}}$ | **M1** | Converts radicals to powers of $x$, at least one power correct: $\sqrt{x^3} \to x^{\frac{3}{2}}$ or $\sqrt[3]{8x} \to ...x^{\frac{1}{3}}$ |
| $= \frac{5x^{2-\frac{1}{3}}}{2} + \frac{x^{\frac{3}{2}-\frac{1}{3}}}{2}$ | **M1** | Correct subtraction index law used at least once (may be implied) |
| $= \frac{5}{2}x^{\frac{5}{3}} + \frac{1}{2}x^{\frac{7}{6}}$ | **A1 A1** | First A1: one term correct (coefficient and index) or both indices correct. Second A1: fully correct expression |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dy}{dx} = ...x^{\frac{2}{3}} + ...x^{\frac{1}{6}}$ | **M1** | $x^n \to ...x^{n-1}$ for at least one term where $n \neq 1$ |
| $\frac{dy}{dx} = \frac{25}{6}x^{\frac{2}{3}} + \frac{7}{12}x^{\frac{1}{6}}$ | **A1ft A1** | A1ft: correct differentiation of one term with $n \neq 1$, follow through from (a). A1: fully correct derivative in simplest form. Score A0 if "$+c$" included |\begin{enumerate}
\item In this question you must show all stages of your working.\\
(a) Write
\end{enumerate}
$$y = \frac { 5 x ^ { 2 } + \sqrt { x ^ { 3 } } } { \sqrt [ 3 ] { 8 x } }$$
in the form
$$y = A x ^ { p } + B x ^ { q }$$
where $A , B , p$ and $q$ are constants to be found.\\
(b) Hence find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ giving each coefficient in simplest form.
\hfill \mbox{\textit{Edexcel P1 2023 Q4 [7]}}