| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Find gradient at specific point |
| Difficulty | Easy -1.2 This is a straightforward differentiation question requiring only the power rule for fractional indices, followed by direct substitution. No chain rule is actually needed despite the topic label. This is routine P1 material with no problem-solving element—purely mechanical application of standard techniques. |
| Spec | 1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y = 2 + 10x^{\frac{1}{2}} - 2x^{\frac{3}{2}}\) (given) | ||
| \(\frac{dy}{dx} = 5x^{-\frac{1}{2}} - 3x^{\frac{1}{2}}\) | M1 A1 A1 | M1: For ANY one of: \(10x^{\frac{1}{2}} \to \ldots x^{-\frac{1}{2}}\) or \(-2x^{\frac{3}{2}} \to \ldots x^{\frac{1}{2}}\) or \(2 \to 0\). A1: One simplified term correct e.g. \(10x^{\frac{1}{2}} \to 5x^{-\frac{1}{2}}\) or \(-2x^{\frac{3}{2}} \to -3x^{\frac{1}{2}}\). A1: Full correct answer \(5x^{-\frac{1}{2}} - 3x^{\frac{1}{2}}\); allow e.g. \(\frac{5}{\sqrt{x}} - 3\sqrt{x}\). No extra terms permitted. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x = 2 \Rightarrow \frac{dy}{dx} = \frac{5}{\sqrt{2}} - 3\sqrt{2}\) | M1 | M1: Substitute \(x = 2\) fully into their \(\frac{dy}{dx}\) which is not \(y\); must be a "changed" function. Mark may be implied by decimal answer of awrt \(-0.7\) following correct derivative. Do not award if \(\frac{dy}{dx} = 5x^{-\frac{1}{2}} - 3x^{\frac{1}{2}} + c\) in part (a) AND subsequently tries to find \(c\) using \(x = 2\). |
| \(\frac{5}{\sqrt{2}} - 3\sqrt{2} = \frac{5}{2}\sqrt{2} - 3\sqrt{2} = -\frac{1}{2}\sqrt{2}\) | A1 | A1: \(-\frac{1}{2}\sqrt{2}\) or exact simplified equivalent e.g. \(-\frac{1}{\sqrt{2}},\ -0.5\sqrt{2},\ -2^{-\frac{1}{2}},\ -2^{-0.5}\) following at least one intermediate line of working. |
# Question 1:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = 2 + 10x^{\frac{1}{2}} - 2x^{\frac{3}{2}}$ (given) | | |
| $\frac{dy}{dx} = 5x^{-\frac{1}{2}} - 3x^{\frac{1}{2}}$ | M1 A1 A1 | M1: For ANY one of: $10x^{\frac{1}{2}} \to \ldots x^{-\frac{1}{2}}$ or $-2x^{\frac{3}{2}} \to \ldots x^{\frac{1}{2}}$ or $2 \to 0$. A1: One simplified term correct e.g. $10x^{\frac{1}{2}} \to 5x^{-\frac{1}{2}}$ or $-2x^{\frac{3}{2}} \to -3x^{\frac{1}{2}}$. A1: Full correct answer $5x^{-\frac{1}{2}} - 3x^{\frac{1}{2}}$; allow e.g. $\frac{5}{\sqrt{x}} - 3\sqrt{x}$. No extra terms permitted. |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x = 2 \Rightarrow \frac{dy}{dx} = \frac{5}{\sqrt{2}} - 3\sqrt{2}$ | M1 | M1: Substitute $x = 2$ fully into their $\frac{dy}{dx}$ which is not $y$; must be a "changed" function. Mark may be implied by decimal answer of awrt $-0.7$ following correct derivative. Do **not** award if $\frac{dy}{dx} = 5x^{-\frac{1}{2}} - 3x^{\frac{1}{2}} + c$ in part (a) AND subsequently tries to find $c$ using $x = 2$. |
| $\frac{5}{\sqrt{2}} - 3\sqrt{2} = \frac{5}{2}\sqrt{2} - 3\sqrt{2} = -\frac{1}{2}\sqrt{2}$ | A1 | A1: $-\frac{1}{2}\sqrt{2}$ or exact simplified equivalent e.g. $-\frac{1}{\sqrt{2}},\ -0.5\sqrt{2},\ -2^{-\frac{1}{2}},\ -2^{-0.5}$ **following at least one intermediate line of working**. |
---\begin{enumerate}
\item A curve $C$ has equation
\end{enumerate}
$$y = 2 + 10 x ^ { \frac { 1 } { 2 } } - 2 x ^ { \frac { 3 } { 2 } } \quad x > 0$$
(a) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ giving your answer in simplest form.\\
(b) Hence find the exact value of the gradient of the tangent to $C$ at the point where $x = 2$ giving your answer in simplest form.\\
(Solutions relying on calculator technology are not acceptable.)
\hfill \mbox{\textit{Edexcel P1 2023 Q1 [5]}}