| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find equation of normal |
| Difficulty | Moderate -0.3 This is a straightforward application of differentiation rules. Part (a) requires substituting x=4 into the given derivative, finding the gradient, then using the perpendicular gradient formula to find the normal - all routine procedures. Part (b) involves integrating a polynomial expression after expanding, which is standard technique. The algebraic manipulation is slightly involved but follows textbook methods with no novel insight required. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.08a Fundamental theorem of calculus: integration as reverse of differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f'(4) = 6(4) - \frac{7 \times 14}{4} = -\frac{1}{2}\) | B1 | Correct gradient at \(P\) (may be implied) |
| \(m_T = -\frac{1}{2} \Rightarrow m_N = \frac{-1}{-\frac{1}{2}} = 2\) | M1 | Attempts perpendicular gradient rule; look for \(m_N = \frac{-1}{m_T}\) or \(m_T \times m_N = -1\) |
| \(y - 12 = 2(x-4)\) or \(y = mx + c \Rightarrow y = 2x + c \Rightarrow 12 = 2(4) + c \Rightarrow c = \ldots\) | M1 A1 | Attempts equation of normal using "changed" gradient with \(x=4\), \(y=12\) correctly placed |
| \(y = 2x + 4\) | A1 | Correct equation in required form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{(2x-1)(3x+2)}{2\sqrt{x}} = \frac{6x^2+x-2}{2\sqrt{x}} = 3x^{\frac{3}{2}} + \frac{1}{2}x^{\frac{1}{2}} - x^{-\frac{1}{2}}\) | M1 | Expands numerator and attempts to split; score for one correct index from correct work |
| \(f(x) = \frac{6x^2}{2} - \frac{6}{5}x^{\frac{5}{2}} - \frac{1}{3}x^{\frac{3}{2}} + 2x^{\frac{1}{2}} (+c)\) | M1 A1 A1 | M1: attempts to integrate a fractional power (e.g. \(x^{\frac{3}{2}} \to x^{\frac{5}{2}}\)); A1: any 2 correct terms; A1: all correct (unsimplified ok, \(+c\) not required here) |
| \(12 = \frac{6(4)^2}{2} - \frac{6}{5}(4)^{\frac{5}{2}} - \frac{1}{3}(4)^{\frac{3}{2}} + 2(4)^{\frac{1}{2}} + c \Rightarrow c = \ldots\) | M1 | Uses \(x=4\), \(y=12\) after increasing at least one power by 1; \(c\) must be a numerical expression |
| \(f(x) = 3x^2 - \frac{6}{5}x^{\frac{5}{2}} - \frac{1}{3}x^{\frac{3}{2}} + 2x^{\frac{1}{2}} + \frac{16}{15}\) | A1 | Correct simplified form; accept equivalent simplified forms |
## Question 10:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f'(4) = 6(4) - \frac{7 \times 14}{4} = -\frac{1}{2}$ | B1 | Correct gradient at $P$ (may be implied) |
| $m_T = -\frac{1}{2} \Rightarrow m_N = \frac{-1}{-\frac{1}{2}} = 2$ | M1 | Attempts perpendicular gradient rule; look for $m_N = \frac{-1}{m_T}$ or $m_T \times m_N = -1$ |
| $y - 12 = 2(x-4)$ or $y = mx + c \Rightarrow y = 2x + c \Rightarrow 12 = 2(4) + c \Rightarrow c = \ldots$ | M1 A1 | Attempts equation of normal using "changed" gradient with $x=4$, $y=12$ correctly placed |
| $y = 2x + 4$ | A1 | Correct equation in required form |
**(4 marks)**
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{(2x-1)(3x+2)}{2\sqrt{x}} = \frac{6x^2+x-2}{2\sqrt{x}} = 3x^{\frac{3}{2}} + \frac{1}{2}x^{\frac{1}{2}} - x^{-\frac{1}{2}}$ | M1 | Expands numerator and attempts to split; score for one correct index from correct work |
| $f(x) = \frac{6x^2}{2} - \frac{6}{5}x^{\frac{5}{2}} - \frac{1}{3}x^{\frac{3}{2}} + 2x^{\frac{1}{2}} (+c)$ | M1 A1 A1 | M1: attempts to integrate a fractional power (e.g. $x^{\frac{3}{2}} \to x^{\frac{5}{2}}$); A1: any 2 correct terms; A1: all correct (unsimplified ok, $+c$ not required here) |
| $12 = \frac{6(4)^2}{2} - \frac{6}{5}(4)^{\frac{5}{2}} - \frac{1}{3}(4)^{\frac{3}{2}} + 2(4)^{\frac{1}{2}} + c \Rightarrow c = \ldots$ | M1 | Uses $x=4$, $y=12$ after increasing at least one power by 1; $c$ must be a numerical expression |
| $f(x) = 3x^2 - \frac{6}{5}x^{\frac{5}{2}} - \frac{1}{3}x^{\frac{3}{2}} + 2x^{\frac{1}{2}} + \frac{16}{15}$ | A1 | Correct simplified form; accept equivalent simplified forms |
**(6 marks)**
---\begin{enumerate}
\item The curve $C$ has equation $y = \mathrm { f } ( x )$ where $x > 0$
\end{enumerate}
Given that
\begin{itemize}
\item $\mathrm { f } ^ { \prime } ( x ) = 6 x - \frac { ( 2 x - 1 ) ( 3 x + 2 ) } { 2 \sqrt { x } }$
\item the point $P ( 4,12 )$ lies on $C$\\
(a) find the equation of the normal to $C$ at $P$, giving your answer in the form $y = m x + c$ where $m$ and $c$ are integers to be found,\\
(b) find $\mathrm { f } ( x )$, giving each term in simplest form.
\end{itemize}
\hfill \mbox{\textit{Edexcel P1 2024 Q10 [10]}}