| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Equation of normal line |
| Difficulty | Moderate -0.8 This is a straightforward P1 differentiation question requiring term-by-term differentiation (including a negative power), finding a gradient at a point, and writing the normal equation. All steps are routine procedures with no problem-solving insight needed, making it easier than average but not trivial due to the fractional x-coordinate requiring careful arithmetic. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
| VILU SIHI NI JIIIM ION OC | VIUV SIHI NI III M M I ON OO | VIAV SIHI NI JIIIM I ION OC |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y = 2x^3 - 5x^2 - \frac{3}{2x} + 7 \Rightarrow \frac{dy}{dx} = 6x^2 - 10x + \frac{3}{2x^2}\) | M1 | Reducing any power by 1 at least once, \(x^n \to x^{n-1}\) including \(7 \to 0\); allow from incorrect index when manipulating \(\frac{3}{2x}\) |
| Two of three correct terms (simplified) \(6x^2 - 10x + \frac{3}{2x^2}\) | A1 | Do not allow \(-10x^1\) |
| All three: \(6x^2 - 10x + \frac{3}{2x^2}\) or \(6x^2 - 10x + \frac{3}{2}x^{-2}\) on one line | A1 | Do not accept \(6x^2 + -10x + \frac{3}{2}x^{-2}\); do not allow \(-10x^1\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x = \frac{1}{2} \Rightarrow y = 3\) | B1 | Correct coordinates for \(P\); ignore labelling of \(y\) |
| Substitutes \(x = \frac{1}{2}\) into \(\frac{dy}{dx} = 6x^2 - 10x + \frac{3}{2x^2} = \frac{5}{2}\) | M1 | Must find a value; ignore labelling of \(\frac{dy}{dx}\) |
| Uses perpendicular gradient rule: \(\frac{5}{2} \to -\frac{2}{5}\) | dM1 | Finds/uses gradient of normal via \(m \to -\frac{1}{m}\); dependent on previous M mark |
| Attempts equation of normal at \(P\): \(y - 3 = -\frac{2}{5}\left(x - \frac{1}{2}\right)\) | M1 | Sight of embedded values sufficient; if \(y = mx+c\) used must reach \(c = \ldots\) |
| \(2x + 5y - 16 = 0\) | A1 | Allow integer multiples; must equal 0; accept \(2x + 5y + -16 = 0\) |
# Question 1:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y = 2x^3 - 5x^2 - \frac{3}{2x} + 7 \Rightarrow \frac{dy}{dx} = 6x^2 - 10x + \frac{3}{2x^2}$ | M1 | Reducing any power by 1 at least once, $x^n \to x^{n-1}$ including $7 \to 0$; allow from incorrect index when manipulating $\frac{3}{2x}$ |
| Two of three correct terms (simplified) $6x^2 - 10x + \frac{3}{2x^2}$ | A1 | Do not allow $-10x^1$ |
| All three: $6x^2 - 10x + \frac{3}{2x^2}$ or $6x^2 - 10x + \frac{3}{2}x^{-2}$ on one line | A1 | Do not accept $6x^2 + -10x + \frac{3}{2}x^{-2}$; do not allow $-10x^1$ |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x = \frac{1}{2} \Rightarrow y = 3$ | B1 | Correct coordinates for $P$; ignore labelling of $y$ |
| Substitutes $x = \frac{1}{2}$ into $\frac{dy}{dx} = 6x^2 - 10x + \frac{3}{2x^2} = \frac{5}{2}$ | M1 | Must find a value; ignore labelling of $\frac{dy}{dx}$ |
| Uses perpendicular gradient rule: $\frac{5}{2} \to -\frac{2}{5}$ | dM1 | Finds/uses gradient of normal via $m \to -\frac{1}{m}$; dependent on previous M mark |
| Attempts equation of normal at $P$: $y - 3 = -\frac{2}{5}\left(x - \frac{1}{2}\right)$ | M1 | Sight of embedded values sufficient; if $y = mx+c$ used must reach $c = \ldots$ |
| $2x + 5y - 16 = 0$ | A1 | Allow integer multiples; must equal 0; accept $2x + 5y + -16 = 0$ |
---\begin{enumerate}
\item A curve has equation
\end{enumerate}
$$y = 2 x ^ { 3 } - 5 x ^ { 2 } - \frac { 3 } { 2 x } + 7 \quad x > 0$$
(a) Find, in simplest form, $\frac { \mathrm { d } y } { \mathrm {~d} x }$
The point $P$ lies on the curve and has $x$ coordinate $\frac { 1 } { 2 }$\\
(b) Find an equation of the normal to the curve at $P$, writing your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers to be found.
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VILU SIHI NI JIIIM ION OC & VIUV SIHI NI III M M I ON OO & VIAV SIHI NI JIIIM I ION OC \\
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\hfill \mbox{\textit{Edexcel P1 2021 Q1 [8]}}