| Exam Board | Edexcel |
|---|---|
| Module | P4 (Pure Mathematics 4) |
| Year | 2018 |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find tangent equation at point |
| Difficulty | Standard +0.3 This is a straightforward implicit differentiation question requiring standard application of the chain rule and product rule, followed by substituting a point to find a tangent equation. While implicit differentiation is a P4/Further Maths topic making it slightly above average difficulty, the execution is mechanical with no conceptual challenges or novel problem-solving required. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation |
| VIII SIHI NI I IIIM I O N OC | VIIN SIHI NI JIHMM ION OO | VI4V SIHI NI JIIYM ION OO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x^3 + 2xy - x - y^3 - 20 = 0\) | ||
| \(3x^2 + \left(2y + 2x\frac{dy}{dx}\right) - 1 - 3y^2\frac{dy}{dx} = 0\) | M1 A1 B1 | M1: Differentiates implicitly to include either \(2y\frac{dx}{dy}\) or \(x^3 \to \pm kx^2\frac{dx}{dy}\) or \(-x \to -\frac{dx}{dy}\). A1: \(x^3 \to 3x^2\frac{dx}{dy}\) and \(-x - y^3 - 20 = 0 \to -\frac{dx}{dy} - 3y^2 = 0\). B1: \(2xy \to 2y\frac{dx}{dy} + 2x\) |
| \(3x^2 + 2y - 1 + \left(2x - 3y^2\right)\frac{dy}{dx} = 0\) | dM1 | Dependent on first M1. Attempt to factorise out all terms in \(\frac{dy}{dx}\) with at least two such terms |
| \(\frac{dy}{dx} = \frac{3x^2 + 2y - 1}{3y^2 - 2x}\) or \(\frac{1 - 3x^2 - 2y}{2x - 3y^2}\) | A1 (cso) | For \(\frac{1-2y-3x^2}{2x-3y^2}\) or equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| At P(3, -2): \(m(T) = \frac{3(3)^2 + 2(-2) - 1}{3(-2)^2 - 2(3)} = \frac{22}{6} = \frac{11}{3}\); and either \(y - (-2) = \frac{11}{3}(x-3)\) or \((-2) = \left(\frac{11}{3}\right)(3) + c\) | M1 | Some attempt to substitute both \(x=3\) and \(y=-2\) into \(\frac{dy}{dx}\); either applies \(y+2 = m_T(x-3)\) or finds \(c\) |
| \(T: 11x - 3y - 39 = 0\) or \(K(11x - 3y - 39) = 0\) | A1 (cso) | Accept any integer multiple of \(11x - 3y - 39 = 0\) or \(11x - 39 - 3y = 0\) or \(-11x + 3y + 39 = 0\) |
## Question 2(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^3 + 2xy - x - y^3 - 20 = 0$ | | |
| $3x^2 + \left(2y + 2x\frac{dy}{dx}\right) - 1 - 3y^2\frac{dy}{dx} = 0$ | M1 A1 B1 | M1: Differentiates implicitly to include either $2y\frac{dx}{dy}$ or $x^3 \to \pm kx^2\frac{dx}{dy}$ or $-x \to -\frac{dx}{dy}$. A1: $x^3 \to 3x^2\frac{dx}{dy}$ and $-x - y^3 - 20 = 0 \to -\frac{dx}{dy} - 3y^2 = 0$. B1: $2xy \to 2y\frac{dx}{dy} + 2x$ |
| $3x^2 + 2y - 1 + \left(2x - 3y^2\right)\frac{dy}{dx} = 0$ | dM1 | Dependent on first M1. Attempt to factorise out all terms in $\frac{dy}{dx}$ with at least two such terms |
| $\frac{dy}{dx} = \frac{3x^2 + 2y - 1}{3y^2 - 2x}$ or $\frac{1 - 3x^2 - 2y}{2x - 3y^2}$ | A1 (cso) | For $\frac{1-2y-3x^2}{2x-3y^2}$ or equivalent |
## Question 2(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| At P(3, -2): $m(T) = \frac{3(3)^2 + 2(-2) - 1}{3(-2)^2 - 2(3)} = \frac{22}{6} = \frac{11}{3}$; and either $y - (-2) = \frac{11}{3}(x-3)$ or $(-2) = \left(\frac{11}{3}\right)(3) + c$ | M1 | Some attempt to substitute both $x=3$ and $y=-2$ into $\frac{dy}{dx}$; either applies $y+2 = m_T(x-3)$ or finds $c$ |
| $T: 11x - 3y - 39 = 0$ or $K(11x - 3y - 39) = 0$ | A1 (cso) | Accept any integer multiple of $11x - 3y - 39 = 0$ or $11x - 39 - 3y = 0$ or $-11x + 3y + 39 = 0$ |
---2. A curve $C$ has the equation
$$x ^ { 3 } + 2 x y - x - y ^ { 3 } - 20 = 0$$
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
\item Find an equation of the tangent to $C$ at the point $( 3 , - 2 )$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
\begin{center}
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VIII SIHI NI I IIIM I O N OC & VIIN SIHI NI JIHMM ION OO & VI4V SIHI NI JIIYM ION OO \\
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\hfill \mbox{\textit{Edexcel P4 2018 Q2 [7]}}