| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| Session | October |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Find stationary points and nature |
| Difficulty | Standard +0.3 This is a structured multi-part question with clear scaffolding. Part (a) involves algebraic manipulation to expand and simplify (routine). Part (b) requires differentiation using power rule and setting equal to zero (standard calculus). Part (c) is factorising a cubic given one root, then solving a quadratic (standard A-level technique). While it has multiple steps, each individual step is straightforward and the question guides students through the process, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{12x^3(x-7)+14x(13x-15)}{21\sqrt{x}} = \frac{12x^4 - 84x^3 + 182x^2 - 210x}{21\sqrt{x}}\) | M1 | Attempts to multiply out numerator (at least 2 correct terms) |
| \(\frac{4}{7}x^{\frac{7}{2}},\ -4x^{\frac{5}{2}},\ +\frac{26}{3}x^{\frac{3}{2}},\ -10x^{\frac{1}{2}}\) | A1 | Any two of these four terms correct (coefficient may be unsimplified but index must be processed) |
| \(y = \frac{4}{7}x^{\frac{7}{2}} - 4x^{\frac{5}{2}} + \frac{26}{3}x^{\frac{3}{2}} - 10x^{\frac{1}{2}}\) | A1 | All four terms correct; allow as a list |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{dy}{dx} = 2x^{\frac{5}{2}} - 10x^{\frac{3}{2}} + 13x^{\frac{1}{2}} - 5x^{-\frac{1}{2}}\) | M1A1ft | M1: differentiates to form \(...x^{\frac{5}{2}} \pm ...x^{\frac{3}{2}} \pm ...x^{\frac{1}{2}} \pm ...x^{-\frac{1}{2}}\); A1ft: correct with simplified coefficients, follow through their \(a,b,c,d\) |
| \(2x^3 - 10x^2 + 13x - 5 = 0\) * | A1* | Must reach printed answer from correct derivative; "= 0" must appear at least once before final answer; allow \(x^{-\frac{1}{2}}(2x^3 - 10x^2 + 13x - 5) = 0\) with minimal conclusion |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((2x^3 - 10x^2 + 13x - 5) \div (x-1) = (2x^2 \pm ...x \pm ...)\) | M1 | Uses \(x-1\) as factor; by inspection or long division |
| \(2x^2 - 8x + 5\) | A1 | Correct quadratic factor |
| \(x = \frac{4 \pm \sqrt{6}}{2}\) | A1 | Or exact equivalents e.g. \(2 \pm \frac{1}{2}\sqrt{6}\) or \(\frac{8\pm\sqrt{24}}{4}\); roots may be found using calculator but M1A1 must be awarded |
## Question 7(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{12x^3(x-7)+14x(13x-15)}{21\sqrt{x}} = \frac{12x^4 - 84x^3 + 182x^2 - 210x}{21\sqrt{x}}$ | M1 | Attempts to multiply out numerator (at least 2 correct terms) |
| $\frac{4}{7}x^{\frac{7}{2}},\ -4x^{\frac{5}{2}},\ +\frac{26}{3}x^{\frac{3}{2}},\ -10x^{\frac{1}{2}}$ | A1 | Any two of these four terms correct (coefficient may be unsimplified but index must be processed) |
| $y = \frac{4}{7}x^{\frac{7}{2}} - 4x^{\frac{5}{2}} + \frac{26}{3}x^{\frac{3}{2}} - 10x^{\frac{1}{2}}$ | A1 | All four terms correct; allow as a list |
## Question 7(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dy}{dx} = 2x^{\frac{5}{2}} - 10x^{\frac{3}{2}} + 13x^{\frac{1}{2}} - 5x^{-\frac{1}{2}}$ | M1A1ft | M1: differentiates to form $...x^{\frac{5}{2}} \pm ...x^{\frac{3}{2}} \pm ...x^{\frac{1}{2}} \pm ...x^{-\frac{1}{2}}$; A1ft: correct with simplified coefficients, follow through their $a,b,c,d$ |
| $2x^3 - 10x^2 + 13x - 5 = 0$ * | A1* | Must reach printed answer from correct derivative; "= 0" must appear at least once before final answer; allow $x^{-\frac{1}{2}}(2x^3 - 10x^2 + 13x - 5) = 0$ with minimal conclusion |
## Question 7(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(2x^3 - 10x^2 + 13x - 5) \div (x-1) = (2x^2 \pm ...x \pm ...)$ | M1 | Uses $x-1$ as factor; by inspection or long division |
| $2x^2 - 8x + 5$ | A1 | Correct quadratic factor |
| $x = \frac{4 \pm \sqrt{6}}{2}$ | A1 | Or exact equivalents e.g. $2 \pm \frac{1}{2}\sqrt{6}$ or $\frac{8\pm\sqrt{24}}{4}$; roots may be found using calculator but M1A1 must be awarded |\begin{enumerate}
\item The curve $C$ has equation
\end{enumerate}
$$y = \frac { 12 x ^ { 3 } ( x - 7 ) + 14 x ( 13 x - 15 ) } { 21 \sqrt { x } } \quad x > 0$$
(a) Write the equation of $C$ in the form
$$y = a x ^ { \frac { 7 } { 2 } } + b x ^ { \frac { 5 } { 2 } } + c x ^ { \frac { 3 } { 2 } } + d x ^ { \frac { 1 } { 2 } }$$
where $a , b , c$ and $d$ are fully simplified constants.
The curve $C$ has three turning points.\\
Using calculus,\\
(b) show that the $x$ coordinates of the three turning points satisfy the equation
$$2 x ^ { 3 } - 10 x ^ { 2 } + 13 x - 5 = 0$$
Given that the $x$ coordinate of one of the turning points is 1\\
(c) find, using algebra, the exact $x$ coordinates of the other two turning points.\\
(Solutions based entirely on calculator technology are not acceptable.)
\hfill \mbox{\textit{Edexcel P2 2022 Q7 [9]}}