| Exam Board | Edexcel |
|---|---|
| Module | P4 (Pure Mathematics 4) |
| Year | 2021 |
| Session | October |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Partial fractions then differentiate |
| Difficulty | Standard +0.3 This is a straightforward partial fractions question with polynomial division followed by routine differentiation. Part (a) is standard algebraic manipulation, part (b) requires differentiating simple terms, and part (c) involves basic inequality reasoning with positive terms. Slightly easier than average due to the mechanical nature of all steps. |
| Spec | 1.02y Partial fractions: decompose rational functions1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(3x^3+8x^2-3x-6 \equiv (Ax+B)x(x+3)+C(x+3)+Dx\) | M1 | Attempts a correct identity; condone slips but intention must be correct |
| Correct method for two constants e.g. \(x=0,\ x=-3 \Rightarrow C=\ldots,\ D=\ldots\) | dM1 | Attempt at correct equation; may set up 4 simultaneous equations |
| Two correct constants e.g. \(C=-2\) and \(D=2\) | A1 | |
| Correct method to find all four constants | ddM1 | e.g. by comparing coefficients |
| \(A=3,\ B=-1,\ C=-2,\ D=2\) | A1 | Allow embedded within the identity |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts to divide \(3x^3+8x^2-3x-6\) by \(x^2+3x\) forming a linear quotient | M1 | |
| Correct method for two constants implied by quotient of \(3x+\ldots\) | dM1 | |
| Correct quotient \(3x-1\) | A1 | |
| Correct method to find all four constants using \(\frac{\text{rem}}{x(x+3)}\) | ddM1 | |
| \(A=3,\ B=-1,\ C=-2,\ D=2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(g(x) = 3x-1-\frac{2}{x}+\frac{2}{x+3} \Rightarrow g'(x) = 3+\frac{2}{x^2}-\frac{2}{(x+3)^2}\) | M1, A1ft | M1: attempts to differentiate achieving \(\frac{C}{x}\rightarrow\frac{\ldots}{x^2}\) or \(\frac{D}{x+3}\rightarrow\frac{\ldots}{(x+3)^2}\). A1ft: correct differentiation following through on their constants |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Explains that (if \(x>0\)) \(\frac{2}{x^2}>\frac{2}{(x+3)^2}\) so \(\frac{2}{x^2}-\frac{2}{(x+3)^2}>0\) and \(\Rightarrow g'(x)>3\) | B1 | Requires correct parts (a) and (b); needs reason and conclusion. Accept: \(3+\frac{2}{x^2}-\frac{2}{(x+3)^2} = 3+\frac{2(6x+9)}{x^2(x+3)^2}\) which is \(3+(+ve)>3\) |
# Question 3(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3x^3+8x^2-3x-6 \equiv (Ax+B)x(x+3)+C(x+3)+Dx$ | M1 | Attempts a correct identity; condone slips but intention must be correct |
| Correct method for two constants e.g. $x=0,\ x=-3 \Rightarrow C=\ldots,\ D=\ldots$ | dM1 | Attempt at correct equation; may set up 4 simultaneous equations |
| Two correct constants e.g. $C=-2$ and $D=2$ | A1 | |
| Correct method to find all four constants | ddM1 | e.g. by comparing coefficients |
| $A=3,\ B=-1,\ C=-2,\ D=2$ | A1 | Allow embedded within the identity |
**Alternative (via division):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts to divide $3x^3+8x^2-3x-6$ by $x^2+3x$ forming a linear quotient | M1 | |
| Correct method for two constants implied by quotient of $3x+\ldots$ | dM1 | |
| Correct quotient $3x-1$ | A1 | |
| Correct method to find all four constants using $\frac{\text{rem}}{x(x+3)}$ | ddM1 | |
| $A=3,\ B=-1,\ C=-2,\ D=2$ | A1 | |
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# Question 3(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $g(x) = 3x-1-\frac{2}{x}+\frac{2}{x+3} \Rightarrow g'(x) = 3+\frac{2}{x^2}-\frac{2}{(x+3)^2}$ | M1, A1ft | M1: attempts to differentiate achieving $\frac{C}{x}\rightarrow\frac{\ldots}{x^2}$ or $\frac{D}{x+3}\rightarrow\frac{\ldots}{(x+3)^2}$. A1ft: correct differentiation following through on their constants |
---
# Question 3(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Explains that (if $x>0$) $\frac{2}{x^2}>\frac{2}{(x+3)^2}$ so $\frac{2}{x^2}-\frac{2}{(x+3)^2}>0$ and $\Rightarrow g'(x)>3$ | B1 | Requires correct parts (a) and (b); needs reason and conclusion. Accept: $3+\frac{2}{x^2}-\frac{2}{(x+3)^2} = 3+\frac{2(6x+9)}{x^2(x+3)^2}$ which is $3+(+ve)>3$ |
---3.
$$\mathrm { g } ( x ) = \frac { 3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 6 } { x ( x + 3 ) } \equiv A x + B + \frac { C } { x } + \frac { D } { x + 3 }$$
\begin{enumerate}[label=(\alph*)]
\item Find the values of the constants $A , B , C$ and $D$.
A curve has equation
$$y = g ( x ) \quad x > 0$$
Using the answer to part (a),
\item find $\mathrm { g } ^ { \prime } ( x )$.
\item Hence, explain why $\mathrm { g } ^ { \prime } ( x ) > 3$ for all values of $x$ in the domain of g .
\end{enumerate}
\hfill \mbox{\textit{Edexcel P4 2021 Q3 [8]}}