| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2015 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Improper fraction with linear factors – division then partial fractions |
| Difficulty | Standard +0.3 This is a structured multi-part question that guides students through standard C4 techniques: partial fractions (routine), polynomial division (standard), parametric-to-cartesian conversion (algebraic manipulation), and integration using partial fractions. While it requires multiple steps and careful algebra, each part uses well-practiced methods with no novel insight required. The scaffolding makes it slightly easier than average. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02y Partial fractions: decompose rational functions1.03g Parametric equations: of curves and conversion to cartesian1.08f Area between two curves: using integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{A}{x} + \frac{B}{x+2}\) | B1 | Award if only implied by answer |
| \(x + 8 = A(x+2) + Bx\) soi | M1 | Allow one sign error; clearing fractions successfully |
| \(A = 4\) and \(B = -3\) | A1 [3] | If M0, B1 for each value www |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Quotient \((P)\) is \(7\) | B1 | |
| \(2x + 16\) seen | B1 | If B0, B1 for \(Q=8\) and B1 for \(R=-6\) www; e.g. as remainder or in division chunking |
| \(7 + \frac{8}{x} - \frac{6}{x+2}\) | B1 [3] | Or allow \(P=7\), \(Q=8\), \(R=-6\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(t = f(x)\) | M1* | From \(x = \frac{2t}{1-t}\); M0 for \(t = g(y)\) |
| \(t = \frac{x}{x+2}\) | A1 | Or B2 if unsupported |
| \(y = 3\times\frac{x}{x+2} + \frac{4}{\frac{x}{x+2}}\) | M1dep* | At least one correct, constructive, intermediate step shown; if M0M0, SC2 for substitution of \(x=\frac{2t}{1-t}\) in RHS of given equation and completion with at least two correct, constructive intermediate steps to \(y = 3t + \frac{4}{t}\) www |
| e.g. \(\frac{3x^2+(8+4x)(x+2)}{x(x+2)}\) and completion to | ||
| \(y = \frac{7x^2+16x+16}{x(x+2)}\) www AG | A1 [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\int \text{their} \left(P + \frac{Q}{x} + \frac{R}{x+2}\right) dx\) | M1\* | Where \(P\), \(Q\) and \(R\) are constants obtained in (ii); allow omission of \(dx\) |
| \(F[x] = 7x + 8\ln x - 6\ln(x+2)\) | A1FT | Allow recovery from omission of brackets in subsequent working. If M0, SC1 for \(Px + Q\ln x + R\ln(x+2)\) where constants are unspecified or arbitrary |
| \(F[2] - F[1]\) | M1dep\* | |
| \(7 - 4\ln 2 + 6\ln 3\) | A1 | |
| [4] |
## Question 10(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{A}{x} + \frac{B}{x+2}$ | B1 | Award if only implied by answer |
| $x + 8 = A(x+2) + Bx$ soi | M1 | Allow one sign error; clearing fractions successfully |
| $A = 4$ and $B = -3$ | A1 **[3]** | If M0, B1 for each value www |
---
## Question 10(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Quotient $(P)$ is $7$ | B1 | |
| $2x + 16$ seen | B1 | If B0, B1 for $Q=8$ and B1 for $R=-6$ www; e.g. as remainder or in division chunking |
| $7 + \frac{8}{x} - \frac{6}{x+2}$ | B1 **[3]** | Or allow $P=7$, $Q=8$, $R=-6$ |
---
## Question 10(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $t = f(x)$ | M1* | From $x = \frac{2t}{1-t}$; M0 for $t = g(y)$ |
| $t = \frac{x}{x+2}$ | A1 | Or B2 if unsupported |
| $y = 3\times\frac{x}{x+2} + \frac{4}{\frac{x}{x+2}}$ | M1dep* | At least one correct, constructive, intermediate step shown; if M0M0, SC2 for substitution of $x=\frac{2t}{1-t}$ in RHS of given equation and completion with at least two correct, constructive intermediate steps to $y = 3t + \frac{4}{t}$ www |
| e.g. $\frac{3x^2+(8+4x)(x+2)}{x(x+2)}$ and completion to | | |
| $y = \frac{7x^2+16x+16}{x(x+2)}$ www **AG** | A1 **[4]** | |
## Question 10(iv):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int \text{their} \left(P + \frac{Q}{x} + \frac{R}{x+2}\right) dx$ | **M1\*** | Where $P$, $Q$ and $R$ are constants obtained in (ii); allow omission of $dx$ |
| $F[x] = 7x + 8\ln x - 6\ln(x+2)$ | **A1FT** | Allow recovery from omission of brackets in subsequent working. If **M0**, **SC1** for $Px + Q\ln x + R\ln(x+2)$ where constants are unspecified or arbitrary |
| $F[2] - F[1]$ | **M1dep\*** | |
| $7 - 4\ln 2 + 6\ln 3$ | **A1** | |
| | **[4]** | |10 (i) Express $\frac { x + 8 } { x ( x + 2 ) }$ in partial fractions.\\
(ii) By first using division, express $\frac { 7 x ^ { 2 } + 16 x + 16 } { x ( x + 2 ) }$ in the form $P + \frac { Q } { x } + \frac { R } { x + 2 }$.
A curve has parametric equations $x = \frac { 2 t } { 1 - t } , y = 3 t + \frac { 4 } { t }$.\\
(iii) Show that the cartesian equation of the curve is $y = \frac { 7 x ^ { 2 } + 16 x + 16 } { x ( x + 2 ) }$.\\
(iv) Find the area of the region bounded by the curve, the $x$-axis and the lines $x = 1$ and $x = 2$. Give your answer in the form $L + M \ln 2 + N \ln 3$.
\hfill \mbox{\textit{OCR C4 2015 Q10 [14]}}