| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Separable variables - partial fractions |
| Difficulty | Standard +0.3 This is a standard separable variables question with straightforward algebraic manipulation. Part (i) provides a helpful hint for the factorization needed in part (ii). The separation, integration (using partial fractions or log rules), and application of initial conditions are all routine C4 techniques, making this slightly easier than average. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Long division: Evidence of division process as far as 1st stage incl. subtraction | M1 | Identity method: \(\equiv Q(x-1)+R\) |
| Quotient \(= x-4\) | A1 | \(Q = x-4\) |
| Remainder \(= 2\) ISW | A1 | 3 marks \(R=2\); N.B. might be B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Separate variables: \(\int \frac{1}{y-5}\,dy = \int \frac{x^2-5x+6}{x-1}\,dx\) | M1 | '\(\int\)' may be implied later |
| Change \(\frac{x^2-5x+6}{x-1}\) into their \(\left(\text{Quotient} + \frac{\text{Rem}}{x-1}\right)\) | M1 | |
| \(\ln(y-5) = \sqrt{\phantom{x}}\) (integration of their previous result) \((+c)\) ISW | \(\checkmark\)A1 | 3 marks f.t. if using Quot \(+ \frac{\text{Rem}}{x-1}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Substitute \(y=7\), \(x=8\) into their equation containing '\(c\)' | M1 | & attempt '\(c\)' \((-3.2, \ln\frac{2}{49})\) |
| Substitute \(x=6\) and their value of '\(c\)' | M1 | & attempt to find \(y\) |
| \(y = 5.00\) (\(5.002529\)). Also \(5 + \frac{50}{49}e^{-6}\) | A2 | 4 marks Accept 5, 5.0 |
# Question 8(i):
| Working/Answer | Mark | Guidance |
|---|---|---|
| **Long division:** Evidence of division process as far as 1st stage incl. subtraction | M1 | **Identity method:** $\equiv Q(x-1)+R$ |
| Quotient $= x-4$ | A1 | $Q = x-4$ |
| Remainder $= 2$ ISW | A1 | **3 marks** $R=2$; N.B. might be B1 |
# Question 8(ii)(a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Separate variables: $\int \frac{1}{y-5}\,dy = \int \frac{x^2-5x+6}{x-1}\,dx$ | M1 | '$\int$' may be implied later |
| Change $\frac{x^2-5x+6}{x-1}$ into their $\left(\text{Quotient} + \frac{\text{Rem}}{x-1}\right)$ | M1 | |
| $\ln(y-5) = \sqrt{\phantom{x}}$ (integration of their previous result) $(+c)$ ISW | $\checkmark$A1 | **3 marks** f.t. if using Quot $+ \frac{\text{Rem}}{x-1}$ |
# Question 8(ii)(b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Substitute $y=7$, $x=8$ into their equation containing '$c$' | M1 | & attempt '$c$' $(-3.2, \ln\frac{2}{49})$ |
| Substitute $x=6$ and their value of '$c$' | M1 | & attempt to find $y$ |
| $y = 5.00$ ($5.002529$). Also $5 + \frac{50}{49}e^{-6}$ | A2 | **4 marks** Accept 5, 5.0 |
*Beware: any wrong working anywhere $\rightarrow$ A0 even if answer is one of the acceptable ones.*
**Total: 10 marks**
---8
\begin{enumerate}[label=(\roman*)]
\item Find the quotient and the remainder when $x ^ { 2 } - 5 x + 6$ is divided by $x - 1$.
\item (a) Find the general solution of the differential equation
$$\left( \frac { x - 1 } { x ^ { 2 } - 5 x + 6 } \right) \frac { \mathrm { d } y } { \mathrm {~d} x } = y - 5 .$$
(b) Given that $y = 7$ when $x = 8$, find $y$ when $x = 6$.
\end{enumerate}
\hfill \mbox{\textit{OCR C4 2010 Q8 [10]}}