| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2012 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Separable variables - standard (polynomial/exponential x-side) |
| Difficulty | Standard +0.3 This is a standard separable differential equation with straightforward separation and integration. Part (i) is routine differentiation using chain rule. Part (ii) requires separating variables, integrating both sides (the left side uses the result from part (i), the right side is a simple logarithmic integral), and applying the boundary condition. While it involves multiple steps, all techniques are standard C4 material with no novel insight required, making it slightly easier than average. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{2}(y^2+1)^{-\frac{1}{2}} \cdot 2y\) or better | B1 [1] | Tolerate "\(\frac{dy}{dx} = \ldots\)" but, otherwise, no \(\frac{dy}{dx}\) or \(\frac{dx}{dy}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Separate variables: \(\int \frac{y}{\sqrt{y^2+1}}\,dy = \int \frac{x-1}{x}\,dx\) | \*M1 | \(\int\) may be implied later |
| Change \(\frac{x-1}{x}\) into \(1 - \frac{1}{x}\) | M1 | |
| RHS \(= x - \ln x\) | A1 | |
| LHS \(= \sqrt{y^2+1}\) | B1 | Quoted or derived |
| Subst \(y = \sqrt{e^2-2e}\), \(x = e\) into their eqn. with '\(c\)' | Dep\*M1 | |
| \(\sqrt{y^2+1} = \sqrt{(e-1)^2} = e - 1\) | A1 | Ignore lack of/no ref to \(1-e\) |
| \(c = 0\) | A1 | Ignore any ref to \(c = 2 - 2e\) |
| \(\sqrt{y^2+1} = x - \ln x\) | A1 | ISW |
| [8] |
## Question 10(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{2}(y^2+1)^{-\frac{1}{2}} \cdot 2y$ or better | B1 **[1]** | Tolerate "$\frac{dy}{dx} = \ldots$" but, otherwise, no $\frac{dy}{dx}$ or $\frac{dx}{dy}$ |
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## Question 10(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Separate variables: $\int \frac{y}{\sqrt{y^2+1}}\,dy = \int \frac{x-1}{x}\,dx$ | \*M1 | $\int$ may be implied later |
| Change $\frac{x-1}{x}$ into $1 - \frac{1}{x}$ | M1 | |
| RHS $= x - \ln x$ | A1 | |
| LHS $= \sqrt{y^2+1}$ | B1 | Quoted or derived |
| Subst $y = \sqrt{e^2-2e}$, $x = e$ into their eqn. with '$c$' | Dep\*M1 | |
| $\sqrt{y^2+1} = \sqrt{(e-1)^2} = e - 1$ | A1 | Ignore lack of/no ref to $1-e$ |
| $c = 0$ | A1 | Ignore any ref to $c = 2 - 2e$ |
| $\sqrt{y^2+1} = x - \ln x$ | A1 | ISW |
| **[8]** | | |10 (i) Write down the derivative of $\sqrt { y ^ { 2 } + 1 }$ with respect to $y$.\\
(ii) Given that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( x - 1 ) \sqrt { y ^ { 2 } + 1 } } { x y }$ and that $y = \sqrt { \mathrm { e } ^ { 2 } - 2 \mathrm { e } }$ when $x = \mathrm { e }$,\\
find a relationship between $x$ and $y$.
\hfill \mbox{\textit{OCR C4 2012 Q10 [9]}}