| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | March |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Substitution method |
| Difficulty | Standard +0.3 This is a standard differential equations question involving rates of change in a tank problem. Part (i) requires forming a separable DE from given rates (routine setup), then rearranging to the given integral form. Part (ii) uses a provided substitution to evaluate the integral. All steps are straightforward applications of standard techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.07t Construct differential equations: in context1.08h Integration by substitution1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(\dfrac{\mathrm{d}V}{\mathrm{d}t} = 2\dfrac{\mathrm{d}h}{\mathrm{d}t}\) | B1 | |
| State or imply \(\dfrac{\mathrm{d}V}{\mathrm{d}t} = 1 - 0.2\sqrt{h}\) | B1 | |
| Obtain the given answer correctly | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(\mathrm{d}u = -\dfrac{1}{2\sqrt{h}}\,\mathrm{d}h\), or equivalent | B1 | |
| Substitute for \(h\) and \(\mathrm{d}h\) throughout | M1 | |
| Obtain \(T = \displaystyle\int_{3}^{5} \dfrac{20(5-u)}{u}\,\mathrm{d}u\), or equivalent | A1 | |
| Integrate and obtain terms \(100\ln u - 20u\), or equivalent | A1 | |
| Substitute limits \(u=3\) and \(u=5\) correctly | M1 | |
| Obtain answer 11.1, with no errors seen | A1 |
## Question 7(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $\dfrac{\mathrm{d}V}{\mathrm{d}t} = 2\dfrac{\mathrm{d}h}{\mathrm{d}t}$ | B1 | |
| State or imply $\dfrac{\mathrm{d}V}{\mathrm{d}t} = 1 - 0.2\sqrt{h}$ | B1 | |
| Obtain the given answer correctly | B1 | |
**Total: 3**
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## Question 7(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $\mathrm{d}u = -\dfrac{1}{2\sqrt{h}}\,\mathrm{d}h$, or equivalent | B1 | |
| Substitute for $h$ and $\mathrm{d}h$ throughout | M1 | |
| Obtain $T = \displaystyle\int_{3}^{5} \dfrac{20(5-u)}{u}\,\mathrm{d}u$, or equivalent | A1 | |
| Integrate and obtain terms $100\ln u - 20u$, or equivalent | A1 | |
| Substitute limits $u=3$ and $u=5$ correctly | M1 | |
| Obtain answer 11.1, with no errors seen | A1 | |
**Total: 6**
---7\\
\includegraphics[max width=\textwidth, alt={}, center]{e26f21c5-3776-4c86-8440-6959c5e37486-12_444_382_258_886}
A water tank has vertical sides and a horizontal rectangular base, as shown in the diagram. The area of the base is $2 \mathrm {~m} ^ { 2 }$. At time $t = 0$ the tank is empty and water begins to flow into it at a rate of $1 \mathrm {~m} ^ { 3 }$ per hour. At the same time water begins to flow out from the base at a rate of $0.2 \sqrt { } h \mathrm {~m} ^ { 3 }$ per hour, where $h \mathrm {~m}$ is the depth of water in the tank at time $t$ hours.\\
(i) Form a differential equation satisfied by $h$ and $t$, and show that the time $T$ hours taken for the depth of water to reach 4 m is given by
$$T = \int _ { 0 } ^ { 4 } \frac { 10 } { 5 - \sqrt { } h } \mathrm {~d} h$$
(ii) Using the substitution $u = 5 - \sqrt { } h$, find the value of $T$.\\
\hfill \mbox{\textit{CAIE P3 2017 Q7 [9]}}