| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2008 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Separable variables - standard (applied/contextual) |
| Difficulty | Standard +0.3 This is a standard separable differential equations problem with straightforward setup and integration. The model formulation is direct (dx/dt = -k√x), separation of variables is routine, and solving for the constant using given conditions follows a well-practiced method. Slightly above average difficulty due to requiring multiple steps and careful algebraic manipulation, but this is a textbook-style C4 question with no novel insight required. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y)1.08l Interpret differential equation solutions: in context |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{dx}{dt}\) or \(-kx^{\frac{1}{2}}\) or \(kx^{\frac{1}{2}}\) seen | M1 | k non-numerical; i.e. 1 side correct |
| \(\frac{dx}{dt} = -kx^{\frac{1}{2}}\) or \(\frac{dx}{dt} = kx^{\frac{1}{2}}\) | A1 | 2 |
| (ii) Separate variables or invert, + attempt to integrate | M1 | Based only on above eqns or \(\frac{dt}{dx} = x^{-\frac{1}{2}}, -x^{-\frac{1}{2}}\) |
| Correct result for their equation after integration Subst \((t,x) = (0,2)\) into eqn containing k &/or c dep*M1 | Other than omission of '\(c\)' or substitution (5,1) or substitute (5,1) | |
| Subst \((t,x) = (5,1)\) into eqn containing k & c subst \(x = 0.5\) into eqn with their k & c subst \(t = 8.5\) (8.5355339) | A1 | 6 |
(i) $\frac{dx}{dt}$ or $-kx^{\frac{1}{2}}$ or $kx^{\frac{1}{2}}$ seen | M1 | k non-numerical; i.e. 1 side correct |
$\frac{dx}{dt} = -kx^{\frac{1}{2}}$ or $\frac{dx}{dt} = kx^{\frac{1}{2}}$ | A1 | 2 | i.e. both sides correct |
(ii) Separate variables or invert, + attempt to integrate | M1 | Based only on above eqns or $\frac{dt}{dx} = x^{-\frac{1}{2}}, -x^{-\frac{1}{2}}$ |
Correct result for their equation after integration Subst $(t,x) = (0,2)$ into eqn containing k &/or c dep*M1 | | Other than omission of '$c$' or substitution (5,1) or substitute (5,1) |
Subst $(t,x) = (5,1)$ into eqn containing k & c subst $x = 0.5$ into eqn with their k & c subst $t = 8.5$ (8.5355339) | A1 | 6 | [1 d.p. requested in question] |8 Water flows out of a tank through a hole in the bottom and, at time $t$ minutes, the depth of water in the tank is $x$ metres. At any instant, the rate at which the depth of water in the tank is decreasing is proportional to the square root of the depth of water in the tank.\\
(i) Write down a differential equation which models this situation.\\
(ii) When $t = 0 , x = 2$; when $t = 5 , x = 1$. Find $t$ when $x = 0.5$, giving your answer correct to 1 decimal place.
\hfill \mbox{\textit{OCR C4 2008 Q8 [8]}}