| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2006 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Geometric curve properties |
| Difficulty | Moderate -0.3 This is a straightforward separable variables question with standard integration and algebraic manipulation. Part (i) requires routine separation and integration, part (ii) is completing the square (a standard technique), and part (iii) is recognizing and sketching a circle. While multi-part, each step follows textbook procedures 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/Working | Marks | Guidance |
| \(\int(y-3)\,dy = \int(2-x)\,dx\) or equiv | M1 | For separation & integration of both sides |
| \(\frac{1}{2}y^2-3y = 2x-\frac{1}{2}x^2\) | A1 | or \(\frac{1}{2}(y-3)^2 = -\frac{1}{2}(x-2)^2\) |
| For an arbitrary const on one/both sides | *B1 | (or + M2 for equiv statement using limits) |
| Substituting \((x,y)=(5,4)\) or \((4,5)\) & finding \(c\) | dep*M1 | |
| \(\frac{1}{2}y^2-3y = -\frac{1}{2}x^2+2x-\frac{3}{2}\) AEF ISW | A1 | 5 or \(\frac{1}{2}(y-3)^2=-\frac{1}{2}(x-2)^2+5\) AEF |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Attempt to clear fracts (if nec) & complete square | M1 | |
| \(a=2, b=3, k=10\) | A2 | 3 For all 3; SR: A1 for 1 or 2 correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Circle clearly indicated in a sketch | B1 | |
| Centre \((2,3)\) or their \((a,b)\) | B1\(\sqrt{}\) | |
| Radius \(\sqrt{10}\) or their \(\sqrt{k}\) | B1\(\sqrt{}\) | 3 \(\sqrt{}\) provided \(k>0\) |
# Question 8:
## Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int(y-3)\,dy = \int(2-x)\,dx$ or equiv | M1 | For separation & integration of both sides |
| $\frac{1}{2}y^2-3y = 2x-\frac{1}{2}x^2$ | A1 | or $\frac{1}{2}(y-3)^2 = -\frac{1}{2}(x-2)^2$ |
| For an arbitrary const on one/both sides | *B1 | (or + M2 for equiv statement using limits) |
| Substituting $(x,y)=(5,4)$ or $(4,5)$ & finding $c$ | dep*M1 | |
| $\frac{1}{2}y^2-3y = -\frac{1}{2}x^2+2x-\frac{3}{2}$ AEF ISW | A1 | **5** or $\frac{1}{2}(y-3)^2=-\frac{1}{2}(x-2)^2+5$ AEF |
## Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Attempt to clear fracts (if nec) & complete square | M1 | |
| $a=2, b=3, k=10$ | A2 | **3** For all 3; SR: A1 for 1 or 2 correct |
## Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Circle clearly indicated in a sketch | B1 | |
| Centre $(2,3)$ or their $(a,b)$ | B1$\sqrt{}$ | |
| Radius $\sqrt{10}$ or their $\sqrt{k}$ | B1$\sqrt{}$ | **3** $\sqrt{}$ provided $k>0$ |8 (i) Solve the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 - x } { y - 3 }$$
giving the particular solution that satisfies the condition $y = 4$ when $x = 5$.\\
(ii) Show that this particular solution can be expressed in the form
$$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
where the values of the constants $a , b$ and $k$ are to be stated.\\
(iii) Hence sketch the graph of the particular solution, indicating clearly its main features.
\hfill \mbox{\textit{OCR C4 2006 Q8 [11]}}