| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | Specimen |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Connected Rates of Change |
| Type | Equal rate condition |
| Difficulty | Standard +0.3 This is a straightforward connected rates of change problem requiring differentiation of a rational function using the chain rule, then applying dx/dt = dy/dt to find x-coordinates. The algebra is routine and the problem-solving approach is standard for this topic, making it slightly easier than average. |
| Spec | 1.07a Derivative as gradient: of tangent to curve1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(-12(3-2x)^{-2} \times -2\) | 2 B1B1 | B1 for \(-12(3-2x)^{-2}\), B1 for \(-2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dt}{dt} = 0.4 \div 0.15\) | 1 M1 | OE; chain rule used correctly |
| \(-\frac{24}{(3-2x)^2} = \frac{8}{3}\) | 1 M1 | Equates their \(\frac{dy}{dx}\) with their \(\frac{8}{3}\) or \(\frac{3}{8}\) and method seen for solution of quadratic equation |
| \(x = 0\) or \(3\) | 2 A1A1 | |
| Total | 4 |
## Question 8(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $-12(3-2x)^{-2} \times -2$ | 2 B1B1 | B1 for $-12(3-2x)^{-2}$, B1 for $-2$ |
## Question 8(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dt}{dt} = 0.4 \div 0.15$ | 1 M1 | OE; chain rule used correctly |
| $-\frac{24}{(3-2x)^2} = \frac{8}{3}$ | 1 M1 | Equates their $\frac{dy}{dx}$ with their $\frac{8}{3}$ or $\frac{3}{8}$ and method seen for solution of quadratic equation |
| $x = 0$ or $3$ | 2 A1A1 | |
| **Total** | **4** | |8 A curve has equation $y = \frac { 12 } { 3 - 2 x }$.\\
(a) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.\\
A point moves along this curve. As the point passes through $A$, the $x$-coordinate is increasing at a rate of 0.15 units per second and the $y$-coordinate is increasing at a rate of 0.4 units per second.\\
(b) Find the possible $x$-coordinates of $A$.\\
\hfill \mbox{\textit{CAIE P1 2020 Q8 [6]}}