| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find curve equation from derivative (extended problem with normals, stationary points, or further geometry) |
| Difficulty | Moderate -0.8 This is a straightforward two-part question requiring basic differentiation concepts. Part (i) applies the chain rule with given rates (dy/dt = dy/dx × dx/dt), and part (ii) is standard integration with a boundary condition to find the constant. Both are routine textbook exercises with no problem-solving insight required. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt} = 7 \times -0.05\) | M1 | Multiply numerical gradient at \(x = 2\) by \(\pm 0.05\) |
| \(-0.35\) (units/s) or Decreasing at a rate of \((+)\ 0.35\) | A1 | Ignore notation and omission of units |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((y) = \frac{x^4}{4} + \frac{4}{x}\ (+c)\) oe | B1 | Accept unsimplified |
| Uses \((2, 9)\) in an integral to find \(c\) | M1 | The power of at least one term increase by 1 |
| \(c = 3\) or \((y =)\frac{x^4}{4} + \frac{4}{x} + 3\) oe | A1 | A0 if candidate continues to a final equation that is a straight line |
## Question 3(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt} = 7 \times -0.05$ | M1 | Multiply numerical gradient at $x = 2$ by $\pm 0.05$ |
| $-0.35$ (units/s) **or** Decreasing at a rate of $(+)\ 0.35$ | A1 | Ignore notation and omission of units |
## Question 3(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $(y) = \frac{x^4}{4} + \frac{4}{x}\ (+c)$ oe | B1 | Accept unsimplified |
| Uses $(2, 9)$ in an integral to find $c$ | M1 | The power of at least one term increase by 1 |
| $c = 3$ **or** $(y =)\frac{x^4}{4} + \frac{4}{x} + 3$ oe | A1 | A0 if candidate continues to a final equation that is a straight line |3 A curve is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 3 } - \frac { 4 } { x ^ { 2 } }$. The point $P ( 2,9 )$ lies on the curve.\\
(i) A point moves on the curve in such a way that the $x$-coordinate is decreasing at a constant rate of 0.05 units per second. Find the rate of change of the $y$-coordinate when the point is at $P$. [2]\\
(ii) Find the equation of the curve.\\
\hfill \mbox{\textit{CAIE P1 2019 Q3 [5]}}