| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | March |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find curve equation from derivative (reverse chain rule / composite functions) |
| Difficulty | Moderate -0.3 Part (a) requires straightforward application of the chain rule dy/dt = (dy/dx)(dx/dt) with given values. Part (b) is a standard reverse chain rule integration to find y, using the boundary condition to find the constant. Both parts are routine A-level techniques with no novel problem-solving required, making this slightly easier than average. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08a Fundamental theorem of calculus: integration as reverse of differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| (a) At \(x = 1\), \(\frac{dy}{dx} = 6\) | B1 | |
| (a) \(\frac{dx}{dt} = \left(\frac{dx}{dy} \times \frac{dy}{dt}\right) = \frac{1}{6} \times 3 = \frac{1}{2}\) | M1 A1 | Chain rule used correctly. Allow alternative and minimal notation |
| (b) \([y =]\ \left(\frac{6(3x-2)^{-2}}{-2}\right) \div (3)\ [+c]\) | B1 B1 | |
| (b) \(-3 = -1 + c\) | M1 | Substitute \(x = 1\), \(y = -3\). \(c\) must be present |
| (b) \(y = -(3x-2)^{-2} - 2\) | A1 | OE. Allow \(f(x) =\) |
## Question 6:
| Answer | Marks | Guidance |
|--------|-------|----------|
| (a) At $x = 1$, $\frac{dy}{dx} = 6$ | B1 | |
| (a) $\frac{dx}{dt} = \left(\frac{dx}{dy} \times \frac{dy}{dt}\right) = \frac{1}{6} \times 3 = \frac{1}{2}$ | M1 A1 | Chain rule used correctly. Allow alternative and minimal notation |
| (b) $[y =]\ \left(\frac{6(3x-2)^{-2}}{-2}\right) \div (3)\ [+c]$ | B1 B1 | |
| (b) $-3 = -1 + c$ | M1 | Substitute $x = 1$, $y = -3$. $c$ must be present |
| (b) $y = -(3x-2)^{-2} - 2$ | A1 | OE. Allow $f(x) =$ |
---6 A curve is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { ( 3 x - 2 ) ^ { 3 } }$ and $A ( 1 , - 3 )$ lies on the curve. A point is moving along the curve and at $A$ the $y$-coordinate of the point is increasing at 3 units per second.
\begin{enumerate}[label=(\alph*)]
\item Find the rate of increase at $A$ of the $x$-coordinate of the point.
\item Find the equation of the curve.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q6 [7]}}