| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find curve equation from derivative (find unknown constant in derivative first) |
| Difficulty | Moderate -0.8 This is a straightforward integration question requiring students to (i) substitute x=1 into the derivative to find k using the given gradient, then (ii) integrate using standard power rule and apply the boundary condition. Both parts are routine applications of basic calculus techniques with no problem-solving insight required, making it easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(6+k=2 \rightarrow k=-4\) | B1 [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((y)=\frac{6x^3}{3}-\frac{4}{-2}x^{-2}\ (+c)\) | B1B1\(\checkmark\) | ft on *their* \(k\). Accept \(+\frac{k}{-2}x^{-2}\) |
| \(9=2+2+c\), \(c\) must be present | M1 | Sub \((1,9)\) with numerical \(k\). Dep on attempt \(\int\) |
| \((y)=2x^3+2x^{-2}+5\) | A1 [4] | Equation needs to be seen. Sub \((2,3)\rightarrow c=-13\tfrac{1}{2}\) scores M1A0 |
## Question 3(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $6+k=2 \rightarrow k=-4$ | B1 [1] | |
## Question 3(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(y)=\frac{6x^3}{3}-\frac{4}{-2}x^{-2}\ (+c)$ | B1B1$\checkmark$ | ft on *their* $k$. Accept $+\frac{k}{-2}x^{-2}$ |
| $9=2+2+c$, $c$ must be present | M1 | Sub $(1,9)$ with numerical $k$. Dep on attempt $\int$ |
| $(y)=2x^3+2x^{-2}+5$ | A1 [4] | Equation needs to be seen. Sub $(2,3)\rightarrow c=-13\tfrac{1}{2}$ scores M1A0 |
---3 A curve is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } + \frac { k } { x ^ { 3 } }$ and passes through the point $P ( 1,9 )$. The gradient of the curve at $P$ is 2 .\\
(i) Find the value of the constant $k$.\\
(ii) Find the equation of the curve.
\hfill \mbox{\textit{CAIE P1 2016 Q3 [5]}}