| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | First-order integration |
| Difficulty | Standard +0.3 This question requires finding a point on a curve using the normal line condition, then integrating to find the curve equation. While it involves multiple steps (finding gradient from normal, solving for point P, integration with boundary condition), each step uses standard AS-level techniques with no novel insight required. Slightly above average due to the multi-step nature and need to connect the normal condition with the derivative. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.08a Fundamental theorem of calculus: integration as reverse of differentiation |
**(i)**
- B1: $\frac{dy}{dx} = 5 - \frac{8}{x^2}$, Normal $3y + x = 17$
- M1: Gradient of line $= -\frac{1}{3}$. Use of $m_1 m_2 = -1$
- DM1: $\frac{dy}{dx} = 3 \Rightarrow x = 2$, $y = 5$. DM1 solution. A1 co.
- A1: co
**(ii)**
- B1: $y = 5x + 8x^{-1} + c$. co
- B1: co. doesn't need $+c$
- M1: Uses $(2, 5) \Rightarrow c = -9$. Use of $+c$ following integration. co.
- A1: co
---7 A curve is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 - \frac { 8 } { x ^ { 2 } }$. The line $3 y + x = 17$ is the normal to the curve at the point $P$ on the curve. Given that the $x$-coordinate of $P$ is positive, find\\
(i) the coordinates of $P$,\\
(ii) the equation of the curve.
\hfill \mbox{\textit{CAIE P1 2011 Q7 [8]}}