| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2009 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Connected Rates of Change |
| Type | Curve motion: find dy/dt |
| Difficulty | Standard +0.3 This is a straightforward connected rates of change question requiring chain rule application (dy/dt = dy/dx × dx/dt). Part (i) is routine differentiation using quotient/chain rule, part (ii) is standard normal line work, and part (iii) involves direct substitution into the chain rule formula with given values. All steps are mechanical with no novel insight required, making it slightly easier than average. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
B1: $y = \frac{12}{x^2 +3}$ — Without the "$\times 2x$"
B1: — For "$\times 2x$"
(i) $\frac{dy}{dx} = –12(x^2 + 3)^{-2} \times 2x$ — Accept unsimplified answer
[2]
(ii) M1: At $x = 1$, $m = –\frac{3}{2}$ — Uses $m_1 m_2 = −1$ algebraic ok
M1: $m$ of normal = $\frac{2}{3}$ — Uses $m_1 m_2 = −1$
M1: Eqn of normal $y – 3 = \frac{2}{3}(x – 1)$ — Correct form of equation
A1: — Correct form of equation
[3]
(iii) M1: $\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt} = −\frac{3}{2} \times 0.012$ — Correct link between differentials
A1: $\to −0.018$ — Co unsimplified; co to his $\frac{dy}{dx}$
[2]
(Omission of $x$ in part (i) causes fortuitous results in (ii) and (iii).)7 The equation of a curve is $y = \frac { 12 } { x ^ { 2 } + 3 }$.\\
(i) Obtain an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$.\\
(ii) Find the equation of the normal to the curve at the point $P ( 1,3 )$.\\
(iii) A point is moving along the curve in such a way that the $x$-coordinate is increasing at a constant rate of 0.012 units per second. Find the rate of change of the $y$-coordinate as the point passes through $P$.
\hfill \mbox{\textit{CAIE P1 2009 Q7 [7]}}