| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find dy/dx expression in terms of parameter |
| Difficulty | Standard +0.3 This is a standard parametric differentiation question requiring the chain rule (dy/dx = (dy/dt)/(dx/dt)) with exponential functions, followed by finding the parameter value where x=0. The differentiation is routine (quotient rule for x, chain rule for y), and finding t when x=0 involves basic exponential equation solving. Slightly above average due to the quotient rule and algebraic manipulation, but still a textbook exercise with no novel insight required. |
| Spec | 1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Differentiate \(x\) using quotient rule or correct equivalent | *M1 | |
| Obtain \(\dfrac{(e^{2t}+1)2e^{2t}-(e^{2t}-2)2e^{2t}}{(e^{2t}+1)^2}\) or equivalent | A1 | |
| Attempt expression for \(\dfrac{dy}{dx}\) in terms of \(t\) | DM1 | |
| Obtain \(\frac{1}{2}e^t(e^{2t}+1)^2\) or (unsimplified) equivalent | A1 | No fractions within fractions. Attempt to simplify \((e^{2t}+1)2e^{2t}-(e^{2t}-2)2e^{2t}\) must be seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Identify \(t = \frac{1}{2}\ln 2\) at point where curve crosses \(y\)-axis | B1 | |
| Substitute non-zero value of \(t\) in *their* expression for \(\dfrac{dy}{dx}\) and attempt simplification | M1 | |
| Obtain \(\frac{9}{2}\sqrt{2}\) or exact equivalent | A1 |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Differentiate $x$ using quotient rule or correct equivalent | *M1 | |
| Obtain $\dfrac{(e^{2t}+1)2e^{2t}-(e^{2t}-2)2e^{2t}}{(e^{2t}+1)^2}$ or equivalent | A1 | |
| Attempt expression for $\dfrac{dy}{dx}$ in terms of $t$ | DM1 | |
| Obtain $\frac{1}{2}e^t(e^{2t}+1)^2$ or (unsimplified) equivalent | A1 | No fractions within fractions. Attempt to simplify $(e^{2t}+1)2e^{2t}-(e^{2t}-2)2e^{2t}$ must be seen |
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## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Identify $t = \frac{1}{2}\ln 2$ at point where curve crosses $y$-axis | B1 | |
| Substitute non-zero value of $t$ in *their* expression for $\dfrac{dy}{dx}$ and attempt simplification | M1 | |
| Obtain $\frac{9}{2}\sqrt{2}$ or exact equivalent | A1 | |
---6 A curve has parametric equations
$$x = \frac { \mathrm { e } ^ { 2 t } - 2 } { \mathrm { e } ^ { 2 t } + 1 } , \quad y = \mathrm { e } ^ { 3 t } + 1$$
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$.\\
\includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-10_2718_42_107_2007}\\
\includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-11_2725_35_99_20}
\item Find the exact gradient of the curve at the point where the curve crosses the $y$-axis.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q6 [7]}}