| Exam Board | CAIE |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2023 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric integration |
| Type | Parametric arc length calculation |
| Difficulty | Standard +0.8 This is a Further Maths parametric arc length question requiring computation of √((dx/dt)² + (dy/dt)²) and integration. Part (a) involves differentiating fractional powers, simplifying a surd expression, and integrating—standard technique but algebraically demanding. Part (b) requires the second derivative formula d²y/dx² = d/dt(dy/dx) ÷ dx/dt with careful handling of negative powers. More challenging than typical A-level due to the fractional indices and multi-step algebraic manipulation, but follows established methods without requiring novel insight. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation4.08e Mean value of function: using integral |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\dot{x} = t^{\frac{1}{2}} - t^{-\frac{1}{2}}\), \(\dot{y} = 2\) | B1 | Differentiates \(x\) and \(y\) with respect to \(t\) |
| \(\dot{x}^2 + \dot{y}^2 = \left(t^{\frac{1}{2}} - t^{-\frac{1}{2}}\right)^2 + 4 = t + 2 + t^{-1} = \left(t^{\frac{1}{2}} + t^{-\frac{1}{2}}\right)^2\) | M1 A1 | Factorises \(\dot{x}^2 + \dot{y}^2\) |
| \(\int_0^3 t^{\frac{1}{2}} + t^{-\frac{1}{2}}\,dt = \left[\frac{2}{3}t^{\frac{3}{2}} + 2t^{\frac{1}{2}}\right]_0^3 = 4\sqrt{3}\) | M1 A1 | Applies correct formula for arc length. M1 for \(\int_0^3\sqrt{\dot{x}^2+\dot{y}^2}\,dt\). Answer must be simplified to \(4\sqrt{3}\) for A1 |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} = \frac{\dot{y}}{\dot{x}} = \frac{2}{t^{\frac{1}{2}} - t^{-\frac{1}{2}}}\) | B1 | Finds first derivative |
| \(\frac{d}{dt}\left(\frac{2}{t^{\frac{1}{2}}-t^{-\frac{1}{2}}}\right) = \frac{-2\left(\frac{1}{2}t^{-\frac{1}{2}}+\frac{1}{2}t^{-\frac{3}{2}}\right)}{\left(t^{\frac{1}{2}}-t^{-\frac{1}{2}}\right)^2}\) | B1 | Differentiates \(\frac{dy}{dx}\) with respect to \(t\) |
| \(\frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{2}{t^{\frac{1}{2}}-t^{-\frac{1}{2}}}\right)\times\frac{dt}{dx} = \frac{-2\left(\frac{1}{2}t^{-\frac{1}{2}}+\frac{1}{2}t^{-\frac{3}{2}}\right)}{\left(t^{\frac{1}{2}}-t^{-\frac{1}{2}}\right)^3} = -\frac{t+1}{(t-1)^3}\) | M1 A1 | Applies chain rule. OE. Does not have to be simplified for A1 |
| \(0 < t < 1\) | A1 | Accept \(-1 < t < 1\). CWO |
| 5 |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dot{x} = t^{\frac{1}{2}} - t^{-\frac{1}{2}}$, $\dot{y} = 2$ | B1 | Differentiates $x$ and $y$ with respect to $t$ |
| $\dot{x}^2 + \dot{y}^2 = \left(t^{\frac{1}{2}} - t^{-\frac{1}{2}}\right)^2 + 4 = t + 2 + t^{-1} = \left(t^{\frac{1}{2}} + t^{-\frac{1}{2}}\right)^2$ | M1 A1 | Factorises $\dot{x}^2 + \dot{y}^2$ |
| $\int_0^3 t^{\frac{1}{2}} + t^{-\frac{1}{2}}\,dt = \left[\frac{2}{3}t^{\frac{3}{2}} + 2t^{\frac{1}{2}}\right]_0^3 = 4\sqrt{3}$ | M1 A1 | Applies correct formula for arc length. M1 for $\int_0^3\sqrt{\dot{x}^2+\dot{y}^2}\,dt$. Answer must be simplified to $4\sqrt{3}$ for A1 |
| | **5** | |
---
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = \frac{\dot{y}}{\dot{x}} = \frac{2}{t^{\frac{1}{2}} - t^{-\frac{1}{2}}}$ | B1 | Finds first derivative |
| $\frac{d}{dt}\left(\frac{2}{t^{\frac{1}{2}}-t^{-\frac{1}{2}}}\right) = \frac{-2\left(\frac{1}{2}t^{-\frac{1}{2}}+\frac{1}{2}t^{-\frac{3}{2}}\right)}{\left(t^{\frac{1}{2}}-t^{-\frac{1}{2}}\right)^2}$ | B1 | Differentiates $\frac{dy}{dx}$ with respect to $t$ |
| $\frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{2}{t^{\frac{1}{2}}-t^{-\frac{1}{2}}}\right)\times\frac{dt}{dx} = \frac{-2\left(\frac{1}{2}t^{-\frac{1}{2}}+\frac{1}{2}t^{-\frac{3}{2}}\right)}{\left(t^{\frac{1}{2}}-t^{-\frac{1}{2}}\right)^3} = -\frac{t+1}{(t-1)^3}$ | M1 A1 | Applies chain rule. OE. Does not have to be simplified for A1 |
| $0 < t < 1$ | A1 | Accept $-1 < t < 1$. CWO |
| | **5** | |
---5 The curve $C$ has parametric equations
$$\mathrm { x } = \frac { 2 } { 3 } \mathrm { t } ^ { \frac { 3 } { 2 } } - 2 \mathrm { t } ^ { \frac { 1 } { 2 } } , \quad \mathrm { y } = 2 \mathrm { t } + 5 , \quad \text { for } 0 < t \leqslant 3$$
\begin{enumerate}[label=(\alph*)]
\item Find the exact length of $C$.
\item Find the set of values of $t$ for which $\frac { d ^ { 2 } y } { d x ^ { 2 } } > 0$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 2 2023 Q5 [10]}}