| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Gradient condition leads to trig equation |
| Difficulty | Standard +0.3 This is a standard parametric differentiation question requiring dy/dx = (dy/dt)/(dx/dt), followed by routine harmonic form conversion and solving. All techniques are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain \(\frac{dx}{dt} = 3 - 2\cos t\) and \(\frac{dy}{dt} = 5 - 4\sin t\) | B1 | |
| Equate expression for \(\frac{dy}{dx}\) to \(\frac{5}{2}\) | M1 | |
| Obtain \(10\cos t - 8\sin t = 5\) | A1 | AG – sufficient working to be shown |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(R = \sqrt{164}\) or exact equivalent | B1 | |
| Use appropriate trigonometry to find \(\alpha\) | M1 | |
| Obtain \(0.675\) with no errors seen | A1 | AWRT |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out correct method to find one value of \(t\) | M1 | Must be using the result from (b) |
| Obtain \(0.495\) | A1 | AWRT |
| Carry out correct method to find second value of \(t\) | M1 | |
| Obtain \(4.44\) | A1 | AWRT |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain $\frac{dx}{dt} = 3 - 2\cos t$ and $\frac{dy}{dt} = 5 - 4\sin t$ | B1 | |
| Equate expression for $\frac{dy}{dx}$ to $\frac{5}{2}$ | M1 | |
| Obtain $10\cos t - 8\sin t = 5$ | A1 | AG – sufficient working to be shown |
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $R = \sqrt{164}$ or exact equivalent | B1 | |
| Use appropriate trigonometry to find $\alpha$ | M1 | |
| Obtain $0.675$ with no errors seen | A1 | AWRT |
## Question 7(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out correct method to find one value of $t$ | M1 | Must be using the result from (b) |
| Obtain $0.495$ | A1 | AWRT |
| Carry out correct method to find second value of $t$ | M1 | |
| Obtain $4.44$ | A1 | AWRT |7 A curve is defined by the parametric equations
$$x = 3 t - 2 \sin t , \quad y = 5 t + 4 \cos t$$
where $0 \leqslant t \leqslant 2 \pi$. At each of the points $P$ and $Q$ on the curve, the gradient of the curve is $\frac { 5 } { 2 }$.
\begin{enumerate}[label=(\alph*)]
\item Show that the values of $t$ at $P$ and $Q$ satisfy the equation $10 \cos t - 8 \sin t = 5$.
\item Express $10 \cos t - 8 \sin t$ in the form $R \cos ( t + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$. Give the exact value of $R$ and the value of $\alpha$ correct to 3 significant figures.
\item Hence find the values of $t$ at the points $P$ and $Q$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2020 Q7 [10]}}