| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2008 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find tangent equation at parameter |
| Difficulty | Standard +0.3 This is a straightforward parametric tangent question requiring standard differentiation (dy/dx = 2t/3t² = 2/3t), substitution into point-slope form, and solving a cubic equation (3p·7 - 2(-10) = p³ gives p³ - 21p - 20 = 0). While it has multiple steps and a cubic to solve, all techniques are routine for C4 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use \(\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}\) or \(\frac{dy/dt}{dx/dt}\) or \(\frac{dy/dp}{dx/dp}\) | M1 | Or conv to cartes form & att to find \(\frac{dy}{dx}\) at P |
| \(= \frac{2t}{3t^2}\) or \(\frac{2p}{3p^2}\) | A1 | |
| Find eqn tgt thro \(\left(p^3, p^2\right)\) or \(\left(t^3, t^2\right)\) ,their gradient | M1 | Using \(y - y_1 = m(x - x_1)\) or \(y = mx + c\) Do not accept t here |
| \(3py - 2x = p^3\) | AG | 4 |
| (ii) Substitute \((-10,7)\) into given equation | *M1 | to produce a cubic equation in p |
| Satis attempt to find at least 1 root/factor | dep*M1 | Inspection/factor theorem/rem theorem/t&i \(-1\) or \(-4\) or 5 |
| Any one root | A1 | |
| All 3 roots | A1 | |
| \((-1,1), (-4,16)\) and \((125,25)\) | A1 | 5 |
(i) Use $\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}$ or $\frac{dy/dt}{dx/dt}$ or $\frac{dy/dp}{dx/dp}$ | M1 | Or conv to cartes form & att to find $\frac{dy}{dx}$ at P |
$= \frac{2t}{3t^2}$ or $\frac{2p}{3p^2}$ | A1 | |
Find eqn tgt thro $\left(p^3, p^2\right)$ or $\left(t^3, t^2\right)$ ,their gradient | M1 | Using $y - y_1 = m(x - x_1)$ or $y = mx + c$ Do not accept t here |
$3py - 2x = p^3$ | AG | 4 |
(ii) Substitute $(-10,7)$ into given equation | *M1 | to produce a cubic equation in p |
Satis attempt to find at least 1 root/factor | dep*M1 | Inspection/factor theorem/rem theorem/t&i $-1$ or $-4$ or 5 |
Any one root | A1 | |
All 3 roots | A1 | |
$(-1,1), (-4,16)$ and $(125,25)$ | A1 | 5 | All 3 sets; no f.t. |9 The parametric equations of a curve are $x = t ^ { 3 } , y = t ^ { 2 }$.\\
(i) Show that the equation of the tangent at the point $P$ where $t = p$ is
$$3 p y - 2 x = p ^ { 3 } .$$
(ii) Given that this tangent passes through the point ( $- 10,7$ ), find the coordinates of each of the three possible positions of $P$.
\hfill \mbox{\textit{OCR C4 2008 Q9 [9]}}