| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Tangent/normal meets curve again |
| Difficulty | Standard +0.8 This is a multi-part parametric question requiring standard differentiation (part i), verification of a tangent equation (part ii), and then solving a non-trivial intersection problem (part iii) where students must substitute the tangent equation back into parametric equations and solve a cubic, excluding the known point. The final part requires careful algebraic manipulation and is above average difficulty. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Produce at least 2 of the 3 relevant eqns in \(\lambda\) and \(\mu\) | M1 | e.g. \(1 + 3\lambda = -8 + \mu,\, -2 + \lambda = 2 - 2\mu\) |
| Solve the 2 eqns in \(\lambda\) & \(\mu\) as far as \(\lambda = \ldots\) or \(\mu = \ldots\) | M1 | |
| 1st solution: \(\lambda = -2\) or \(\mu = 3\) | A1 | |
| 2nd solution: \(\mu = 3\) or \(\lambda = -2\) | A1 √ | |
| Substitute their \(\lambda\) and \(\mu\) into 3rd eqn and find 'a' | M1 | |
| Obtain \(a = 2\) & clearly state that \(a\) cannot be 2 | A1 6 | |
| (ii) Subst their \(\lambda\) or \(\mu\) (& poss \(a\)) into either line eqn; Point of intersection is \(−5\mathbf{i}−4\mathbf{j}\) | M1 A1 2 | Accept any format; No f.t. here |
(i) Produce at least 2 of the 3 relevant eqns in $\lambda$ and $\mu$ | M1 | e.g. $1 + 3\lambda = -8 + \mu,\, -2 + \lambda = 2 - 2\mu$
Solve the 2 eqns in $\lambda$ & $\mu$ as far as $\lambda = \ldots$ or $\mu = \ldots$ | M1 |
1st solution: $\lambda = -2$ or $\mu = 3$ | A1 |
2nd solution: $\mu = 3$ or $\lambda = -2$ | A1 √ |
Substitute their $\lambda$ and $\mu$ into 3rd eqn and find 'a' | M1 |
Obtain $a = 2$ & clearly state that $a$ cannot be 2 | A1 6 |
(ii) Subst their $\lambda$ or $\mu$ (& poss $a$) into either line eqn; Point of intersection is $−5\mathbf{i}−4\mathbf{j}$ | M1 A1 2 | Accept any format; No f.t. here7 A curve is given parametrically by the equations
$$x = t ^ { 2 } , \quad y = \frac { 1 } { t }$$
(i) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$, giving your answer in its simplest form.\\
(ii) Show that the equation of the tangent at the point $P \left( 4 , - \frac { 1 } { 2 } \right)$ is
$$x - 16 y = 12$$
(iii) Find the value of the parameter at the point where the tangent at $P$ meets the curve again.
June 2005\\
\hfill \mbox{\textit{OCR C4 Q7 [10]}}