| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2018 |
| Session | June |
| Marks | 10 |
| Topic | Parametric differentiation |
| Type | Normal passes through specific point verification |
| Difficulty | Standard +0.8 This is a two-part parametric differentiation question requiring (i) finding dy/dx using the chain rule, computing the normal at a specific point, and verifying an equation form; (ii) proving the normal doesn't intersect the curve again by solving a cubic equation and showing t=2 is the only real solution. Part (ii) requires algebraic manipulation and reasoning about cubic roots, elevating this above routine normal-finding exercises. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.03g Parametric equations: of curves and conversion to cartesian1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation |
**Question 7(i) and 7(ii)**
**(i)**
- $\frac{dx}{dt} = 2t$, $\frac{dy}{dt} = 3t^2 - 2$; Hence $\frac{dy}{dx} = \frac{3t^2-2}{2t}$ **M1** Attempt to find $\frac{dy}{dx}$ either from parametric equations or a Cartesian equation
- **A1** Obtain correct $\frac{dy}{dx}$. Could be $\frac{dy}{dx} = \frac{3}{2}(x-1)^{\frac{1}{2}} - (x-1)^{-\frac{1}{2}}$
- $m = \frac{5}{2}$; $m' = -\frac{2}{5}$ **M1** Attempt to find gradient of normal when $t=2$
- $y - 4 = -\frac{2}{5}(x-5)$ **M1** Attempt to find equation of normal when $t=2$
- $2x + 5y = 30$ **A.G.** **A1** Obtain $2x+5y=30$ **A.G.**
**(ii)**
- $2(t^2+1) + 5(t^3-2t) = 30$ **M1** Attempt to solve equations for normal and curve simultaneously to obtain an equation in a single variable
- $5t^3 + 2t^2 - 10t - 28 = 0$ **A1** Obtain correct 4 term cubic. Could be $25x^3 - 179x^2 + 495x - 1125 = 0$
- $(t-2)(5t^2+12t+14)=0$ **M1** Attempt to factorise cubic, using $(t-2)$ or $(x-5)$ as a factor
- Obtain $(t-2)(5t^2+12t+14)=0$ or $(x-5)(25x^2-54x+225)=0$ **A1**
- $\Delta = -136$, hence no other roots **A1** Show that quadratic has no real roots and conclude – detail needed. NB allow other methods, such as gradients of the two functions, as long as fully convincing7 A curve is given parametrically by $x = t ^ { 2 } + 1 , y = t ^ { 3 } - 2 t$ where $t$ is any real number.\\
(i) Show that the equation of the normal to the curve at the point where $t = 2$ can be written in the form $2 x + 5 y = 30$.\\
(ii) Show that this normal does not meet the curve again.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2018 Q7 [10]}}