| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2017 |
| Session | June |
| Marks | 7 |
| Topic | Implicit equations and differentiation |
| Type | Find normal equation at point |
| Difficulty | Standard +0.3 This is a straightforward implicit differentiation question requiring students to differentiate both sides, substitute the given point to find dy/dx, then find the normal gradient and write the equation. While it involves multiple steps (implicit differentiation, substitution, finding perpendicular gradient, forming line equation), each step is routine and the question explicitly tells students what answer to reach, making it slightly easier than average. |
| Spec | 1.07s Parametric and implicit differentiation |
Obtain $3y^2\dfrac{\text{d}y}{\text{d}x} + 12y\dfrac{\text{d}y}{\text{d}x}$ **B1**
Obtain $-2\dfrac{\text{d}y}{\text{d}x} = 6x + 2$ **B1**
Substitute $(1, 1)$ into their $\dfrac{\text{d}y}{\text{d}x}$ as long as valid implicit differentiation used **M1**
Use $m_1 m_2 = -1$ **M1**
Obtain $\dfrac{-13}{8}$ **A1**
Use $(y-1) = m(x-1)$ **M1**
Obtain $8y + 13x - 21 = 0$ **A1**
(Unclear notation used or apparent slips in working but otherwise correct. Award final A0)
**Total: 7 marks**8 The curve $C$ has equation $y ^ { 3 } + 6 y ^ { 2 } - 2 y = 3 x ^ { 2 } + 2 x$. Show that the equation of the normal to $C$ at the point $( 1,1 )$ can be written in the form $8 y + 13 x - 21 = 0$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2017 Q8 [7]}}