| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Topic | Implicit equations and differentiation |
| Type | Find stationary points |
| Difficulty | Standard +0.3 This is a straightforward implicit differentiation problem requiring students to find dy/dx, set it to zero, and use the second derivative test. While it involves multiple steps (implicit differentiation, solving simultaneous equations, classification), these are all standard techniques with no novel insight required. The algebra is manageable and the question type is common in textbooks, making it slightly easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07p Points of inflection: using second derivative1.07s Parametric and implicit differentiation |
**Question 7:**
$18x + 8y\frac{\text{d}y}{\text{d}x} - 6 - 4\frac{\text{d}y}{\text{d}x} = 0$ **M1** (Obtain 4 term expression from implicit differentiation but must include $ky\frac{\text{d}y}{\text{d}x}$)
$\frac{\text{d}y}{\text{d}x} = \frac{6-18x}{8y-4} = 0$ **M1** (Set their $\frac{\text{d}y}{\text{d}x} = 0$ and solve to obtain a value for $x$)
$x = \frac{1}{3}$ **A1**
$1 + 4y^2 - 2 - 4y = 34$ **M1** (Substitute their value for $x$ and obtain a quadratic in $y$)
$\left(\frac{1}{3}, \frac{7}{2}\right),\ \left(\frac{1}{3}, \frac{-5}{2}\right)$ **A1** (Obtain correct coordinates)
$\frac{\text{d}^2y}{\text{d}x^2} = \frac{-18(8y-4) - 8\frac{\text{d}y}{\text{d}x}(6-18x)}{(8y-4)^2} = \frac{-18}{(8y-4)}$ **M1** (A generous attempt at quotient rule, allowing some slips)
$\frac{\text{d}^2y}{\text{d}x^2} = -0.75$ at $\left(\frac{1}{3}, \frac{7}{2}\right)$ so maximum **A1** (Final value must be shown)
$\frac{\text{d}^2y}{\text{d}x^2} = 0.75$ at $\left(\frac{1}{3}, \frac{-5}{2}\right)$ so minimum **A1** (Final value must be shown)
**Total: 8 marks**7 Find the coordinates of the two stationary points of the curve
$$9 x ^ { 2 } + 4 y ^ { 2 } - 6 x - 4 y = 34$$
showing that one is a maximum and one is a minimum.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2018 Q7 [8]}}