| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Topic | Parametric differentiation |
| Type | Parametric curve crosses axis, find gradient there |
| Difficulty | Standard +0.3 This is a straightforward parametric differentiation question requiring students to differentiate x with respect to y (using product rule), then find where x=0 to locate points A and B, and evaluate the gradient at those points. The product rule application and solving ln(2y+3)=0 are standard techniques, making this slightly easier than average. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation |
**(i)**
Attempt use of product rule to produce an expression of the form $k\ln(2y+3) + \frac{\text{linear in }y}{\text{linear in }y}$ **M1**
Obtain $\ln(2y+3)$ **A1**
Obtain $\ldots + \frac{2(y-4)}{2y+3}$ or unsimplified equiv **A1**
**[3]**
*Alternative method:*
Attempt use of product rule to produce $1 = \frac{\mathrm{d}y}{\mathrm{d}x}\left(\ln(2y+3) + \frac{(y-4)\frac{2\mathrm{d}y}{\mathrm{d}x}}{2y+3}\right)$ **M1**
Obtain $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{2y+3}{2y-8+(2y+3)\ln(2y+3)}$ **A1**
Obtain $\frac{\mathrm{d}x}{\mathrm{d}y} = \frac{2y-8+(2y+3)\ln(2y+3)}{2y+3}$ **A1**
**(ii)**
Attempt to find value of $y$ for which $x = 0$ **M1**
Obtain $y = -1$ and $y = 4$ **A1**
Substitute $y = -1$ into attempt from part (i) or into their attempt (however poor) at its reciprocal **M1**
SR: $-10$ without working M1A0. Other incorrect answers with no working M0
Obtain $-0.1$ (dependent on correct answer from (i)) **depA1**
Substitute $y = 4$ into attempt from part (i) or into their attempt (however poor) at its reciprocal **M1**
SR: ln 11 without working M1A0. Other incorrect answers with no working M0
Obtain $\frac{1}{\ln 11}$ (dependent on correct answer from (i)) **depA1**
**[6]**10 The diagram shows the curve with equation
$$x = ( y - 4 ) \ln ( 2 y + 3 ) .$$
The curve crosses the $y$-axis at $A$ and $B$.\\
\includegraphics[max width=\textwidth, alt={}, center]{afc8561d-94ae-42c0-bc6c-e9b091938368-3_588_780_1087_680}\\
(i) Find an expression for $\frac { \mathrm { d } x } { \mathrm {~d} y }$ in terms of $y$.\\
(ii) Find the exact gradient of the curve at each of the points $A$ and $B$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2016 Q10 [6]}}