# Question 8:

## Part (i):

| Answer/Working | Marks | Guidance |
|---|---|---|
| Sub parametric eqns into $y=3x$ & produce $t=-2$ | | |
| OR sub $t=-2$ into para eqns, obtain $(-1,-3)$ & state $y=3x$ | | |
| OR other similar methods producing (or verifying) $t=-2$ | B1 | |
| Value of $t$ at other point is $2$ | B1 **2** | $t=\pm2$ is sufficient for B1+B1 |

## Part (ii):

| Answer/Working | Marks | Guidance |
|---|---|---|
| Use (not just quote) $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ | M1 | |
| $= -(t+1)^2$ | A1 | or $\frac{-1}{x^2}$ or $\frac{-(2+y)}{x}$ |
| Attempt to use $-\frac{1}{\frac{dy}{dx}}$ for gradient of normal | M1 | |
| Gradient of normal $= 1$ cao | A1 | |
| Subst $t=-2$ into the parametric eqns | M1 | to find pt at which normal is drawn |
| Produce $y=x-2$ as equation of the normal WWW | A1 **6** | 'A' marks in (ii) are dep on prev 'A' |

## Part (iii):

| Answer/Working | Marks | Guidance |
|---|---|---|
| Substitute the parametric values into their eqn of normal | M1 | |
| Produce $t=0$ as final answer cao | A1 **2** | This is dep on final A1 in (ii) |
| N.B. If $y=x-2$ is found fortuitously in (ii) (& given A0 in (ii)), you must award A0 here in (iii) | | |

## Part (iv):

| Answer/Working | Marks | Guidance |
|---|---|---|
| Attempt to eliminate $t$ from the parametric equations | M1 | |
| Produce any correct equation | A1 | e.g. $x=\frac{1}{y+2}$ |
| Produce $y=\frac{1}{x}-2$ or $y=\frac{1-2x}{x}$ ISW | A1 **3** | Must be seen in (iv) |
| {N.B. Candidate producing only $y=\frac{1}{x}-2$ is awarded both A1 marks.} | | |