## Question 4:

$C_1: x = 10\cos t,\ y = 4\sqrt{2}\sin t,\ 0 \leq t < 2\pi;\ C_2: x^2 + y^2 = 66$

**Way 1:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| $(10\cos t)^2 + (4\sqrt{2}\sin t)^2 = 66$ | M1 | Begins by substituting parametric equations into Cartesian equation to give equation in one variable $t$ only |
| $100(1-\sin^2 t) + 32\sin^2 t = 66$ OR $100\cos^2 t + 32(1-\cos^2 t) = 66$ | M1 | Uses identity $\sin^2 t + \cos^2 t \equiv 1$ |
| Correct equation in $\sin^2 t$ only or $\cos^2 t$ only | A1 | |
| $\sin^2 t = \frac{1}{2} \Rightarrow \sin t = \ldots$ OR $\cos^2 t = \frac{1}{2} \Rightarrow \cos t = \ldots$ | dM1 | Dependent on both previous M marks; rearranges to make $\sin t = \ldots$ where $-1 \leq \sin t \leq 1$ or $\cos t = \ldots$ where $-1 \leq \cos t \leq 1$. Condone $\sin^2 t = \frac{1}{2} \Rightarrow \sin t = \frac{1}{4}$ |
| Substitutes solution back into relevant original equation(s) to get $x$-coordinate and $y$-coordinate | M1 | These may not be in the correct quadrant |
| $S = (5\sqrt{2},\ -4)$ or $x = 5\sqrt{2},\ y = -4$ or $S = (\text{awrt } 7.07,\ -4)$ | A1 | |

**Way 2:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\left(\frac{x}{10}\right)^2 + \left(\frac{y}{4\sqrt{2}}\right)^2 = 1\ \{\Rightarrow 32x^2 + 100y^2 = 3200\}$ | M1 | Complete process using $\cos^2 t + \sin^2 t \equiv 1$ to convert parametric to Cartesian for $C_1$ |
| $\frac{x^2}{100} + \frac{66-x^2}{32} = 1$ OR $\frac{66-y^2}{100} + \frac{y^2}{32} = 1$ | M1 | Complete valid attempt to write equation in $x$ only or $y$ only |
| Correct equation in $x$ only or $y$ only | A1 | |
| $x^2 = 50 \Rightarrow x = \ldots$ OR $y^2 = 16 \Rightarrow y = \ldots$ | dM1 | Dependent on both M marks; their $x^2$ or $y^2$ must be $> 0$ |
| Substitutes back to get corresponding coordinate | M1 | Their $x^2$ and $y^2$ must be $> 0$ |
| $S = (5\sqrt{2},\ -4)$ or $x = 5\sqrt{2},\ y = -4$ or $S = (\text{awrt } 7.07,\ -4)$ | A1 | Allow $S = (\sqrt{50},\ -4)$ or $S = \left(\frac{10}{\sqrt{2}},\ -4\right)$ |

**Way 3:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| $x = \sqrt{66}\cos\alpha,\ y = \sqrt{66}\sin\alpha$; combining with $C_1$: $10\cos t = \sqrt{66}\cos\alpha,\ 4\sqrt{2}\sin t = \sqrt{66}\sin\alpha$; applying $\cos^2\alpha + \sin^2\alpha = 1$: $\left(\frac{10\cos t}{\sqrt{66}}\right)^2 + \left(\frac{4\sqrt{2}\sin t}{\sqrt{66}}\right)^2 = 1$ | M1 | Complete process of writing $C_2$ in parametric form, combining with $C_1$, applying identity to give equation in one variable $t$ only |
| *then continue with Way 1 mark scheme* | | |

**Way 4:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| $(10\cos t)^2 + (4\sqrt{2}\sin t)^2 = 66$ | M1 | Complete process substituting into Cartesian equation |
| $100\left(\frac{1+\cos 2t}{2}\right) + 32\left(\frac{1-\cos 2t}{2}\right) = 66$ | M1 | Uses identities $\cos 2t \equiv 2\cos^2 t - 1$ and $\cos 2t \equiv 1 - 2\sin^2 t$ |
| Correct equation in $\cos 2t$ only | A1 | At least one of the double angle identities must be correct |
| $34\cos 2t + 66 = 66 \Rightarrow \cos 2t = \ldots$ | dM1 | Dependent on both M marks; $-1 \leq \cos 2t \leq 1$ |
| Substitutes back to get coordinates | M1 | |
| $S = (5\sqrt{2},\ -4)$ or $x = 5\sqrt{2},\ y = -4$ or $S = (\text{awrt } 7.07,\ -4)$ | A1 | |

**Way 5:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| $(10\cos t)^2 + (4\sqrt{2}\sin t)^2 = 66(\sin^2 t + \cos^2 t)$ | M1 | Complete process substituting into Cartesian equation |
| $100\cos^2 t + 32\sin^2 t = 66\sin^2 t + 66\cos^2 t \Rightarrow 34\cos^2 t = 34\sin^2 t \Rightarrow \tan t = \ldots$ | M1, A1 | Uses identity to achieve equation in $\sin^2 t$ and $\cos^2 t$ with no constant term; correct equation containing no constant term |
| $\tan t = \ldots$ | dM1 | Dependent on both M marks |
| Substitutes back | M1 | |
| $S = (5\sqrt{2},\ -4)$ or $x = 5\sqrt{2},\ y = -4$ or $S = (\text{awrt } 7.07,\ -4)$ | A1 | Allow $S = (\sqrt{50},\ -4)$ or $S = \left(\frac{10}{\sqrt{2}},\ -4\right)$ |

**Note:** Give final A0 for writing $x = 5\sqrt{2},\ y = -4$ followed by $S = (-4,\ 5\sqrt{2})$

**(6 marks)**

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