8. (i) Solve the equation $$\pi - 3 \cos ^ { - 1 } \theta = 0$$ (ii) Sketch on the same diagram the curves \(y = \cos ^ { - 1 } ( x - 1 ) , 0 \leq x \leq 2\) and \(y = \sqrt { x + 2 } , x \geq - 2\). Given that \(\alpha\) is the root of the equation $$\cos ^ { - 1 } ( x - 1 ) = \sqrt { x + 2 }$$ (iii) show that \(0 < \alpha < 1\),
(iv) use the iterative formula $$x _ { n + 1 } = 1 + \cos \sqrt { x _ { n } + 2 }$$ with \(x _ { 0 } = 1\) to find \(\alpha\) correct to 3 decimal places.
You should show the result of each iteration.

**(i)** $\cos^{-1} \theta = \frac{\pi}{5}$, $\theta = \cos \frac{\pi}{5} = \frac{1}{2}$ | M1, A1 |

**(ii)** [Sketch showing $y = \sqrt{x+2}$ and $y = \cos^{-1}(x-1)$ curves] | B3 |

**(iii)** let $f(x) = \cos^{-1}(x-1) - \sqrt{x+2}$ | M1 |
$f(0) = 1.7$, $f(1) = -0.16$ | M1 |
sign change, $f(x)$ continuous $\therefore$ root | A1 |

**(iv)** $x_1 = 0.83944$, $x_2 = 0.88598$, $x_3 = 0.87233$, $x_4 = 0.87632$, $x_5 = 0.87515$, $x_6 = 0.87549$ | M1, A1 |
$\therefore \alpha = 0.875$ (3dp) | A1 | **(10 marks)** |

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8. (i) Solve the equation

$$\pi - 3 \cos ^ { - 1 } \theta = 0$$

(ii) Sketch on the same diagram the curves $y = \cos ^ { - 1 } ( x - 1 ) , 0 \leq x \leq 2$ and $y = \sqrt { x + 2 } , x \geq - 2$.

Given that $\alpha$ is the root of the equation

$$\cos ^ { - 1 } ( x - 1 ) = \sqrt { x + 2 }$$

(iii) show that $0 < \alpha < 1$,\\
(iv) use the iterative formula

$$x _ { n + 1 } = 1 + \cos \sqrt { x _ { n } + 2 }$$

with $x _ { 0 } = 1$ to find $\alpha$ correct to 3 decimal places.\\
You should show the result of each iteration.\\

\hfill \mbox{\textit{OCR C3  Q8 [10]}}