$\sin \theta = BC/AC$, $\cos \theta = AB/AC$ | M1 | or $a/b$, $c/b$ condone taking $AC = 1$ but must be stated

$AB^2 + BC^2 = AC^2$ | |

$\Rightarrow \quad (AB/AC)^2 + (BC/AC)^2 = 1$ | |

$\Rightarrow \quad \cos^2 \theta + \sin^2 \theta = 1$ | A1, B1 [3] | Must use Pythagoras allow $\leq$, or 'between 0 and 90°' or $< 90$ allow $< \pi/2$ or 'acute'

Valid for $(0° <) \theta (< 90°)$ | |

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# Question 5(i)

Graph showing: | B1, B1, B1 [3] | for first and second B1s graphs must include negative $x$ values condone no asymptote $y = -1$ shown asymptote to $x$-axis (shouldn't cross)

shape of $y = e^x - 1$ and through O | |

shape of $y = 2e^{-x}$ | |

through $(0, 2)$ (not $(2,0)$) | |

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# Question 5(ii)

$e^x - 1 = 2e^{-x}$ | M1 | equating

$\Rightarrow \quad e^{2x} - e^x = 2$ | |

$\Rightarrow \quad (e^x)^2 - e^x - 2 = 0$ | M1 | re-arranging into a quadratic in $e^x = 0$ | allow one error but must have $e^{2x} = (e^x)^2$ (soi)

$\Rightarrow \quad (e^x - 2)(e^x + 1) = 0$ | |

$\Rightarrow \quad e^x = 2$ (or $-1$) | B1, B1 | stated www www

$\Rightarrow \quad x = \ln 2$ | |

$\Rightarrow \quad y = 1$ | B1cao [5] | www | award even if not from quadratic method (i.e. by 'fitting') provided www allow for unsupported answers, provided www need not have used a quadratic, provided www

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