4th Extension answers

Answers to 4th year extension

1. \[ \\13000 \times 0.85^? = 1000 \\0.85^? = \frac{1}{13} \\?=log_{0.85}{\frac{1}{13}}= 15.8 \]

   i.e. 16 years is needed (or you can do it by trial and error on the calculator) YOU DO NOT NEED TO KNOW THE LOG METHOD, trial and error using a calculator is sufficient at this stage

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2. a. \begin{align*} \\y^2 &= 5-3x^2 \\x^2 &= 5-y^2 \\x &=\sqrt{\frac{5-y^2}{3}} \end{align*}

   b. \begin{align*} \\ y(2x^2-1) &= 2x^2+3 && \textit{multiply by denominator} \\ 2x^2y-y&=2x^2+3 && \textit{expand the brackets} \\ 2x^2y-2x^2&=y+3 && \textit{get all the }x^2 \textit{ stuff on the left} \\ x^2(2y-2)&=y+3 && \textit{take out common factor of }x^2 \\ x^2&=\frac{y+3}{2y-2} \\ x&=\sqrt{\frac{y+3}{2y-2}} \end{align*}

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3. \[ \\3x -2(30-x)=55 \\3x -60+2x=55 \\5x -60=55 \\x =23 \]

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4. Rearranging gives \[y=\frac{3}{2}x + \frac{5}{2} \] Gradient is $\dfrac{3}{2}$, so for the new line: \[ y=\frac{3}{2}x+c \] Now use (2,-4) in here to get c ie \begin{align*} -4 = \frac{3}{2}\times 2 + c \\ \therefore c = -7 \\ \therefore y=\frac{3}{2}x-7 && \times \textit{both sides by 2 and subtract 3}x \\2y -3x = -14 && \textit{to get the required format} \\3x-2y = 14 && \textit{or this, either is fine} \end{align*}

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5.