<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">$x=2\theta -\sin 2\theta ,$ $y=2{{\sin }^{2}}\theta $,</p>
<p>for $0\le \theta \le \pi $.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\cot \theta .$</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">The normal to the curve at the point where $\theta =\alpha $ meets the $x$-axis at the point $A$. Show that the $x$-coordinate of $A$ is $k\alpha $, where $k$ is a constant to be found.</div><p style="text-align: right;">[4]</p></li></ol>
<p><strong>Do not use a calculator in answering this part.</strong></p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">The distance between two points along a curve is the arc-length. Scientists and engineers need to use arc-length in applications such as finding the work done in moving an object along the path described by a curve or the length of the cabling used on a suspension bridge.</div><div class="list-item-paragraph">The arc-length between two points on $C$, where $\theta =\beta $ and $\theta =\gamma $, is given by the formula</div><div class="list-item-paragraph" style="text-align: center">$\int_{\beta }^{\gamma }{\sqrt{{{\left( \frac{\mathrm{d}x}{\mathrm{d}\theta } \right)}^{2}}+{{\left( \frac{\mathrm{d}y}{\mathrm{d}\theta } \right)}^{2}}}\mathrm{d}\theta }$</div><div class="list-item-paragraph">Find the total length of $C$.</div><p style="text-align: right;">[5]</p></li></ol>

$x=2\theta -\sin 2\theta ,$ $y=2{{\sin }^{2}}\theta $,

Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\cot \theta .$

The normal to the curve at the point where $\theta =\alpha $ meets the $x$-axis at the point $A$. Show that the $x$-coordinate of $A$ is $k\alpha $, where $k$ is a constant to be found.

**Do not use a calculator in answering this part.**

The distance between two points along a curve is the arc-length. Scientists and engineers need to use arc-length in applications such as finding the work done in moving an object along the path described by a curve or the length of the cabling used on a suspension bridge.

The arc-length between two points on $C$, where $\theta =\beta $ and $\theta =\gamma $, is given by the formula

$\displaystyle\displaystyle \int_{\beta }^{\gamma }{\sqrt{{{\left( \frac{\mathrm{d}x}{\mathrm{d}\theta } \right)}^{2}}+{{\left( \frac{\mathrm{d}y}{\mathrm{d}\theta } \right)}^{2}}}\mathrm{d}\theta }$

N2018 P1 Q9

N2018 P1 Q9

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