$\"\"$

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(a)

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The parametric equations are $\"\"$ and $\"\"$ .

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Graph:

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Graph the parametric equations $\"\"$ and $\"\"$ .

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Represent the orientation for different values of $\"\"$ .

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$\"\"$

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Observe the graph,

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Above graph resembles the cycloid.

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$\"\"$

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(b)

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Observe the graph:

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Distance between the $\"\"$ intercepts is $\"\"$ and it represents the path of the points on a circle rotating one full cycle on a straight line without slippage.

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$\"\"$

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(c)

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Observe the graph,

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Maximum value of $\"\"$ is $\"\"$ since the minimum value of $\"\"$ is $\"\"$ .

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It represents the maximum height of the cycloid.

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For circles of different radii, $\"\"$ will not effect since the term $\"\"$ does not contain the radius value.

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$\"\"$

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(a)

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$\"\"$

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Above graph resembles the cycloid.

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(b)

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Distance between the $\"\"$ intercepts is $\"\"$ and it represents the path of the points on a circle rotating one full cycle on a straight line without slippage.

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(c)

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Maximum value of $\"\"$ is $\"\"$ since the minimum value of $\"\"$ is $\"\"$ .

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For circles of different radii, $\"\"$ will not effect since the term $\"\"$ doesnot contain the radius value.