3.5.4 Differentiation (A-level only)

Medium Hard

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Question 51

A precision laser cutting tool follows a path defined by the relationship between its lateral displacement, $x$ mm, and the angle of its guide-rail, $y$ radians, given by

16\sin^2 y + 5, \quad 0 \leqslant y \leqslant \frac{\pi}{2}

The point $P(k,π3)P\left(k, \frac{\pi}{3}\right)$ lies on this path.

Verify that $k = 17$ .

[2]

bi.

Find $dxdy\frac{dx}{dy}$ in terms of $y$ .

[2]

bii.

Hence show that $dydx=12x−521−x\frac{dy}{dx} = \frac{1}{2\sqrt{x-5}\sqrt{21-x}}$ .

[3]

The normal to the path at $P$ cuts the $x$ -axis at the point $N$ .

Find the exact area of triangle $O P N$ , where $O$ is the origin. Give your answer in the form $aπ+bπ2a\pi + b\pi^2$ where $a$ and $b$ are constants.

[5]

3.5.4 Differentiation (A-level only) Questions

A Level
Maths
3.5.4 Differentiation (A-level only)

A Level
Maths CCEA
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