A robotic arm which is attached to a flat surface at the origin O, is used to draw a graphic design - AQA - A-Level Maths Mechanics - Question 9 - 2021 - Paper 2

From the previous equation, we can apply the identity for cosine: $\cos(2\theta) = 2\cos^2 \theta - 1$ Substituting this into our equation gives: $x = d \cos \theta - d(2\cos^2 \theta - 1)$ Simplifying further yields: $x = d(1 + \cos \theta - 2\cos^2 \theta)$

The equation provided indicates that the value of x can be maximized when (\cos \theta) is at its minimum, which occurs at (\frac{1}{4}). Substituting (\cos \theta = \frac{1}{4}) into the equation: $x = \frac{9d}{8}$ Thus, the greatest possible value of x is (\frac{9d}{8}) and the corresponding value of (\cos \theta) is (\frac{1}{4}).

To find the distance OQ, we can use the law of cosines. Since the maximum x-coordinate is known, we can use: $OQ^2 = d^2 + d^2 - 2d^2\cos \theta$ Substituting (\cos \theta = \frac{1}{4}): $OQ^2 = 2d^2(1 - \frac{1}{4}) = \frac{3}{2}d^2$ Thus, the distance OQ becomes: $OQ = d\sqrt{\frac{3}{2}} = \frac{d\sqrt{6}}{2}$