A high-precision industrial laser tracks a path C C\,C on a component, defined by the parametric equations
x=3cos2t−6sint,y=6sint+5cost,0≤t≤2π x = 3\cos^2 t - 6\sin t, \quad y = 6\sin t + 5\cos t, \quad 0 \le t \le 2\pi x=3cos2t−6sint,y=6sint+5cost,0≤t≤2πShow that dydx=−1\displaystyle \frac{dy}{dx} = -1dxdy=−1 at the point where t=πt = \pit=π.
The point P P\,P lies on the path where t=πt = \pit=π.
Find the equation of the tangent to the laser's path at P P\,P in the form y=mx+cy = mx + cy=mx+c, where m m\,m and c c\,c are constants to be determined.
The laser's path is such that the tangent at P P\,P intersects the path C C\,C again at the point QQQ.
Show that the value of t t\,t at point Q Q\,Q satisfies the equation
3cos2t+5cost+2=0 3\cos^2 t + 5\cos t + 2 = 0 3cos2t+5cost+2=0Hence find the exact possible values of the yyy-coordinate of point QQQ.
Practise Edexcel A Level Maths Differentiation with exam-style questions for A Level Maths. 327 questions covering 12.1 Gradients of Curves, 12.2 Differentiation from first principles, 12.3 Differentiating x^n, 12.4 Differentiating Quadratics, 12.5 Differentiating functions with two or more terms, 12.6 Gradients, Tangents and Normals, 12.7 Increasing and Decreasing Functions, 12.8 Second Order Derivatives, 12.9 Stationary Points, 12.10 Sketching Gradient Functions, and 12.11 Modelling with Differentiation, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
