A robotic welding head follows a path C C\,C in the xyxyxy-plane described by the parametric equations
x=2sinθ+5cosθ,y=4cos2θ+2sinθ,0≤θ≤π x = 2\sin \theta + 5\cos \theta, \quad y = 4\cos^2 \theta + 2\sin \theta, \quad 0 \le \theta \le \pi x=2sinθ+5cosθ,y=4cos2θ+2sinθ,0≤θ≤πShow that dydx=1\displaystyle \frac{dy}{dx} = 1dxdy=1 where θ=0\theta = 0θ=0.
The point P P\,P lies on C C\,C where θ=0\theta = 0θ=0.
Find the equation of the tangent to the path C C\,C at the point PPP, giving your answer in the form y=mx+cy = mx + cy=mx+c.
The tangent to the path at P P\,P intersects the curve C C\,C again at the point QQQ.
Show that the value of θ \theta\,θ at point Q Q\,Q satisfies the equation
4cos2θ−5cosθ+1=0 4\cos^2 \theta - 5\cos \theta + 1 = 0 4cos2θ−5cosθ+1=0Hence find the exact value of the yyy-coordinate of 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.
