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3.6 Differentiation (A-level only)

3.6 Differentiation (A-level only)

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Question 110

A robotic welding head follows a path C C\,C in the xyxyxy-plane described by the parametric equations

x=2sin⁡θ+5cos⁡θ,y=4cos⁡2θ+2sin⁡θ,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≤π
a.

Show that dydx=1\displaystyle \frac{dy}{dx} = 1dxdy​=1 where θ=0\theta = 0θ=0.

[3]
b.

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.

[3]
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

4cos⁡2θ−5cos⁡θ+1=0 4\cos^2 \theta - 5\cos \theta + 1 = 0 4cos2θ−5cosθ+1=0
[3]
d.

Hence find the exact value of the yyy-coordinate of QQQ.

[3]
Markscheme

3.6 Differentiation (A-level only) Questions

  1. A Level
  2. /Maths
  3. /3.6 Differentiation (A-level only)

333 exam-style questions on WJEC A Level Maths 3.6 Differentiation (A-level only), covering 3.6.1 Differentiation (A-level only), 3.6.2 Differentiation (A-level only), 3.6.3 Differentiation (A-level only), 3.6.4 Differentiation (A-level only), 3.6.5 Differentiation (A-level only), 3.6.6 Differentiation (A-level only), 3.6.7 Differentiation (A-level only), and 3.6 Differentiation (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.

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