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Differentiation

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Question 433

A high-precision industrial laser tracks a path C C\,C on a component, defined by the parametric equations

x=3cos⁡2t−6sin⁡t,y=6sin⁡t+5cos⁡t,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≤2π
a.

Show that dydx=−1\displaystyle \frac{dy}{dx} = -1dxdy​=−1 at the point where t=πt = \pit=π.

[2]
b.

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.

[3]
c.

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

3cos⁡2t+5cos⁡t+2=0 3\cos^2 t + 5\cos t + 2 = 0 3cos2t+5cost+2=0
[3]
d.

Hence find the exact possible values of the yyy-coordinate of point QQQ.

[2]
Markscheme

Differentiation Questions

  1. A Level
  2. /Maths
  3. /Differentiation

649 exam-style questions on Edexcel A Level Maths Differentiation, covering 9.1 Differentiating sin x and cos x, 9.2 Differentiating exponentials and logarithms, 9.3 The Chain Rule, 9.4 The Product Rule, 9.5 The Quotient Rule, 9.6 Differentiating Trigonometric Functions, 9.7 Parametric Differentiation, 9.8 Implicit Differentiation, 9.9 Using Second Derivatives, and 9.10 Rates of Change. Each one has a worked solution and a mark scheme showing where the marks go.

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