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7 marksDifferentiation II

CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2 · Question 1(a)(ii)

SECTION A - Module 1

A curve is defined parametrically as x = a\cos^3 t, y = a\sin^3 t. Show that the tangent at the point P(x, y) is the line y\cos t + x\sin t = a\sin t\cos t.

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Other parts of this question

  1. 1(a)(i)Differentiate, with respect to x, y = \ln(x^2 + 4) - x \tan^{-1}\left(\frac{x}{2}\right).[5 marks]
  2. 1(b)(i)Determine the nature of the roots of the equation.[2 marks]
  3. 1(b)(ii)Express \alpha and \beta in the form r e^{i\theta}, where r is the modulus and \theta is the argument, where -\pi < \theta \le \pi.[4 marks]
  4. 1(b)(iii)Using de Moivre's theorem, or otherwise, compute \alpha^3 + \beta^3.[4 marks]
  5. 1(b)(iv)Hence, or otherwise, obtain the quadratic equation whose roots are \alpha^3 and \beta^3.[3 marks]

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