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5 marksDifferentiation II

CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2 · Question 1(b)(ii)

A function w is defined as w(x, y) = \ln \left| \frac{2x + y}{x - 1} \right|.

Show that \frac{\partial^2 w}{\partial y \partial x} - 2 \frac{\partial^2 w}{\partial y^2} = 0.

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Other parts of this question

  1. 1(a)Find the first derivative of the function f(x) = \cos^{-1}(\sin^{-1} x).[3 marks]
  2. 1(b)(i)Given that \frac{\partial w}{\partial x} = -\frac{1}{9} at the point (4, y_0), calculate the value of y_0.[5 marks]
  3. 1(c)(i)Find the complex numbers u = x + \mathrm{i}y such that x and y are real and u^2 = -15 + 8\mathrm{i}.[7 marks]
  4. 1(c)(ii)Hence, or otherwise, solve the equation z^2 - (3 + 2\mathrm{i})z + (5 + \mathrm{i}) = 0, for z.[5 marks]

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