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6 marksMatrices and Systems of Equations

CAPE Integrated Mathematics · 2016 · Paper 2 · Question 2(d)

A system of linear equations models the cost of computers from three manufacturers: 2x + y + z = 1200 x + 4y + z = 1900 3x + 2z = 1600 Using Cramer's rule or otherwise, where the determinant of the coefficient matrix of the system is 5, find z.

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  1. 2(a)(i)Given that f(x) = 2 ln x^2 - ln x, show that f(x) can be rewritten as f(x) = ln x^3.[3 marks]
  2. 2(a)(ii)Hence, solve the equation 2 ln x^2 = 2 + ln x, giving your answer correct to 2 decimal places.[5 marks]
  3. 2(b)(i)Solve the equation 2 sin(theta + pi/3) - 1 = 0 for 0 <= theta <= pi.[5 marks]
  4. 2(b)(ii)Determine the minimum value of f(theta) = 2 sin(theta + pi/3) - 1.[3 marks]
  5. 2(c)A length of string is cut into 10 pieces such that the shortest and longest pieces measure 0.2 metres and 1 metre, respectively. When the 10 pieces are…[3 marks]

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