4.02g Conjugate pairs: real coefficient polynomials

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SPS SPS FM Pure 2025 February Q6
10 marks Standard +0.8
$$f(z) = 3z^3 + pz^2 + 57z + q$$ where \(p\) and \(q\) are real constants. Given that \(3 - 2\sqrt{2}i\) is a root of the equation \(f(z) = 0\)
  1. show all the roots of \(f(z) = 0\) on a single Argand diagram, [7]
  2. find the value of \(p\) and the value of \(q\). [3]
OCR FP1 AS 2021 June Q3
10 marks Moderate -0.3
In this question you must show detailed reasoning.
  1. Express \((2 + 3i)^3\) in the form \(a + ib\). [3]
  2. Hence verify that \(2 + 3i\) is a root of the equation \(3z^3 - 8z^2 + 23z + 52 = 0\). [3]
  3. Express \(3z^3 - 8z^2 + 23z + 52\) as the product of a linear factor and a quadratic factor with real coefficients. [4]
Pre-U Pre-U 9795/1 2015 June Q11
13 marks Challenging +1.3
  1. The cubic equation \(x^3 + 2x^2 + 3x - 4 = 0\) has roots \(p\), \(q\) and \(r\). A second cubic equation has roots \(qr\), \(rp\) and \(pq\). Show how the substitution \(y = \frac{4}{x}\) can be used to determine this second equation. Hence, or otherwise, find this equation in the form \(y^3 + ay^2 + by + c = 0\). [6]
  2. The cubic equation \(x^3 - 4x^2 + 5x - 4 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\). You are given that \(\alpha\) is real and positive, and that \(\beta\) and \(\gamma\) are complex.
    1. Describe the relationship between \(\beta\) and \(\gamma\). [1]
    2. Explain why \(|\beta| = \frac{2}{\sqrt{\alpha}}\). [2]
    3. Verify that \(\alpha = 2.70\) correct to 3 significant figures, and deduce that \(\text{Re}(\beta) = 0.65\) correct to 2 significant figures. [4]