N2018 P2 Q2

N2018 P2 Q2

Junior College 2
12 marks
Free
  1. One of the roots of the equation \(4x^4-20x^3+sx^2-56x+t=0\), where \(s\) and \(t\) are real, is \(2-3\mathrm{i}\). Find the other roots of the equation and the values of \(s\) and \(t\).[5]
  2. The complex number \(w\) is such that \(w^3=27\).
    1. Given that one possible value of \(w\) is \(3\), use a non-calculator method to find the other possible values of \(w\). Give your answers in the form \(a+\mathrm{i}b\), where \(a\) and \(b\) are exact values.[3]
    2. Write these values of \(w\) in modulus-argument form and represent them on an Argand diagram.[2]
    3. Find the sum and the product of all the possible values of \(w\), simplifying your answers.[2]

Default solution

(a) Real coefficients give the conjugate root \(2+3\mathrm{i}\); their factor is \(x^2-4x+13\).

\(4x^4-20x^3+sx^2-56x+t=(x^2-4x+13)(4x^2-4x+1)\).

\(4x^2-4x+1=(2x-1)^2\Rightarrow x=\frac12\) twice; \(s=69,\ t=13\).

(b)(i) \(w^3-27=(w-3)(w^2+3w+9)=0\).

\(w=\frac{-3\pm\sqrt{-27}}2=-\frac32\pm\frac{3\sqrt3}{2}\mathrm{i}\).

(b)(ii) \(w=3e^{0\mathrm{i}},\ 3e^{\frac{2\pi}{3}\mathrm{i}},\ 3e^{-\frac{2\pi}{3}\mathrm{i}}\). Plot \((3,0)\) and \(\left(-\frac32,\pm\frac{3\sqrt3}{2}\right)\) on the Argand plane.

(b)(iii) The root sum is \(0\), and the product is \(27\).

Answer:(a) \(2+3\mathrm{i},\ \frac12\text{ (double)};\ s=69,\ t=13\); (b)(i) \(-\frac32\pm\frac{3\sqrt3}{2}\mathrm{i}\); (ii) modulus \(3\), arguments \(0,\pm2\pi/3\); (iii) sum \(0\), product \(27\)

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