The complex number $$\(u\)$$ is given by $$\(u=10-4 \sqrt{6}\)$$ i. Find the two square roots of $$\(u\)$$, giving your answers in the form $$\(a+\mathrm{i} b\)$$, where $$\(a\)$$ and $$\(b\)$$ are real and exact. ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................
Exam No：9709_s21_qp_32 Year：2021 Question No：5
Answer:
Square \(a+\mathrm{i} b\), use \(\mathrm{i}^{2}=-1\) and equate real and imaginary parts to 10 and \(-4 \sqrt{6}\) respectively
Obtain \(a^{2}-b^{2}=10\) and \(2 a b=-4 \sqrt{6}\)
Eliminate one unknown and find an equation in the other
Obtain \(a^{4}-10 a^{2}-24[=0]\), or \(b^{4}+10 b^{2}-24[=0]\), or 3-term equivalent
Obtain final answers \(\pm(2 \sqrt{3}-\sqrt{2} i)\), or exact equivalents
Alternative method for Question 5
Use the correct method to find the modulus and argument of \(\sqrt{u}\)
Obtain modulus \(\sqrt{14}\)
Obtain argument \(\tan ^{-1} \frac{-1}{\sqrt{6}}\) using an exact method
Convert to the required form
Obtain answers \(\pm(2 \sqrt{3}-\sqrt{2} \mathrm{i})\), or exact equivalents
Obtain \(a^{2}-b^{2}=10\) and \(2 a b=-4 \sqrt{6}\)
Eliminate one unknown and find an equation in the other
Obtain \(a^{4}-10 a^{2}-24[=0]\), or \(b^{4}+10 b^{2}-24[=0]\), or 3-term equivalent
Obtain final answers \(\pm(2 \sqrt{3}-\sqrt{2} i)\), or exact equivalents
Alternative method for Question 5
Use the correct method to find the modulus and argument of \(\sqrt{u}\)
Obtain modulus \(\sqrt{14}\)
Obtain argument \(\tan ^{-1} \frac{-1}{\sqrt{6}}\) using an exact method
Convert to the required form
Obtain answers \(\pm(2 \sqrt{3}-\sqrt{2} \mathrm{i})\), or exact equivalents
Knowledge points:
3.9.6 find the two square roots of a complex number
Solution:
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