Standard +0.3 This is a standard complex square root problem requiring the method of equating real and imaginary parts (setting √u = a + bi, squaring, and solving simultaneous equations). While it involves some algebraic manipulation with surds, it's a routine textbook exercise with a well-established procedure, making it slightly easier than average.
5 The complex number \(u\) is given by \(u = 10 - 4 \sqrt { 6 } \mathrm { i }\).
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.
Square \(a + ib\), use \(i^2 = -1\) and equate real and imaginary parts to 10 and \(-4\sqrt{6}\) respectively
M1
Obtain \(a^2 - b^2 = 10\) and \(2ab = -4\sqrt{6}\)
A1
Allow \(2abi = -4\sqrt{6}i\)
Eliminate one unknown and find an equation in the other
M1
Must be sensible algebra e.g. use of \(\sqrt{a^2 - b^2} = a - b\) scores M0
Obtain \(a^4 - 10a^2 - 24[=0]\), or \(b^4 + 10b^2 - 24[=0]\), or 3-term equivalent
A1
Or equivalent horizontal equation from correct work
Obtain final answers \(\pm(2\sqrt{3} - \sqrt{2}i)\), or exact equivalents
A1
e.g. \(\pm(\sqrt{12} - \sqrt{2}i)\) from correct work
Alternative method: Use correct method to find modulus and argument of \(\sqrt{u}\)
M1
Obtain modulus \(\sqrt{14}\)
A1
Obtain argument \(\tan^{-1}\frac{-1}{\sqrt{6}}\) using an exact method
A1
e.g. by using half angle formula which gives \(2\sqrt{6}t^2 - 10t - 2\sqrt{6} = 0\)
Convert to the required form
M1
\(\pm\sqrt{14}\!\left(\frac{\sqrt{6}}{\sqrt{7}} - \frac{1}{\sqrt{7}}i\right)\). This mark is available if working in decimals
Obtain answers \(\pm(2\sqrt{3} - \sqrt{2}i)\), or exact equivalents
A1
e.g. \(\pm(\sqrt{12} - \sqrt{2}i)\)
## Question 5:
| Answer | Mark | Guidance |
|--------|------|----------|
| Square $a + ib$, use $i^2 = -1$ and equate real and imaginary parts to 10 and $-4\sqrt{6}$ respectively | M1 | |
| Obtain $a^2 - b^2 = 10$ and $2ab = -4\sqrt{6}$ | A1 | Allow $2abi = -4\sqrt{6}i$ |
| Eliminate one unknown and find an equation in the other | M1 | Must be sensible algebra e.g. use of $\sqrt{a^2 - b^2} = a - b$ scores M0 |
| Obtain $a^4 - 10a^2 - 24[=0]$, or $b^4 + 10b^2 - 24[=0]$, or 3-term equivalent | A1 | Or equivalent horizontal equation from correct work |
| Obtain final answers $\pm(2\sqrt{3} - \sqrt{2}i)$, or exact equivalents | A1 | e.g. $\pm(\sqrt{12} - \sqrt{2}i)$ from correct work |
| **Alternative method:** Use correct method to find modulus and argument of $\sqrt{u}$ | M1 | |
| Obtain modulus $\sqrt{14}$ | A1 | |
| Obtain argument $\tan^{-1}\frac{-1}{\sqrt{6}}$ using an exact method | A1 | e.g. by using half angle formula which gives $2\sqrt{6}t^2 - 10t - 2\sqrt{6} = 0$ |
| Convert to the required form | M1 | $\pm\sqrt{14}\!\left(\frac{\sqrt{6}}{\sqrt{7}} - \frac{1}{\sqrt{7}}i\right)$. This mark is available if working in decimals |
| Obtain answers $\pm(2\sqrt{3} - \sqrt{2}i)$, or exact equivalents | A1 | e.g. $\pm(\sqrt{12} - \sqrt{2}i)$ |
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5 The complex number $u$ is given by $u = 10 - 4 \sqrt { 6 } \mathrm { i }$.\\
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.\\
\hfill \mbox{\textit{CAIE P3 2021 Q5 [5]}}