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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Complex Number

NDA 2023 · Mathematics · Complex Number

Which one of the following is a square root of - -1 ?

Which one of the following is a square root of −−1-\sqrt{-1} ?

Options

  1. A1+i1+i
  2. B1−i2\frac{1-i}{\sqrt{2}}
  3. C1+i2\frac{1+i}{\sqrt{2}}
  4. D12i\frac{1}{\sqrt{2}} i

Correct answer

B. 1−i2\frac{1-i}{\sqrt{2}}

Step-by-step solution

 since, (−−1)12=−i=x+ iy (let) −i=x2−y2+2xyi⇒2xy=−1,x2−y2=0\begin{aligned} & \text{ since, }(-\sqrt{-1})^{\frac{1}{2}}=\sqrt{-i}=x+\text{ iy (let) } \\ & -i=x^2-y^2+2 x y i \Rightarrow 2 x y=-1, x^2-y^2=0 \end{aligned} after solving these equation, we get x=−12,y=±12⇒−i=±(12−i2)±1−i2=1−i2 or  −1+i2\begin{aligned} & x=\frac{-1}{\sqrt{2}}, y=\frac{ \pm 1}{\sqrt{2}} \\ \Rightarrow & \sqrt{-\mathrm{i}}= \pm\left(\frac{1}{\sqrt{2}}-\frac{\mathrm{i}}{\sqrt{2}}\right) \pm \frac{1-\mathrm{i}}{\sqrt{2}}=\frac{1-\mathrm{i}}{\sqrt{2}} \text{\text{ or } } \frac{-1+\mathrm{i}}{\sqrt{2}} \end{aligned}
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