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Square root of complex numbers - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Square root of complex numbers, solution of complex equation is considered one the most difficult concept.

  • 4 Questions around this concept.

Solve by difficulty

If (x+iy)^{2}=7+24i, then a value of (7+\sqrt{-576})^{1/2}-(7-\sqrt{-576})^{1/2} is:

Concepts Covered - 0

Square root of complex numbers, solution of complex equation

Let z = x + iy, is the complex number whose square root we have to find

Since the square root of complex number must be a complex number, 

\\\mathrm{so\; let\; {z}^{1/2}=a+ib}

Now squaring both sides

\\\mathrm{z=x+iy=(a+ib)^2=a^2-b^2+2iab}

Now comparing real and imaginary part and finding the value of a and b in terms of x and y 

\\\mathrm{a^2-b^2=x \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; ...(i)} \\\mathrm{2ab = y \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; ...(ii)} \\\mathrm{a^2+b^2=\sqrt{(a^2-b^2)^2+4a^2b^2}=\sqrt{x^2+y^2}=|z|\;\;\; ...(iii)} \\\mathrm{Solving \;(i)\; and\; (iii)\; we \;get} \\\mathrm{2a^2=x+|z| \Rightarrow a=\pm \sqrt{\frac{x+|z|}{2}}} \\\mathrm{Similarly\; we \;find\; b=\pm \sqrt{\frac{|z|-x}{2}}} \\\mathrm{So\;\; \sqrt{z}=\pm \left ( \sqrt{\frac{\left | z \right |+Re(z)}{2}}+i\sqrt{\frac{\left | z \right |-Re(z)}{2}} \right ) }

 if Im(z) > 0 otherwise there will be -ve sign between real and imaginary part of the square root of z.

Note:

1. Students do not need to remember this formula. But, they are required to know the procedure to find the square root of a complex number.

2. If a + ib is one of the square root of z, then other square root must be -(a+ib) 

 

Complex Equations

To find the solution of the complex equation we substitute z = x + iy and find the value of x and y by comparing real and imaginary parts of the equation obtained. z = x+iy is the required solution.

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