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QJEEM20Ma8771: JEE Main 2020 Mathematics - Class 11 JEEM20Ma8771 Solution: The imaginary part of can be :

The imaginary part of
(3+2−54)12−(3−2−54)12{\left( {3 + 2\sqrt { - 54} } \right)^{{1 \over 2}}} - {\left( {3 - 2\sqrt { - 54} } \right)^{{1 \over 2}}} can be :

Multiple Choice Options

  • A -2 6\sqrt 6
  • B 6
  • C 6\sqrt 6
  • D - 6\sqrt 6

Step-by-Step Verified Solution

3+2−543 + 2\sqrt { - 54}

=9−6+2−54= 9 - 6 + 2\sqrt { - 54}

=9+(6i)2+2.3.6i= 9 + {\left( {\sqrt 6 i} \right)^2} + 2.3.\sqrt 6 i

=32+(6i)2+2.3.6i= {3^2} + {\left( {\sqrt 6 i} \right)^2} + 2.3.\sqrt 6 i

=(3+6i)2= {\left( {3 + \sqrt 6 i} \right)^2}

Similarly, (3−2−54)=(3−6i)2\left( {3 - 2\sqrt { - 54} } \right) = {\left( {3 - \sqrt 6 i} \right)^2}

∴(3+2−54)12−(3−2−54)12\therefore {\left( {3 + 2\sqrt { - 54} } \right)^{{1 \over 2}}} - {\left( {3 - 2\sqrt { - 54} } \right)^{{1 \over 2}}}

=±(3+6i)−[±(3−6i)]= \pm \left( {3 + \sqrt 6 i} \right) - \left[ { \pm \left( {3 - \sqrt 6 i} \right)} \right]

=6,−6,26i,−26i= 6, - 6,2\sqrt 6 i, - 2\sqrt 6 i

∴\therefore Possible imaginary parts are 26i,−26i2\sqrt 6 i, - 2\sqrt 6 i

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