JEE MAIN - Mathematics (2020 - 2nd September Evening Slot - No. 19)
The imaginary part of
$${\left( {3 + 2\sqrt { - 54} } \right)^{{1 \over 2}}} - {\left( {3 - 2\sqrt { - 54} } \right)^{{1 \over 2}}}$$ can be :
$${\left( {3 + 2\sqrt { - 54} } \right)^{{1 \over 2}}} - {\left( {3 - 2\sqrt { - 54} } \right)^{{1 \over 2}}}$$ can be :
-2$$\sqrt 6 $$
6
$$\sqrt 6 $$
-$$\sqrt 6 $$
Explanation
$$3 + 2\sqrt { - 54} $$
$$ = 9 - 6 + 2\sqrt { - 54} $$
$$ = 9 + {\left( {\sqrt 6 i} \right)^2} + 2.3.\sqrt 6 i$$
$$ = {3^2} + {\left( {\sqrt 6 i} \right)^2} + 2.3.\sqrt 6 i$$
$$ = {\left( {3 + \sqrt 6 i} \right)^2}$$
Similarly, $$\left( {3 - 2\sqrt { - 54} } \right) = {\left( {3 - \sqrt 6 i} \right)^2}$$
$$ \therefore {\left( {3 + 2\sqrt { - 54} } \right)^{{1 \over 2}}} - {\left( {3 - 2\sqrt { - 54} } \right)^{{1 \over 2}}}$$
$$ = \pm \left( {3 + \sqrt 6 i} \right) - \left[ { \pm \left( {3 - \sqrt 6 i} \right)} \right]$$
$$ = 6, - 6,2\sqrt 6 i, - 2\sqrt 6 i$$
$$ \therefore $$ Possible imaginary parts are $$2\sqrt 6 i, - 2\sqrt 6 i$$
$$ = 9 - 6 + 2\sqrt { - 54} $$
$$ = 9 + {\left( {\sqrt 6 i} \right)^2} + 2.3.\sqrt 6 i$$
$$ = {3^2} + {\left( {\sqrt 6 i} \right)^2} + 2.3.\sqrt 6 i$$
$$ = {\left( {3 + \sqrt 6 i} \right)^2}$$
Similarly, $$\left( {3 - 2\sqrt { - 54} } \right) = {\left( {3 - \sqrt 6 i} \right)^2}$$
$$ \therefore {\left( {3 + 2\sqrt { - 54} } \right)^{{1 \over 2}}} - {\left( {3 - 2\sqrt { - 54} } \right)^{{1 \over 2}}}$$
$$ = \pm \left( {3 + \sqrt 6 i} \right) - \left[ { \pm \left( {3 - \sqrt 6 i} \right)} \right]$$
$$ = 6, - 6,2\sqrt 6 i, - 2\sqrt 6 i$$
$$ \therefore $$ Possible imaginary parts are $$2\sqrt 6 i, - 2\sqrt 6 i$$
Comments (0)
