JEE MAIN - Mathematics (2020 - 5th September Evening Slot - No. 4)
The value of $${\left( {{{ - 1 + i\sqrt 3 } \over {1 - i}}} \right)^{30}}$$ is :
–215i
–215
215i
65
Explanation
$${\left( {{{ - 1 + i\sqrt 3 } \over {1 - i}}} \right)^{30}}$$
= $${\left( {{{2\omega } \over {1 - i}}} \right)^{30}}$$
= $${{{2^{30}}.{\omega ^{30}}} \over {{{\left( {{{\left( {1 - i} \right)}^2}} \right)}^{15}}}}$$
= $${{{2^{30}}.1} \over {{{\left( {1 + {i^{^2}} - 2i} \right)}^{15}}}}$$
= $${{{2^{30}}.1} \over { - {2^{15}}.{i^{15}}}}$$
= –215i
= $${\left( {{{2\omega } \over {1 - i}}} \right)^{30}}$$
= $${{{2^{30}}.{\omega ^{30}}} \over {{{\left( {{{\left( {1 - i} \right)}^2}} \right)}^{15}}}}$$
= $${{{2^{30}}.1} \over {{{\left( {1 + {i^{^2}} - 2i} \right)}^{15}}}}$$
= $${{{2^{30}}.1} \over { - {2^{15}}.{i^{15}}}}$$
= –215i
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