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Question : 13 of 160
Marks:
+1 ,
-0
Solution:
( √ 2 + i √ 2 ) 2030 = ( + i ) 2022 = ( + ) 2020 = ( cos ‌ 45 ° ‌ cos ‌ 30 ° − sin ‌ 45 ° ‌ sin ‌ 30 ° ) + i ( sin ‌ 45 ° ‌ cos ‌ 30 ° + cos ‌ 45 ° ‌ sin ‌ 30 ° ) ) 2200 = ( cos ‌ 75 ° + i ‌ sin ‌ 75 ° ) 2020 = ( cos ‌ + i ‌ sin ‌ ) 2020 = ( cos ‌ × 2020 + i ‌ sin ‌ × 2020 ) [De-moivre]
= ( cos ‌ + i ‌ sin ‌ ) = ( cos ( 2525 × 60 ) ° + i ‌ sin ( 2525 × 60 ) ° = ( cos ( n π − 60 ) ° + i ‌ sin ( n π − 60 ) ° ) , where
n = 842 = ( ( − 1 ) n ‌ cos ‌ 60 ° + i ( − sin ‌ 60 ° ) ) = ( cos ‌ 60 ° − i ‌ sin ‌ 60 ° ) [ ∵ n = even
] = − i
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