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Question : 9 of 160
Marks:
+1 ,
-0
Solution:
Let
z = 1 + √ 5 + i √ 10 − 2 √ 5 S o , | z | ‌ = √ ( 1 + √ 5 ) 2 + ( √ 10 − 2 √ 5 ) 2 ‌ = √ 1 + 5 + 2 √ 5 + 10 − 2 √ 5 ‌ = √ 16 + 2 √ 5 − 2 √ 5 = √ 16 = 4 ‌ And ‌ θ ‌ = arg ( z ) = tan − 1 ( ‌ ) ‌ = tan − 1 ( tan ( ‌ ) ) = ‌ So,
z = | z | ⋅ ( cos ‌ θ + i s i n ‌ θ ) ‌ = 4 [ cos ( ‌ ) + i s i n ‌ ( ‌ ) ] ∴ z 5 ‌ = 4 5 [ cos ( ‌ ) + i s i n ‌ ( ‌ ) ] 5 ‌ = 1024 [ cos ( 5 ⋅ ‌ ) + i s i n ‌ ( 5 ⋅ ‌ ) ] ‌ = 1024 [ cos ‌ π + i s i n ‌ π ] ‌ = 1024 [ − 1 + 0 ] = − 1024 © examsiri.com
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