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Question

Which of the following are true?

A
¯¯¯¯¯¯¯(¯¯¯z)=z
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B
z¯¯¯zR
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C
z+¯¯¯z=0z is purely real
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D
z¯¯¯z=2Im(z)
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Solution

The correct option is B z¯¯¯zR
Let z=a+bi we will check all the options one bye one

(a)¯¯¯z=abi
¯¯¯¯¯¯¯(¯¯¯z)=¯¯¯¯¯¯¯¯¯¯¯¯¯¯abi
¯¯¯¯¯¯¯(¯¯¯z)=a+bi

(b)z¯¯¯z=(a+bi)(abi)
z¯¯¯z=a2+b2
z¯¯¯zR

(c)z¯¯¯z=(a+bi)(abi)
z¯¯¯z=2bi
z¯¯¯z=2Im(z)i

(c)z+¯¯¯z=0
(a+bi)+(abi)=0
a=0 purely imaginary

Hence, option a,b are correct options.

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