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Question

Let z≠−i be any complex number such that z−iz+i is a purely imaginary number. Then z+1z is :

A
0
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B
Any non-zero real number other than 1.
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C
Any non-zero real number.
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D
A purely imaginary number.
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Solution

The correct option is B Any non-zero real number.Let 'z' be x+iyGiven (z−iz+i) is a purely imaginary number.x+iy−ix+iy+i=x+i(y−1)x+i(y+1)=(x+i(y−1))(x−i(y+1))x2+(y+1)2=x2+(y2−1)−ix(y+1)+ix(y−1)x2+(y+1)2 .=x2+(y2−1)−2ixx2+(y+1)2x2+(y2−1)=0x2+y2=1 .....(1)Now, z+1z=x+iy+1x+iy⇒x+iy+x−iyx2+y2=2x (From 1)Any non zero real number.

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