Complex Numbers 1 Question 1

1. Let zC with Im(z)=10 and it satisfies 2zn2z+n=2i1 for some natural number n, then (2019 Main, 12 April II)

(a) n=20 and Re(z)=10

(b) n=40 and Re(z)=10

(c) n=40 and Re(z)=10

(d) n=20 and Re(z)=10

Show Answer

Answer:

Correct Answer: 1. (c)

Solution:

  1. Let z=x+10i, as Im(z)=10 (given).

Since z satisfies,

2zn2z+n=2i1,nN,(2x+20in)=(2i1)(2x+20i+n)(2xn)+20i=(2xn40)+(4x+2n20)i

On comparing real and imaginary parts, we get

2xn=2xn40 and 20=4x+2n204x=40 and 4x+2n=40x=10 and 40+2n=40n=40 So, n=40 and x=Re(z)=10



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