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Question:

Let z be a complex number such that |z| + z = 3 + i (where i = √-1). Then |z| is equal to:

√343

53

54

√414

Solution:

The correct option is D
53
|z| + z = 3 + i
z = 3 - |z| + i
Let 3 - |z| = a ⇒ |z| = (3 - a) ⇒ z = a + i ⇒ |z| = √a² + 1 ⇒ 9 + a² - 6a = a² + 1 ⇒ -6a = -8 ⇒ a = 8/6 = 4/3 ⇒ |z| = 3 - 4/3 = 5/3