For a non-zero complex number z, let (z) denote the principal argument with - < (z) . Then, which of the following statement(s) is (are) **FALSE?**

JEE Advanced 2018 Mathematics question

For a non-zero complex number , let denote the principal argument with . Then, which of the following statement(s) is (are) **FALSE?**

(A), where Correct
(B)The function , defined by for all , is continuous at all points of , where Correct
(C)For any two non-zero complex numbers and , is an integer multiple of
(D)For any three given distinct complex numbers , and , the locus of the point satisfying the condition , lies on a straight lineCorrect

Answer: (A),(B),(D)

Solution

(A) lies in the third quadrant: , so the statement is false. (B) For , as , while for it tends to ; jumps at , so the statement is false. (C) differs from by a multiple of

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Question ID VK-026736