JEE Advanced2018MathematicsComplex NumberActual
For a non-zero complex number z , let arg ( z ) denote the principal argument with - π < a r g z ≤ π then, which of the following statement(s) is (are) FALSE?
Options
- AArg - 1 - i = π 4 , w h e r e i = - 1
- BThe function f : R → - π , π defined by f t = a r g - 1 + i t f o r a l l t ∈ R , is continuous at all points
- CFor any two non-zero complex numbers z 1 a n d z 2 , a r g z 1 z 2 - a r g z 1 + a r g z 2 is an integer multi
- DFor any three given distinct complex numbers z 1 , z 2 a n d z 3 , the locus of the point z satisfying the con
Correct answer
A. Arg - 1 - i = π 4 , w h e r e i = - 1
Step-by-step solution
A Arg - 1 - i = - 3 π 4 B Arg - 1 + i t = π - tan - 1 t , t ≥ 0 - π + tan - 1 t , t < 0 Discontinuous at t = 0 ( C ) a r g z 1 z 2 - a r g z 1 + a r g z 2 = a r g z 1 - a r g z 2 - a r g z 1 + a r g z 2 =0 D a r g z - z 1 z 2 - z 3 z - z 3 z 2 - z 1 = π ⇒ z - z 1 z 2 - z 3 z - z 3 z 2 - z 1 is real ⇒ z , z 1 , z 2 , z 3 are concyclic