For a non-zero complex number z, let arg(z) denote the principal argument with -π<argz≤π then, which of the following statement(s) is (are) FALSE?

(B) The function f:R→-π,π defined by ft=arg-1+it for all t∈R , is continuous at all points of R, where i=-1

(C) For any two non-zero complex numbers z1 and z2 , arg z1z2-argz1+arg z2 is an integer multiple of 2π

(D) For any three given distinct complex numbers z1, z2 and z3 , the locus of the point z satisfying the condition argz-z1z2-z3z-z3z2-z1=π lies on a straight line

JEE Advanced · Mathematics · 3. Complex Numbers

For a non-zero complex number $z$ , let arg $(z)$ denote the principal argument with $- π < a r g (z) \leq π$ then, which of the following statement(s) is (are) FALSE?

A Arg $(- 1 - i) = \frac{π}{4}, w h e r e i = \sqrt{- 1}$
B The function $f : R \to (- π, π]$ defined by $f (t) = a r g (- 1 + i t) f o r a l l t \in R$ , is continuous at all points of $R, w h e r e i = \sqrt{- 1}$
C For any two non-zero complex numbers $z_{1} a n d z_{2}$ , $a r g (\frac{z_{1}}{z_{2}}) - a r g (z_{1}) + a r g (z_{2})$ is an integer multiple of $2 π$
D For any three given distinct complex numbers $z_{1}, z_{2} a n d z_{3}$ , the locus of the point $z$ satisfying the condition $a r g (\frac{(z - z_{1}) (z_{2} - z_{3})}{(z - z_{3}) (z_{2} - z_{1})}) = π$ lies on a straight line

Verified Solution

Answer & Solution

Correct Answer

(A) Arg $(- 1 - i) = \frac{π}{4}, w h e r e i = \sqrt{- 1}$

Step-by-step Solution

Detailed explanation

(A)

Arg (- 1 - i) = - \frac{3 π}{4}

(B)

Arg

(- 1 + i t) = \{\begin{matrix} π - \tan^{- 1} (t), & t \geq 0 \\ - π + \tan^{- 1} (t), & t < 0 \end{matrix}

Discontinuous at

t = 0

(C)

a r g (\frac{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 (\frac{(z - z_{1}) (z_{2} - z_{3})}{(z - z_{3}) (z_{2} - z_{1})}) = π

\Rightarrow \frac{(z - z_{1}) (z_{2} - z_{3})}{(z - z_{3}) (z_{2} - z_{1})}

is real

\Rightarrow z, z_{1}, z_{2}, z_{3}

are concyclic

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

	List- I		List- II
$(I)$	$2 h + k$	$(P)$	$6$
$(I I)$	$\frac{L e n g t h o f Z W}{L e n g t h o f X Y}$	$(Q)$	$\sqrt{6}$
$(I I I)$	$\frac{A r e a o f t r i a n g l e M Z N}{A r e a o f t r i a n g l e Z M W}$	$(R)$	$\frac{5}{4}$
$(I V)$	$α$	$(S)$	$\frac{21}{5}$
		$(T)$	$2 \sqrt{6}$
		$(U)$	$\frac{10}{3}$

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