For a non-zero complex number z, let arg(z) denote the principal argument with - π

Question

For a non-zero complex number z, let arg(z) denote the principal argument with - π < arg(z) ≤ π. Then, which of the following statement(s) is (are) FALSE? (A) arg(-1 - i) = (π/4), where i = √-1 (B) The function f: mathbb R → (- π , π ], defined by f (t) = arg (-1 + it ) for all t ∈ mathbb R , is continuous at all points of mathbb R, where i = √-1 (C) For any two non-zero complex numbers z1 and z2, arg ((z1/z2)) - arg (z1) + arg (z2) is an integer multiple of 2 π  (D) For any three given distinct complex numbers z1, z2 and z3, the locus of the point z satisfying the condition arg ( ((z - z1)(z2 - z2)/(z - z3)(z2 - z1))) = π , lies on a straight line

Tardigrade Admin · Accepted Answer

a) arg (-1-i)=-(3 π/4) b) f(t)= arg (-1+i t)= begincasesπ- tan -1(t), t ≥ 0 -π+ tan -1 t < 0 endcases Discontinuous at t = 0 arg ((z1/z2))- arg (z1)+ arg (z2) = arg z1- arg (z2)+2 n π- arg (z1)+ arg (z2)=2 n π d) arg (((z-z1)(z2-z3)/(z-z3)(z2-z1)))=π ⇒ ((z-z1)(z2-z3)/(z-z3)(z2-z1)) is real ⇒ z, z1, z2, z3 are concyclic

Q. 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?

$a r g (- 1 - i)$ = $\frac{π}{4}$ , where $i = - 1$

The function $f : R \to (- π, π]$ , defined by $f (t)$ = arg $(- 1 + i t)$ for all $t \in R$ , is continuous at all points of $R$ , where $i = - 1$

For any two non-zero complex numbers $z_{1}$ and $z_{2}, a r g (\frac{z _{1}}{z _{2}}) - a r g (z_{1}) + a r g (z_{2})$ is an integer multiple of $2 π$

For any three given distinct complex numbers $z_{1}, z_{2}$ and $z_{3}$ , the locus of the point $z$ satisfying the condition $a r g (\frac{( z - z _{1} ) ( z _{2} - z _{2} )}{( z - z _{3} ) ( z _{2} - z _{1} )}) = π$ , lies on a straight line

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

arg(−1−i) = 4π​, where i=−1​

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

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

For any three given distinct complex numbers z1​,z2​ and z3​, the locus of the point z satisfying the condition arg((z−z3​)(z2​−z1​)(z−z1​)(z2​−z2​)​)=π, lies on a straight line

Q. 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?

$a r g (- 1 - i)$ = $\frac{π}{4}$ , where $i = - 1$

The function $f : R \to (- π, π]$ , defined by $f (t)$ = arg $(- 1 + i t)$ for all $t \in R$ , is continuous at all points of $R$ , where $i = - 1$

For any two non-zero complex numbers $z_{1}$ and $z_{2}, a r g (\frac{z _{1}}{z _{2}}) - a r g (z_{1}) + a r g (z_{2})$ is an integer multiple of $2 π$

For any three given distinct complex numbers $z_{1}, z_{2}$ and $z_{3}$ , the locus of the point $z$ satisfying the condition $a r g (\frac{( z - z _{1} ) ( z _{2} - z _{2} )}{( z - z _{3} ) ( z _{2} - z _{1} )}) = π$ , lies on a straight line