For a non-zero complex number z , let arg ( z ) denote the principal argument with arg ( z ) . Then, which of the following statement(s) is (are) FALSE? [2018]

Question

For a non-zero complex number $z$ , let $\arg (z)$ denote the principal argument with $- π < \arg (z) \leq π$ . Then, which of the following statement(s) is (are) FALSE? [2018]

Smart Achievers · Accepted Answer

(1, 2, 4)

$(a) \arg (- 1 - i) = - \frac{3 π}{4}$

$∴$ $(a) is false$

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

$\lim_{t \to 0^{-}} f (t) = - π and \lim_{t \to 0^{+}} f (t) = π$

$LHL \neq RHL \Rightarrow f is discontinuous at t = 0$

$∴$ $(b) is false$

$(c) \arg (\frac{z_{1}}{z_{2}}) - \arg z_{1} + \arg z_{2}$

$= 2 n π + \arg z_{1} - \arg z_{2} - \arg z_{1} + \arg z_{2}$

$= 2 n π, multiple of 2 π$

$∴$ $(c) is true$

$(d)$ $\arg (\frac{(z - z_{1}) (z_{2} - z_{3})}{(z - z_{3}) (z_{2} - z_{1})}) = π$

$\Rightarrow \frac{(z - z_{1}) (z_{2} - z_{3})}{(z - z_{3}) (z_{2} - z_{1})} = k, k \in ℝ$

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

$\Rightarrow z, z_{1}, z_{2}, z_{3} are concyclic, i.e. z lies on a circle.$

$∴$ $(d) is false$

Solution

Q.
For a non-zero complex number $z$ , let $\arg (z)$ denote the principal argument with $- π < \arg (z) \leq π$ . Then, which of the following statement(s) is (are) FALSE [2018]

1 $\arg (- 1 - i) = \frac{π}{4}$ , where $i = \sqrt{- 1}$

2 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 $i = \sqrt{- 1}$

3 For any two non-zero complex numbers $z_{1}$ and $z_{2}$ , $\arg (\frac{z_{1}}{z_{2}}) - \arg (z_{1}) + \arg (z_{2})$ is an integer multiple of $2 π$

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

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Solution

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 [2018]

1 arg(-1-i)=π4, where i=-1

2 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

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