isi-entrance 2011 Q4

isi-entrance · India · solved Complex Numbers Argand & Loci Locus Identification from Modulus/Argument Equation

Suppose $z$ is a complex number with $| z | < 1$. Let $w = ( 1 + z ) / ( 1 - z )$. Which of the following is always true? [$\operatorname { Re } ( w )$ is the real part of $w$ and $\operatorname { Im } ( w )$ is the imaginary part of $w$]
(a) $\operatorname { Re } ( w ) > 0$
(b) $\operatorname { Im } ( w ) \geq 0$
(c) $| w | \leq 1$
(d) $| w | \geq 1$

(A) Let $z = re^{i\theta}$. Then $\operatorname{Re}(w) = \left( 1 - r ^ { 2 } \right) / \{ 2 ( 1 - \cos \theta ) \}$. Since $r = |z| < 1$ and $\cos\theta < 1$, we get $\operatorname{Re}(w) > 0$.

Suppose $z$ is a complex number with $| z | < 1$. Let $w = ( 1 + z ) / ( 1 - z )$. Which of the following is always true? [$\operatorname { Re } ( w )$ is the real part of $w$ and $\operatorname { Im } ( w )$ is the imaginary part of $w$]\\
(a) $\operatorname { Re } ( w ) > 0$\\
(b) $\operatorname { Im } ( w ) \geq 0$\\
(c) $| w | \leq 1$\\
(d) $| w | \geq 1$

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