Question

Which of the following statements is/are false

• A. $\left| \frac{z}{|z|} - 1 \right|$£ |arg z|
• B. |z – 1| £ ||z| + 1| + |z| |arg z|
• C. |z| £ |Re(z)| + |Im(z)| £$\sqrt{2}$|z|
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

Sol. z = r e^iq

For (1) |e^iq – 1| £ | q |

AP = £ l( $\overline{AP}$)

For (2) |z – 1| £ | | z | – 1| | + | z | | arg z |

|re^iq – 1| £ |r – 1| + r |q|

Ž B P £ AB + arc PA true

For (3)

$\sqrt{x^{2} + y^{2}}$ £ | x | + | y | £ $\sqrt { 2 }$ $\sqrt{x^{2} + y^{2}}$

Ž true

Hence (4)QW