Question

If $z_1,z_2--,z_n$ are complex numbers such that $|z_i|<land{\lambda }_i>0$ for $i=1,2,---n$ and ${\lambda }_1+{\lambda }_2+--+{\lambda }_n=1$ then $|{\lambda }_1{z}_1+{\lambda }_2{z}_2+--+{\lambda }_n{z}_1|?$

• A. $=1$
• B. $<1$
• C. $>1$
• D. $=n$

Answer 1

Answer: (B)

$<1$

Given:

${\lambda }_i>0$ and ${\lambda }_1+{\lambda }_2+...+{\lambda }_n=1$

$\therefore 0<{\lambda }_i<1$

Also, $|z_i|<1$

$\therefore {\lambda }_1|z_1|+{\lambda }_2|z_2|+.....+{\lambda }_n|z_n|<1$ ......( 1 )

$\therefore |{\lambda }_1{z}_1+{\lambda }_2{z}_2+.....+{\lambda }_n{z}_n|$

$\le |{\lambda }_1{z}_1|+|{\lambda }_2{z}_2|+.....+|{\lambda }_n{z}_n|$

$\le {\lambda }_1|z_1|+{\lambda }_2|z_2|+.....+{\lambda }_n|z_n|$

$<1$