Question

If the point $z=(1+i)(1+2i)(1+3i).......(1+10i)$ lies on a circle with centre at origin and radius $r$, then $r^2=$

• A. $10!$
• B. $2\times 3\times 4\times ......\times 10$
• C. $2\times 5\times 10\times ........101$
• D. $11!$

Answer 1

The correct option is C $2\times 5\times 10\times ........101$

Fact: $|z_1{z}_2{z}_3..........|=|z_1||z_2||z_3|.......$

We have,

$z=(1+i)(1+2i)(1+3i).......(1+10i)$

$\Rightarrow |z|=|1+i||1+2i||1+3i|.......||1+10i|$

$\Rightarrow r^2=|z{|}^2=|1+i{|}^2|1+2i{|}^2|1+3i{|}^2.......||1+10i{|}^2$

$=2\times 5\times 10\times ..............\times 101$

Note: $|x+iy{|}^2=x^2+y^2$