Q49 / 160 · TS EAMCET 2016 · Mathematics

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

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

Solution

Any circle with centre at origin and radius $r$ has the equation $$ \begin{array}{rlrl} & & |z| & =r \\ \Rightarrow & & |1+i||1+2 i||1+3 i| \ldots|1+10 i| & =r \\ \Rightarrow & \sqrt{1^2+1^2} \cdot \sqrt{1^2+2^2} \cdot \sqrt{1^2+3^2} \ldots \sqrt{1^2+10^2} & =r \\ \Rightarrow & & \sqrt{2} \cdot \sqrt{5} \cdot \sqrt{10} \ldots \sqrt{101} & =r \\ & \therefore & r^2=2 \times 5 \times 10 \times \ldots \times 101 \end{array} $$

Hence, the correct answer is option: 'C'.

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