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If (1+I)(1 + 2i)(1+3i)..... (1+ Ni) = A+Ib,Then 2 ×5 ×10 ×...... ×(1+N2) is Equal To. - Mathematics

MCQ

If (1+i)(1 + 2i)(1+3i)..... (1+ ni) = a+ib,then 2 ×5 ×10 ×...... ×(1+n2) is equal to.

Options

  • `sqrt(a^2 +b^2)`

  • `sqrt(a^2 +b^2)`

  • `sqrt(a^2 - b^2)`

  • `a^2 +b^2`

  • `a^2 -b^2`

  • a+b

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Solution

`a^2 +b^2`

(1 + i)(1 + 2i)(1 + 3i) ......(1 + ni) = a + ib

Taking modulus on both the sides, we get:

`|(1+i)(1+2i) (1+3i).............. (1+ni)| = |a+ib|`

`|(1+i)(1+2i)(1+3i)..............(1+ni)|`can be written as `|(1+i)|  |(1+2i)|  |(1+3i)|........|(1+ ni)|`

\[\sqrt{1^2 + 1^2} \times \sqrt{1^2 + 2^2} \times \sqrt{1^2 + 3^2} \times . . . \times \sqrt{1 + n^2} = \sqrt{a^2 + b^2}\]

\[\Rightarrow \sqrt{2} \times \sqrt{5} \times \sqrt{10} \times . . . \times \sqrt{1 + n^2} = \sqrt{a^2 + b^2}\]

Squaring on both the sides, we get:

\[2 \times 5 \times 10 \times . . . \times (1 + n^2 ) = a^2 + b^2\]

  Is there an error in this question or solution?
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 13 Complex Numbers
Q 3 | Page 63
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