Question

The modulus of $\left( {1 + i} \right)\left( {1 + 2i} \right)\left( {1 + 3i} \right)$ is equal to –
A) $\sqrt {10}$
B) $\sqrt 5$
C) $5$
D) $10$

Answer 1

Here we are given a complex number in the form of $\left( {1 + i} \right)\left( {1 + 2i} \right)\left( {1 + 3i} \right)$ and we are asked what is it’s modulus. For approaching we firstly need to simplify the given complex number that can be done by multiplying all the terms in this case then we will find the modulus of the simplified resultant complex number for which let us see the complete step by step solution of this question

Complete step-by-step answer:
Here we are given a complex number (a complex number is the number of the form $x + iy$where $x$is the real part and the $y$ is the imaginary part satisfying the equation ${i^2} = - 1$) in the form of the $\left( {1 + i} \right)\left( {1 + 2i} \right)\left( {1 + 3i} \right)$ and we are asked to find its modulus
First of all we need to simplify the given complex number that is $\left( {1 + i} \right)\left( {1 + 2i} \right)\left( {1 + 3i} \right)$into a simpler form by the opening of the brackets and multiplying the terms to each other that is –
$\left( {1 + 2i + i + 2{i^2}} \right)\left( {1 + 3i} \right)$
Now as we know ${i^2} = - 1$a basic property of the complex numbers
So the putting the value${i^2} = - 1$ in above equation we get-
$\Rightarrow$ $\left( {1 + 3i - 2} \right)\left( {1 + 3i} \right)$
Now further solving the equation the resultant we get is –
$\Rightarrow$ $\left( {3i - 1} \right)\left( {1 + 3i} \right)$
Now further multiplying the resultant we get-
$\Rightarrow$ $\left( {9{i^2} - 1} \right)$
So the putting the value${i^2} = - 1$ in above equation we get-
$\Rightarrow$ $- 10$
The resultant complex number is –
$\Rightarrow$ $- 10 + 0i$
Now we have to find the modulus of this complex number that is done by the formula of the modulus of complex number of the form $x + iy$ is equal to the
$\left| z \right| = \sqrt {{x^2} + {y^2}}$
Where $\left| z \right|$is the modulus of the complex number
$x =$the real part of the complex number
$y =$the imaginary part of the complex number
Now the modulus of the complex number is $- 10 + 0i$

$$\left| z \right| = \sqrt { - {{10}^2} + {0^2}} \\\ \Rightarrow \left| z \right| = \sqrt {100} \\\ \Rightarrow \left| z \right| = 10 \\\$$

So the modulus is $10$ and the option similar to it is option D that is $10$.
So the correct option is option D.

Note: while solving such kind of questions one should take care about the simplifying the given complex number which should be done with the utter concentration as a mistake can cost the whole question being incorrect and also one should be aware of the formula of the complex number which is the main key in solving this question.