If ` z= 1/((1+i)(1-2i))` , then |z| is

Options

A. `2/10`
B. ` sqrt7/10`
C. `9/sqrt10`
D. `1/sqrt10`

Answer

To find \( |z| \) where \( z = \frac{1}{(1+i)(1-2i)} \), we will follow these steps:

### Step 1: Simplify the denominator
First, we need to simplify the expression in the denominator \( (1+i)(1-2i) \).

\[ (1+i)(1-2i) = 1 \cdot 1 + 1 \cdot (-2i) + i \cdot 1 + i \cdot (-2i) \]
\[ = 1 - 2i + i - 2i^2 \]
Since \( i^2 = -1 \), we have:
\[ = 1 - 2i + i + 2 = 3 - i \]

### Step 2: Write \( z \) in simplified form
Now we can rewrite \( z \):

\[ z = \frac{1}{3-i} \]

### Step 3: Rationalize the denominator
To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator \( 3+i \):

\[ z = \frac{1 \cdot (3+i)}{(3-i)(3+i)} = \frac{3+i}{3^2 - (-i^2)} = \frac{3+i}{9 + 1} = \frac{3+i}{10} \]

### Step 4: Find the modulus of \( z \)
The modulus of a complex number \( z = a + bi \) is given by \( |z| = \sqrt{a^2 + b^2} \).

Here, \( a = \frac{3}{10} \) and \( b = \frac{1}{10} \):

\[ |z| = \sqrt{\left(\frac{3}{10}\right)^2 + \left(\frac{1}{10}\right)^2} = \sqrt{\frac{9}{100} + \frac{1}{100}} = \sqrt{\frac{10}{100}} = \sqrt{\frac{1}{10}} = \frac{1}{\sqrt{10}} \]

### Final Answer
Thus, \( |z| = \frac{1}{\sqrt{10}} \).

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