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 the modulus of the complex number \( z = \frac{1}{(1+i)(1-2i)} \), we will follow these steps:

### Step 1: Simplify the denominator
First, we need to simplify the expression in the denominator:

\[ (1+i)(1-2i) \]

Using the distributive property (FOIL method):

\[ = 1 \cdot 1 + 1 \cdot (-2i) + i \cdot 1 + i \cdot (-2i) \]
\[ = 1 - 2i + i - 2i^2 \]

Since \( i^2 = -1 \), we can replace \( -2i^2 \) with \( 2 \):

\[ = 1 - 2i + i + 2 \]
\[ = 3 - i \]

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

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

### Step 3: Rationalize the denominator
To find the modulus, we can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator:

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

Calculating the denominator:

\[ (3 - i)(3 + i) = 3^2 - i^2 = 9 - (-1) = 9 + 1 = 10 \]

Thus, we have:

\[ z = \frac{3 + i}{10} \]

### Step 4: Find the modulus of \( z \)
The modulus of a complex number \( z = a + bi \) is given by:

\[ |z| = \sqrt{a^2 + b^2} \]

In our case, \( a = \frac{3}{10} \) and \( b = \frac{1}{10} \):

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

### Final Answer
Thus, the modulus \( |z| \) is:

\[ |z| = \frac{\sqrt{10}}{10} \]

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