If `Z=1+i`, where `i=sqrt(-1)`, then what is the modulus of `Z+(2)/(Z)`?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the modulus of \( Z + \frac{2}{Z} \) where \( Z = 1 + i \), we can follow these steps:

### Step 1: Calculate \( \frac{2}{Z} \)

Given \( Z = 1 + i \), we need to compute \( \frac{2}{Z} \).

\[ \frac{2}{Z} = \frac{2}{1+i} \]

To simplify this, we can multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{2}{1+i} \cdot \frac{1-i}{1-i} = \frac{2(1-i)}{(1+i)(1-i)} \]

### Step 2: Simplify the denominator

Now we simplify the denominator:

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

So, we have:

\[ \frac{2(1-i)}{2} = 1 - i \]

### Step 3: Calculate \( Z + \frac{2}{Z} \)

Now we can add \( Z \) and \( \frac{2}{Z} \):

\[ Z + \frac{2}{Z} = (1+i) + (1-i) \]

### Step 4: Combine the terms

Combining the real and imaginary parts:

\[ (1 + 1) + (i - i) = 2 + 0i = 2 \]

### Step 5: Find the modulus

The modulus of a complex number \( a + bi \) is given by \( \sqrt{a^2 + b^2} \). Here, our result is \( 2 + 0i \), so:

\[ |Z + \frac{2}{Z}| = |2| = 2 \]

### Final Answer

The modulus of \( Z + \frac{2}{Z} \) is \( 2 \).

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