Step 1: Understanding the Concept:
The fundamental concept behind this problem is the representation of complex numbers and the principle of equality for complex expressions.
A complex number is defined as \( z = x + iy \), where \( x \) and \( y \) are real numbers known as the real and imaginary parts, respectively.
The symbol \( i \) represents the imaginary unit, defined as \( \sqrt{-1} \).
The modulus of a complex number, \( |z| \), is a scalar value representing the magnitude or the distance of the complex number from the origin in the Argand plane.
Crucially, \( |z| = \sqrt{x^2 + y^2} \) is always a non-negative real number.
When an equation contains both complex variables and their moduli, we treat the modulus as a real constant part of the equation's real side.
Equality of two complex numbers states that if \( a + bi = c + di \), then \( a = c \) and \( b = d \).
Step 2: Key Formula or Approach:
We substitute the standard form \( z = x + iy \) and its modulus formula \( |z| = \sqrt{x^2 + y^2} \) into the given equation.
The equation becomes a system of two equations: one for the real parts and one for the imaginary parts.
By solving for the imaginary part first, we simplify the equation for the real part significantly.
Step 3: Detailed Explanation:
Let the complex number be represented by its components, \( z = x + iy \).
The given equation is \( |z| + z = 2 + i \).
Substituting the definitions into the equation:
\[ \sqrt{x^2 + y^2} + (x + iy) = 2 + i \]
Group the real terms and imaginary terms on the left-hand side:
\[ (\sqrt{x^2 + y^2} + x) + i(y) = 2 + i \]
Now, equate the imaginary parts on both sides of the equation:
\[ y = 1 \]
Equate the real parts on both sides of the equation:
\[ \sqrt{x^2 + y^2} + x = 2 \]
Substitute the value \( y = 1 \) into the real part equation:
\[ \sqrt{x^2 + (1)^2} + x = 2 \]
\[ \sqrt{x^2 + 1} = 2 - x \]
To eliminate the radical, we square both sides of the equation.
Note: Squaring can introduce extraneous solutions, so we must ensure that the right side \( 2 - x \) is non-negative, meaning \( x \leq 2 \).
\[ (\sqrt{x^2 + 1})^2 = (2 - x)^2 \]
Expand the right side using the identity \( (a-b)^2 = a^2 - 2ab + b^2 \):
\[ x^2 + 1 = 4 - 4x + x^2 \]
Subtract \( x^2 \) from both sides to isolate the variable \( x \):
\[ 1 = 4 - 4x \]
\[ 4x = 4 - 1 \]
\[ 4x = 3 \]
\[ x = \frac{3}{4} \]
Since \( x = 3/4 \) is less than 2, the solution is valid.
Now, calculate the value of \( |z| \) using the determined values of \( x \) and \( y \):
\[ |z| = \sqrt{\left(\frac{3}{4}\right)^2 + (1)^2} \]
\[ |z| = \sqrt{\frac{9}{16} + 1} = \sqrt{\frac{9 + 16}{16}} \]
\[ |z| = \sqrt{\frac{25}{16}} = \frac{5}{4} \]
Step 4: Final Answer:
The real part of the complex number is \( 3/4 \) and the imaginary part is \( 1 \).
Substituting these into the modulus formula gives \( |z| = 5/4 \).
This confirms that option (3) or (C) is the correct choice.