Number of solutions of the equation `z^(2)+|z|^(2)=0` is (i) 1 (ii) 2 (iii) 3 (iv) infinitely many

Options

A. 1
B. 2
C. 3
D. infinitely many

Answer

To solve the equation \( z^2 + |z|^2 = 0 \), we will follow these steps:

### Step 1: Express \( z \) in terms of its real and imaginary parts
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step 2: Calculate \( z^2 \)
Using the formula for the square of a complex number:
\[ z^2 = (x + iy)^2 = x^2 - y^2 + 2xyi \]

### Step 3: Calculate \( |z|^2 \)
The modulus squared of \( z \) is given by:
\[ |z|^2 = x^2 + y^2 \]

### Step 4: Substitute into the equation
Now substitute \( z^2 \) and \( |z|^2 \) into the equation:
\[ x^2 - y^2 + 2xyi + (x^2 + y^2) = 0 \]

### Step 5: Combine like terms
Combine the real and imaginary parts:
\[ (2x^2 - y^2) + (2xy)i = 0 \]

### Step 6: Set real and imaginary parts to zero
For the equation to hold, both the real part and the imaginary part must equal zero:
1. \( 2x^2 - y^2 = 0 \)
2. \( 2xy = 0 \)

### Step 7: Solve the equations
From the second equation \( 2xy = 0 \), we have two cases:
- Case 1: \( x = 0 \)
- Case 2: \( y = 0 \)

#### Case 1: \( x = 0 \)
Substituting \( x = 0 \) into the first equation:
\[ 2(0)^2 - y^2 = 0 \implies -y^2 = 0 \implies y = 0 \]
This gives us the solution \( z = 0 \).

#### Case 2: \( y = 0 \)
Substituting \( y = 0 \) into the first equation:
\[ 2x^2 - (0)^2 = 0 \implies 2x^2 = 0 \implies x = 0 \]
This again gives us the solution \( z = 0 \).

### Step 8: Analyze the solutions
Since \( z = 0 \) is the only solution found, we need to check if there are other solutions. However, if \( x = 0 \), \( y \) can take any real value, leading to infinitely many solutions of the form \( z = iy \).

### Conclusion
Thus, the number of solutions of the equation \( z^2 + |z|^2 = 0 \) is infinitely many.

### Final Answer
The correct option is (iv) infinitely many.

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