If `x=111"...."(20 digits),y=333"...."(10digits) and z=222".....2"(10 digits), then (x-y^(2))/(z)` equals.

Options

A. `(1)/(2)`
B. 1
C. 2
D. 4

Explanation

To solve the problem, we need to find the value of \((x - y^2) / z\) where:

- \(x = 111...1\) (20 digits)
- \(y = 333...3\) (10 digits)
- \(z = 222...2\) (10 digits)

### Step 1: Express \(x\), \(y\), and \(z\) in a mathematical form.

1. **Finding \(x\)**:
- \(x\) consists of 20 digits of 1.
- This can be expressed as:
\[ x = \frac{10^{20} - 1}{9} \]

2. **Finding \(y\)**:
- \(y\) consists of 10 digits of 3.
- This can be expressed as:
\[ y = 3 \times \frac{10^{10} - 1}{9} = \frac{3(10^{10} - 1)}{9} = \frac{10^{10} - 1}{3} \]

3. **Finding \(z\)**:
- \(z\) consists of 10 digits of 2.
- This can be expressed as:
\[ z = 2 \times \frac{10^{10} - 1}{9} \]

### Step 2: Substitute \(x\), \(y\), and \(z\) into the expression \((x - y^2) / z\).

1. **Calculate \(y^2\)**:
\[ y^2 = \left(\frac{10^{10} - 1}{3}\right)^2 = \frac{(10^{10} - 1)^2}{9} \]

2. **Substituting into the expression**:
\[ \frac{x - y^2}{z} = \frac{\frac{10^{20} - 1}{9} - \frac{(10^{10} - 1)^2}{9}}{2 \times \frac{10^{10} - 1}{9}} \]

3. **Simplifying the numerator**:
\[ = \frac{10^{20} - 1 - (10^{10} - 1)^2}{2(10^{10} - 1)} \]

4. **Expanding \((10^{10} - 1)^2\)**:
\[ (10^{10} - 1)^2 = 10^{20} - 2 \times 10^{10} + 1 \]

5. **Substituting this back into the expression**:
\[ = \frac{10^{20} - 1 - (10^{20} - 2 \times 10^{10} + 1)}{2(10^{10} - 1)} \]
\[ = \frac{10^{20} - 1 - 10^{20} + 2 \times 10^{10} - 1}{2(10^{10} - 1)} \]
\[ = \frac{2 \times 10^{10} - 2}{2(10^{10} - 1)} \]
\[ = \frac{2(10^{10} - 1)}{2(10^{10} - 1)} = 1 \]

### Final Result:
\[ \frac{x - y^2}{z} = 1 \]