When m gm of water at `10^(@)C` is mixed with m gm of ice at `0^(@)C`, which of the following statements are false?

Options

A. The temperature of the system will be given by the equation
`m xx 80 +m xx1xx (T -0) =m xx 1xx (10 -T)`
B. Whole of ice will melt and temperature will be more than `0^(@)C` but lesser than `10^(@)C`
C. Whole of ice will melt and temperature will be `0^(@)C`
D. Whole of ice will not melt and temperature will be `0^(@)C`

Explanation

To solve the problem, we need to analyze the situation when \( m \) grams of water at \( 10^\circ C \) is mixed with \( m \) grams of ice at \( 0^\circ C \). We will determine which statements about the system are false.

### Step-by-step Solution:

1. **Identify Initial Conditions**:
- We have \( m \) grams of water at \( 10^\circ C \).
- We have \( m \) grams of ice at \( 0^\circ C \).

2. **Heat Transfer**:
- The water will lose heat as it cools down from \( 10^\circ C \) to \( 0^\circ C \).
- The ice will gain heat as it melts and turns into water at \( 0^\circ C \).

3. **Calculate Heat Released by Water**:
- The heat released by the water as it cools down can be calculated using the formula:
\[ Q_{\text{released}} = m \cdot c \cdot \Delta T \]
where \( c \) (specific heat of water) = 1 cal/g°C, and \( \Delta T = 10 - 0 = 10^\circ C \).
- Thus,
\[ Q_{\text{released}} = m \cdot 1 \cdot 10 = 10m \text{ calories} \]

4. **Calculate Heat Absorbed by Ice**:
- The heat absorbed by the ice to melt into water can be calculated using the formula:
\[ Q_{\text{absorbed}} = m \cdot L \]
where \( L \) (latent heat of fusion for ice) = 80 cal/g.
- Thus,
\[ Q_{\text{absorbed}} = m \cdot 80 = 80m \text{ calories} \]

5. **Compare Heat Released and Absorbed**:
- The heat released by the water (10m) is less than the heat required to melt the entire ice (80m):
\[ 10m < 80m \]
- This means not all the ice will melt.

6. **Determine the Amount of Ice that Melts**:
- Let \( x \) be the mass of ice that melts. The heat absorbed by the melting ice is:
\[ Q_{\text{absorbed}} = x \cdot 80 \]
- Setting the heat released equal to the heat absorbed:
\[ 10m = x \cdot 80 \]
- Solving for \( x \):
\[ x = \frac{10m}{80} = \frac{m}{8} \]
- Therefore, \( \frac{m}{8} \) grams of ice will melt.

7. **Final Amounts**:
- The total amount of water at \( 0^\circ C \) after mixing:
\[ \text{Total water} = m + \frac{m}{8} = \frac{9m}{8} \text{ grams} \]
- Remaining ice:
\[ \text{Remaining ice} = m - \frac{m}{8} = \frac{7m}{8} \text{ grams} \]

8. **Evaluate Statements**:
- Check each statement against our findings:
1. The temperature of the system will be given by the equation \( m \cdot 80 + m \cdot 1 \cdot (T - 0) \) - **False** (as the equilibrium temperature is \( 0^\circ C \)).
2. Whole of ice will melt and temperature will be more than \( 0^\circ C \) but less than \( 10^\circ C \) - **False** (not all ice melts).
3. Whole of ice will melt and temperature will be \( 0^\circ C \) - **False** (not all ice melts).
4. Whole of ice will not melt, and temperature will be \( 0^\circ C \) - **True** (this is correct).

### Conclusion:
The false statements are:
- The first statement
- The second statement
- The third statement