The variance of `alpha,beta and gamma` is 9, then variance of `5 alpha, 5 beta, 5 gamma` is

Options

A. 45
B. `9//25`
C. 5//9`
D. 225

Answer

To solve the problem, we need to find the variance of the new variables \(5\alpha\), \(5\beta\), and \(5\gamma\) given that the variance of \(\alpha\), \(\beta\), and \(\gamma\) is 9.

### Step-by-Step Solution:

1. **Understand the Variance Scaling Property**:
The variance of a random variable \(X\) scaled by a constant \(c\) is given by:
\[ \text{Var}(cX) = c^2 \cdot \text{Var}(X) \]
This means if we multiply a random variable by a constant, the variance is multiplied by the square of that constant.

2. **Identify the Given Information**:
We are given that:
\[ \text{Var}(\alpha, \beta, \gamma) = 9 \]
This means the variance of the variables \(\alpha\), \(\beta\), and \(\gamma\) is 9.

3. **Apply the Scaling Property**:
We want to find the variance of the new variables \(5\alpha\), \(5\beta\), and \(5\gamma\). Using the scaling property:
\[ \text{Var}(5\alpha, 5\beta, 5\gamma) = 5^2 \cdot \text{Var}(\alpha, \beta, \gamma) \]
Substituting the known variance:
\[ \text{Var}(5\alpha, 5\beta, 5\gamma) = 25 \cdot 9 \]

4. **Calculate the New Variance**:
Now, calculate:
\[ 25 \cdot 9 = 225 \]

5. **Final Result**:
Thus, the variance of \(5\alpha\), \(5\beta\), and \(5\gamma\) is:
\[ \text{Var}(5\alpha, 5\beta, 5\gamma) = 225 \]

### Conclusion:
The variance of \(5\alpha\), \(5\beta\), and \(5\gamma\) is **225**.

---