Problem 18 - A Complete Resource Book in Mathematics for JEE Main

Question

The variance of \(\alpha, \beta\) and \(\gamma\) is 9 , then variance of \(5 \alpha, 5 \beta\) and \(5 \gamma\) is (A) 45 (B) \(9 / 5\) (C) \(5 / 9\) (D) 225

Step-by-Step Solution

Answer

The variance of \(5\alpha, 5\beta, 5\gamma\) is 225, option (D).

Step 1: Write down the variance formula

The variance for a set of numbers is calculated using the formula \( \text{Var}(X) = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2 \) where \( \mu \) is the mean of the numbers.

Step 2: Given variance of \(\alpha, \beta, \gamma\)

We know that the variance of \( \alpha, \beta \), and \( \gamma \) is 9. This means \( \text{Var}(\{\alpha, \beta, \gamma\}) = 9 \).

Step 3: Find formula for new set \(5\alpha, 5\beta, 5\gamma\)

When each value in a set is multiplied by a constant, the new variance is given by \( c^2 \times \text{Var}(X) \), where \( c \) is the constant multiplier.

Step 4: Calculate the new variance

In this case, \( c = 5 \). Therefore, the variance of \( 5\alpha, 5\beta, 5\gamma \) is \( 5^2 \times 9 = 25 \times 9 = 225 \).

Step 5: Conclusion

Based on the calculation, the variance of \( 5\alpha, 5\beta, 5\gamma \) is 225.

Key Concepts

Variance Formula

To understand variance in statistics, we must first grasp its formula. Variance is a measure of how much the values in a data set deviate from the mean (average) of that set. This is important in statistics as it helps us understand data spread. The formula for variance is: \[ \text{Var}(X) = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2 \] In this formula:

• \(n\) is the number of data points,
• \(x_i\) represents each individual data point, and
• \(\mu\) is the mean of the data set.

By using this formula, we can determine how much each point differs from the mean and squaring those differences before averaging them out. This gives us the variance.

Constant Multiplier Effect on Variance

Sometimes, each data point in a set is multiplied by a constant. How does this affect variance? It's an interesting phenomenon due to the formula structure. When all values in a set are multiplied by the same constant \( c \), the variance of the new set becomes \( c^2 \times \text{Var}(X) \). Here’s why:

• As variance calculates the square of deviations from the mean, multiplying each data point by \( c \) multiplies the deviation by \( c \) as well.
• Squaring this deviation multiplies the variance by \( c^2 \) instead of just \( c \).

Thus, if the original variance is high, multiplying by a constant significantly increases this spread. For example, if \(\alpha, \beta,\) and \(\gamma\) have a variance of 9 and are each multiplied by 5, this leads to a large increase in variance, specifically 225 as calculated with the formula.

Calculation of Variance

Let's dive into how we calculate variance in a practical scenario. This process involves applying the variance formula and understanding the impacts of certain transformations.Suppose we have a set of numbers, say \(\alpha, \beta, \gamma\) with a given variance of 9. This number tells us about the spread of their values. We are told to find the variance of the numbers when each is multiplied by 5. Given our understanding of the constant multiplier effect, here's the calculation process:

• Identify the original variance: 9.
• Determine the constant multiplier: 5.
• Apply the multiplier in the variance formula: \(5^2 \times 9 = 25 \times 9 = 225\).

Thus, the variance of the new set \(5\alpha, 5\beta, 5\gamma\) is 225. This raised variance indicates a much larger spread in the data values compared to the original set.