1. Key Concepts and Formulas

• Arithmetic Progression (AP): A sequence where the difference between consecutive terms is constant, known as the common difference ($d$). The $n$-th term is $a_n = a_1 + (n-1)d$. The sum of $N$ terms is $S_N = \frac{N}{2}(2a_1 + (N-1)d)$.
• Mean ($\bar{x}$): For a set of $N$ numbers $x_1, x_2, \dots, x_N$, the mean is $\bar{x} = \frac{\sum_{i=1}^{N} x_i}{N}$.
• Variance ($\sigma^2$): A measure of the spread of data points around the mean. The formula used for calculation is $\sigma^2 = \frac{\sum_{i=1}^{N} x_i^2}{N} - (\bar{x})^2$. For an AP with $N$ terms and common difference $d$, a useful formula for variance is $\sigma^2 = \frac{d^2(N^2-1)}{12}$.

2. Step-by-Step Solution

Step 1: Represent the A.P. and Formulate Equations We are given six numbers in A.P.: $a_1, a_2, a_3, a_4, a_5, a_6$. Let the first term be $a$ and the common difference be $d$. The terms can be written as: $a_1 = a$ $a_2 = a + d$ $a_3 = a + 2d$ $a_4 = a + 3d$ $a_5 = a + 4d$ $a_6 = a + 5d$

We are given two conditions:

• Condition 1: $a_1 + a_3 = 10$ Substitute the terms in terms of $a$ and $d$: $a + (a + 2d) = 10$ $2a + 2d = 10$ Divide by 2: $a + d = 5 \quad \text{... (Equation 1)}$ This equation relates the first term and the common difference.

• Condition 2: The mean of these six numbers is $\frac{19}{2}$ The sum of the six terms in A.P. is $S_6 = \frac{6}{2}(2a + (6-1)d) = 3(2a + 5d) = 6a + 15d$. The mean is $\bar{x} = \frac{S_6}{6}$. We are given $\bar{x} = \frac{19}{2}$. $\frac{6a + 15d}{6} = \frac{19}{2}$ Multiply both sides by 6: $6a + 15d = \frac{19}{2} \times 6$ $6a + 15d = 57$ Divide by 3: $2a + 5d = 19 \quad \text{... (Equation 2)}$ This equation provides another relationship between $a$ and $d$.

Step 2: Solve for $a$ and $d$ We now have a system of two linear equations:

1. $a + d = 5$
2. $2a + 5d = 19$

From Equation 1, express $a$ in terms of $d$: $a = 5 - d \quad \text{... (Equation 3)}$ Substitute this expression for $a$ into Equation 2: $2(5 - d) + 5d = 19$ $10 - 2d + 5d = 19$ $10 + 3d = 19$ $3d = 19 - 10$ $3d = 9$ $d = 3$ Now, substitute $d=3$ back into Equation 3 to find $a$: $a = 5 - 3$ $a = 2$ So, the first term $a=2$ and the common difference $d=3$.

Step 3: Determine the A.P. terms Using $a=2$ and $d=3$, the six numbers are: $a_1 = 2$ $a_2 = 2 + 3 = 5$ $a_3 = 2 + 2(3) = 8$ $a_4 = 2 + 3(3) = 11$ $a_5 = 2 + 4(3) = 14$ $a_6 = 2 + 5(3) = 17$ The numbers are $2, 5, 8, 11, 14, 17$.

Step 4: Calculate the Variance ($\sigma^2$) We use the formula $\sigma^2 = \frac{\sum x_i^2}{N} - (\bar{x})^2$. We know $N=6$ and $\bar{x} = \frac{19}{2}$.

First, calculate the sum of the squares of the numbers ($\sum x_i^2$): $\sum x_i^2 = 2^2 + 5^2 + 8^2 + 11^2 + 14^2 + 17^2$ $= 4 + 25 + 64 + 121 + 196 + 289$ $= 699$

Now, substitute the values into the variance formula: $\sigma^2 = \frac{699}{6} - \left(\frac{19}{2}\right)^2$ Simplify the terms: $\sigma^2 = \frac{233}{2} - \frac{361}{4}$ To subtract, find a common denominator (4): $\sigma^2 = \frac{233 \times 2}{2 \times 2} - \frac{361}{4}$ $\sigma^2 = \frac{466}{4} - \frac{361}{4}$ $\sigma^2 = \frac{466 - 361}{4}$ $\sigma^2 = \frac{105}{4}$

Alternatively, using the formula for variance of an AP: $\sigma^2 = \frac{d^2(N^2-1)}{12}$ With $d=3$ and $N=6$: $\sigma^2 = \frac{3^2(6^2-1)}{12} = \frac{9(36-1)}{12} = \frac{9 \times 35}{12} = \frac{3 \times 35}{4} = \frac{105}{4}$. Both methods confirm $\sigma^2 = \frac{105}{4}$.

Step 5: Calculate $8\sigma^2$ The problem asks for the value of $8\sigma^2$. $8\sigma^2 = 8 \times \frac{105}{4}$ $8\sigma^2 = 2 \times 105$ $8\sigma^2 = 210$

3. Common Mistakes & Tips

• Arithmetic Errors: Be extremely careful with calculations, especially squaring numbers and adding them. Double-check your sums.
• Formula Application: Ensure you use the correct variance formula. For population variance, divide by $N$.
• System of Equations: Solving for $a$ and $d$ is crucial. Any error here will propagate through the entire problem.
• Alternative Variance Formula: Remember the shortcut formula for the variance of an A.P., $\sigma^2 = \frac{d^2(N^2-1)}{12}$, as it can save time and provide a quick check for your calculations.

4. Summary

We first established the terms of the A.P. using the given conditions, forming a system of linear equations to find the first term ($a=2$) and common difference ($d=3$). This allowed us to list the six numbers. Then, we calculated the sum of the squares of these numbers and used the variance formula $\sigma^2 = \frac{\sum x_i^2}{N} - (\bar{x})^2$ along with the given mean to find the variance. Finally, we multiplied the variance by 8 to get the required value. The calculated value for $8\sigma^2$ is 210.

5. Final Answer

The final answer is $\boxed{210}$ which corresponds to option (B).