To solve this problem, we need to find the product of two unknown numbers in a dataset, given its mean and variance. This involves leveraging the fundamental definitions and computational formulas for mean and variance, along with a crucial algebraic identity that connects sums and products of numbers.

1. Key Concepts and Formulas

1. Mean ($\bar{x}$): The mean represents the average value of a dataset. For a set of $n$ observations $x_1, x_2, \dots, x_n$, the mean is defined as the sum of all observations divided by the total number of observations. $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$ Why it's important: This formula directly links the sum of all observations (including our unknowns) to the total count and the given mean.

2. Variance ($\sigma^2$): Variance quantifies the spread or dispersion of data points around the mean. For ungrouped data, the computational (or shortcut) formula is often preferred for ease of calculation: $\sigma^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - (\bar{x})^2$ Why this formula is useful: This formula is particularly efficient when the mean ($\bar{x}$) is a simple number, as it avoids calculating and squaring multiple differences $(x_i - \bar{x})$, thereby simplifying calculations and reducing potential arithmetic errors.

3. Algebraic Identity: The identity $(a+b)^2 = a^2 + b^2 + 2ab$ (or $(x+y)^2 = x^2 + y^2 + 2xy$) is fundamental. Why it's important: This identity provides a direct link between the sum of two numbers $(x+y)$, the sum of their squares $(x^2+y^2)$, and their product $(xy)$. In this problem, we will first find $(x+y)$ and $(x^2+y^2)$ using the mean and variance formulas, respectively, and then use this identity to solve for $xy$.

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2. Step-by-Step Solution

We are given the following information:

• Eight numbers: $3, 7, 9, 12, 13, 20, x, y$.
• Total number of observations, $n = 8$.
• Mean, $\bar{x} = 10$.
• Variance, $\sigma^2 = 25$.

Our ultimate goal is to find the value of $x \cdot y$.

Step 1: Using the Mean to find the sum of unknowns $(x + y)$

The mean of a dataset is the sum of its observations divided by the total number of observations. We use this definition to establish a linear equation for $x$ and $y$.

• Calculation: The sum of all observations is: $\sum x_i = 3 + 7 + 9 + 12 + 13 + 20 + x + y$ Summing the known numerical observations: $3 + 7 + 9 + 12 + 13 + 20 = 64$ So, the total sum is $64 + x + y$. Applying the mean formula $\bar{x} = \frac{\sum x_i}{n}$ with the given values $\bar{x} = 10$ and $n=8$: $10 = \frac{64 + x + y}{8}$ To clear the denominator, multiply both sides of the equation by 8: $10 \times 8 = 64 + x + y$ $80 = 64 + x + y$ Subtracting 64 from both sides to isolate $(x+y)$: $x + y = 80 - 64$ $x + y = 16 \quad \dots(1)$ This gives us our first crucial relationship between $x$ and $y$.

Step 2: Using the Variance to find the sum of squares of unknowns $(x^2 + y^2)$

The computational formula for variance relates the sum of the squares of all observations, the number of observations, and the mean. This formula allows us to form another equation, this time involving $x^2$ and $y^2$.

• Calculation: First, calculate the squares of the known numbers and sum them: $3^2 = 9$ $7^2 = 49$ $9^2 = 81$ $12^2 = 144$ $13^2 = 169$ $20^2 = 400$ Sum of squares of known numbers: $\sum_{i=1}^{6} x_i^2 = 9 + 49 + 81 + 144 + 169 + 400 = 852$ The sum of squares of all observations is $\sum x_i^2 = 852 + x^2 + y^2$. The square of the mean is $\bar{x}^2 = 10^2 = 100$. Applying the variance formula $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$ with the given values $\sigma^2 = 25$, $\bar{x} = 10$, and $n=8$: $25 = \frac{852 + x^2 + y^2}{8} - 100$ To begin isolating the term with $x^2+y^2$, add 100 to both sides of the equation: $25 + 100 = \frac{852 + x^2 + y^2}{8}$ $125 = \frac{852 + x^2 + y^2}{8}$ Next, multiply both sides by 8 to clear the denominator: $125 \times 8 = 852 + x^2 + y^2$ $1000 = 852 + x^2 + y^2$ Finally, subtract 852 from both sides to isolate $(x^2 + y^2)$: $x^2 + y^2 = 1000 - 852$ $x^2 + y^2 = 148 \quad \dots(2)$ This is our second crucial relationship.

**Step 3: Combining Results using an Algebraic Identity to find $x \cdot y$}

We now have the sum of the unknowns $(x+y)$ from Step 1 and the sum of their squares $(x^2+y^2)$ from Step 2. The algebraic identity $(x+y)^2 = x^2 + y^2 + 2xy$ perfectly connects these two quantities with their product $xy$, which is what we need to find.

• Calculation: From equation (1): $x+y = 16$. From equation (2): $x^2+y^2 = 148$. Substitute these values into the identity: $(16)^2 = 148 + 2xy$ Calculate $16^2$: $256 = 148 + 2xy$ To isolate the term $2xy$, subtract 148 from both sides: $256 - 148 = 2xy$ $108 = 2xy$ Finally, divide by 2 to find the value of $xy$: $xy = \frac{108}{2}$ $xy = 54$

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3. Common Mistakes & Tips

• Choose the Right Variance Formula: While both variance formulas are correct, the computational formula ($\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$) is generally less prone to calculation errors, especially when the mean is an integer, as it avoids dealing with decimals or fractions from $(x_i - \bar{x})$.
• Careful Arithmetic: This problem involves several squaring operations and sums. Double-check all your arithmetic calculations (e.g., $16^2$, summing the squares of numbers) to prevent small errors from propagating and leading to an incorrect final answer.
• Don't Forget $(\bar{x})^2$ in Variance: A very common mistake is to forget to subtract $(\bar{x})^2$ from $\frac{\sum x_i^2}{n}$ or to subtract just $\bar{x}$ instead of $\bar{x}^2$. Always remember it's the square of the mean that is subtracted.

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4. Summary

This problem is a classic example of how statistical concepts (mean and variance) are often integrated with fundamental algebraic identities to solve for unknown quantities. The solution strategy involves a clear three-step process: first, using the mean formula to establish a relationship for the sum of the unknown numbers $(x+y)$; second, using the computational variance formula to establish a relationship for the sum of the squares of the unknown numbers $(x^2+y^2)$; and finally, applying the algebraic identity $(x+y)^2 = x^2+y^2+2xy$ to combine these two relationships and directly solve for the product $xy$. Mastering these fundamental formulas and understanding their interconnections is crucial for success in JEE Mathematics, especially in the Statistics topic.

The final answer is $\boxed{54}$.