If the mean and variance of eight numbers 3, 7, 9, 12, 13, 20, x and y be 10 and 25 respectively, then x.y is equal to _______.

Solution

Maths sir · 4 steps

Mean gives you x + y, variance gives you x² + y², and one square identity gives xy.

1. Use the mean. The sum of all eight numbers is 8 × 10 = 80. The six known numbers add to 3 + 7 + 9 + 12 + 13 + 20 = 64, so x + y = 16.
2. Use the variance. Variance is (Σ xᵢ²)/n - x̄², so (Σ xᵢ²)/8 - 100 = 25. That gives Σ xᵢ² = 8 × 125 = 1000.
3. Find x² + y². The known squares add to 9 + 49 + 81 + 144 + 169 + 400 = 852. So x² + y² = 1000 - 852 = 148.

   2xy = (x+y)² - (x²+y²) = 256 - 148 = 108

4. So. xy = 108/2 = 54.

The common slip: 108. Forgetting to halve (x+y)² - (x²+y²) gives 108, which is 2xy, not xy.

The idea to keep. With two unknowns in a data set, the mean fixes their sum and the variance fixes the sum of their squares. Then (x+y)² = x² + y² + 2xy finishes the job.

Answer from the NTA final answer key. Solution by PrepWiser.