If the mean and the variance of 6,4, a, 8, b, 12,10,13 are 9 and 9.25 respectively, then a+b+a b is equal to :

Solution

Maths sir · 4 steps

Mean gives you a + b, variance gives you a² + b², and from those two ab follows at once.

1. Use the mean. There are 8 values with mean 9, so their total is 72. The known six add to 6+4+8+12+10+13 = 53, so a+b = 19.

   σ² = (Σ xᵢ²)/n - x̄²

2. Use the variance. 9.25 = (Σ xᵢ²)/8 - 81, so Σ xᵢ² = 8 × 90.25 = 722. The known squares add to 36+16+64+144+100+169 = 529, so a²+b² = 193.
3. Find ab. (a+b)² = a²+b²+2ab gives 361 = 193 + 2ab, so ab = 84. As a check, a and b are 7 and 12.
4. So. a+b+ab = 19 + 84 = 103, which is option (a).

The idea to keep. With two unknowns in a data set, the mean gives their sum and the variance gives their sum of squares. Use (a+b)² to get the product without solving for each one.

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