Two dice A and B are rolled. Let the numbers obtained on A and B be α and β respectively. If the variance of α-β is p/q, where p and q are co-prime, then the sum of the positive divisors of p is equal to :

Solution

Maths sir · 4 steps

Independent dice: variances add even for a difference, so double one die's variance.

1. One die. For a fair die, E[α] = 21/6 = 7/2 and E[α²] = (1+4+9+16+25+36)/6 = 91/6. So Var(α) = 91/6 - 49/4 = (182 - 147)/12 = 35/12, and Var(β) is the same.
2. Variance of difference. The two dice are independent, so their covariance is 0. Since Var(-β) = (-1)² Var(β), the minus sign turns into a plus.

   Var(α-β) = Var(α) + Var(β) = 35/12 + 35/12 = 35/6

3. Divisors of p. 35 and 6 are co-prime, so p = 35 = 5 × 7. Its positive divisors are 1, 5, 7 and 35.
4. So. Their sum is 1 + 5 + 7 + 35 = 48, which is option (a).

The idea to keep. For independent variables, Var(aX + bY) = a²Var(X) + b²Var(Y). Squaring the coefficient means a minus sign never reduces the variance.

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