Let P(A) and P(B) be the roots of the equation 6x^2 - 5x + 1 = 0 . If the probability that exactly one of the events A or B occurs is 2 3 , and P(A B) and P(B A) are the roots of the equation kx^2 - mx + 1 = 0 , then the value of k + m is :

Step-by-step solution

The given equation is 6x^2 - 5x + 1 = 0 (2x - 1)(3x - 1) = 0 Let P(A) = 1 2 and P(B) = 1 3 The probability that exactly one of the events occurs is given by P(A) + P(B) - 2P(A B) 1 2 + 1 3 - 2P(A B) = 2 3 5 6 - 2P(A B) = 4 6 2P(A B) = 1 6 P(A B) = 1 12 Now, P(A B) = P(A B) P(B) = 1 12 1 3 = 1 4 P(B A) = P(A B) P(A) = 1 12 1 2 = 1 6 The quadratic equation with roots 1 4 and 1 6 is x^2 - ( 1 4 + 1 6 )x + ( 1 4 1 6 ) = 0 x^2 - 5 12 x + 1 24 = 0 To match the constant term 1 in kx^2 - mx + 1 = 0 , multiply the equation