Let α > 0 , β > 0 be such that α 3 + β 2 = 4 . If the maximum value of the term independent of x in the binomial expansion of α x 1 9 + β x - 1 6 10 is 10 k , then k is equal to

Step-by-step solution

T r + 1 = C r 10 α x 1 9 10 - r β x - 1 6 r T r + 1 = C r 10 α 10 - r β r x 10 - r 9 - r 6 Term independent of x   ⇒ Power of x = 0 10 - r 9 - r 6 = 0 ⇒    r = 4 T 5 = C 4 10 α 6 β 4 Now Let α 3 , β 2 are two numbers. Clearly both are positive. A . M . ≥ G . M . ⇒    α 3 + β 2 2 ≥ α 3 β 2 1 / 2 ⇒    α 3 β 2 ≤ 4 ⇒    α 6 β 4 ≤ 16 ⇒ &#