JEE Main Mathematics Question (2026) — Solution

Step-by-step solution

The general term in the expansion of (9x - 1 3 x )^ 18 is given by: T_ r+1 = ^ 18 C_ r (9x)^ 18-r (- 1 3 x )^r T_ r+1 = ^ 18 C_ r 9^ 18-r (- 1 3 )^r x^ 18-r x^ -r/2 T_ r+1 = ^ 18 C_ r 9^ 18-r (- 1 3 )^r x^ 18 - 3r 2 For the term independent of x, the exponent of x must be zero: 18 - 3r 2 = 0 3r 2 = 18 r = 12 Substituting r = 12 into the general term: T_ 13 = ^ 18 C_ 12 9^ 18-12 (- 1 3 )^ 12 T_ 13 = ^ 18 C_ 12 9^6 ( 1 3 )^ 12 T_ 13 = ^ 18 C_ 12 (3^2)^6 1 3^ 12 T_ 13 = ^ 18 C_ 12 3^ 12 1 3^ 12 = ^ 18 C_ 12 Now, calculating ^ 18 C_ 12 : ^ 18 C_ 12 = ^ 18 C_ 6 = 18 17 16 15 14 13 6 5 4 3 2 1 ^ 18 C_ 12 = 17 13 3 2 14 ^ 18 C_ 12 = 221 84 Given that the term independent of x is 221k: 221k = 221 84 k = 84 Answer: 84