NDA Mathematics Question (2025) — Solution

Step-by-step solution

The general term in the expansion of (3^ 1 2 + 5^ 1 4 )^ 12 is given by: T_ r+1 = ^ 12 C_ r (3^ 1 2 )^ 12-r (5^ 1 4 )^ r T_ r+1 = ^ 12 C_ r 3^ 12-r 2 5^ r 4 For the term to be rational, the exponents of 3 and 5 must be integers. Therefore, 12-r 2 and r 4 must be integers. This requires r to be a multiple of 4. Since 0 r 12, the possible values of r are 0, 4, 8, 12. There are 4 such values of r, which means there are 4 rational terms in the expansion. Answer: 4