BITSAT Mathematics Question (2021) — Solution

Step-by-step solution

The (r+1) th term in the expansion of ( x 3 + 3 2 x^ 2 )^ 10 is given by array l T _ r+1 = ^ 10 C _ r ( x 3 )^ 10-r ( 3 2 x^ 2 )^ r \\ = ^ 10 C _ r x^ 5- ( r 2 ) 3^ 5- ( r 2 ) 3^ r 2^ r x^ 2 r \\ = ^ 10 C _ r 3^ ( 3 r 2 )-5 -5- ( 5 r 2 ) 2^ r x array T _ r+1 = ^ 10 C _ r ( x 3 )^ 10-r ( 3 2 x^ 2 )^ r = ^ 10 C _ r x^ 5- ( r 2 ) 3 3^ r 2^ r x^ 2 r = ^ 10 C _ r 3^ ( 3 r 2 )-5 - ( 5 2 )- ( 5 r 2 ) 2^ r x For T _ r+1 to be independent of x, we must have 5- ( 5 r 2 )=0 or r=2. Thus, the 3 rd term is independent of x and is equal to ^ 10 C _ 2 3^ 3-5 2^ 2 = 10 9 2 3^ -2 4 = 5 4