prove that there is no term involving x^5 int he expansion of (2x^2-3/x)^11

let's do it by contradiction :

let r+1^th the term contain X⁵ in the expansion of (2x² -3/x)¹¹

so T_r+1 = ¹¹c_r(2x²)^11-r (-3/x)^r

(now remove the coefficients and focus on the variable 'x' and equate it with x⁵ since we are assuming this term to contain x⁵)

x^22-2r-r = x⁵

22-3r = 5 (cancel out x)

r = 17/3

which is not possible because n belongs to whole numbers only .

HENCE PROOVED !