Suppose k∈R. Let a be the coefficient of the middle term in the expansion of kx+xk10, and b be the term independent of x in the expansion of k2x+xk10. If ab=1, then k is equal to

Middle term of kx+xk10is  102+1th = 6th term. We have        T6=10C5kx10−5xk3=10C5∴ a=10C5Also, (r+1)th term of kx+xk10is     Tr+1=10Crk2x10−rxkr=10Crk20−3rx2r−10For term independent of x, set 2r−10=0 or r=5. Thus, term independent of x in the expansion of k2x+xk10is  10C5k5=b.As, ab=1, we get k5=1 or k=1