The correct options are
A (a−b)2 is a constant and is equal to 64.
B For particular values of a and b, 2 is one of the root.
D Roots are always in A.P.
x4−8x3+ax2−bx+16=0
x1+x2+x3+x4=8 ...(1)
∑xixj=a ...(2)
∑xixjxk=b ...(3)
x1.x2.x3.x4=16 ...(4)
A.M. of x1,x2,x3,x4=x1+x2+x3+x44
=2
G.M. of x1,x2,x3,x4=4√x1.x2.x3.x4
=4√16
=2
∵A.M. of x1,x2,x3,x4=G.M. of x1,x2,x3,x4
Hence, the roots are equal.
⇒x1=x2=x3=x4=2
From eqn(2), a=24
From eqn(3), b=32
(a−b)2=(24−32)2=64
f(x)=x4−8x3+24x2−32x+16
=(x−2)4
f(3)=1