The correct option is D 2
If ax2+bx+c=0 has exactly one root at infinity, then it implies that a=0 and b≠0.
Given (P3−3P2+2P)x2+(P3−P)x+P3+3P2+2P=0
Comparing both, we get
P3−3P2+2P=0
⇒P(P2−3P+2)=0
⇒P(P2−2P−P+2)=0
⇒P(P−1)(P−2)=0
⇒P=0,1,2⋯⋯(i)
Also,
P3−P≠0
⇒P(P−1)(P+1)≠0
⇒P≠0,±1⋯⋯(ii)
From (i) and (ii), we get
P=2