For (A)
We know that Fg=GMeM5/r2
Dimension of Fg is [MLT−2]
Dimension of r2 is[L2]
Therefore, putting in eq. we get
[MLT−2]=dim(GMeM5)/[L2]
dim(GMeM5)=[ML3T−2]
therefore, A→2
Now we know that, for a gas average kinetic energy is given by
KE=3nRT2
KE=3wRT2M
Therefore, dim(KE)=dim(3wRT2M)
dim(3RT2M)=dim(KEw)
dim(3RT2M)=[ML2T−2][M]
dim(3RT2M)=[L2T−2]
Therefore, B→3
We know that magnetic force is given by,
F=qvBsinθ
F=qvB when sinθ=1
F2=(qvB)2
v2=F2(qB)2
dim(v2)=dim(F2(qB)2)
dim(F2(qB)2)=[L2T−2]
therefore, C→3
We know that gravitational potentialon earth's surface is given by
Vg=GMeRe
Ugm=GMeRe
dim(GMeRe)=dim(Ugm)
dim(GMeRe)=[L2T−2]
therefore, D→3