The correct options are
A n(P∪Q∪R)=10
B n(P∩Q∩R)=1
C n(P∩Q′∩R′)=1
D n(P∩Q∩R′)=1
For set P
14xx+1−(9x−30x−4)≤0
For the inequality to be defined, x≠−1,4
(14x2−56x)−(9x2−21x−30)(x+1)(x−4)≤0
⇒5x2−35x+30(x+1)(x−4)≤0
⇒5(x−1)(x−6)(x+1)(x−4)≤0
⇒x∈(−1,1]∪(4,6]
Since x∈N,P={1,5,6}
For set Q
|x−1|≤5 and |x−1|≥2
⇒2≤|x−1|≤5
⇒−5≤x−1≤−2 or 2≤x−1≤5
⇒−4≤x≤−1 or 3≤x≤6
⇒x∈[−4,−1]∪[3,6]
Since x∈Z,Q={−4,−3,−2,−1,3,4,5,6}
For set R
Clearly, x>0 and x≠1
log6x+2log6x=3
⇒(log6x)2−3log6x+2=0
⇒(log6x−1)(log6x−2)=0
⇒log6x=1 or log6x=2
⇒x=6,36
∴R={6,36}
∴P∪Q∪R={1,5,6,−4,−3,−2,−1,3,4,36}
P∩Q∩R={6}
P∩Q′={1}, ∴P∩Q′∩R′={1}
P∩Q={5,6}, ∴P∩Q∩R′={5}
So, n(P∪Q∪R)=10
n(P∩Q∩R)=1
n(P∩Q′∩R′)=1
n(P∩Q∩R′)=1