Question

For three positive integers $p , q , r , x^{p q^2}=y^{y r}=z^{p^2 r }$ and $r = pq +1$ such that $3,3 \log _y x, 3 \log _z y$ $7 \log _x z$ are in AP with common difference $\frac{1}{2}$ Then $r-p-q$ is equal to

• A. 6
• B. 2
• C. 12
• D. $-6$

Answer 1

Answer: (B)

2

Answer 2

pq2=logx​λ
qr=logy​λ
p2r=logz​λ
logy​x=pq2qr​=pqr​…….(1)
logx​z=p2rpq2​=prq2​………(2)
logz​y=qrp2r​=qp2​………(3)
3,pq3r​,q3p2​,pr7q2​ in A.P
pq3r​−3=21​
r=67​pq………(4)
r=pq+1
pq=6………(5)
r=7…………(6)
q3p2​=4
After solving p=2 and q=3