CAT 2017 Quantitative Ability

log62x2+log6y

1

2

+3log63y

1

2

z=6
⇒‌‌

2‌logx

2‌log‌6

+‌‌

1

2

‌‌

logy

log‌6

+‌‌

logy 1 2 z

log‌6

=6
⇒2‌logx+logy+2‌log(y

1

2

z)=6‌log‌62
  ⇒logx2+logy+logyz2=log‌612
     ⇒log(x2y⋅y⋅z2)=log‌612
         ⇒(xyz)2=(66)2
           ⇒xyz=66
Given x, y, z is in G.P Let x = a, y = ab, z = ab2
⇒xyz=a3b3=(ab)3
⇒(ab)3=(62)3
Possible values of (a, b) satisfying the equation
(1,36) (2, 18) (3. 12). (4, 9). (9, 4), (12. 3). (18. 2). (36, 1)
Given, y - x is a perfect cube
=>ab - a is perfect cube
⇒a(b - 1) is perfect cube
Only possible when (a, b) =(9, 4)
∴ x = 9, y = 36, z= 144
∴ x + y + z = 9 + 36 + 144= 189