Let us begin with simplifying the equation:-
=> log_{62}x^{2} + log_{6}y^{1/2} + 3log_{63}y^{1/2}z = 6
* log_{6}x + * log_{6}y + 3 * * log_{6}y^{1/2}z = 6
log_{6}x + log_{6}y^{1/2}y^{1/2}z = 6
log_{6}xyz = 6
xyz = 6^{6}
Given x,y,z is in G.P. Let x =a, y= ab, z= ab^{2}
=) xyz = a^{3}b^{3} = (ab)^{3}
(ab)^{3} = (6^{2})^{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
Correct Answer is (A)
Correct Answer: 189
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