(d) obtuse angled
x, y, z are in G.P. ⇒ y2 = xz ...(i)
(log x – log 2y), (log 2y – log 3z) and (log 3z – log x) are in A.P. ⇒ 2(log 2y – log 3z) = (log x – log 2y) + (log 3z – log x)
⇒ 3 log 2y = 3 log 3z ⇒ log 2y = log 3z ⇒ y = \(\frac{3}{2}\)z
∴ Putting the value of y in (i), we have
\(\big(\frac{3}{2}z\big)^2\) = xz ⇒ x = \(\frac{9}{4}\)z.
Now, by the cosine rule of triangles,
Cos A = \(\frac{y^2+z^2-x^2}{2yz}\),
where x is the length of the side opposite ∠A.
∵ cos A is less than 0, i.e, negative, ∠A is obtused and the triangle is obtuse angled.