If \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are in \(\mathrm{A} . \mathrm{P}\). and geometric means of ac and \(\mathrm{ab}, \mathrm{ab}\) and \(b c\), ba nad \(c b\) are \(d\), e, f respectively then \(d^{2}, e^{2}, f^{2}\) are in (A) A. P. (B) G. P. (C) H. P. (D) A. G P.

We have, \(a\), \(b\), \(c\) are in AP. Geometric means of: 1. \(ac\) is \(d\) 2. \(ab\) is \(e\) 3. \(bc\) is \(f\)

The difference between the arithmetic progression terms is constant, so we have: \(b - a = c - b\)

Using the definition of geometric mean: 1. \(d = \sqrt{ac}\) 2. \(e = \sqrt{ab}\) 3. \(f = \sqrt{bc}\)

Now we need to square the geometric means which will be: \(d^2 = ac\), \(e^2 = ab\), and \(f^2 = bc\).

Now let's check which progression \(d^2\), \(e^2\), \(f^2\) are in: A) Arithmetic Progression (AP): To be in AP, the difference between terms should be constant, i.e.: \(e^2 - d^2 = f^2 - e^2\) Substituting the values of \(d^2\), \(e^2\), \(f^2\): \((ab - ac) = (bc - ab)\) Expanding and simplifying: \(b(a-c) = c(b-a)\) Using the result from Step 2: \(b - a = c - b\), we get: \(b(a-c) = c(c-b)\) \(c^2 - cb = bc - b^2\) \(c^2 - b^2 = c^2 - 2bc\) B) Geometric Progression (GP): To be in GP, the ratio between terms should be constant, i.e.: \(\frac{e^2}{d^2} = \frac{f^2}{e^2}\) Substituting the values of \(d^2\), \(e^2\), \(f^2\): \(\frac{ab}{ac} = \frac{bc}{ab}\) \(b^2c^2 = a^2b^2\) \(ac = ab\) Clearly, this does not hold true always, since the condition \(ac = ab\) leads to \(a = b\) or \(c = b\), which is not guaranteed for arbitrary values a, b, and c. C) Harmonic Progression (HP): To be in HP, the reciprocals of the terms should be in AP, i.e.: \(\frac{1}{d^2} - \frac{1}{e^2} = \frac{1}{e^2} - \frac{1}{f^2}\) Substituting the values of \(d^2\), \(e^2\), \(f^2\): \(\frac{1}{ac} - \frac{1}{ab} = \frac{1}{ab} - \frac{1}{bc}\) This expression does not simplify to a true statement like in AP case, so the terms are not in HP. D) A.G.P: For an A.G.P., the terms should be in neither AP, GP, nor HP, which leaves us with A.G.P. So the terms \(d^2\), \(e^2\), \(f^2\) are in A.G.P. with \(f^2 = 2e^2 - d^2\) as derived from the AP expression in option A. So, the correct answer is: (D) A.G.P.