NIMCET 2015 — Mathematics PYQ

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Question

NIMCET 2015 — Mathematics PYQ

NIMCET | Mathematics | 2015

If $a, b$ and $c$ are in geometric progression, then $\log_{ax} x, \log_{bx} x$ and $\log_{cx} x$ are in:

Choose the correct answer:

Explanation

Given $a, b, c$ are in G.P., therefore:

b^2 = ac

Taking $\log$ with base $x$ on both sides:

\log_x(b^2) = \log_x(ac)

2\log_x b = \log_x a + \log_x c

This shows that $\log_x a, \log_x b, \log_x c$ are in A.P.

Now, add $2\log_x x$ (which is $2$ ) to the equation:

2\log_x b + 2\log_x x = \log_x a + \log_x c + 2\log_x x

2(\log_x b + \log_x x) = (\log_x a + \log_x x) + (\log_x c + \log_x x)

2\log_x(bx) = \log_x(ax) + \log_x(cx)

This implies $\log_x(ax), \log_x(bx), \log_x(cx)$ are in A.P.

We know that $\log_n m = \frac{1}{\log_m n}$ . Therefore:

\frac{1}{\log_{ax} x}, \frac{1}{\log_{bx} x}, \frac{1}{\log_{cx} x} \text{ are in A.P.}

By definition, if the reciprocals of a sequence are in A.P., the sequence itself is in Harmonic Progression (H.P.).

Correct Option: (c)