JEE Mathematics 2024 Question with Solution

Given: $\log a, \log b, \log c$ are in A.P. Also, $\log a - \log 2b, \log 2b - \log 3c, \log 3c - \log a$ are in A.P.

Find: The ratio $a : b : c$.

Since $\log a, \log b, \log c$ are in A.P., the middle term property gives

So,

Hence,

Now rewrite the second A.P. terms using logarithm properties:

Therefore, $\log\left(\frac{a}{2b}\right), \log\left(\frac{2b}{3c}\right), \log\left(\frac{3c}{a}\right)$ are in A.P.

Again using the middle term property for A.P.,

So,

Thus,

Simplifying,

Using $b^2 = ac$ and $b = \frac{3}{2}c$,

Hence,

Therefore, the correct option is A, and $a : b : c = 9 : 6 : 4$.

Given: Both sets of logarithmic terms are in A.P.

Find: $a : b : c$.

From $\log a, \log b, \log c$ in A.P., we get

Now let

Since their logarithms are in A.P., $x, y, z$ satisfy

But

So,

which gives

Then from $b^2 = ac$,

Therefore,

So the correct option is A.