NIMCET 2011 — Mathematics PYQ

1. Property of Harmonic Progression (HP):

If $a,b,c$ are in HP, then $b$ is the harmonic mean of $a$ and $c$:

From this, we can also write:

2. Simplify the Logarithmic Expression:

Let the given expression be $E$:

Using the property $\log M+\log N=\log (MN)$:

3. Substitute the value of $2b$:

From the HP formula $b=\frac{2ac}{a+c}$, we can express $2b$ as:

Substitute this into the expression inside the logarithm:

Multiply $(a+c)$ into the bracket:

4. Use Algebraic Identity:

Recall the identity $(x+y{)}^2-4xy=(x-y{)}^2$.

5. Final Calculation:

Substitute this back into the logarithm:

Using the property $\log M^n=n\log M$:

The correct option is (c) $2\log (c-a)$.