NIMCET 2012 — Mathematics PYQ

Step 1: Set the expressions equal to a constant

Let $2^a=3^b=6^{-c}=k$.

Step 2: Express 2, 3, and 6 in terms of $k$

• From $2^a=k$, we get: $2=k^{1/a}$

• From $3^b=k$, we get: $3=k^{1/b}$

• From $6^{-c}=k$, we get: $6=k^{-1/c}$

Step 3: Establish a relationship between the bases

We know that:

Step 4: Substitute the values of $k$

Using the law of indices ($x^m\times x^n=x^{m+n}$):

Step 5: Equate the exponents

Since the bases are the same, we can equate the powers:

Step 6: Simplify to find the required expression

To get a common denominator, multiply by $abc$:

Conclusion:

The value of $ab+bc+ca$ is $0$.

Correct Option: (c)