NIMCET 2011 — Mathematics PYQ

1. Define the Domain:

For the logarithms to be defined:

• x + 2 > 0 \Rightarrow x > -2

• x + 4 > 0 \Rightarrow x > -4

  The common domain is x > -2.

2. Simplify the Inequality:

Use the base change property ${\log }_{a^n}b=\frac{1}{n}{\log }_{a}b$ and the product rule.

Left Side (LHS):

• ${\log }_{1/3}(x+2)={\log }_{3^{-1}}(x+2)=-{\log }_3(x+2)$

• Combining with the first term: ${\log }_3[(x+2)(x+4)]-{\log }_3(x+2)$

• Using $\log M-\log N=\log (\frac{M}{N})$:

  $$\text{LHS}={\log }_3\left(\frac{(x+2)(x+4)}{x+2}\right)={\log }_3(x+4)$$

Right Side (RHS):

• $\frac{1}{2}{\log }_{\sqrt{3}}7=\frac{1}{2}{\log }_{3^{1/2}}7$

• Using ${\log }_{3^{1/2}}7=\frac{1}{1/2}{\log }_{3}7=2{\log }_{3}7$:

  $$\text{RHS}=\frac{1}{2}(2{\log }_{3}7)={\log }_{3}7$$

3. Solve the resulting inequality:

Since the base ($3$) is greater than 1, the inequality direction remains the same:

4. Intersection with Domain:

We must satisfy both x > -2 (from the domain) and x < 3.

In interval notation, this is $(-2,3)$.

The correct option is (b) $(-2,3)$.