NIMCET 2008 — Mathematics PYQ

1. Sum of Infinite GP:

   The series is $\frac{1}{3}+\frac{1}{3^2}+\dots$ where $a=\frac{1}{3}$ and $r=\frac{1}{3}$.

   $$S_{\infty }=\frac{a}{1-r}=\frac{1/3}{1-1/3}=\frac{1/3}{2/3}=0.5$$

2. Simplify Logarithm:

   The expression becomes $y=0.36{\log }_{0.25}(0.5)$.

   Since $0.25=(0.5{)}^2$:

   $${\log }_{0.25}(0.5)={\log }_{(0.5{)}^2}(0.5)=\frac{1}{2}{\log }_{0.5}(0.5)=0.5$$

3. Final Value:

   $$y=0.36\times 0.5=0.18$$

Conclusion: The value of $y$ is $0.18$. Correct choice is (b).