GATE 2025 EC GA Q.3: solution and answer

Official answer: BSource: Official answer key, GATE 2025 EC (IIT Roorkee)

1. Each note is $\sqrt[12]{2}=2^{1/12}$ times the note before it, so the frequencies form a geometric progression with common ratio $r=2^{1/12}$.

2. In a geometric progression, the term $n$ places after a given term is $f_n=f_0{r}^n$. So the ratio of two notes depends only on how many steps apart they are: $\frac{f_n}{f_0}=r^n$.

3. Count the steps from $C$ to $F^{\#}$ in the printed order: $C\to C^{\#}\to D\to D^{\#}\to E\to F\to F^{\#}$. That is $n=6$ steps. Count the arrows, not the notes: there are 7 notes but only 6 steps.

4. $$\frac{f_{F^{\#}}}{f_C}={\left(2^{1/12}\right)}^6=2^{6/12}=2^{1/2}=\sqrt{2}$$

5. The given frequency of $C$ (130.8 $\text{Hz}$) cancels in the ratio, so you do not need it.

6. The ratio $F^{\#}:C$ is $\sqrt{2}$, which is option (B).