Let $f(t)$ be a periodic signal with fundamental period $T_0>0$. Consider the signal $y(t)=f(\alpha t)$, where $\alpha>1$.

The Fourier series expansions of $f(t)$ and $y(t)$ are given by

$$ f(t)=\sum\limits_{k = - \infty }^\infty c_k e^{j \frac{2 \pi}{T_0} k T} \text { and } y(t)=\sum\limits_{k = - \infty }^\infty d_k e^{j \frac{2 \pi}{T_0} \alpha k T} . $$

Which of the following statements is/are TRUE?

Here are two analogous groups, Group-I and Group-II, that list words in their decreasing order of intensity. Identify the missing word in Group-II.

Group-I: Abuse $\rightarrow$ Insult $\rightarrow$ Ridicule

Group-II: __________$\rightarrow$ Praise $\rightarrow$ Appreciate

The 12 musical notes are given as $C, C^{\#}, D, D^{\#}, E, F, F^{\#}, G, G^{\#}, A, A^{\#}$. Frequency of each note is $\sqrt[12]{2}$ times the frequency of the previous note. If the frequency of the note $C$ is 130.8 Hz , then the ratio of frequencies of notes $F^{\#}$ and $C$ is: