Consider the infinite-length, discrete-time sequence $x[n]=0.9^{|n|}$, where $n$ is an integer. The region of convergence of its Z-transform $X(z)$ is given by:

(Note: $z$ is a complex variable)

Let $x_C(t)$ be any continuous-time periodic signal with period $T$. It is sampled uniformly with a sampling period $T_s$ where $T_s \neq T$, resulting in the discrete sequence $x[n]=x_c\left(n T_s\right)$, where $n$ is an integer. Which one of the following statements is correct about $x[n]$ ?

Consider the following differential equation:

$$ t^2 \frac{d^2 y}{d t^2}+7 t \frac{d y}{d t}+8 t y=10 \sin (t) $$

Which one of the following options is correct?

A time-limited waveform $g(x)$ is specified as follows:

$$ g(x)=\left\{\begin{array}{cc} -k, & -\pi

A new waveform $f(x)$ is constructed from $g(x)$ as follows:

$$ f(x)=\sum_{m=-\infty}^{\infty} g(x+2 \pi n), \text { for all } x \in R $$

The sum of the coefficients of the third harmonics of the sine and cosine terms in the trigonometric Fourier series expansion of $f(x)$ is $\frac{2}{3 \pi}$. What is the value of $k$ ?