A sinusoidal message signal having root mean square value of 4 V and frequency of 1 kHz is fed to a phase modulator with phase deviation constant $2 \mathrm{rad} /$ volt. If the carrier signal is $c(t)=2 \cos \left(2 \pi 10^6 t\right)$, the maximum instantaneous frequency of the phase modulated signal (rounded off to one decimal place) is $\_\_\_\_$ Hz.

$S_{P M}(t)$ and $S_{F M}(t)$ are defined below, are the phase modulated and the frequency modulated waveforms, respectively, corresponding to the message signal $m(t)$ shown in the figure.

$$ \begin{aligned} & S_{P M}(t)=\cos \left[1000 \pi t+k_p m(t)\right] \\ & S_{F M}(t)=\cos \left[1000 \pi t+k_f \int_{-\infty}^t m(\tau) d \tau\right] \end{aligned} $$

Where $k_p$ is the phase deviation constant in radians/volt and $k_f$ is the frequency deviation constant in radians/second/volt. If the highest instantaneous frequencies of $S_{P M}(t)$ and $S_{F M}(t)$ are same, then the value of the ratio $\frac{k_p}{k_f}$ is $\_\_\_\_$ seconds.

For the modulated signal $x(t)=m(t) \cos \left(2 \pi f_c t\right)$, the message signal $m(t)=4 \cos (1000 \pi t)$ and the carrier frequency $f_c$ is 1 MHz . The signal $x(t)$ is passed through a demodulator, as shown in figure below. The output $y(t)$ of the demodulator is