$1.$ Determine the dependence of the transverse magnetisation M$$ on the pulse length tP $($in microseconds$)$ and sketch this function in the interval between $0$ and $10$ microseconds. The angular precession frequency $$ of the considered spins in the RCS under the influence of B$1($t$)$ amounts to $5\backslash$times $105$ s$-1.$ How long is the resulting $\backslash$pi $/2$ pulse?

$2.$ Assume that an NMR spectrum exhibits signals covering $50$ kHz frequency range. You are using a rectangular pulse at a carrier frequency in the middle of the spectrum and a pulse length of $4$ microseconds. Is the spectrum fully excited? Explain your answer!