Waves practice · Mixed difficulty

A $100 \mathrm{~Hz}$ sinusoidal wave is travelling in the positive $x$-direction along a string with a linear mass density of $3.5 \times 10^{-3} \mathrm{~kg} / \mathrm{m}$ and a tension of $35 \mathrm{~N}$. At time $t=0$, the point $x=0$, has maximum displacement in the positive $y$-direction. Next when this point has zero displacement, the slope of the string is $\pi / 20$. Which of the following expression represent (s) the displacement of string as a function of $x$ (in metre) and $t$ (in second)

Out of the given waves (1), (2), (3) and (4)
$y = a\sin \left( {kx + \omega t} \right)$ ...($1$)
$y = a\sin \left( {\omega t - kx} \right)$ …($2$)
$y = a\cos \left( {kx + \omega t} \right)$ …($3$)
$y = a\cos \left( {\omega t - kx} \right)$ …($4$)
Emitted by four different sources ${S_1},{S_2},{S_3}\,\,and\,\,{S_4}$ respectively, interference phenomena would be observed in space under appropriate conditions when

A string of length $2 \mathrm{x}$ is stretched by $0.1 \mathrm{x}$ and the velocity of a transverse wave along it is $v$. When it is stretched by $0.4 \mathrm{x}$ the velocity of the wave is