The equation of a stationary wave is $y= 2A sin \omega t cos K x$. Comparing this equation with the equation given in the problem, we have
$2 \pi f = 50 \implies f = 25 Hz$

The correct option is (c)

During propagation of a plane progressive mechanical wave all the particles are vibrating with different phases.
While all other statement are correct.

The amplitude of the stationary wave is $10 cos 3 x$ and this fluctuates as x increases

The correct option is (d)

At nodes, displacement $y=0$

$v = 250 \times 1 = 250  m  s^{-1}$
$t = \displaystyle \frac{100}{250} s = 0.45$
Distance travelled by wave in the time taken to produce 100 waves $=$ v $\times$ time
$= $ 250 $\times$ 0.45
$=112.5m \approx 100m$

The waves in which the displacement of the medium is in the same direction as, or the opposite direction to the direction of the propagation of the wave, are called lognitudinal waves. Ex$:-$ Sound waves, sewmic waves

$y=2Asin(\dfrac{2\pi ct}{\lambda}). cos(\dfrac{2\pi x}{\lambda})$

$w=2\Pi f$

progressive waves are waves after generation from the source the keep on propagating on the direction of propagation .

Comparing the given wave equation with standard standing wave equation
$y (x, t) = A \sin (\omega t)\cos (kx)$, 

The equation of the progressive wave is given as, $y=4\,sin(4\pi t-0.04x+\dfrac{\pi}{3})$.

The velocity of the wave would be equal to

$\dfrac{\omega}{k}=\dfrac{4\pi}{0.04}=100\pi\;m/s$

Let two waves be $y _1=A \sin\ (wt-kx)$
$y _2=A \sin\ (wt+kx)$
$y=y _1+y _2$
$=(2A \cos\ kx)\sin\ wt.$ 
For all point in one loop i.e as $x$ varies in $2A \cos k x$, the phase is same. The phase changes only after crossing a node.

Standing wave produces when two waves of identical frequency interfere with one another while travelling in opposite directions and this coincidence directions and this coincidence is not possible in fluids or gases.

In a string which is connected at both ends (similarity to sine wave), anti-nodes appear at odd multiples of $\dfrac{\lambda}{4}$.

Equ of standing wave is $y=2A \sin wt \cos Kx$. The quantity $\dfrac { W }{ K } $ always represent the speed of the wave.

Stationary wave is also known as standing wave. It remains in a constant position. Two opposing waves combine to form a standing wave. Hence energy is not propagated in stationary wave.

The main difference between stationary and progressive waves is: Progressive waves transfer energy from one place to another, without transferring matter and Stationary waves do not transfer energy from one place to another. Clearly only one option has stationary waves.

Stationary waves do not carry energy with it as it is stationary or does not change position.

a ) Two particles in alternate loops refers to particles having same phase and direction
$\therefore $ phase difference is $2\pi $
b ) Two particles in successive loops differs in phase by $\pi $ 
c ) Two particles in same loop are always in phase $\Rightarrow $ phase difference is $0$.
d ) $y _1= a  \sin  (\omega t-kx)$
$y _2= a  \cos  (\omega t-kx)$
$= a  \sin (\omega t-kx +\dfrac{\pi}{2})$
$\Rightarrow $ phase difference is $ \dfrac{\pi }{2}$.

(A) Magnitude of strain is maximum at antinode because medium particles at antinodes have minimum possible velocity.
(B) Nodes and antinodes form in case of travelling waves also.
(C) In case of stationary waves maximum pressure change occurs at node.
(D) Due to propagation of longitudinal wave in air maximum pressure change is equal to $2\pi f s _{0}\rho V$.

In stationary wave, total energy associated with it is twice the energy of each of incident and reflected wave. large amount of energy are stored equally in standing wave and became trapped with wave. Hence there is no transmission of energy through the waves.

We know that frequency of a wave is characterized by the source of wave . In both the cases wave in the slinky has a constant frequency but not same in both cases  , 

Standing waves are obtained when two waves with same angular frequencies and velocity are superimposed, (They are however moving in the opposite directions).
$x(t) = A\sin(\omega t - kx) + A\sin(\omega t + kx + \delta)$
$x(t) = 2A\cos(kx)\sin(\omega t +\dfrac{\delta}{2})$
For all particles to be simultaneously at rest, the value of the sine function must be equal to zero.
i.e. $\omega t + \dfrac{\delta}{2} = n\pi$
$\Rightarrow$ $t = \dfrac{1}{\omega}(n\pi - \dfrac{\delta}{2})$
$\omega = \dfrac{2\pi}{T}$
$\Rightarrow$ $t = \dfrac{T}{2\pi}(n\pi - \dfrac{\delta}{2})$
$t _{1} =  \dfrac{T}{2\pi}(n\pi - \dfrac{\delta}{2})$
$t _{2} =  \dfrac{T}{2\pi}((n+1)\pi - \dfrac{\delta}{2})$
$t _{2} - t _{1} = \dfrac{T}{2}$

$y=1A\sin\omega t\cos kx\k=\cfrac{2\pi}{\lambda}\ \Rightarrow \cfrac{\omega}{K}=\cfrac{\omega\times \lambda}{2\pi}=\cfrac{\lambda}{2\pi/\omega}=\cfrac{\lambda}{T}=\lambda f=V$