At a free end, there's no transverse force, so the slope \( \frac{\text{dy}}{\text{dx}} = 0 \). The resultant pulse is formed such that the free end is a displacement antinode. This implies zero slope, ensuring no transverse force. Both Assertion and Reason are true, and R correctly explains A.

At the open end of an organ pipe, the pressure is approximately atmospheric pressure, meaning there is no excess pressure variation, hence it is a pressure node. When a pressure wave reflects from an open end (a boundary to a less dense medium), it undergoes a \( \pi \) (180 degrees) phase shift. This phase shift causes the incident and reflected waves to destructively interfere at the open end, creating a pressure node. Both A and R are true, and R explains A.

The fundamental frequency of an open organ pipe is \(f = \frac{v}{2L}\). Velocity of sound \(v\) increases with temperature as \(v \propto \sqrt{T}\). While the pipe's length \(L\) also increases with temperature, the increase in \(v\) is proportionally greater than \(L\). Thus, \(f\) increases. Both A and R are true, and R correctly explains A.

For a closed organ pipe, the resonance lengths are in the ratio \(L_1 : L_2 : L_3 = 1 : 3 : 5\). If \(L_1 = 60 text{ cm}\), then \(L_2 = 3 times 60 \text{ cm} = 180 \text{ cm}\). So (A) is true. The resonant frequencies for a closed pipe are \(f_n = (2n-1)f_1\), thus the second resonance (third harmonic) is \(f_2 = 3f_1\). (R) is true and correctly explains (A).

At an open end, the pressure must remain atmospheric, establishing a pressure node. An incident high pressure pulse reflects as a low pressure pulse ((pi) phase change) to maintain this. Thus (A) is true. At a pressure node, particles have maximum displacement, which is a displacement antinode. Thus (R) is true and explains the reflection in (A).

Our ears perceive sound based on pressure variations. At a displacement node, particles have zero displacement, but pressure variations are maximum (a pressure antinode). Therefore, maximum sound is heard at a displacement node. Both (A) and (R) are true, and (R) provides the direct physical reason for (A).

Assertion (A) is true. If a transverse wave pulse propagates along the rotating hoop in a direction opposite to the hoop's rotation, and its speed relative to the string is equal to the string's linear speed, the wave appears stationary to a ground observer. Reason (R) provides this exact condition: if \(v_{\text{wave relative to string}} = -v_{\text{string}}\), then the net velocity relative to the lab frame is zero. Both are true and (R) correctly explains (A).

Assertion (A) is true by definition. Wavelength (\(lambda\)) in a longitudinal wave is the distance between two consecutive compressions or rarefactions. Reason (R) is false; a standing wave is a form of wave resulting from the superposition of two progressive waves. While it does not transfer net energy, it still represents a wave phenomenon with energy oscillation within its segments.

In a harmonic traveling wave, all particles oscillate with the same amplitude but their phases vary with position. So, (A) is true. In a stationary wave, particles between two successive nodes vibrate in phase, but particles on either side of a node are \(180^\circ\) out of phase. Their amplitudes also vary with position. So, (R) is false.