Assertion (A) is false. An 80  dB sound has an intensity \(10^4\) times greater than a \(40 \text{ dB}\) sound, not twice. Reason (R) is true as it correctly defines loudness in decibels.

Since (A) is false and (R) is true, and the option for 'A is false, R is true' is not provided, option (4) is selected as it states (A) is false.

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).

When two sound waves of frequencies (f_1) and (f_2) interfere, the perceived average frequency is \((f_1 + f_2)/2 = (256 + 260)/2 = 258 \text{ Hz}\). So (A) is true. The beat frequency is \(|f_1 - f_2| = |256 - 260| = 4 \text{ Hz}\).

So (R) is also true. However, the beat phenomenon does not explain the average perceived frequency.

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 as mechanical waves transfer energy via the elastic interaction (coupling) of particles. Reason (R) is false because wave propagation in a medium depends on both its elastic properties (like bulk modulus) and its inertial properties (density), not 'only' elastic properties. The speed of a mechanical wave is given by \(v = \sqrt{\frac{\text{Elasticity}}{\text{Inertia}}}\).

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. Transverse waves involve shear deformation perpendicular to the direction of propagation. Reason (R) is true; ideal liquids have a modulus of rigidity (shear modulus) of zero. Since liquids cannot sustain shear stress, they cannot propagate transverse waves internally. Thus, (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.