For first minimum, \(a sin(30^\circ) = \lambda \Rightarrow a = 2\lambda\). For first secondary maximum, \[a sin\theta = \frac{3}{2}\lambda \Rightarrow sin\theta = \frac{3\lambda}{2(2\lambda)} = \frac{3}{4}\].

The width of the central maximum in single slit diffraction is given by \(w = \frac{2\lambda D}{d}\). Substituting the values: \(w = \frac{2 \times 500 \times 10^{-9} \times 2}{10^{-3}} = 2.0 \times 10^{-3} \text{ m} = 2.0 \text{ mm}\).

The position of \(n^{\text{th}}\) minimum in single-slit diffraction is \(y_n = \frac{n\lambda D}{a}\). The distance between the 1st and 3rd minima is \(Delta y = \frac{2\lambda D}{a} ⇒ a = \frac{2\lambda D}{\Delta y} = \frac{2 \times 6000 \times 10^{-10} \times 0.50}{3 \times 10^{-3}} = 0.2\text{ mm}\).

The distance between the first and third minima on the same side of central maximum is \(\Delta y = \frac{2\lambda D}{d}\). Substituting the values: \(3 \times 10^{-3}\text{ m} = \frac{2 \times 6000 \times 10^{-10}\text{ m} \times 0.5\text{ m}}{d}\), we get \(d = 2 \times 10^{-4}\text{ m} = 0.2\text{ mm}\).

Assertion (A) is true; the intensity of higher order maxima in single slit diffraction decreases rapidly. Reason (R) is also true; higher order maxima occur at larger distances from the central maximum. However, R is not the correct explanation for A, as the decrease in intensity is due to the smaller effective aperture contributing to these maxima, not their distance.

Assertion (A) is false. Resolving power of optical instruments (e.g., telescope, microscope) is inversely proportional to the wavelength (\(RP \propto 1/\lambda\)). So, increasing wavelength decreases resolving power. Reason (R) is also false. In diffraction, the width of the central maximum is directly proportional to the wavelength (\(w \propto \lambda\)). Thus, increasing wavelength increases the width of the central maximum.

Assertion (A) is true. Using a converging lens to focus the diffraction pattern at its focal plane is standard for Fraunhofer diffraction, simulating far-field conditions. Reason (R) is false. A lens works by introducing varying optical path lengths across its aperture to achieve focusing, thus creating path differences.

Assertion (A) is true. This is a common observation due to the differing wavelengths of sound and light.
Reason (R) is true. Sound waves have longer wavelengths than light waves, causing them to diffract (bend) significantly around common obstacles. Light waves also diffract, but negligibly so for large obstacles like walls.
Reason (R) correctly explains Assertion (A).

Both Assertion (A) and Reason (R) are true. Diffraction is a universal wave phenomenon, occurring for all wave types. Its effects are most noticeable when the wavelength is comparable to the obstacle's size.

However, (R) states the condition for observation, not the fundamental reason why diffraction occurs for all waves (A).