Speed of sound wave depends on medium and is independent of wavelength and frequency.

Speed of sound in a fluid is given by

\[ v = \sqrt{\frac{B}{\rho}} \]

The speed of sound in a gas is \(v = \sqrt{\frac{\gamma RT}{M}}\), meaning \(v \propto \sqrt{T}\). Since temperature is higher on a hot summer day than a cold winter day, sound travels faster in summer. Both statements are true and Reason is the correct explanation.

Velocity of sound \(v = \sqrt{\frac{\gamma P}{\rho}}\). Since water vapor has a lower density than dry air, moist air has a lower density, raising the sound velocity. Thus, (A) is true but (R) is false.

For wind instruments, pitch \( f \) is proportional to the speed of sound in air \( v_{\text{air}} propto \sqrt{T_{\text{temp}}} \). As temperature rises, \( v_{\text{air}} \) increases, so pitch rises. For string instruments, \( f propto v_{\text{string}} = \sqrt{T_{\text{tension}}/\mu} \). As temperature rises, the string expands, reducing tension \( T_{\text{tension}} \), so \( v_{\text{string}} \) decreases and pitch falls. Reason R accurately explains this behavior for both cases.

The speed of sound in a gas is \( v = \sqrt{\gamma P/\rho} \). Moist air (humid air) has a lower average molecular mass and thus lower density \( \rho \) than dry air at the same pressure and temperature. A lower density leads to a higher speed of sound. Thus, A is true. Reason R is false; increasing humidity does not necessarily increase total pressure, and the primary factor for faster sound is reduced density.

Intensity of a sound wave is \( I = \frac{1}{2} \rho v \omega^2 A^2 \), where \( \omega = 2\pi f \). If \( I_1 = I_2 \) and the medium is the same (so \( \rho \) and \( v \) are constant), then \( \omega_1^2 A_1^2 = \omega_2^2 A_2^2 \). Given \( f_1 : f_2 = 1 : 2 \), so \( \omega_1 : \omega_2 = 1 : 2 \). This yields \( (1)^2 A_1^2 = (2)^2 A_2^2 \), so \( A_1^2 = 4 A_2^2 \), meaning \( A_1 : A_2 = 2 : 1 \). Thus, A is true. Reason R is false because intensity also depends on frequency (\( \omega \)), so waves with the same velocity and amplitude but different frequencies will have different intensities.