Introduction:
In modulation, a carrier wave is used to transmit a message signal. The modulation index determines the extent to which the carrier wave is modified by the message signal. To achieve a modulation index of 75%, we need to determine the peak voltage of the modulating signal.

Modulation Index:
The modulation index, denoted by "m," is the ratio of the peak amplitude of the modulating signal to the peak amplitude of the carrier signal. It determines the degree of modulation in a system.

Modulation Index Calculation:
To calculate the modulation index, we use the formula:

m = (Vmax - Vmin) / (Vmax + Vmin)

where Vmax is the peak voltage and Vmin is the minimum voltage of the modulating signal.

Given Information:
- Peak voltage of the carrier wave: 12V
- Modulation index required: 75%

Calculating Peak Voltage of the Modulating Signal:
To achieve a modulation index of 75%, we need to determine the peak voltage of the modulating signal.

Let's assume the peak voltage of the modulating signal as Vm.

Using the modulation index formula, we have:

0.75 = (Vm - Vmin) / (Vm + Vmin)

Since the minimum voltage of the modulating signal can be assumed as zero, the equation simplifies to:

0.75 = Vm / (Vm + 0)

Simplifying further, we get:

0.75 = Vm / Vm

Therefore, the peak voltage of the modulating signal is 0.75 times the peak voltage of the carrier wave.

Calculating the Peak Voltage of the Modulating Signal:
Given that the peak voltage of the carrier wave is 12V, we can calculate the peak voltage of the modulating signal.

Vm = 0.75 * 12V

Vm = 9V

Therefore, the peak voltage of the modulating signal should be 9V in order to achieve a modulation index of 75%.

Summary:
To achieve a modulation index of 75% with a carrier wave of peak voltage 12V, the peak voltage of the modulating signal should be 9V. The modulation index is calculated by taking the ratio of the peak voltage difference between the modulating signal and the carrier signal to the sum of their peak voltages.