AMU 2026 — science PYQ

1. Circuit Fundamentals

An RC Integrator is a low-pass filter circuit consisting of a resistor ($R$) in series with a capacitor ($C$), where the output voltage ($V_{\text{out}}$) is taken across the capacitor.

Mathematically, an ideal integrator produces an output voltage that is directly proportional to the time integral of its input voltage ($V_{\text{in}}$):

$$V_{\text{out}}(t)\propto \int V_{\text{in}}(t)dt$$

2. Mathematical Response to a Ramp Input

A standard ramp input signal is defined mathematically as a function that increases linearly with time ($t$) for $t\ge 0$:

$$V_{\text{in}}(t)=k\cdot t$$

Where:

• $k$ is a constant representing the slope of the linear ramp.

Now, substituting this linear ramp function into the ideal integration expression:

$$V_{\text{out}}(t)\propto \int (k\cdot t)dt$$

Using the fundamental rules of integration ($\int t^{n}dt=\frac{t^{n+1}}{n+1}$), we integrate the variable $t$:

$$V_{\text{out}}(t)\propto k\cdot \left(\frac{t^2}{2}\right)$$

$$V_{\text{out}}(t)=\frac{k}{2}\cdot t^2$$

3. Conclusion

The mathematical expression contains a quadratic time term ($t^2$). A wave shape whose amplitude varies in proportion to the square of time represents a classic parabolic waveform.

Correct Answer

The correct option is (d) Parabolic waveform.