= Solution
The ideal operational-amplifier rules are \b[zero input currents] and, when stable <negative feedback> maintains unsaturated linear operation, \b[equal input voltages]. Infinite <open-loop gain> motivates the latter; it must not be assumed to guarantee <asymptotic stability> for every external network.
For the numerical sketch and usefulness discussion, use the <transfer function> derived in part (c). The chosen values give $K=5$ and $\tau=RC=10^{-4}\,\mathrm s$. With $x=\omega\tau$,
$$
G(i\omega)=\frac5{1-x^2-2ix}=\frac5{(1-ix)^2},\qquad
\boxed{|G|=\frac5{1+x^2},\quad\arg G=2\arctan x}.
$$
The formal amplitude falls from 5 to zero, with high-frequency slope $-40$ dB per decade. Its unwrapped <phase> rises from zero through $+90^\circ$ at $\omega=10^4\,\mathrm{s^{-1}}$ to $+180^\circ$; the corresponding characteristic <frequency> is 1.59 kHz. At that <frequency> the amplitude is 2.5.
\b[These are formal sinusoidal particular-solution curves, not a stable settled filter response.] For the <formal sinusoidal response of an unstable linear system>, replacing $i\omega$ by the Laplace variable $s$ gives $G(s)=5/(1-s\tau)^2$, with a repeated <pole> at $s=+1/\tau$. A homogeneous output contains $(A+Bt)e^{t/\tau}$ and grows, so real perturbations lead to supply-limited nonlinear behaviour. With equal positive <resistors> and <capacitors> in the frequency-setting network, the general denominator is $1+(3-K)s\tau+s^2\tau^2$: the <Sallen-Key equal-component stability bound> is $K<3$. For example $K=3-\sqrt2$ gives a <Butterworth filter> response. Thus the <Sallen-Key low-pass filter> architecture is useful for active filtering with suitable gain and damping, but the stated $K=5$ component set is unsuitable for linear filtering. The component values are retained in this diagnosis.<Analysis of the Sallen-Key Architecture by James Karki>[Sections 3 and 3.4, low-pass transfer function and equal-component gain/Q relation]{ref}
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2012/paper-1-amplifier-response.png]
{title=Original formal gain and phase curves for the stated circuit, with its unstable positive Laplace pole explicitly identified}
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