GATE IN 2025 Question Paper (01-Feb-2025) (Shift 2)

The circuits mentioned in the following options are realized using ideal opamp. Among these, the circuit(s) performing non-linear operation on the input signal is/are

If one of the eigenvectors of the matrix $A=\left[\begin{array}{cc}-1& -1\\ x& -4\end{array}\right]$ is along the direction of $\left[\begin{array}{c}\alpha \\ 2\alpha \end{array}\right]$, where $\alpha$ is any non-zero real number, then the value of $x$ is ________ (in integer).

Consider the function $f(z)=\frac{2z+1}{z^2-z}$, where $z$ is a complex variable. The sum of the residues at singular points of $f(z)$ is ________ (in integer).

A dual-slope ADC has a fixed integration time of 100 ms. The reference voltage of the ADC is – 5 V. The time taken by the ADC to measure an input voltage of 1.25 V is ________ ms (rounded off to the nearest integer).

In the circuit shown, assume that the BJT in the circuit has very high $\beta$ and $V_{BE}=0.7$ V, and the Zener diode has $V_Z=4.7$ V. The current $I$ through the LED is ________ mA (rounded off to two decimal places).