GATE IN 2024 Question Paper (04-Feb-2024) (Shift 2)

A $3\times 3$ matrix $P$ with all real elements has eigenvalues $\frac{1}{4},1$, and $-2$. The value of $|P^{-1}|$ is ________ (rounded off to nearest integer).

The Nyquist sampling frequency for $x(t)=10{\sin }^2(200\pi t)$ is ________ Hz (rounded off to nearest integer).

The resistance of a 20 k$\Omega$ resistor is measured six consecutive times using an LCR meter. The first five readings are 19 k$\Omega$, 18 k$\Omega$, 23 k$\Omega$, 21 k$\Omega$ and 17 k$\Omega$. If the mean of the measurements and the true value are equal, the last reading is ________ k$\Omega$ (rounded off to nearest integer).

Consider the readout circuit of a piezoelectric sensor shown in the figure.



When the piezoelectric sensor generates a charge $q_p$, the resulting change in voltage $V_x$ is -2 V. Then the corresponding change in the voltage $V_{out}$ is ________ V (rounded off to nearest integer).
Note: Assume all components are ideal.

The voltage applied and the current drawn by a circuit are
$v(t)=95+200\cos (120\pi t)+90\cos (360\pi t\text{–}60^{\circ })\text{ V}$
$i(t)=4\cos (120\pi t\text{–}60^{\circ })+1.5\cos (240\pi t\text{–}75^{\circ })\text{ A}$
The average power absorbed by the circuit is ________ W (rounded off to nearest integer).