List of practice Questions

In the circuit shown, the switch is opened at $t = 0$ s. The current $i(t)$ at $t = 2$ ms is ________ mA (rounded off to two decimal places).

In the circuit shown, the galvanometer (G) has an internal resistance of $100 \Omega$. The galvanometer current $I_G$ is ________ $\mu A$ (rounded off to the nearest integer).

The circuit given in the figure is driven by a voltage source $V_s = 25\sqrt{2}\angle 30^\circ V$. The system is operating at a frequency of 50 Hz. The transformers are assumed to be ideal. The average power dissipated, in W, in the $50 k\Omega$ resistance is ________ (rounded off to two decimal places).

A feedback control system is shown in the figure.

The maximum allowable value of $ n $ such that the output $ y(t) $, due to any step disturbance signal $ d(t) $, becomes zero at steady-state, is ________ (in integer).

The resistance of a thermistor is measured to be 2.25 k$\Omega$ at 30 Â°C and 1.17 k$\Omega$ at 60 Â°C. Its material constant $ \beta $ is ________ K (rounded off to two decimal places).

An air filled parallel plate electrostatic actuator is shown in the figure. The area of each capacitor plate is $100 \mu m \times 100 \mu m$. The distance between the plates $d_0 = 1 \mu m$ when both the capacitor charge and spring restoring force are zero as shown in Figure (a). A linear spring of constant $k = 0.01 N/m$ is connected to the movable plate. When charge is supplied to the capacitor using a current source, the top plate moves as shown in Figure (b). The magnitude of minimum charge (Q) required to momentarily close the gap between the plates is ________ $\times 10^{-14} C$ (rounded off to two decimal places). Note: Assume a full range of motion is possible for the top plate and there is no fringe capacitance. The permittivity of free space is $\epsilon_0 = 8.85 \times 10^{-12} F/m$ and relative permittivity of air ($\epsilon_r$) is 1.

Consider an AC bridge shown in the figure with $R = 300 \Omega$, $R_1 = 1000 \Omega$, $R_2 = 500 \Omega$, $L = 30 mH$, and a detector D. At the bridge balance condition, the frequency of the excitation source $V_s$ is ________ kHz (rounded off to two decimal places).

A schematic of a Michelson interferometer, used for the measurement of refractive index of gas, is shown in the figure. The transparent chamber is filled with a gas of refractive index $n_g$, where $n_g \neq 1$, at atmospheric pressure. If a 532 nm laser beam produces 30 interference fringes on the screen, then the number of fringes produced by a 632.8 nm laser beam will be ________ (rounded off to one decimal place). Note: Assume that the effect of beamsplitter width is negligible. The setup is placed in air medium with refractive index equal to 1.
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Let $ X(e^{j\omega}) $ represent the discrete-time Fourier transform of a 4-length sequence $ x[n] $, where $ x[0] = 1 $, $ x[1] = 2 $, $ x[2] = 2 $, and $ x[3] = 4 $. $ X(e^{j\omega}) $ is sampled at $ \omega = \frac{2\pi k}{3} $ to generate a periodic sequence in $ k $ with period 3, where $ k $ represents an integer. Let $ y[n] $ represent another sequence such that its discrete Fourier transform $ Y[k] $ is given as $ Y[k] = X(e^{j\omega}) $ for $ 0 \leq k \leq 2 $. The value of $ y[0] $ is ________ (in integer).

A metallic strain-gauge (SG) with resistance $R_{SG}$ is connected as shown in the figure, where $R_{L1}$, $R_{L2}$, $R_{L3}$ represent the lead wire resistances. The SG has a gauge factor of 2 and nominal resistance $R_N$ of $125 \Omega$. When the SG is subjected to a tensile strain of $2 \times 10^{-3}$, the resulting change in $R_{SG}$ is $\Delta R$. The $\Delta R$ value is measured as $\Delta R_{MEAS} = R_{EQ2} - R_{EQ1}$. The $R_{EQ1}$ and $R_{EQ2}$ are the equivalent resistances measured between the terminals 1 and 2, and terminals 2 and 3, respectively. If $R_{L1} = R_{L2} = 5 \Omega$, and $R_{L3} = 4.95 \Omega$, the measured value of tensile strain is ________ $\times 10^{-3}$ (rounded off to two decimal places).

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).

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

If one of the eigenvectors of the matrix

\[ A = \begin{bmatrix} 1 & 1 \\ -4 & x \end{bmatrix} \]

is along the direction of

\[ \begin{bmatrix} 2\alpha \\ \alpha \end{bmatrix} \]

where $ \alpha $ is any non-zero real number, then the value of $ x $ is __________ (in integer).

While doing Bayesian inference, consider estimating the posterior distribution of the model parameter (m), given data (d). Assume that Prior and Likelihood are proportional to Gaussian functions given by \[ {Prior} \propto \exp(-0.5(m - 1)^2) \] \[ {Likelihood} \propto \exp(-0.5(m - 3)^2) \]

The mean of the posterior distribution is (Answer in integer)

Consider a medium of uniform resistivity with a pair of source and sink electrodes separated by a distance $ L $, as shown in the figure. The fraction of the input current $ (I) $ that flows horizontally $ (I_x) $ across the median plane between depths $ z_1 = \frac{L}{2} $ and $ z_2 = \frac{L\sqrt{3}}{2} $, is given by $ \frac{I_x}{I} = \frac{L}{\pi} \int_{z_1}^{z_2} \frac{dz}{(L^2/4 + z^2)} $. The value of $ \frac{I_x}{I} $ is equal to

Suppose a mountain at location A is in isostatic equilibrium with a column at location B, which is at sea-level, as shown in the figure. The height of the mountain is 4 km and the thickness of the crust at B is 1 km. Given that the densities of crust and mantle are 2700 kg/m$^3$ and 3300 kg/m$^3$, respectively, the thickness of the mountain root (r1) is km. (Answer in integer)

If the lowest Digital Number (DN) value in an image of 10-bit radiometric resolution is 0, then the maximum DN value of that image is ________

A watershed has an area of 74 km$^2$. The stream network within this watershed consists of three different stream orders. The stream lengths in each order are as follows: Ist order streams: 3 km, 2.5 km, 4 km, 3 km, 2 km, 5 km
IInd order streams: 10 km, 15 km, 7 km
IIIrd order streams: 30 km
The drainage density of the watershed is _________km/km$^2$ (Round off to two decimal places)

Two players $ A $ and $ B $ are playing a game. Player $ A $ has two available actions $ a_1 $ and $ a_2 $. Player $ B $ has two available actions $ b_1 $ and $ b_2 $. The payoff matrix arising from their actions is presented below:

Let $ p $ be the probability that player $ A $ plays action $ a_1 $ in the mixed strategy Nash equilibrium of the game.
Then the value of p is ________ (round off to one decimal place).

The installation cost (IC) of a solar power plant is INR 89,000. The plant shall be operational for 5 years. The recurring costs for maintenance of the solar plant per year is INR 5,000 but the benefits it creates including reduction in emissions amounts to INR 25,000 per year. These are the only costs and benefits associated with this project. The social discount rate (r) considered is 4% per year. The yearwise information is presented below.

There are two firms in an industry producing a homogeneous product. The market demand function is given by $ P = 1 - (q_1 + q_2) $, where $ q_1 $ and $ q_2 $ are the output levels of Firm 1 and Firm 2, respectively. Firm 1’s cost function is common knowledge and equals zero. Firm 2’s cost function is private information. Firm 1 believes that Firm 2’s cost function is $ 0.5q_2 $ with probability 0.5 and that Firm 2’s cost function is $ 0.25q_2 $ with probability 0.5. The firms choose their quantities simultaneously. Let $ q_1^* $ denote the quantity produced by Firm 1 in the Bayesian Nash equilibrium of this game. Then, the value of $ 24q_1^* $ is _____________ (round off to one decimal place).

If $ X $ is a continuous random variable whose probability density function is given by

\[ f_X(x) = \begin{cases} c x^3 + 0.25, & \text{for } 0 \leq x \leq 1,\ c \in \mathbb{R} \\ 0, & \text{elsewhere} \end{cases} \]

Then the value of $ c $ is ________________ (in integer).

If $ X $ is a continuous random variable whose probability density function is given by

\[ f_X(x) = \begin{cases} \dfrac{1}{x^2}, & \text{for } 1 < x < \infty \\ 0, & \text{elsewhere} \end{cases} \]

Then the median of $ X $ is ________________ (in integer).

Consider a lottery with three possible outcomes:

The maximum amount that a risk-neutral person would be willing to pay to play the above lottery is INR ____________.

Water from a hand pump located near a landfill has 1 mg/L arsenic (oral carcinogenic potency factor = 1.75 (kg-day)/mg). A person who lives nearby drinks 2 L/day of water from this hand pump for 10 years. Assume a body weight of 70 kg and an average life duration of 70 years. The chances of this person getting an excess risk of cancer is ________ $\times 10^{-3$ (rounded off to three decimal places).}