List of practice Questions

A full adder and an XOR gate are used to design a digital circuit with inputs $ X, Y, Z $, and output $ F $, as shown below. The input $ Z $ is connected to the carry-in input of the full adder.
If the input $ Z $ is set to logic ‘1’, then the circuit functions as _________ with $ X $ and $ Y $ as inputs.

A simplified small-signal equivalent circuit of a BJT-based amplifier is given below.
The small-signal voltage gain $ \frac{V_o}{V_S} $ (in V/V) is _________.

Let $ i_C, i_L, $ and $ i_R $ be the currents flowing through the capacitor, inductor, and resistor, respectively, in the circuit given below. The AC admittances are given in Siemens (S).
Which one of the following is TRUE?

Consider a continuous-time, real-valued signal $ f(t) $ whose Fourier transform \[ F(\omega) = \int_{-\infty}^{\infty} f(t) \exp(-j \omega t) \, dt { exists.} \] Which one of the following statements is always TRUE?

The Nyquist plot of a system is given in the figure below. Let $ \omega_P, \omega_Q, \omega_R, $ and $ \omega_S $ be the positive frequencies at the points $ P, Q, R, $ and $ S $, respectively. Which one of the following statements is TRUE?

Consider an additive white Gaussian noise (AWGN) channel with bandwidth $ W $ and noise power spectral density $ \frac{N_0}{2} $. Let $ P_{{av}} $ denote the average transmit power constraint. Which one of the following plots illustrates the dependence of the channel capacity $ C $ on the bandwidth $ W $ (keeping $ P_{{av}} $ and $ N_0 $ fixed)?

Consider a frequency-modulated (FM) signal \[ f(t) = A_c \cos(2\pi f_c t + 3 \sin(2\pi f_1 t) + 4 \sin(6\pi f_1 t)), \] where $ A_c $ and $ f_c $ are, respectively, the amplitude and frequency (in Hz) of the carrier waveform. The frequency $ f_1 $ is in Hz, and assume that $ f_c>100 f_1 $. The peak frequency deviation of the FM signal in Hz is _________.

A pot contains two red balls and two blue balls. Two balls are drawn from this pot randomly without replacement. What is the probability that the two balls drawn have different colours?

Consider the following series:
(i) $ \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} $
(ii) $ \sum_{n=1}^{\infty} \frac{1}{n(n+1)} $
(iii) $ \sum_{n=1}^{\infty} \frac{1}{n!} $
Choose the correct option.

Consider the matrix $ A $ below: \[ A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & \alpha & \beta \\ 0 & 0 & 0 & \gamma \end{bmatrix} \] For which of the following combinations of $ \alpha, \beta, $ and $ \gamma $, is the rank of $ A $ at least three? (i) $ \alpha = 0 $ and $ \beta = \gamma \neq 0 $.
(ii) $ \alpha = \beta = \gamma = 0 $.
(iii) $ \beta = \gamma = 0 $ and $ \alpha \neq 0 $.
(iv) $ \alpha = \beta = \gamma \neq 0 $.

An ideal p-n junction germanium diode has a reverse saturation current of 10 $\mu A$ at 300 K. The voltage (in Volts, rounded off to two decimal places) to be applied across the junction to get a forward bias current of 100 mA at 300 K is _________. (Consider the Boltzmann constant $ k_B = 1.38 \times 10^{-23} { J/K} $ and the charge of an electron $ e = 1.6 \times 10^{-19} { C} $.

The random variable $ X $ takes values in $ \{-1, 0, 1\} $ with probabilities \[ P(X = -1) = P(X = 1) = \alpha \quad {and} \quad P(X = 0) = 1 - 2\alpha, \quad 0<\alpha<\frac{1}{2}. \] Let $ g(\alpha) $ denote the entropy of $ X $ (in bits), parameterized by $ \alpha $. Which of the following statements is/are TRUE?

Let $ f(t) $ be a periodic signal with fundamental period $ T_0>0 $. Consider the signal \[ y(t) = f(\alpha t), \quad {where} \, \alpha>1. \] The Fourier series expansions of $ f(t) $ and $ y(t) $ are given by \[ f(t) = \sum_{k=-\infty}^{\infty} c_k e^{j \frac{2\pi}{T_0} k t}, \quad y(t) = \sum_{k=-\infty}^{\infty} d_k e^{j \frac{2\pi}{T_0 \alpha} k t}. \] Which of the following statements is/are TRUE?

The electron mobility $ \mu_n $ in a non-degenerate germanium semiconductor at 300 K is 0.38 m$^2$/Vs. The electron diffusivity $ D_n $ at 300 K (in cm$^2$/s, rounded off to the nearest integer) is _________.

Consider a non-negative function $ f(x) $ which is continuous and bounded over the interval [2, 8]. Let $ M $ and $ m $ denote, respectively, the maximum and the minimum values of $ f(x) $ over the interval. Among the combinations of $ \alpha $ and $ \beta $ given below, choose the one(s) for which the inequality \[ \beta \leq \int_2^8 f(x) \, dx \leq \alpha \] is guaranteed to hold.

A 10-bit analog-to-digital converter (ADC) has a sampling frequency of 1 MHz and a full scale voltage of 3.3 V. For an input sinusoidal signal with frequency 500 kHz, the maximum SNR (in dB, rounded off to two decimal places) and the data rate (in Mbps) at the output of the ADC are _________, respectively.

A source transmits symbol $ S $ that takes values uniformly at random from the set $ \{-2, 0, 2\} $. The receiver obtains $ Y = S + N $, where $ N $ is a zero-mean Gaussian random variable independent of $ S $. The receiver uses the maximum likelihood decoder to estimate the transmitted symbol $ S $. Suppose the probability of symbol estimation error $ P_e $ is expressed as follows: \[ P_e = \alpha P(N>1), \] where $ P(N>1) $ denotes the probability that $ N $ exceeds 1. What is the value of $ \alpha $?

Which of the following statements is/are TRUE with respect to an ideal op-amp?

The ideal BJT in the circuit given below is biased in the active region with a $ \beta $ of 100. If $ I_B $ is 10 µA, then $ V_{CE} $ (in Volts, rounded off to two decimal places) is _________.

A 3-input majority logic gate has inputs $ X $, $ Y $, and $ Z $. The output $ F $ of the gate is logic ‘1’ if two or more of the inputs are logic ‘1’. The output $ F $ is logic ‘0’ if two or more of the inputs are logic ‘0’.
Which one of the following options is a Boolean expression of the output $ F $?

Consider the discrete-time system below with input $ x[n] $ and output $ y[n] $. In the figure, $ h_1[n] $ and $ h_2[n] $ denote the impulse responses of LTI Subsystems 1 and 2, respectively. Also, $ \delta[n] $ is the unit impulse, and $ b>0 $.
Assuming $ h_2[n] \neq \delta[n] $, the overall system (denoted by the dashed box) is _________.
\includegraphics[width=0.5\linewidth]{17.png}

What will be the corresponding mRNA sequence based on the given DNA template strand?

The cardiac output of a person at rest is 5 litres per minute and the mean aortic pressure is 100 mmHg.
During exercise, if the cardiac output doubles and mean aortic pressure rises to 110 mmHg, then the peripheral resistance in the systemic circulation will decrease by _______%.
(rounded off to the nearest integer). Assume venous pressure in systemic circulation to be negligible.

Polylactic-co-glycolic acid (PLGA) is combined with an equal amount (by weight) of hydroxyapatite (HA) to make a bone scaffold. If the porosity of the scaffold is 80%, what is the scaffold density in g/cm$^3$? (rounded off to two decimal places)
Assume densities of PLGA and HA to be 1 g/cm$^3$ and 3 g/cm$^3$, respectively.

The plot of $ \log_{10} ({BMR}) $ as a function of $ \log_{10} (M) $ is a straight line with slope 0.75, where $ M $ is the mass of the person and BMR is the Basal Metabolic Rate. If a child with $ M = 10 \, {kg} $ has a BMR = 600 kcal/day, the BMR for an adult with $ M = 100 \, {kg} $ is _______ kcal/day. (rounded off to the nearest integer)