List of practice Questions

Five years ago, the ratio of Aman's age to his father's age was 1:4, and five years from now, the ratio will be 2:5. What was his father's age when Aman was born?
For a real number \(x>1\), solve the equation: \[ \frac{1}{\log_2x} +\frac{1}{\log_3x} + \frac{1}{\log_4x} = 1 \] The value of x is:
The greatest prime factor of \((3^{199} - 3^{196})\) is:
Sequence the following sentences (P, Q, R, S) in a coherent passage: P: Shifu’s student exclaimed, “Why do you run since the bull is an illusion?”
Q: Shifu said, “Surely my running away from the bull is also an illusion.”
R: Shifu once proclaimed that all life is illusion.
S: One day, when a bull gave him chase, Shifu began running for his life.

Four identical cylindrical chalk-sticks, each of radius \( r = 0.5 \, {cm} \) and length \( l = 10 \, {cm} \), are bound tightly together using a duct tape as shown in the following figure.  

The width of the duct tape is equal to the length of the chalk-stick. The area (in cm\(^2\)) of the duct tape required to wrap the bundle of chalk-sticks once, is:

The bar chart shows the data for the percentage of population falling into different categories based on Body Mass Index (BMI) in 2003 and 2023. Based on the data provided, which one of the following options is INCORRECT?

Examples of mirror and water reflections are shown in the figures below: 

 An object appears as the following image after first reflecting in a mirror and then reflecting on water: 

The original object is:

Two identical sheets A and B, of dimensions \( 24 \, {cm} \times 16 \, {cm} \), can be folded into half using two distinct operations, FO1 or FO2. In FO1, the axis of folding remains parallel to the initial long edge, and in FO2, the axis of folding remains parallel to the initial short edge. If sheet A is folded twice using FO1, and sheet B is folded twice using FO2, the ratio of the perimeters of the final shapes of A and B is:
The general form of the complementary function of a differential equation is given by: \[ y(t) = (At + B)e^{-2t}, \] where \(A\) and \(B\) are real constants determined by the initial condition. The corresponding differential equation is:
In the context of Bode magnitude plots, \(40 \, {dB/decade}\) is the same as:

In the feedback control system shown in the figure below, \[ G(s) = \frac{6}{s(s+1)(s+2)}. \]\(R(s)\), \(Y(s)\), and \(E(s)\) are the Laplace transforms of \(r(t)\), \(y(t)\), and \(e(t)\), respectively. If the input \(r(t)\) is a unit step function, then:

 

A digital communication system transmits through a noiseless bandlimited channel \([-W, W]\). The received signal \(z(t)\) at the output of the receiving filter is given by: \[ z(t) = \sum_n b[n] x(t - nT), \] where \(b[n]\) are the symbols and \(x(t)\) is the overall system response to a single symbol. The received signal is sampled at \(t = mT\). The Fourier transform of \(x(t)\) is \(X(f)\). The Nyquist condition that \(X(f)\) must satisfy for zero intersymbol interference at the receiver is:
Consider a lossless transmission line terminated with a short circuit as shown in the figure below. As one moves towards the generator from the load, the normalized impedances \(z_{inA}\), \(z_{inB}\), \(z_{inC}\), and \(z_{inD}\) (indicated in the figure) are: \begin{center} \includegraphics[width=8cm]{15.png} \end{center}
For the Boolean function: \[ F(A, B, C, D) = \Sigma m(0, 2, 5, 7, 8, 10, 12, 13, 14, 15), \] the essential prime implicants are:
A white Gaussian noise \(w(t)\) with zero mean and power spectral density \(\frac{N_0}{2}\), when applied to a first-order RC low-pass filter produces an output \(n(t)\). At a particular time \(t = t_k\), the variance of the random variable \(n(t_k)\) is:
For the closed-loop amplifier circuit shown below, the magnitude of open-loop low-frequency small signal voltage gain is 40. All the transistors are biased in saturation. The current source \(I_{SS}\) is ideal. Neglect body effect, channel length modulation, and intrinsic device capacitances. The closed-loop low-frequency small signal voltage gain \(\frac{v_{out}}{v_{in}}\) (rounded off to three decimal places) is: \begin{center} \includegraphics[width=6cm]{18.png} \end{center}
A causal and stable LTI system with impulse response \(h(t)\) produces an output \(y(t)\) for an input signal \(x(t)\). A signal \(x(0.5t)\) is applied to another causal and stable LTI system with impulse response \(h(0.5t)\). The resulting output is:

For non-degenerately doped n-type silicon, which one of the following plots represents the temperature (\(T\)) dependence of free electron concentration (\(n\))?

In the circuit shown, the \(n:1\) step-down transformer and the diodes are ideal. The diodes have no voltage drop in forward-biased condition. If the input voltage (in Volts) is \(V_s(t) = 10\sin\omega t\) and the average value of load voltage \(V_L(t)\) (in Volts) is \(2.5/\pi\), the value of \(n\) is \(\_\_\_\_\). 

For a causal discrete-time LTI system with transfer function: \[ H(z) = \frac{2z^2 + 3}{(z + \frac{1}{3})(z - \frac{1}{3})}, \] which of the following statements is/are true?

In the circuit below, assume that the long channel NMOS transistor is biased in saturation. The small signal transconductance of the transistor is \(g_m\). Neglect body effect, channel length modulation, and intrinsic device capacitances. The small signal input impedance \(Z_{in}(j\omega)\) is: 

Let \(\hat{i}\) and \(\hat{j}\) be the unit vectors along \(x\) and \(y\) axes, respectively, and let \(A\) be a positive constant. Which one of the following statements is true for the vector fields: \[ \vec{F_1} = A(\hat{i}y + \hat{j}x) \quad {and} \quad \vec{F_2} = A(\hat{i}y - \hat{j}x)? \]
Let \(\rho(x, y, z, t)\) and \(\mathbf{u}(x, y, z, t)\) represent density and velocity, respectively, at a point \((x, y, z)\) and time \(t\). Assume \(\frac{\partial \rho}{\partial t}\) is continuous. Let \(V\) be an arbitrary volume in space enclosed by the closed surface \(S\), and \(\mathbf{\hat{n}}\) be the outward unit normal of \(S\). Which of the following equations is/are equivalent to: \[ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0? \]
An amplitude modulator has output (in Volts): \[ s(t) = A \cos(400\pi t) + B \cos(360\pi t) + B \cos(440\pi t). \] The carrier power normalized to \(1 \, \Omega\) resistance is 50 Watts. The ratio of the total sideband power to the total power is \(1/9\). The value of \(B\) (in Volts, rounded off to two decimal places) is \(\_\_\_\_\).
In a number system of base \(r\), the equation \(x^2 - 12x + 37 = 0\) has \(x = 8\) as one of its solutions. The value of \(r\) is \(\_\_\_\_\).