List of top Engineering Mathematics Questions

Consider three Boolean variables \( x, y, z \). A majority function outputs 1 if the majority of its inputs are 1, otherwise, it outputs 0. Derive the Boolean expression for the majority function and simplify it using Boolean algebra.
Find the next term in the sequence: 3, 9, 19, 33, _
Given the matrix: \[A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & \alpha & \beta \\ 0 & 0 & 0 & \gamma \end{bmatrix}\]

 If rank(A) is at least 3, then what are the possible values of \( \alpha, \beta, \gamma \)?

If a complex function \( f(z) \) is analytic everywhere inside a closed contour \( C \) with anti-clockwise direction, then which of the following statements are correct?
Two fair dice are rolled, and the random variable \( X \) denotes the sum of the outcomes. What is the expected value of \( X \)?

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.

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 function \( f: \mathbb{R} \to \mathbb{R} \), defined as \[ f(x) = 2x^3 - 3x^2 - 12x + 1. \] Which of the following statements is/are correct? (Here, \( \mathbb{R} \) is the set of real numbers.)
Consider the polynomial \[ p(s) = s^5 + 7s^4 + 3s^3 - 33s^2 + 2s - 40. \] Let \( (L, I, R) \) be defined as follows: \[ L \text{ is the number of roots of } p(s) \text{ with negative real parts.} \] \[ I \text{ is the number of roots of } p(s) \text{ that are purely imaginary.} \] \[ R \text{ is the number of roots of } p(s) \text{ with positive real parts.} \] Which one of the following options is correct?

In the circuit below, \( M_1 \) is an ideal AC voltmeter and \( M_2 \) is an ideal AC ammeter. The source voltage (in Volts) is \( v_s(t) = 100 \cos(200t) \). What should be the value of the variable capacitor \( C \) such that the RMS readings on \( M_1 \) and \( M_2 \) are 25 V and 5 A, respectively?


 

Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle \( C \) in the complex plane is/are TRUE?
Consider a system where \( x_1(t), x_2(t), \) and \( x_3(t) \) are three internal state signals and \( u(t) \) is the input signal. The differential equations governing the system are given by: \[ \frac{d}{dt} \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} = \begin{bmatrix} 2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} + \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} u(t). \] 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?
Find the order and degree of the following differential equation: \[ \frac{d^3 y}{dx^3} + \frac{d^2 y}{dx^2} + \frac{dy}{dx} + y = 0 \]
Eigenvalue question: \[\begin{pmatrix} 6 & 8 \\ 4 & 2 \end{pmatrix} \quad \text{Eigenvalue options:} \quad \text{(i) 1, (ii) 2, (iii) 3, (iv) 4}\]
Solve the cubic equation: \[ x^3 - \frac{15}{2} x^2 + 18x + 20 = 0 \]
Given the matrix equation: \[A = \begin{pmatrix} 2 & 3 & 4 \\ 1 & 4 & 5 \\ 4 & 3 & 2 \end{pmatrix} \quad A \cdot X = B\]
Where \( A \) is the matrix, \( X \) is the unknown vector, and \( B \) is a constant vector. Solve for \( X \).
Solve the second-order differential equation: \[ y'' + 0.8 y' + 0.16 y = 0, \quad y(0) = 3, \quad y'(0) = 4.5 \]

Let \( (x, y) \in \mathbb{R}^2 \). The rate of change of the real-valued function
\[ V(x, y) = x^2 + x + y^2 + 1 \] at the origin in the direction of the point \( (1, 2) \) is _____________ (round off to the nearest integer). 

Consider ordinary differential equations given by \[ \frac{dx_1(t)}{dt} = 2x_2(t), \quad \frac{dx_2(t)}{dt} = r(t) \] with initial conditions \( x_1(0) = 1 \) and \( x_2(0) = 0 \). If 

The divergence of the curl of a vector field is:
In the closed interval [0, 3], the minimum value of the function \( f \) given below is: \[ f(x) = 2x^3 - 9x^2 + 12x \]

If \( C \) is the unit circle in the complex plane with its center at the origin, then the value of \( n \) in the equation given below is (rounded off to 1 decimal place). \[ \int_C \frac{z^3}{(z^2 + 4)(z^2 - 4)} \, dz = 2 \pi i n \]

The directional derivative of the function \( f \) given below at the point \( (1, 0) \) in the direction of \( \frac{1}{2} (\hat{i} + \sqrt{3} \hat{j}) \) is (rounded off to 1 decimal place). \[ f(x, y) = x^2 + xy^2 \]

The values of a function \( f \) obtained for different values of \( x \) are shown in the table below.

Using Simpson’s one-third rule, approximate the integral \[ \int_0^1 f(x) \, dx \quad {(rounded off to 2 decimal places)}. \]