List of top Engineering Mathematics Questions

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?
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 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) \( \alpha = \beta = \gamma \neq 0 \).
 

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?
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.
Two systems \( U \) and \( V \) are defined as shown below. Which of the following statement(s) is/are CORRECT? \[ x(t) \quad {System} \, U \quad y(t) = x(t)^2 + 1 \] \[ x(t) \quad {System} \, V \quad y(t) = x(t) + 1 \]
Which of the following is/are CORRECT about the value of \( y = \log_e(-e^x) \), where \( x \) is a real number.
Gelatin solutions P and Q of concentrations 3.5% and 0.5%, respectively, need to be mixed to obtain a 3% solution R. How much volume (in ml) of P needs to be mixed with 100 ml of Q to obtain R? (Choose the correct option)
The elements of the dataset \( \{-5, 1, a, 5, b\} \) are in ascending order. If both the mean and median of the dataset are equal to 3, what is the value of \( b \)?
The eigenvalues of the matrix: \[ \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \] are ________
\( N \) independent trials of a binomial random variable yield an expectation value of 16 and variance of 12. What is the value of \( N \)?
If \( u = x^4 + y^4 + 3x^2 y^2 \), the value of \( \left( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} \right)^{-1} \) is:
The value of \( \lim_{x \to 0} \frac{x \cos x - \sin x}{x^2 \sin x} \) is:
The median of the following set of numbers: 154, 130, 144, 137, 156, 146, 138, 149, 160, 138 is:

The equation \[ y'' + p(x)y' + q(x)y = r(x) \] is a _________ ordinary differential equation.

If the matrix \[ \begin{pmatrix} x & -3 \\ 2 & x - 5 \end{pmatrix} \] is singular, sum of the values of \( x \) is:
The value of \( y \) at \( x = 0.3 \) and \( y(x = 0) = 0 \) for the differential equation \[ \frac{dy}{dx} = 3y + 2e^x \] is _________. (Rounded off to 3 decimal places)
The series \[ \sum_{n=0}^{r} q^n = 1 + q + q^2 + \cdots + q^r \] has the sum:
The Laplace transform of \( \frac{s}{s^2 + a^2} \) is:

Let \[ A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & k & 0 \\ 3 & 0 & -1 \end{pmatrix}. \] If the eigenvalues of \( A \) are -2, 1, and 2, then the value of \( k \) is _. 
(Answer in integer)

Let \( a_0 = 0 \) and define \( a_n = \frac{1}{2} (1 + a_{n-1}) \) for all positive integers \( n \geq 1 \). The least value of \( n \) for which \( |1 - a_n|<\frac{1}{2^{10}} \) is ______.

The value of \( k \), for which the linear equations \( 2x + 3y = 6 \) and \( 4x + 6y = 3k \) have at least one solution, is ________. (Answer in integer)