List of top Engineering Mathematics Questions

Solution of the differential equation \[ \frac{dy}{dx} - y = \cos x \] is

In the complex z-domain, the value of the integral \[ \oint_{C} \frac{z^{3} - 9}{3z - i} \, dz \] is

Due to the current COVID pandemic conditions, assume that positive or negative status of any individual are equally likely. There are 3 members in a family. If one of the members has tested COVID positive, the conditional probability that at least 2 members are COVID positive is ______ (rounded off to three decimal places).

Given $x$ is real, identify all the even-functions among the following:

If \(\vec{V} = a\hat{i} + b\hat{j} + c\hat{k}\), identify the INVALID operation on \(\vec{V}\).

Evaluation of the integral \[ \int \frac{dx}{\sqrt{2x - x^2}} \] results in

The activities of a project are given in the following table along with their durations and dependency.

The total float of the activity \( E \) (in days) is _________ (in integer).

Let \(y\) be a non-zero vector of size \(2022 \times 1\). Which of the following statement(s) is/are TRUE?

A pair of six-faced dice is rolled thrice. The probability that the sum of the outcomes in each roll equals 4 in exactly two of the three attempts is _________ (round off to three decimal places).

Consider the polynomial \( f(x) = x^3 - 6x^2 + 11x - 6 \) on the domain \( S \) given by \( 1 \leq x \leq 3 \). The first and second derivatives are \( f'(x) \) and \( f''(x) \).
Consider the following statements:
I. The given polynomial is zero at the boundary points \( x = 1 \text{ and } x = 3. \)
II. There exists one local maxima of \( f(x) \text{ within the domain } S. \)
III. The second derivative \( f''(x)>0 \text{ throughout the domain } S. \)
IV. There exists one local minima of \( f(x) \text{ within the domain } S. \)
The correct option is:

P and Q are two square matrices of the same order. Which of the following statement(s) is/are correct?

The components of pure shear strain in a sheared material are given in the matrix form:
\[ \epsilon = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \] Here, \(\text{Trace}(\epsilon) = 0\). Given, P = \text{Trace}(\epsilon^8) and Q = \text{Trace}(\epsilon^{11}). \text{The numerical value of} (P + Q) \text{ is} _________ \text{. (in integer)}.

The error in measuring the radius of a 5 cm circular rod was 0.2%. If the cross-sectional area of the rod was calculated using this measurement, then the resulting absolute percentage error in the computed area is _________ % . (round off to two decimal places).

The integral \( \int \left( \frac{x}{2} - \frac{x^2}{3} + \frac{x^3}{4} - \cdots \right) \, dx \) is equal to

The function \( f(x, y) \) satisfies the Laplace equation
\[ \nabla^2 f(x, y) = 0 \] on a circular domain of radius \( r = 1 \) with its center at point P with coordinates \( x = 0, y = 0 \). The value of this function on the circular boundary of this domain is equal to 3.
The numerical value of \( f(0, 0) \) is:

Let \(y(x)\) be the solution of the differential equation \(y'' - 4y' - 12y = 3e^{5x}\) satisfying \(y(0)=\frac{18{7}\) and \(y'(0)=-\frac{1}{7}\). Then \(y(1)\) is ______________ (rounded off to nearest integer).}

We have two coins. One is biased with probability of head = 1.0 and the other is a fair coin. One coin is chosen at random and tossed twice. If both outcomes are heads, the probability that the chosen coin is fair is ____________ (correct to one decimal place).

Let \[ M = \begin{pmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{pmatrix} \] Which of the following are TRUE?

The value of the line integral \[ \oint_{C} (-3y\,dx + 3x\,dy + z\,dz) \] along the circle \( C: x^{2} + y^{2} = 1,\ z = 1 \) oriented in the clockwise sense as seen from the origin, is

A simply supported beam is loaded with a uniformly distributed load (UDL). Which statements about strain energy are TRUE?

Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Consider the complex functions: \[ f(z) = (x^2 - y^2) + i 2xy \] and \[ g(z) = 2xy + i(x^2 - y^2) \] Then on the complex plane,

Let \( A \) be a real non-zero square matrix of order \( n \). If the homogeneous system of linear equations \( Ax = 0 \) has only the trivial solution, then:

If a population has exponential distribution with mean 1, then its median is:

Following observation equations are obtained in a survey task.
\[ x + y = 3 \\ 2x + y = 6 \\ x + 2y = 4 \] Using least square method, the most probable values of x and y will be

Using the following regression equations, the correlation coefficient between two survey quantities x and y will be __________________ (round off to 2 decimal places). \[ 2x - 5y + 98 = 0 \] \[ 6x - 7y + 114 = 0 \]