List of top Engineering Mathematics Questions

Consider the partial differential equation $\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0$ where $x, y$ are real. If $f(x, y) = a(x)b(y)$, where $a(x)$ and $b(y)$ are real functions, which one of the following statements can be true?

The height of a right circular cone of maximum volume that can be enclosed within a hollow sphere of radius $R$ is

The arc length of the parametric curve: $ x=\cos\theta,\; y=\sin\theta,\; z=\theta $ from $ \theta=0 $ to $ \theta=2\pi $ is equal to _________ (round off to one decimal place).

If $\hat{a}, \hat{b}, \hat{c}$ are three mutually perpendicular unit vectors, then $\hat{a} \cdot (\hat{b} \times \hat{c})$ can take the value(s):

Let $\hat{i}, \hat{j}, \hat{k}$ be the unit vectors in the x, y, and z directions, respectively. If the vector $\hat{i} + \hat{j}$ is rotated about positive $\hat{k}$ by $135^\circ$, one gets

The equation of the straight line representing the tangent to the curve $y = x^2$ at the point (1, 1) is

The signal $ x(t) = (t - 1)^2 u(t - 1) $, where $ u(t) $ is the unit-step function, has the Laplace transform $ X(s) $. The value of $ X(1) $ is_________

Evaluate the limit:

\[ \lim_{x \to 0} \frac{x^2}{\sin x} \]

= _______.

The 2×2 matrices P and Q satisfy the following relations:

The matrix Q is equal to _______.

The table below shows the carbon content of four samples of powdered coal. If these four samples are mixed completely, what would be the resultant carbon percentage of the mixture by weight?

Consider a function \[ y = f(x) \text{ which satisfies the following equation:} \] \[ \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0. \] As $ x \to -\infty, y = 1 $, and at $ x = 0, y = 2 $. The value of \[ \frac{dy}{dx} \text{ at } x = 0 \text{ is } _______ \text{ (answer in integer).}

A biased die has six faces numbered as $ k = 1, 2, 3, 4, 5, 6 $. On rolling the die, the probability of the number $ k $ appearing is proportional to $ k^2 $. What is the probability that an even number will appear on rolling the die?

Consider the following equation: \[ x^3 - 10x^2 + 31x - 30 = 0 \] Which of the following is/are the root(s) of the above equation?

The ordinary differential equation \[ \frac{dy}{dx} = x^2 y \] has $ y $ as the dependent variable and $ x $ as the independent variable. Which of the following classification(s) is/are applicable to the equation?

A tangent is drawn on the curve of the function $ y = x^2 $ at the point $ (x, y) = (3, 9) $. The slope of the tangent is _______.

A real $ 2 \times 2 $ non-singular matrix $ A $ with repeated eigenvalue is given as

where $ x $ is a real positive number. The value of $ x $ (rounded off to one decimal place) is _________.

Two continuous random variables $ X $ and $ Y $ are related as
$ Y = 2X + 3 $.
Let $ \sigma_X^2 $ and $ \sigma_Y^2 $ denote the variances of $ X $ and $ Y $, respectively. The variances are related as

Let A and B be two events with probabilities $ P(A) = \frac{3}{4} $ and $ P(B) = \frac{1}{3} $; then which of the following options is true?

Consider that a sample of size 3 is randomly drawn from a population that takes only two values, equally likely: -1 and 1. Let $ z = \max(x_1, x_2, x_3) $ where $ x_1, x_2, x_3 $ are the sample observations. The expected value of $ z $, $ E(z) $ is _________ (round off to two decimal places).

Consider a vector $ p $ in 2-dimensional space. Let its direction (counter-clockwise angle with the positive x-axis) be $ \theta $. Let $ p $ be an eigenvector of a 2 $ \times $ 2 matrix $ A $ with corresponding eigenvalue $ \lambda $, where $ \lambda>0 $. If we denote the magnitude of a vector $ v $ by $ \|v\| $, identify the VALID statement regarding $ p' $, where $ p' = Ap $.

Truth table of a logic gate is given below:
\[ \begin{array}{|c|c|c|} \hline \text{Input A} & \text{Input B} & \text{Output Y} \\ \hline 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 1 & X \\ \hline \end{array} \]

The value of X in the above table is ..............

Let $ A = \begin{pmatrix} \sin \theta & \cos \theta \\ \cos \theta & \sin \theta \end{pmatrix} $ and $ A + A^T - 2I = 0 $, where $ A^T $ is the transpose of $ A $ and $ I $ is the identity matrix. The value of $ \theta $ (in degrees) is ............