List of top Engineering Mathematics Questions

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):

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

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 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_________

Let $ f(z) = \dfrac{1}{z^2 + 6z + 9} $ defined in the complex plane. The integral $ \oint_{c} f(z) \, dz $ over the contour of a circle $ c $ with center at the origin and unit radius is $\underline{\hspace{2cm}}$.

The determinant of the matrix $ M $ shown below is $\underline{\hspace{2cm}}$. \[ M = \begin{bmatrix} 1 & 2 & 0 & 0 \\ 3 & 4 & 0 & 0 \\ 0 & 0 & 4 & 3 \\ 0 & 0 & 0 & 1 \end{bmatrix} \]

Given $ A = \begin{pmatrix} 2 & 5 \\ 0 & 3 \end{pmatrix} $, the value of the determinant $ |A^4 - 5A^3 + 6A^2 + 2I| $ is $\underline{\hspace{2cm}}$.

Consider that $ X $ and $ Y $ are independent continuous valued random variables with uniform PDF given by $ X \sim U(2, 3) $ and $ Y \sim U(1, 4) $. Then $ P(Y \leq X) $ is equal to $\underline{\hspace{2cm}}$ (rounded off to two decimal places).

The function \[ f(z) = (z - 1)^{-1} - 1 + (z - 1) - (z - 1)^2 + \cdots \] is the series expansion of

Consider the function $ f(x) = -x^2 + 10x + 100 $. The minimum value of the function in the interval $ [5, 10] $ is $\underline{\hspace{2cm}}$.

Consider the sequence \[ x_n = 0.5x_{n-1} + 1, n = 1, 2, \ldots \text{ with } x_0 = 0. \text{ Then } \lim_{n \to \infty} x_n \text{ is} \]

Consider the row vectors $ v = (1, 0) $ and $ w = (2, 0) $. The rank of the matrix \[ M = 2v^T v + 3w^T w \] where the superscript $ T $ denotes the transpose, 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?

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?

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