List of top Engineering Mathematics Questions

Cholesky decomposition is carried out on the following square matrix [A]. \[ [A] = \begin{bmatrix} 8 & -5 \\ -5 & a_{22} \end{bmatrix} \] Let $ l_{ij} $ and $ a_{ij} $ be the (i,j)\textsuperscript{th elements of matrices [L] and [A], respectively. If the element $ l_{22} $ of the decomposed lower triangular matrix [L] is 1.968, what is the value (rounded off to the nearest integer) of the element $ a_{22} $?}

Two vectors $[2 \, 1 \, 0 \, 3]^T$ and $[1 \, 0 \, 1 \, 2]^T$ belong to the null space of a $4 \times 4$ matrix of rank 2. Which one of the following vectors also belongs to the null space?

For the matrix \[ \begin{bmatrix} 4 & 2 \\ 3 & 3 \end{bmatrix} \] , eigenvalue corresponding to the eigenvector \[ \begin{bmatrix} 2 \\ -3 \end{bmatrix} \] is ................. (in integer).

Two cards are drawn one after the other from a regular deck of 52 playing cards without replacement. The probability that the drawn cards are of different suits is

Three unbiased coins are tossed. Provided that at least two outcomes are tails, the probability of having all three outcomes as tails is

What is $\lim_{x \to 0} f(x)$, where $f(x) = x \sin \frac{1}{x}$?

The rank of the matrix $A$ given below is one. The ratio $\dfrac{\alpha}{\beta}$ is ______ (rounded off to the nearest integer). \[ A = \begin{bmatrix} 1 & 4 \\ -3 & \alpha \\ \beta & 6 \end{bmatrix} \]

Consider the real-valued function $ g(x)=\max\{(x-2)^2,\,-2x+7\}$, $x\in(-\infty,\infty)$. The minimum value attained by $g(x)$ is _____(rounded off to one decimal place).

Let $f(z)=j\,\dfrac{1-z}{1+z}$, where $z$ is complex and $j=\sqrt{-1}$. The inverse function $f^{-1}(z)$ maps the {real axis} to the ______.

$X$ is a discrete random variable taking values $0,1,2$ with $P(X=0)=0.25$, $P(X=1)=0.5$, and $P(X=2)=0.25$. With $E[ . ]$ denoting expectation, compute $E[X]-E[X^2]$ (rounded off to one decimal place).

The solution $x(t)$, $t\ge 0$, to $\ddot{x}=-k\dot{x}$ ($k>0$) with $x(0)=1$ and $\dot{x}(0)=0$ is:

$F(z)=\dfrac{1}{1-z}$, when expanded as a power series around $z=2$, would result in $F(z)=\sum_{k=0}^{\infty} a_k (z-2)^k$ with ROC $|z-2|<1$. The coefficients $a_k,\;k\ge 0$, are given by the expression ________.

Choose solution set $S$ corresponding to the system of two equations \[ x-2y+z=0,\qquad x-z=0 \] (Note: $\mathbb{R}$ denotes the set of real numbers).

Which one among the following statements must be TRUE about the mean and the median of the scores of all candidates appearing for GATE 2023?

A ‘frabjous’ number is defined as a 3 digit number with all digits odd, and no two adjacent digits being the same. For example, 137 is a frabjous number, while 133 is not. How many such frabjous numbers exist?

Given vectors \[ \vec{A} = 9\hat{i} - 5\hat{j} + 2\hat{k}, \vec{B} = 11\hat{i} + 4\hat{j} + \hat{k}, \vec{C} = -7\hat{i} + 14\hat{j} - 3\hat{k} \] which of the following statements are TRUE?

The sum of eigenvalues of the matrix $ \begin{bmatrix} 4 & 3 & 2 \\ 0 & -1 & 2 \\ 0 & 0 & -3 \end{bmatrix} $ is ................... (in integer).

Order (O) and degree (D) of the differential equation $ \left(\frac{dy}{dx}\right)^3 = \sqrt{\frac{d^2y}{dx^2} + 10} $ are

Match the plots in Section I with the corresponding functions in Section II.
Section I

Section II
(1) $ y = \frac{\sin^2x}{x} $
(2) $ y = x \sin^2x $
(3) $ y = \frac{\sin x}{x} $
(4) $ y = x \sin x $

The maximum value of function $f(x) = 4x^3 - 24x^2 + 36$ in the domain [-1, 5] is ................... (round off to nearest integer).

A fluid flow field is given by the velocity vector $ \vec{V} = e^{xyz}(x\hat{i} + z\hat{k}) $. The curl of velocity at (1, 2, 3) is

Given, $ \vec{\phi} = xy\hat{i} + yz\hat{j} + xz\hat{k} $. S is a surface bounded by the planes x = 0, y = 0, z = 0, x = 3, y = 2, and z = 1. If $ \hat{n} $ is the unit vector normal to S, then $ \iint_S \vec{\phi} . \hat{n} dS $ is

The probability of setting an easy exam paper by three setters are $ \frac{1}{2}, \frac{1}{3}, $ and $ \frac{1}{4} $. If all three are setting one paper each, then the probability that at least one of the papers will be easy is ................... (round off to 2 decimal places).

The value of $ \lim_{x\to 1} \frac{7x^7-20x^5+13x}{3x^3+x-4} $ is