List of top Engineering Mathematics Questions asked in GATE Chemical Engineering

Consider the matrix:

\[ A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} \]

The eigenvalues of the matrix are 0.27 and ____ (rounded off to 2 decimal places).

Consider the differential equation \(\frac{dy}{dx} + \frac{y}{x} = 0\). Choose the CORRECT option(s) for the solution \(y\).
Consider a Cartesian coordinate system with orthogonal unit basis vectors \( \hat{i}, \hat{j} \) defined over a domain: \( x, y \in [0,1] \). Choose the condition for which the divergence of the vector field \( \mathbf{v} = ax\hat{i} - by\hat{j} \) is zero.
Given the random variable \( X \) which takes the values 0, 1, 2, 7, 11, and 12 with the following probabilities: \[ P(X = 0) = 0.4, \quad P(X = 1) = 0.3, \quad P(X = 2) = 0.1, \quad P(X = 7) = 0.1, \quad P(X = 11) = ? \]
Find the maximum value of the function \( f(x) = -x^3 + 2x^2 \) in the interval \( [-1, 1.5] \).
Given that \( \lambda \) is an eigenvalue of matrix \( A \) with the corresponding eigenvector \( x \), and \( x \) is also an eigenvector of \( B = A - 2I \), find the relationship between \( \lambda \) and the eigenvalue of \( B \).
Find the sum of the series: \[ 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots \]

The position \(x(t)\) of a particle, at constant \(\omega\), is described by \(\dfrac{d^{2}x}{dt^{2}}=-\omega^{2}x\) with initial conditions \(x(0)=1\) and \(\left.\dfrac{dx}{dt}\right|_{t=0}=0\). The position of the particle at \(t=\dfrac{3\pi}{\omega}\) is (in integer).

If a matrix \(M\) is defined as \(M=\begin{bmatrix}10 & 6 \\ 6 & 10\end{bmatrix}\), the sum of all the eigenvalues of \(M^3\) is equal to ________________ (in integer).
The first derivative of the function \[ U(r)=4\left[\left(\frac{1}{r}\right)^{12}-\left(\frac{1}{r}\right)^{6}\right] \] evaluated at \(r=1\) is ______________ (in integer).
Simpson’s one-third rule is used to estimate the definite integral \(I=\displaystyle\int_{-1}^{1}\sqrt{1-x^{2}}\,dx\) with an interval length of \(0.5\). Which one of the following is the CORRECT estimate of \(I\) obtained using this rule?
An exhibition was held in a hall on 15 August 2022 between 3 PM and 4 PM during which any person was allowed to enter only once. Visitors who entered before 3:40 PM exited the hall exactly after 20 minutes from their time of entry. Visitors who entered at or after 3:40 PM, exited exactly at 4 PM. The probability distribution of the arrival time of any visitor is uniform between 3 PM and 4 PM. Two persons \(X\) and \(Y\) entered the exhibition hall independent of each other. Which one of the following values is the probability that their visits to the exhibition overlapped with each other?
Which one of the following is the CORRECT value of \(y\), as defined by the expression given below?
\[ y = \lim_{x \to 0} \frac{2x}{e^x - 1} \]
The vector \(\vec{v}\) is defined as \[ \vec{v} = zx\,\hat{i} + 2xy\,\hat{j} + 3yz\,\hat{k}. \] Which one of the following is the CORRECT value of divergence of \(\vec{v}\), evaluated at the point \((x,y,z) = (3,2,1)\)?
For a two-dimensional plane, the unit vectors \((\hat{e}_r,\hat{e}_\theta)\) of the polar coordinate system and \((\hat{i},\hat{j})\) of the cartesian coordinate system are related by \[ \hat{e}_r=\cos\theta\,\hat{i}+\sin\theta\,\hat{j}, \qquad \hat{e}_\theta=-\sin\theta\,\hat{i}+\cos\theta\,\hat{j}. \] Which one of the following is the CORRECT value of \(\dfrac{\partial(\hat{e}_r+\hat{e}_\theta){\partial\theta}\)?}
Given that \( F=\dfrac{|z_1+z_2|}{|z_1|+|z_2|} \), where \(z_1=2+3i\) and \(z_2=-2+3i\) with \(i=\sqrt{-1}\), which one of the following options is CORRECT?