List of top Engineering Mathematics Questions asked in GATE Civil Engineering

Eigenvalue question: \[\begin{pmatrix} 6 & 8 \\ 4 & 2 \end{pmatrix} \quad \text{Eigenvalue options:} \quad \text{(i) 1, (ii) 2, (iii) 3, (iv) 4}\]

Given the matrix equation: \[A = \begin{pmatrix} 2 & 3 & 4 \\ 1 & 4 & 5 \\ 4 & 3 & 2 \end{pmatrix} \quad A \cdot X = B\]
Where \( A \) is the matrix, \( X \) is the unknown vector, and \( B \) is a constant vector. Solve for \( X \).

Solve the second-order differential equation: \[ y'' + 0.8 y' + 0.16 y = 0, \quad y(0) = 3, \quad y'(0) = 4.5 \]

Solve the cubic equation: \[ x^3 - \frac{15}{2} x^2 + 18x + 20 = 0 \]

Find the order and degree of the following differential equation: \[ \frac{d^3 y}{dx^3} + \frac{d^2 y}{dx^2} + \frac{dy}{dx} + y = 0 \]

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?

The solution of the differential equation \[ \frac{d^3 y}{dx^3}-5.5\,\frac{d^2 y}{dx^2}+9.5\,\frac{dy}{dx}-5\,y=0 \] is expressed as $y=C_1 e^{2.5x}+C_2 e^{\alpha x}+C_3 e^{\beta x}$, where $C_1,C_2,C_3,\alpha,\beta$ are constants, with $\alpha$ and $\beta$ distinct and not equal to $2.5$. Which of the following options is correct for the values of $\alpha$ and $\beta$?

For the matrix \[ A = \begin{bmatrix} 1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1 \end{bmatrix} \] which of the following statements is/are TRUE? \\

Which of the following probability distribution functions (PDFs) has the mean greater than the median? \includegraphics[width=0.75\linewidth]{image3.png}

Let $\phi$ be a scalar field, and $\mathbf{u$ be a vector field. Which of the following identities is true for $\nabla \cdot (\phi \mathbf{u})$?}

For the matrix \[ [A] = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 2 & 1 \\ 3 & 1 & 2 \end{bmatrix} \\ which of the following statements is/are TRUE?

A function $f(x)$, that is smooth and convex-shaped (concave downward) on the interval $(x_l,x_u)$ is shown. The function is observed at an odd number of regularly spaced points. If the area under the function is computed numerically, then \underline{\hspace{1cm}.} \includegraphics[width=0.5\linewidth]{image44.png}