List of top Engineering Mathematics Questions asked in GATE Petroleum Engineering

The eigenvalues of the matrix \[ A = \begin{bmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3 \end{bmatrix} \] are $\lambda_1, \lambda_2, \lambda_3$. The value of $\lambda_1 \lambda_2 \lambda_3 (\lambda_1 + \lambda_2 + \lambda_3)$ is:

A stationary tank is cylindrical in shape with two hemispherical ends and is horizontal, as shown in the figure. $R$ is the radius of the cylinder as well as of the hemispherical ends. The tank is half filled with an oil of density $\rho$ and the rest of the space in the tank is occupied by air. The air pressure, inside the tank as well as outside it, is atmospheric. The acceleration due to gravity ($g$) acts vertically downward. The net horizontal force applied by the oil on the right hemispherical end (shown by the bold outline in the figure) is:

Let $f(x) = \ln x$. The first derivative $f'(x)$ is to be calculated at $x=1$ using numerical differentiation. $f'(1)$ is calculated using: - First-order forward difference ($f'_{FD}$), - First-order backward difference ($f'_{BD}$), - Second-order central difference ($f'_{CD}$), with interval width $h = 0.1$. The correct order of the values of $f'_{FD}, f'_{BD}, f'_{CD}$ is:

If $\dfrac{dy}{dx} + y = x$, and $y(0) = 0$, then the value of $y(1)$ is:

The product of the roots of the equation $x^4 + 1 = 0$ is ............ (answer in integer).

Which of the following statement(s) is/are CORRECT?

Let $\vec{A} = 2\hat{i} - \hat{j} + \hat{k}$ and $\vec{B} = \hat{i} + \hat{j}$, where $\hat{i}, \hat{j}, \hat{k}$ are unit vectors. The projection of $\vec{B}$ on $\vec{A}$ is:

The eigenvalues of the matrix

are $ \lambda_1, \lambda_2, \lambda_3 $. The value of $ \lambda_1 \lambda_2 \lambda_3 ( \lambda_1 + \lambda_2 + \lambda_3 ) $ is:

Which of the following statement(s) is/are CORRECT?

Four fair coins are tossed simultaneously. The probability that at least one tail turns up is:

If a function $ f(x) $ is continuous in the closed interval $ [a, b] $ and the first derivative $ f'(x) $ exists in the open interval $ (a, b) $, then according to the Lagrange's mean value theorem:
\[ \frac{f(b) - f(a)}{b - a} = f'(c) \] If $ a = 0 $, $ b = 1.5 $, and $ f(x) = x(x - 1)(x - 2) $, then the value(s) of $ c $ in $ [a, b] $ is/are:

Let $ f(x) = \ln x $. The first derivative $ f'(x) $ is to be calculated at $ x = 1 $ using numerical differentiation. $ f'(1) $ is calculated using first order forward difference ($ f'_{FD} $), first order backward difference ($ f'_{BD} $), and second order central difference ($ f'_{CD} $), using interval width $ h = 0.1 $. The CORRECT order of the values of $ f'_{FD} $, $ f'_{BD} $, and $ f'_{CD} $ is:

The product of the roots of the equation $ x^4 + 1 = 0 $ is ....... (answer in integer).

If $ \frac{dy}{dx} + y = x $, and $ y(0) = 0 $, then the value of $ y(1) $ is:

Let $ \vec{A} = 2\ell - \mathbf{j} + \mathbf{k} $ and $ \vec{B} = \ell + \mathbf{f} $, where $ \ell, \mathbf{f}, $ and $ \mathbf{k} $ are unit vectors. The projection of $ \vec{B} $ on $ \vec{A} $ is:

The value of $ \lim_{x \to \frac{\pi}{2}} \left( \frac{\cos x}{x - \frac{\pi}{2}} \right) $ is:

The directional derivative of $f = x^{3} + 4y^{2} + z^{2}$ at the point $P(2,1,3)$ in the direction of the vector $\vec{V} = 3\hat{\imath} - 4\hat{k}$ is ............. (rounded to one decimal place).

Using Simpson’s one-third rule (with step size $h=0.25$), the area under the curve $y = e^{-x^{3}}$, from $x=0$ to $x=1$ is ............. (rounded to two decimal places).

The value of $\displaystyle \int_{0}^{\pi} \int_{-1}^{1} r^{2}\sin^{2}\theta \, dr \, d\theta$ is

Consider a vector field $\vec{V} = x^{3}\,\hat{\imath} + 2y^{2}x\,\hat{\jmath} + 0.5\,z\,\hat{k}$, where $\hat{\imath}$, $\hat{\jmath}$, and $\hat{k}$ are the unit vectors in $x$, $y$, and $z$ directions, respectively. The divergence of $\vec{V}$ at the point $(1,2,1)$ is ............. (rounded to one decimal place).

In the given diagram, ovals are marked at different heights (h) of a hill. Which one of the following options P, Q, R, and S depicts the top view of the hill?

Let $\mathbf{X}=\begin{bmatrix} x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33} \end{bmatrix}$ be a $3\times 3$ matrix. The determinant of $\mathbf{X}$ is $5$. The determinant of matrix $\mathbf{Y}=\begin{bmatrix} x_{11} & x_{12} & x_{13} \\ 2x_{21} & 2x_{22} & 2x_{23} \\ 3x_{31} & 3x_{32} & 3x_{33} \end{bmatrix}$ is __________.

Let $z_1$ and $z_2$ be two arbitrary complex numbers with non-zero modulus. Which of the following conditions is FALSE?

In the 4th order Runge-Kutta method for solving ordinary differential equations with step size $h<1$, the ratio of the order of local error to the order of global error is

Residency is a famous housing complex with many well-established individuals among its residents. A recent survey conducted among the residents of the complex revealed that all of those residents who are well established in their respective fields happen to be academicians. The survey also revealed that most of these academicians are authors of some best-selling books. Based only on the information provided above, which one of the following statements can be logically inferred with certainty?