List of top Engineering Mathematics Questions asked in GATE Petroleum Engineering

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