List of top Engineering Mathematics Questions asked in GATE Biotechnology

Let \[ A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & k & 0 \\ 3 & 0 & -1 \end{pmatrix}. \] If the eigenvalues of \( A \) are -2, 1, and 2, then the value of \( k \) is _. 
(Answer in integer)

Let \( a_0 = 0 \) and define \( a_n = \frac{1}{2} (1 + a_{n-1}) \) for all positive integers \( n \geq 1 \). The least value of \( n \) for which \( |1 - a_n|<\frac{1}{2^{10}} \) is ______.

The value of \( k \), for which the linear equations \( 2x + 3y = 6 \) and \( 4x + 6y = 3k \) have at least one solution, is ________. (Answer in integer)

Consider a nonlinear algebraic equation, \( e^x - 2 = 0 \). Using the Newton-Raphson method, with the initial guess of \( x_0 = 1 \), the approximated value of the root of the equation after one iteration is ________.