List of top Numerical Analysis Questions asked in GATE MA

If Jacobi method is used to solve the following system of linear equations
\[ \begin{pmatrix} 1 & 2 & 1 \\ 0 & 2 & 2 \\ 1 & 1 & 1 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 2 \\ 2 \\ 2 \end{pmatrix} \]
with the initial guess \( x^{(0)} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} \) and \( x^{(i)} = \begin{pmatrix} x_1^{(i)} \\ x_2^{(i)} \\ x_3^{(i)} \end{pmatrix} \), \( i = 1,2,3,\ldots \), denotes the \( i^{th} \) iterate, then the value of \( \left| x_1^{(2)} + x_2^{(2)} + x_3^{(2)} \right| \) is equal to ______.

Consider the differential equation \(\frac{dy}{dx} = x + y\) with the initial condition \(y(0) = 1\). Using the modified Euler's method, the second approximation to \(y(h)\), where \(h = 0.05\) (step size), is equal to ______. (rounded off to TWO decimal places)