PYQs

1) Explain the main steps of the Gauss-Jordan method and apply this method to find the inverse of the matrix $\left[\begin{array}{ccc}2& 6& 6\\ 2& 8& 6\\ 2& 6& 8\end{array}\right]$.

[2017, 10M]

2) Find the values of the two-valued variables $A$, $B$, $C$ and $D$ by solving the set of simultaneous equations $$\begin{gathered} A^{\mathrm{\prime }}+A.B=0 \\ A.B=A.C \\ \text{A.B}+A.C^{\mathrm{\prime }}+C.D=C^{\mathrm{\prime }}.D \end{gathered}$$.

[2001, 15M]

1) Apply Gauss-Seidel iteration method to solve the following system of equation:
$2x+y-2z=17$
$3x+20y-z=-18$
$2x-3y+20z=25$, correct to three decimal places.

[2019, 15M]

2) Find the solution of the system
$10x_1-2x_2-x_3-x_4=3$
$-2x_1+10x_2-x_3-x_4=15$ $-x_1-x_2+10x_3-2x_4=27$
$-x_1-x_2-2x_3+10x_4=-9$
using Gauss-Seidel method (make four iterations).

[2015, 15M]

3) Solve the system of equations $2x_1-x_2=7$
$-x_1+2x_2-x_3=1$
$-x_2+2x_3=1$
using Gauss-Seidel iteration method (perform three iterations).

[2014, 15M]

4) Solve the following system of simultaneous equations, using Gauss-Seidel iterative method:

$3x+20y-z=-18$
$20x+y-2z=17$
$2x-3y+20z=25$

[2012, 20M]

5) Given the system of equations:

$2x+3y=1$
$2x+4y+z=2$
$2x+6z+Aw=4$
$4z+Bw=C$

State the solvability and uniqueness conditions for the system. Give the solution when it exists.

[2010, 20M]

6) The equation $x^2+ax+b=0$ has two real roots $\alpha$ and $\beta$. Show that the iterative method given by: $x_{k+1}=-{\displaystyle \frac{(a{x}_k+b)}{x_k}}$, $k=0,1,2\dots$ is convergent near $x=\alpha$, if $|\alpha |>|\beta |$.

[2009, 6M]

7) Apply Gauss-Seidel method to calculate $x$, $y$, $z$ from the system:
$-x-y+6z=42$
$6x-y-z=11.33$
$-x+6y-z=32$
with initial values (4.67, 7.62, 9.05). Carry out computations for two iterations.

[2008, 15M]

8) Using Gauss-Seidel iterative method, find the solution of the following system:
$4x-y+8z=26$ $5x+2y-z=6$ $x-10y+2z=-13$
up to three iterations.

[2004, 15M]

9) Using Gauss Seidel iterative method and the starting solution $x_1=x_2=0$, determine the solution of the following system of equations in two iterations:
$10x_1-x_2-x_3=8$
$x_1+10x_2+x_3=12$
$x_1-x_2+10x_3=10$

Compare the approximate solution with the exact solution.

[2001, 30M]