ALCCS
NOTE:
· Question 1 is compulsory and
carries 28 marks. Answer any FOUR questions from the rest. Marks are indicated against each question.
· Parts of a question should be
answered at the same place.
· All
calculations should be up to three places of decimals.
Q.1 a. Find
a real root of the equation by Regula-Falsi method
correct to 4 decimal places.
b. Solve using Gauss
elimination method.
c. Find all the eigenvalues and the corresponding
eigenvectors of the matrix .
d. Find the missing terms in the
following table
x |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
f(x) |
103.4 |
97.6 |
122.9 |
? |
179.0 |
? |
195.8 |
e. Obtain the least square polynomial approximation of degree
two for on [0, 1].
f. Evaluate
.
g. Evaluate by using the Simpson’s
rule, dividing the
interval into 3 parts. Hence find an
approximate value of log . (7
4)
Q.2 a. Solve the matrix equation
using
the Cholesky method.
b. Solve the equations by
relaxation method . (9+9)
Q.3 a. Transform the matrix to tridiagonal form by Given’s
method. Use exact arithmetic.
b. Find all the eigenvalues and the
eigenvectors of the matrix A using Jacobi method where (8+10)
Q.4 a. Find the cubic polynomial which takes the following values
using
x |
0 |
1 |
2 |
3 |
f(x) |
1 |
2 |
1 |
10 |
b. Using Hermite interpolation determine the
values of f(-0.5) and f(0.5) for the following given values of f(x) and (9+9)
x |
f(x) |
|
-1 |
1 |
-5 |
0 |
1 |
1 |
1 |
3 |
7 |
Q.5 a. A slider in a machine moves
along a fixed straight rod. Its distance
x cm along the rod is given below for various values of the time t seconds. Find the velocity of the slider and its
acceleration when t = 0.3 second.
t |
0 |
0.1 |
0.2 |
0.3 |
0.4 |
0.5 |
0.6 |
x |
30.13 |
31.62 |
32.87 |
33.64 |
33.95 |
33.81 |
33.24 |
b. The following table gives the temperature (in degree Celsius) of a cooling body at different instants
of time t (in secs)
t |
1 |
3 |
5 |
7 |
9 |
|
85.3 |
74.5 |
67.0 |
60.5 |
54.3 |
Find
approximately the rate of cooling at t = 8 secs (9+9)
Q.6 a. Evaluate the integral by subdividing the
interval [0, 1] into 2 equal parts and then applying the Gauss-Legendre three
point formula.
b. Evaluate the integral using the Gauss-Hermite
two point and three point formulas. (9+9)
Q.7 a. Solve the initial value problem using the Euler method with stepsize h=0.1 to find
y(0.2).
b. Solve the initial value problemusing
fourth order Runge-Kutta method on the interval [0, 0.4] with stepsize h=0.2.
Compare your result with the exact solution. (8+10)