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
the number of terms of the exponential series such that their sum gives the
value of ex correct to six decimal places at x = 1.
b. Find a root of the equation, by the secant method upto two iterations.
c. Factorize
the matrix using LU decomposition.
d. Given the matrix A = I + L +
U, where , L and U are the lower and upper triangular matrices
respectively, decide whether Jacobi
method converges to the solution of Ax = b.
e. Evaluate using Gauss formula
for n = 2.
f. Solve
at x = 0.1 using Euler method.
g. Show that . (7
4)
Q.2 a. Find a real root of correct to four
decimal places using Newton’s Method.
b. One entry in the following
table is incorrect and y is a cubic polynomial in x. Use the difference table
to locate and correct the error: (9+9)
x |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
y |
25 |
21 |
18 |
18 |
27 |
45 |
76 |
123 |
Q.3 a. Solve by Gauss-Seidel method, the following
system of equations:
b. Solve by Relaxation method,
the system of equations:
(9+9)
Q.4 a. Using Jacobi’s method, find all the eigenvalues and the
eigenvectors of the matrix
b. Using inverse interpolation, find the real
root of the equation which is close
to 1.2. (9+9)
Q.5 a. The following table gives the
distance in nautical miles of the visible horizon for the given heights in feet
above the earth’s surface :
x
= height |
100 |
150 |
200 |
250 |
300 |
350 |
400 |
y
= distance |
10.63 |
13.03 |
15.04 |
16.81 |
18.42 |
19.90 |
21.27 |
Find the value of y when x = 218 ft and x = 410 ft.
b. From the given data, find the maximum value of y: (9+9)
x |
-1 |
1 |
2 |
3 |
y |
-21 |
15 |
12 |
3 |
Q.6 a. Use Romberg’s method to compute correct to four decimal places.
b. Determine f(x) as a polynomial in x for the following data: (9+9)
x |
-4 |
-1 |
0 |
2 |
5 |
y |
1245 |
33 |
5 |
9 |
1335 |
Q.7 a. Using Runge-Kutta method of fourth order,
solve for y(0.1), y(0.2) given that .
b. Using Taylor’s series method, solve at x = 0.1, 0.2. (12+6)