CBCS Scheme
Sixth Semester B.E. Degree Model Question Paper
Numerical Methods and Applications (15CV663)
Time: 3 hrs. Max. Marks: 80
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module
1 a. Using Newton-Raphson method 2x cos x 3 with initial value of x 1.5. (05 Marks)
b. Solve the following equations by Gauss-elimination method: (06 Marks)
1 2 3 9 x x x
1 2 2 3 3 8 x x x
2 1 2 3 3 x x x
c. Solve the following equations by Gauss-Jacobi method: (05 Marks)
20 1 2 2 3 17 x x x
3 1 20 2 3 18 x x x
2 1 3 2 20 3 25 x x x
OR
2 a. Use Fixed point iteration technique to solve cos x xex with initial value of x 0.5. (06 Marks)
b. Solve the following equations by Gauss-Jordan method: (05 Marks)
2x y z =10
3x 2y +3z =18
x 4y 9z =16
c. Find the inverse of the matrix using by Gauss-Jordan method: A








3 2 2
1 3 1
2 1 4
(05 Marks)
Module
3 a. From the following estimate the number of students who obtained marks between 40 and 45:
(05 Marks)
Marks 30-40 40-50 50-60 60-70 70-80
No. of students 31 42 51 35 31
b. Find Lagrangian interpoplation polynomial from: 132.
Find (06 Marks)
c. Evaluate from the following data: (05 Marks)
x 10 20 30 40 50
46 66 81 93 101
OR
4 a. Using Newton's divided difference method evaluate and from data: (06 Marks)
x 4 5 7 10 11 13
48 100 297 900 1210 2028
b. Fit the cubic spline from the 11). Evaluate f(1.5) . (10 Marks)
Module
5 a. Determine
dx
dy
and 2
2
dx
d y
at x 1.1 and x 1.6 from the following data. (07 Marks)
x 1.0 1.1 1.2 1.3 1.4 1.5 1.6
y 7.989 8.403 8.781 9.129 9.451 9.750 10.031
b Estimate ex dx
2
0
2 taking 10 intervals by Trapezoidal formula Simpson's 1/3rd Fformula.
(09 Marks)
OR
6 a. Evaluate
1
0
1 x2
dx
using Romberg's method. (07 Marks)
b. Compute x y dxdy
2
0
2
0
sin(
p p
using Trapezoidal method. (09 Marks)
Module
7 a. Using Taylor's series solve x2 y
dx
dy
given =1. Compute y(0.1) and y(0.2) (07 Marks)
b. Given x2
dx
dy with 1. Compute y(1.4) using Adams Bashforth method
(09 Marks)
OR
8 a. Using Runge-Kutta forth order method solve 2 2
2 2
y x
y x
dx
dy

with y=1 when x=0. Find taking
h=0.2 (08 Marks)
b. Using Modified Euler's method find y(20.2) and y(20.4) given that





y
x
dx
dy
10 log with 5 c
and taking h=0.2. (08 Marks)
Module
9 a. Find y(0.5) and y(0.75) satisfying y x
dx
d y 2
2
with boundary conditions 0 and
2. (07 Marks)
b. Solve 0 xx yy u u for the following square mesh with boundary values given in Fig. Q
Compute u1 to u9 up to 3 iterations. (09 Marks)
Fig.
OR
10 a. Solve the equation ut uxx subjected to the conditions sin for
0 t 0.1 by taking h 0.2 (08 Marks)
b. Using finite difference equation, solve 2
2
2
2
4
dx
d u
dt
d u subjected to
0 and upto 4 steps. Choose h 1 and k 0.5. (08 Marks)