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Practical Examination April-2015

Subject: Numerical Methods and Optimization Time: 2.00 Hrs

Marks: 50 Class: T.E. (Mechanical) Examination Seat No:

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Instructions:

1. Draw the flowcharts of all methods.2. Solve any one question from 1 & 2 , 3 & 4.3. Submit attached (a) Question paper (b) printout pages to the supervisor.4. Write Examination seat number in printout page as well

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SET-A

Q1. Write a program to solve root of equation f(x)=cosx-1.3x with accuracy of

0.01 using bi-section method. [20]

OR

Q 2. Solve the integral using trapezoidal rule ∫1

4

4 x+2 dx [20]

Q 3. Using Langranges interpolation formula for given set of values find

y(1.5) [30]

OR

Q4 Solve following set of equation using Gauss elimination method

X+3y+z=10 X+2y+5z=12 4X+y+2z=16 [30]

X 0 1 2 5

Y=f(x) 2 3 12 147




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Instructions:


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SET-B

Q1. Write a program to solve root of equation f(x)=-0.9x2+1.7x+2.5. With

initial guesses x1= 2.8 and x2 =3, using three iteration of bi-section method.

[20]

OR

Q2. Solve the integral using simpsons 1/3rd rule ∫0

1

(cosx−x)/(1+x )dx [20]

Q3. Q1. Write the program on Gauss-Elimination Method with Solver. Also draw the Flowchart for the same. Given: Solve 1) 10x+y+z=12; 2)2 x+10y+z=13; 3)

2x+2y+10z=14. [30]

OR

Que 4. Using newtons forward interpolation formula for given set of values

find y (3.5) [30]

X 2 3 4 5 6 7 8 9

Y=F(x) 19 48 99 178 291 444 643 894




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Instructions:


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SET-C

Q1. Write a program to solve root equation f(x)=xexcos3x-0.5 between the

limits of x1=0 and x2=1 with accuracy of 0.01 using false position method.

[20]

OR

Q2. Solve the integral using simpsons 3/8 th rule∫0

1

sinx /(2+3 sinx)dx [20]

Que 3. Solve following set of equation using Gauss Seidal Method.

4x+y+2z=16 x+3y+z=10 x+2y+5z=12 [30]

OR

Q2. Write the program on Lagrange’s interpolation with Solver. Also draw the Flowchart for the same. Given: Find y at x=1.5 for the following data. [30]

x 0 1 2 5y 2 3 12 47




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Instructions:


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SET-D

Que 1. Write a program to solve root equation f(x)= excosx-1.2sinx-0.5. With

initial guesses x1= 0 and x2 =1, using three iteration of regula falsi method [20]

OR

Q2. Write the program on Newton-Raphson Method with Solver. Given: Find root of

f(x)=0.51x-sin(x)=0; do 3 iterations. [20]

Que 3. Find dy/dx at x=0.1 using Newtons Forwards Differentiation [30]

X 0.1 0.2 0.3 0.4 0.5

Y -2.3 -1.6 -1.2 -0.91 -0.69

OR

Que-4 . Write the program on Gauss-Elimination Method Given: 1) 10x+y+z=12; 2)2

x+10y+z=13; 3) 2x+2y+10z=14. [30]




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Instructions:


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SET-E

Q1. Write the program on Newton-Raphson Method with Solver. Also draw the Flowchart

for the same. Given: Find root of f(x)=cos(x)-xex =0; accuracy=0.0001. [20]

OR

Que 2. Solve the double integral using trapezoidal rule.∬ x+ ydxdy for x = 1 to

3 and y= 0 to 2. With step size for both x and y is 1 [20]

Que 3. Using Langranges interpolation formula for given set of values find y(1.5) [30]

OR

Que-4 .Solve following set of equation using Gauss elimination method with partial pivoting

X+3y+z=10 X+2y+5z=12 4X+y+2z=16 [30]

X 0 1 2 5

Y=f(x) 2 3 12 147




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Instructions:


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SET-F

Que-1. Write a program to solve root of equation f(x)=excosx-1.4 with accuracy of 0.001 using newton raphson method. [20]

OR

Que 2. Solve the integral using Guass Legender 2 point formula ∫1

4

x3+x−1 dx

[20]

Que 3. Solve following set of equation using Tridigonal matrix algoritham (Thomas method) method

X+2y=3 2x+3y+z=4 2y-z=1 [30]

OR

Que 4. Using newtons forward interpolation formula for given set of values find y (3.5) [30]

X 2 3 4 5 6 7 8 9Y=F(x) 19 48 99 178 29

1444 64

3894




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Instructions:


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SET-G

Que-1. Write a program to solve root of equation x3 -15x+8=0 using Successive Iteration Method [20]

OR

Que 2. . Write the program on Simpson’s 1/3rd rule with Solver. Given: Find integration of

f(x)=ex-x3-2x+1; with limits 1 to 4. n=6 [20]

Que-3. Solve the following equations by Gauass Seidal Method. x1+20x2+9x3= -23 , 2x1-7x2-20x3= -57 and 20x1+2x2+6x3=28 . Write a Computer program .Also solve it by MATLAB .Assume Acc=0.001 [30]

OR

Que-4. Solve Elliptic Laplace equation w.r.t. grid shown in fig. with specified BCs. Find Temp.,T1, T2, T3, T4 H=K=250.Write a Computer program [30]




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Instructions:


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SET-H

Que 1. Write a program to solve root of equation x3 -15x+8=0 using successive iteration method [20]

OR

Que-2. Solve dy/dx = x+2y for given boundary condition that x=1,y=1 find y(1.4) take h=0.1. using Eulers Method [20]

Que 3. Using newtons forward interpolation formula for given set of values find y (3.5) [30]

X 2 3 4 5 6 7 8 9Y=F(x) 19 48 99 178 29

1444 64

3894

OR




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Instructions:


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Que-4 . Solve the parabolic equation for the following condition using explicit finite difference method at t=0 U= sin πx (0<x<1) at x=0 and x=1, u=0 for all values of t. taking increment in t as 0.02 and increment in x as 0.2. tabulate values of u for t=0 to 0.006 and x=0 to 1. [20]

SET-I

Que-1 . Fit a straight line using following data. Find values of a and b. [20]

X 1 2 3 4 5 6 7Y 0.5 2.5 2 4 3.5 6 5.5

OR

Que-2 . Write the program on Simpson’s 1/3rd rule with Solver. Given: Find integration of

f(x)=ex-x3-2x+1; with limits 1 to 4. n=6 [20]

Que 3. Find dy/dx at x=0.1 using newtons forwards differentiation [30]

X 0.1 0.2 0.3 0.4 0.5




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Instructions:


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Y -2.3

-1.6 -1.2 -0.91

-0.69

OR

Que4. Write the program on Gauss-Elimination Method Given: 1) 10x+y+z=12;

2)2 x+10y+z=13; 3) 2x+2y+10z=14. [20]

SET-KQue-1. Fit a exponential equation y=aebx . find value of a and b. [20]

X 0.1 0.2 0.3 0.4Y 1.8 2.23

82.733 3.338

OR

Que-2. Solve dy/dx = x+y for given boundary condition that x=0,y=1 find y(0.2) take h=0.1. using RK2 method. [20]

Que-3. Solve the following equations by Gauass Seidal Method. x1+20x2+9x3= -23 , 2x1-7x2-20x3= -57 and 20x1+2x2+6x3=28 . Write a Computer program .Also solve it by MATLAB .Assume Acc=0.001 [30]




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Instructions:


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ORQue-4. Solve Elliptic Laplace equation w.r.t. grid shown in fig. with specified BCs. Find Temp.,T1, T2, T3, T4. H=K=250.Write a Computer program [30]

SET-LQue-1. For given data fit a Power equation y=axb . find value of a and b. [20]

X 1 2 3 4 5Y 0.5 2 4.5 8 12.5

OR

Que 2. Solve the simultaneous method using RK2 method [20]

Y’=x+y, y(0)=1, Find y(0.2). Take h=0.1

Que 3. Using Langranges interpolation formula for given set of values find y(1.5) [30]

X 0 1 2 5Y=f(x) 2 3 12 147




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Instructions:


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ORQue-4. Write the program on Gauss-Elimination Method with Solver. Also draw the Flowchart for the same. Given: Solve 1) 10x+y+z=12; 2)2 x+10y+z=13; 3)

2x+2y+10z=14. [30]

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