ap assignment 1

7/31/2019 AP Assignment 1

1/17

Q#01: Solve the following system of linear equations by using crammers rule.

10x+4y-z+2w+u=112

x-10y+z-3w+2u=120

2x-3y+10z+w-2u=135

-x+3y+z-10w+u=145

3x-y+3z-w+10u=99

A B C D E F G H I

1 Matrix

A B X

2 10 4 -1 2 1 112 x

3 1 -10 1 -3 2 120 y

4 2 -3 10 1 -2 135 z

5 -1 3 1 -10 1 145 w

6 3 -1 3 -1 10 99 u

78 det A= 111993

9

10 D1

11 112 4 -1 2 1

12 120 -10 1 -3 2 det D1= 1941372

13 135 -3 10 1 -2 x= 17.33476

14 145 3 1 -10 1

15 99 -1 3 -1 10

16

17 D218 10 112 -1 2 1

19 1 120 1 -3 2

20 2 135 10 1 -2 det D2= -488857

21 -1 145 1 -10 1 y= -4.36507

22 3 99 3 -1 10

23


2/17

24 D3

25 10 4 112 2 1

26 1 -10 120 -3 2 det D3= 1151746

27 2 -3 135 1 -2 z= 10.28409

28 -1 3 145 -10 1

29 3 -1 99 -1 10

30

31 D4

32 10 4 -1 112 1

33

1 -10 1 120 2 det D4=

-

1854876

34 2 -3 10 135 -2 w= -16.5624

35 -1 3 1 145 1

36 3 -1 3 99 10

37

38 D539 10 4 -1 2 112

40 1 -10 1 -3 120 det D5= -53578

41 2 -3 10 1 135 u= -0.4784

42 -1 3 1 -10 145

43 3 -1 3 -1 99

X=|D1|/|A|

Y=|D2|/|A|

Z=|D3|/|A|

w=|D4|/|A|

u=|D5|/|A|


3/17

Q#1(b): Solve the following system of equation by using inversion of matrices.

10x+4y-z+2w+u=112

X-10y+z-3w+2u=120

2x-3y+10z+w-2u=135

-x+3y+z-10w+u=145

3x-y+3z-w+10u=99

A B C D E F G H I

1 Matrix A B X2 10 4 -1 2 1 112 x

3 1 -10 1 -3 2 120 y

4 2 -3 10 1 -2 135 z

5 -1 3 1 -10 1 145 w

6 3 -1 3 -1 10 99 u

7

8 Inverse

A

9 0.100051 0.041815 0.010018 0.010206 -0.01739 x= 17.33476

10 0.006786 -0.09132 0.002464 0.027466 0.015331 y= -4.3650711 -0.02155 -0.03506 0.092175 0.012795 0.026323 z= 10.28409

12 -0.01254 -0.03656 0.005974 -0.09285 0.019046 w= -16.5624

13 -0.02413 -0.01481 -0.02981 -0.01344 0.100756 u= -0.4784

14

BAX1


4/17

Q#1(c): Solve the following system of equation by using Jacobi Method.

10x+4y-z+2w+u=112

X-10y+z-3w+2u=120

2x-3y+10z+w-2u=135

-x+3y+z-10w+u=145

3x-y+3z-w+10u=99

A B C D E F G H

1 Matrix

A B

2 10 4 -1 2 1 1123 1 -10 1 -3 2 120

4 2 -3 10 1 -2 135

5 -1 3 1 -10 1 145

6 3 -1 3 -1 10 99

7 k x y z w u

8 0 0 0 0 0 0

9 1 11.2 -12 13.5 -14.5 9.9

10 2 16.56 -6.1 18.29 -3.03 5.14

11 3 11.903 -7.184 13.349 -4.047 0.358

124 13.5123 -8.9985 13.7509

-11.1022 3.4475

13

5 15.3

-

7.47396 15.29681

-

8.27555 3.73111

14

6 13.9419

-

7.36654 14.25597

-

6.89204 2.295908

15

7 13.86984

-

8.03183 14.06997

-

8.74307 2.866498

16

8 14.4677

-

7.75841 14.58319

-

8.43508 3.195549

17

9 14.21251

-

7.61229 14.4166

-

7.81025 2.80408318

10 14.0849

-

7.79525 14.28303

-

8.23337 2.853522

19

11 14.25112

-

7.76917 14.41564

-

8.29165 2.992484

20

12 14.22519

-

7.70566 14.40819

-

8.08846 2.906055

21 13 14.16853 - 14.35672 - 2.889401


5/17

7.74661 8.16067

22

14 14.20616

-

7.75353 14.38422

-

8.21071 2.933152

23

15 14.21182

-

7.73326 14.39253

-

8.15702 2.919308

24 16 14.19353 -7.74 14.37718-

8.16173 2.907725

25

17 14.19986

-

7.74521 14.38101

-

8.18177 2.918962

26

18 14.20444

-

7.73994 14.38556

-

8.17031 2.918437

27

19 14.19964

-

7.74028 14.38181

-

8.16743 2.914025

28

20 14.20002

-

7.74231 14.3817

-

8.17355 2.916335

29

The following formulas are used in Jacobi Method.

=($G$2-$B$2*C8+$C$2*D8-$D$2*E8-$E$2*F8)/$A$2

=($G$3-$A$3*B8-$C$3*D8+$D$4*E8-$E$3*F8)/$B$3

=($G$4-$A$4*B8+$B$4*C8-$D$4*E8-$E$4*F8)/$C$4

=($G$5+$A$5*B8-$B$5*C8-$C$6*D8-$E$6*F8)/$D$5

=($G$6-$A$6*B8+$B$6*C8-$C$6*D8+$D$6*E8)/$E$6


6/17

Q#1(d): Solve the following system of equation by using Gauss seidal Method.

10x+4y-z+2w+u=112

X-10y+z-3w+2u=120

2x-3y+10z+w-2u=135

-x+3y+z-10w+u=145

3x-y+3z-w+10u=99

A B C D E F G

1 Gauss

seidal

method

2 Matrix A B

3 10 4 -1 2 1 112

41 -10 1 -3 2 120

5 2 -3 10 1 -2 135

6 -1 3 1 -10 1 145

7 3 -1 3 -1 10 99

8 K x y z w u

9 0 0 0 0 0 0

10

1 11.2 -10.88 14.524-

15.1916 4.78996

11

2 16.65892-

12.4812 16.38973-

14.4605 2.679574

12

3 17.17765-

12.4455 15.78008-

14.6699 2.724222

13

4 17.26175

-

12.5519 15.82507

-

14.6845 2.69759614

5 17.30541-

12.5528 15.81272-

14.6843 2.688266

15

6 17.30786-

12.5556 15.81118-

14.6859 2.688438

16

7 17.30945 -12.556 15.8112-

14.6859 2.687997

17

8 17.30967-

12.5561 15.81108 -14.686 2.687978

18

9 17.30972-

12.5561 15.81108 -14.686 2.687967

19

10 17.30973-

12.5561 15.81108 -14.686 2.687965

=($G$3-$B$3*C9+$C$3*D9-$D$3*E9-$E$3*F9)/$A$3

=($G$4-$A$4*B10-$C$4*D9+$D$4*E9-$E$4*F9)/$B$4

=($G$5-$A$5*B10+$B$5*C10-$D$5*E9-$E$5*F9)/$C$5

=($G$6+$A$6*B10-$B$6*C10-$C$6*D10-$E$6*F9)/$D$6

=($G$7-$A$7*B10+$B$7*C10-$C$7*D10+$D$7*E10)/$E$7


7/17

Q#02(a): If y=f(x), then ()

=

[+2(++) +]

Where, h= ( ) , n= 10 given, = f ()

Then evaluate the following integral.

1:

A B C D E F G

1 Trapezoidal

Rule

2 n Xo Xn h X f(x) Integral

3

10 0 1 0.1 0 **0 *0.691478424 0.1 0.198025 0.2 0.384615

6 0.3 0.550459

7 0.4 0.689655

8 0.5 0.8

9 0.6 0.882353

10 0.7 0.939597

11 0.8 0.97561

12 0.9 0.994475

13 1 1

The following formulas have been used:

*=$D$3*(F3+2*SUM (F4:F12) +F13)/2

**=2*E3/ (1+E3^2)


8/17

(b): If y=f(x), then ()

=

[+2(++) +]

Where, h= ( ) , n= given, = f ()


With n=16

A B C D E F G

1 Trapezoidal

Rule


3

16 0 1 0.0625 0 **1 *0.7852354034 0.0625 0.9961095 0.125 0.984615

6 0.1875 0.966038

7 0.25 0.941176

8 0.3125 0.911032

9 0.375 0.876712

10 0.4375 0.839344

11 0.5 0.8

12 0.5625 0.759644

13 0.625 0.71910114 0.6875 0.679045

15 0.75 0.64

16 0.8125 0.602353

17 0.875 0.566372

18 0.9375 0.532225

19 1 0.5

The Following formulas have been used:

*=$D$3*(F3+2*SUM(F4:F18) +F19)/2

**=1/(1+E3^2)


9/17

(c): If y=f(x), then ()

=

[+2(++) +]

Where, h= ( ) , n= given, = f ()


dx and n=16

A B C D E F G

1 Trapezoidal

Rule


316 0 90 5.625 0 0 68.2819728

4 5.625 0.313077

5 11.25 0.441696 16.875 0.538781

7 22.5 0.618614

8 28.125 0.686583

9 33.75 0.745366

10 39.375 0.796488

11 45 0.840896

12 50.625 0.87921

13 56.25 0.91185

14 61.875 0.939107

15 67.5 0.96118716 73.125 0.978233

17 78.75 0.990346

18 84.375 0.997589

19 90 1


*=$D$3*(F3+2*SUM(F4:F18)+F19)/2

**=SQRT(SIN(RADIANS(E3)))


10/17

Q#03: Find the value of sinx.

Sinx= x-

+

-

x=10

A B C D E F G

1

n

x in

degrees x in radians Terms

2 0 10 0.17453293 *0.174533

3 1 -0.00089

4 2 1.35E-06

5

3 -9.8E-10 sinx= 0.1736486 4 4.14E-137 5 -1.1E-16

8 6 2.24E-20

9 7 -3.2E-24

10 8 3.64E-28

11 9 -3.2E-32

12 10 2.35E-36

13 11 -1.4E-40

14 12 7.18E-45

15 13 -3.1E-4916 14 1.17E-53

17 15 -3.8E-58

18 16 1.1E-62

19 17 -2.8E-67

20 18 6.47E-72

21 19 -1.3E-76

22 20 2.47E-81


*=((-1)^A2)*($C$2^(2*A2+1))/FACT(2*A2+1)


11/17

(b): X=15

A B C D E F G

1

n

x in


2 0 15 0.26179939 0.261799

3 1 -0.00299

4 2 1.02E-05

53 -1.7E-08 sinx= 0.258819

6 4 1.59E-11

7 5 -9.9E-15

8 6 4.36E-189 7 -1.4E-21

10 8 3.58E-25

11 9 -7.2E-29

12 10 1.17E-32

13 11 -1.6E-36

14 12 1.81E-40

15 13 -1.8E-44

16 14 1.49E-48

17 15 -1.1E-52

18 16 7.15E-5719 17 -4.1E-61

20 18 2.12E-65

21 19 -9.8E-70

22 20 4.1E-74


*=((-1)^A2)*($C$2^(2*A2+1))/FACT(2*A2+1)


12/17

(c): x=30

A B C D E F G

1n

x indegrees x in radians Terms

2 0 30 0.52359878 0.523599

3 1 -0.02392

4 2 0.000328

53 -2.1E-06 sinx= 0.5

6 4 8.15E-09

7 5 -2E-11

8 6 3.57E-14

97 -4.7E-1710 8 4.7E-20

11 9 -3.8E-23

12 10 2.46E-26

13 11 -1.3E-29

14 12 6.09E-33

15 13 -2.4E-36

16 14 8.02E-40

17 15 -2.4E-43

18 16 6.14E-47

19 17 -1.4E-5020 18 2.91E-54

21 19 -5.4E-58

22 20 9.01E-62


*=((-1)^A2)*($C$2^(2*A2+1))/FACT(2*A2+1)


13/17

Q#03(b): Find the value of cosx.

Cosx= 1 -

+

-.

(1) X=10

A B C D E F G

1

n

x in


2 0 10 0.17453293 *1

3

1

-

0.01523

4

2

3.87E-

05

5 3 -3.9E-08

6

4

2.14E-

11

75 -7.2E-15 cosx= 0.984807753

8

6

1.67E-

18

9 7 -2.8E-22

10

8

3.54E-

26

11 9 -3.5E-30

12

10

2.83E-

3413 11 -1.9E-38

14

12

1.03E-

42

15 13 -4.8E-47

16

14

1.94E-

51

17 15 -6.8E-56

18

19

20

21

22


*=(-1)^(A2)*($C$2^(2*A2))/FACT(2*A2)


14/17

(2) X=15

A B C D E F G

1

n

x in


2 0 15 0.26179939 1

3 1 -0.03427

4 2 0.000196

5 3 -4.5E-07

6 4 5.47E-10

75 -4.2E-13 cosx= 0.965925826

8 6 2.16E-16

9 7 -8.1E-20

10 8 2.33E-23

11 9 -5.2E-27

12 10 9.4E-31

13 11 -1.4E-34

14 12 1.73E-38

15 13 -1.8E-42

16 14 1.66E-46

17 15 -1.3E-50


*= (-1)^(A2)*($C$2^(2*A2))/FACT(2*A2)


15/17


16/17

Q#04: If ( )

,then find the value offrom the following data.

12,29,34,37,25,60,15,20,30,35,42,47,48,50,57,59,24,53,51,58.

A B C D E F

1 x x- ( )2 12 39.3 -27.3 745.29

3 15 -24.3 590.49

4 20 -19.3 372.49

5 24 -15.3 234.09

6 25 -14.3 204.49

7 29 -10.3 106.09

8 30 -9.3 86.49 ( )= 4452.29 34 -5.3 28.09

1035 -4.3 18.49 = 222.61

11 37 -2.3 5.29

1242 2.7 7.29 = 14.92012

13 47 7.7 59.29

14 48 8.7 75.69

15 50 10.7 114.49

16 51 11.7 136.89

17 53 13.7 187.69

16 57 17.7 313.29

19 58 18.7 349.69

20 59 19.7 388.09

60 20.7 428.49


17/17

Q#05: Find the area of circle, perimeter of circle, volume of cylinder, with given

height (h), volume of the sphere for radius r. (radius for circle and sphere is the

same).radius is taken as 4cm, and height is taken as 6cm.

a) Area of circle= = PI()*A2^2b) Perimeter of circle= p= =2*PI()*A2c) Volume of cylinder= v= =PI()*A2^2*B2

d) Volume of sphere= v=

=(4*PI()*A2^3)/3

A B C D E F

1 Radius

(r)

Height

(h)

Area of

circle

perimeter of

circle

volume of

cylinder

volume of

sphere2 4 6 50.26548246 25.13274123 301.5928947 268.0825731

3

4

5

6

7

8

9

ap assignment 1

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