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Department of Computer Numerical Analysis Coursework – Part I

Gursewak Singh Question – 01

Answers a) b) c) d) e) f)

Audrey speaks French or Danish Audrey speaks French and Danish Audrey speaks French and not that she speaks Danish Audrey can’t speak either French nor Danish Not Audrey Speaks Not Danish Not that Audrey not Speaks French and not speaks Danish

Question – 02

Answers a) b) c) d) e) f)

p¬q ¬p∨¬q ¬p→q ¬p↔¬q ¬p→q ¬p↔¬q

Question – 03 Answer a) p T T F

Q T F T

PV ¬q T T T

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Numerical Analysis Coursework – Part I F

F

F

b) p T T F F

q T F T F

¬ p F F T T

¬ q F T F T

¬p ⋀¬q F F F T

Question – 04 Answer

p T T F F

q T F T F

p⋀ q T F T F

¬ (pVq ) F F F T

(p⋀q)⋀¬( pVq) F F F F

Question – 05

Answers Let p be the statement: if it rains Let q be the statement: Erik will be sick Let r be If did not rain

Questions – 06 Page 2 of 20

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Numerical Analysis Coursework – Part I

Answers p = I study q = fail in mathematics r = play basket ball statement 1 -> if p then ¬q statement 2 -> if ¬r then p statement 3 and 4 -> q hence r from these logical statements Negate Statement 1 : p(¬p) -> q Statement 3,4 " q -> r Negate statement 2 : r - > p

Question – 07 Answer a) p T T F F

q T F T F

p⋀ q T F F F

p T T F F

q p q T T F F T F F F

Logically its Equall

b) p q

¬

p

p q

p→

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Numerical Analysis Coursework – Part I T T T F F T F F

q F T F T

¬q F T F F

T T F F

T F T F

q T F T T

Logically its not equall

Question – 08

Question – 09

Answers a) If the teacher is present, then all the students will complete their homework. b) Some of the students will not complete their homework and the teacher is absent c) All the students complete their homework or the teacher is present.

Question – 10

Answers a) {3} because the counterpart doesn’t agree with 3, with the

equation b) {} because all are odd numbers c) {3,5,7} because {9} is the only prime number d) There cannot be a determinant because the element is itself the same, and for determinant you should have equal row and columns.

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Department of Computer Numerical Analysis Coursework – Part I

Question – 11

Answers a) 3(1,-2,4)-2(3,5,1) = (3,-6,12)-(-6,-10,-2) = (9,4,5) b) 4(3,5,1)-(3,5,1)-3(2,1,-3) =(12,20,4)-(3,5,1)-(-6,-3,9) =(9,15,3)-(-6,-3,9) = (15,18,-6)

Question – 12

Answers a) (1,-2,4)*(3,5,1) =(3,-10,4) (1,-2,4)*(2,1,-3) =(2,-2,-12) (3,5,1)*(2,1,-3) = (6,5,-3) b) √1,-2,4 =√1,4,16 ||u|| = √21 √3,5,1 =√9,25,1 ||v||= √35

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Department of Computer Numerical Analysis Coursework – Part I

√2,1,-3 =√4,1,9 ||w||= √14

Question – 13

Answers a) 2u-3v =2(2,-1,0,-3)-3(1,-1,-1,3) = (4,-2,0,6)-(-3,1,1,-9) =(7,-3,0,15) b) 5u-3v-4w =5(2,-1,0,3)-3(1,-1,-1,3)-4(1,3,-2,2) =(10,-5,0,15)-(-3,3,3,-9)-(4,12,8,-8) =(13,-2,0,24)-(4,12,8,-8) =(9,-14,0,32) c) –u-2v-2w =-(2,-1,0,-3)-2(1,-1,-1,3)-2(1,3,-2,2) =-(2,-1,0,-3)-(-2,2,2,-6)-(-2,-6,4,-4) =(0,-1,0,9)- (-2,-6,4,-4) = (0,7,0,13) d) u.v , u.w , v.w u.v = (2*1,-1*-1,0*-1,-3*3) = (2,1,0,-9) u.w=(2*1,-1*3,0*-2,-3*2) =(2,-3,0,-6) v.w = (1*1,-1*3,-1*-2,3*2) =(1,-3,2,6)

e) ||u|| =√2,1,0,3 =√4,1,0,9 =√14

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Numerical Analysis Coursework – Part I ||v|| =√1,1,1,3 =√1,1,1,9 =v12 ||w|| =√1,3,2,2 =√1,9,4,4 =√18

Question – 14

Answers a) 5u-3v =5

-3

=515-12--6-3-15 = 11183

b) 2u+4v-6w =2

+4

-6

=23-12+8420--1812-36 =1078--1812-36 =-28-5-44

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Department of Computer Numerical Analysis Coursework – Part I

c) u.v , u.w , v.w

u.v = 13-4 X 215 = 23-20

u.w = 13-4 X 3-26 =3-6-24 v.w = 215 X 3-26 =6-230

Question – 15

a) x(2,5)+y(4,-3) = (8,33)

Question – 16

Use the following for Question – 17 to Question – 21

Question – 17

Answers

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Department of Computer Numerical Analysis Coursework – Part I

a) 5A-2B = 5 123-4- 2 50-67 =51015-20 - -10012-7 = 1503-13

b) C+D =1-3426-5 + 37-14-89 = 1+3-3+74+-12+46+-8-5+9 =3438-24 c) 2c – 3D =21-3426-5 - 3 37-14-89 =2-38412-10 - -9-14-3-12-16-27 = 11-17111628-37

Question – 18

a) AB = 123-4 X 50-67 =1*5+2*-61*0+2*73*5+-4*(-6)3*0+-4*7 =5+(-12)0+1415+240+(-28)

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Department of Computer Numerical Analysis Coursework – Part I

=-41439-28

b) BA =

50-67 X 123-4

= 5*1+0*35*2+0*(-4)-6*1+7*37*2+0*(-4) =5+010+0-6+1414+0 =510814

Question – 19

a) 123-4X 1-3426-5

= 1*1+2*21*-3+2*61*4+2*-53*1+-4*23*-3+-4*63*4+-4*(-5) =1+4-3+124+(-10)3+(-8)-9+(-24)12+20 =59-6-5-3322 b) 123-4 X 37-14-89

= 1*3+2*41*7+2*-81*-1+2*93*3+-4*43*7+-4*(-8)3*-1+-4*9 = 3+87+(-8)-1+189+(-16)21+32-3+(-36) =11-117-753-39 c) 50-67 X 1-3426-5

=5*1+0*65*-3+0*65*4+0*(-5)-6*1+7*2-6*-3+7*6-6*4+0*(-5) = 5+0-15+020+0-6+1418+42-24+0

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Department of Computer Numerical Analysis Coursework – Part I

=5-1520-860-24 d) 50-67 X 37-14-89

=5*3+0*45*7+0*(-8)5*-1+0*9-6*3+7*4-6*7+7*(-8)-6*-1+7*9 =15+035+0-5+0-18+28-42+(-56)6+63 =1535-5-8-9869

Question – 20

Answers a) AT

=132-4 b) CT

=12-364-5 c) CTC

=12-364-5 X 1-3426-5 =1*1+2*21*-3+2*61*4+2*(-5)-3*1+6*2-3*-3+6*6-3*4+6*(-5)4*1+-5*24*-3+5*64*4+-5*(-5) =1+4-3+124+(-10)-3+129+36-12+-304+(-10)-12+(-30)16+25 =5-9-6-945-32-6-4241 d) CCT

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Department of Computer Numerical Analysis Coursework – Part I

=1-3426-5 X 12-364-5 =1*1+-3*-3+4*41*2+-3*6+4*-52*1+6*-3+-5*42*2+6*6+-5*(-5) = 1+9+162+-18+(-20)2+-18+(-20)4+36+25 =26-36-3665

Question – 21

Answers a) A2 = AA

=123-4X 123-4 =1*1+2*31*2+2*(-4)3*1+-4*33*2+-4*(-4) =1+62+(-8)3+(-12)6+16 =7-6-922 AA =123-4X 123-4 =1*1+2*31*2+2*(-4)3*1+-4*33*2+-4*(-4) =1+62+(-8)3+(-12)6+16 =7-6-922

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Department of Computer Numerical Analysis Coursework – Part I

∴ A2=AA b) B2=BB

B2 =50-67 X 50-67 =5*5+0*(-6)0*0+0*7-6*5+7*(-6)-6*0+7*7 =25+00+0-30+(-42)0+49 =250-7249 BB =50-67 X 50-67 =5*5+0*(-6)0*0+0*7-6*5+7*(-6)-6*0+7*7 =25+00+0-30+(-42)0+49 =250-7249 ∴ B2=BB c) C2=CC

=1-3426-5 X 1-3426-5 =1*1+-3*-3+4*41*2+-3*6+4*(-5) 1*2+-3*6+4*(-5)2*1+6*(-3)+-

5*42*2+6*6+-5*(-5)2*2+6*6+(

Use the following for Question – 22 to Question – 25 Page 13 of 20

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Department of Computer Numerical Analysis Coursework – Part I

Question – 22

Answers a) A+B =1-12034 + 40-3-1-23 =50-1017 b) 1-12034 +2-3015-1-42-1003

You can’t add this matrix because A=2x3 matrix and C=3x3 matrix c) 3A – 4B = 3 1-12034 - 4 40-3-1-23 = 3-360912 - 160-12-4-812 =-13018017-3

Question – 23

Answers a) AB

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Department of Computer Numerical Analysis Coursework – Part I

=1-12034 X 40-3-1-23 You can’t multiply this matrix because A=2x3 matrix and B=2x3 matrix. b) AC 1-12034

X 2-3015-1-42-1003

=1*2+-1*5+2*(-1)1*-3+-1*-1+2*01*0+-1*-4+2*01*1+-1*2+2*30*2+3*5+4*(1)0*-3+3*-1+4*00*0+3*-4+4*00*1+3*2+4*3

=2+-5+(-2)-3+1+00+4+01+-2+60+15+(-1)0+-3+00+-12+00+6+12 =-5-24514-3-1218 c) AD =1-12034 X 2-13 =1*2+-1*-1+2*30*2+3*-1+4*3 =2+1+60+-3+12 =9-9

Question – 24

Answers a) BC =40-3-1-23 X 2-3015-1-42-1003 =4*2+0*5+(-3)*(-1)4*-3+0*-1+(-3)*04*0+0*-4+(-3)*04*1+0*2+(-3)*31*2+(-2)*5+3*(-1)-1*-3+(-2)*-1+3*0-1*0+(-2)*-4+3*0-1*1+(-2)*2+3*3

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Department of Computer Numerical Analysis Coursework – Part I

=8+0+3-12+0+00+0+04+0+(-9)-2+-10+(-3)-3+2+00+8+0-1+-4+9 =11-120-5-15-1814 b) BD =40-3-1-23 X 2-13 =4*2+0*-1+-3*3-1*2+-2*-1+3*3 =8+0+(-9)-2+2+9 =-19 c) CD =2-3015-1-42-1003 X 2-13 You cannot do this multiplication because C=3x4 matrix and D=3x1

Question – 25

a) AT

= 1-12034 =10-1324 b) ATB

= 10-1324 x 40-3-1-23 =1*4)+(0*-1)+(0*0)(1*0)+(0*-2)+(0*0)(1*-3)+(0*3)+(0*0)(-1*4)+(3*1)+(0*0)(-1*0)+(3*-2)+(0*0)

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Numerical Analysis Coursework – Part I

Question – 26

Question – 27

Question – 28

Answers a) 2-531

Question – 31

(a) 2 4 2-20 0 1-0 5 1 0 1 (e)-2

= -18 =1

(b) 6 3 1 -2

-12 -3 =- 15

(c) 4 0

8+10

-5 2

(d) 1

=8

-5 4+40

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Numerical Analysis Coursework – Part I 8 -2

=44

Question – 32

2 0 1 =2 5 -3 2 + 1015-3 1 5 -3 -2 4 1 -2 4 -2 211-3

-0 1-3

+1 1 5

+1 0 1

14

1-2

1 4

+5

2 1

-3 2 01-

1 4

28 -7-2+35+12=66 (b)-320-256-4-11 56-11-2-26-41+025-41+220-11+520-11-6324-1=-420561-1-30-26+1-32-25 49 -122 =-133 (c)-264-1-314-22 -2-37-22-6-1142+4-1-34-2+164-22+(-3)-2442-1-244-2+464-31+224-11+226-1-3 154-24=130 (d) 713-62-25-21 72+4-16+10+36-1061-3+27-15+27-5+5-2-6+1-14-18+114-6 86-72=14

Question – 33

(a)

7453=7543=21-20=1 A-1=1/[1]7543A-17543ans

(b)2435 |b|=10-12

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Department of Computer Numerical Analysis Coursework – Part I

|b|-2 |b|-1 -1-22/-25/-2ans (c)4-2-63 |c|=12-12 |c|=0 C-1=1/|C| 4-2-6-3 4-2-63ANS (D) 4-2-63 |D| -15+12 =-3 TRANSPOSE OF MATERX 1/|d| 56-2-3 1/-3 56-2-4ANS

Question – 38

(A)12-426-5311-4 LET B =-3212 AB COMBINED IS CALLED MATRIX M=12-4-325-52311-412 (B)

a= 12-426-5310-6 B=31014 AB=M=12-4326-510310-614ANS

Question – 39

(A)1-32-13-97-12-674 B277 AB=M=1-32-123-97-172-6747ANS (B)1231312653107B=761523AB=M=1237131626515310723ANS Page 19 of 20

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Department of Computer Numerical Analysis Coursework – Part I

Question – 41

(41)XYZT1101=X.1+Y.0X.1+Y.12.1+T.O2.1+T.1 =X+0 Z+0

X+Y 2+T

=XX+YZZ+T

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