i*,5ht.o,I F{ E f : E

I E; -- : 'h

j$ $ : E''L Ir ! r, EG: i-e x: E'tq.i ' :E r, i i, t ai q ; :iq6 S s ! ia'0.+S ;io-i 3': c, ;ti I lr-* I 3 :.-Ei * ' .-.,,i X :. ;' =Ef . , J i :Ef \ O E E ::i6 *; f :EE f F T :E? ^."s u g'iF; tE f F i' FE 6l F E r ,Ff' ss E E; -E\ At E E: !.r.f Frr: g af st;! H,t'

Y. \v/\/\ A eJlSl

uJ-. l+ l-+ 4$l ,3tnl-.fll c)-l /\er"

\FJ cnJl dr"U ii"al Yj r:13 qii*. iJ. .riSl La-ri:+ --------- (\3l c-ur+ll t + -----, G:'J'11 (Y

Switch user (l------ Ju.ij eiLLJl orA ; crUJiJl cF+ {.rJ'i dl ljrJl 1i13 ------ r ---cJ' $i!i P&Yt :ili' (t

ir. ,r..)t3 cEtll a,Bl .i!l (o

JJI +J uF.P word C.U;rL stiL ri. J*lS uc tjpl gl5 lll (1

Juij J ---- .tr.5 Jj.ri"sll ik+ d-,I-J- Jkt:) (v

slc r&l ir++^:ll iL"tJl 6jJ'r Jslll dr' $j:ll -'jB" (A

jiu i",B- ijJirl Jt ,J:.-./1 ErJl lil (1

eLl"i; taskbar buttons CiJ. Jrlr i l.$c (\ '

irid,ill ci.Js-J,Jl elril (t i+r:sl)l i.-. Ul crl-r- (\ i..1' & $c /ld

sin{ (a+2b)/3 } U)l allJr '.--,rJ +jJlF '-nsl /ro'

JrYl '&,ill 015:rl ilill

Y. \V-Y. \1 s-lJJl d'll

i"-ill Us/ 6t,;. LJI r*,!

J-r)l .iJl

'-- - ++,i ,qS=;jS .r

'o$Jl UUr.

al-rnD Ujt^ .)

edl .r+t

61..!:rL,jl !*l /irLJlio+.1t . ; tCx_itt

&'r1 ot-iJvdJ)l Olj!'Yl iLi'*l

t. \ V-\ ' \1s-lJrll alrJl

4+..;,,11 i$ / i;-'rtill ilts

JJYI i.al['jl'+lrJ'll el

. p(-) Qt} cl,6.Ip rJ!+'rfJ 'e;1'r'11 d!i& rJ'&11 l- -\/ 'o*

ris., OE 111 - , -J, v,AulI],g- !fLi' &ie" lsl^ 0U* e+Fi O-" )r ''*-)!dr A^ ilP dE d^s1 -t

(r)t(pvq) +-, -ql-- (2 ) t(-p^-q) *ql -

(2uJeL'-&lJ+:xcrG2rPi"'ilr;1qx'?;s111)i,r''lildl'r\i,.r;!Ln\jtar+lli"!iJ'Lrlii'l-r

(Uu Ai) - (U981) e l) i,11(114(A11- ri'l)loi !a'lf iJv r'rr'oj*;'r e'vt lB)g ' {A;};'7 o's o9-t

-:!r+'ll Jsle'ofu')1 slJ+ll r3i 3t i-s'a;a /'cr

^ - P 'G 5Ul -PAQ 6 Jt'Jr ojllllr -\

.ai$'' a'rsi 1x,vy,p(x'y) -+ Vy' lx' p(x' y) 'o'!all -t

(p v- q), l- p t (qA- pl r (- p v- q;;3'rr" ors 4J' l;,1l fui\AJl -r

dom (T " R) c rlzn(n-1) otl c 4t3"+ll .,J1 B d,' 4i)eT J B ieJ'+..ll ull a io'e..;'ll s"'43)c R dJs lll 't

. A x (B u C) = (A xB) u (A x 6) crG;r'r..+' crB 'A o'ix 6jll lll--c

.P('4 u B) c P(A) u P(B) OG ic'ri B'A .rj|l 1!l -1

i,rLall i*-2u/ l-i-jg ,9 -rl ..r.p,iaJ!3rllj 6\4 &|^+;U"

U niversity of Al-QadisiYah

College of Education

'"r':l:lt of ma1emlcs

1.t SemesterExam

Academic year

_l!9!91?017

Subject: Mathematical analYsis

Time: 1.5 hrs

Daret].6 /l 1201,7

Ql/ Which of the following statements is true and which of them

is false and why? (12 marks)

l-. For any family {Vo} of closed set in a metric space [X,d), then

laeyVa is closed.

2. Ifthe sequence < an> converge to zero Y n e z+ then

) ar, is convergent series.

3. The equation x2=2 has only one real positive root.

4. For any real number r there exist a sequence of irrational

numbers converge to r.

lf an 2 O Y n,and ! on convergent series then ) #; it

convergent series.

IR is uncountable set.

If (X,d) be a metrtc space , prove that: [6 marks)

L.lf A subset o/ X then4fuA) is closed set.

2.lf O + S g X and S is a closed set then (S,dr) complete

subspace ofX.

5.

6.

Q2/

Lecturer

Fatma K Majeed

Good Luck

Head of Dept.

Dr.Mazin Omran Kareem

2/

irJ;tti ayrt'4tud|

qrfll.;t.aJl lJYt ;4ul dn t

L-. Jil I 4ls/i/- r LiJ I ir. 4

ot+;tlll ?-l

aJil 4+UGll fuu.lt J.rl .i-li. &1s3,!b ;ii! /'\rr

(Zxy' - ?y)ilx + (3x2Y '4x)dY - g

furrr.Jt d-nr x=xt*h t y--y1 + k c!^riJt Ol Ol)'t arbz + bra2 dB lill\ o!'

. ir.?6i,.,,.1t *rtjAl.ii (h,k) e+F fuL"ii;r-i:i'lrl'-"ll *9I = f (ffi)

+ (2x+3y)p3+5(x+ y)p|+(3x+2y)p-g tu'lt 4".,r;6ill tuJ*ll d'/iu{- -dY'P- d*

, y + 0. furt*ll luJl jrll q /tg"

i.bJt 14/t

9e u'

o, +r, = 2ydxy-

^.Jl Jl.Li...all

C/.1-.,J1 et 'YlcJ:Yl:gJl 6lsjrlA!*i

(Y . \ V-Y . '\ l)g+,I-,rrJl ptr.lJ

{#Jill +1Js

.:rt;,.l#ll f*9

O! aAu N(p,o2) !,#L l+iul tilJ ef''+'o'r jJ'*' 4lt+lo & xr,...,x, Ol L':Jil /\cr'{

e2J llri.ll a;ll+ll i-$ e11r pr(O < X <6) + o2 = 100, lt:3, 1525 crjts l3l3 .f" -D?7rf @ -t)

_2 0.933 q+ 1.5

;AJii:Jl r3ss' i

ryirri'+Loril:.. Y,'X,6s41. 1to1'/

E(*') rjltvt.l,.B +1f(4, 4) ,*t e,.'l-lr..:t gn2 {+p +1b/3'e & X, crjE l3l1ur

: giYl rjs.&lr.r li-1l.oll A+lt"i.rYl 4.itistt i.tt.r # crr+tt t#:+ Oe ,Jii,,* e+tsl" €lJr'ii^ x1,...,xn & 11 1r on

o--.9I = {: ',:',.,

:c,iYl cJs.&ltrr-i.t*..jr Ydli-ecll J+ii^ll crl cA* f(*,0)=0e-0' x>0,0>0

e-ke -Jiirl^ J+6

slr^l-yl eirttl o^ 5 rc-r^ +teu lbl rS,x # ,#yr1lz<h<lq<ys o! cFjl (:

f*illU/rdJfiJs OlJ e &jt-..t

. 4*rl uii + f (x) #{ 0<x<1

4.llilt Otg 3q; ss.r

zl-:lL ,51, jLi.iu. .\

.,jtll :dJl

i--::eJt: .:,b+ll ,. Lr"o.,L11

rJ3)l ;a;ll;t'';lt ' ' ':r-'oa*triit9l'raga;r|tEJl .i{JrJl ftl.ll c!.lO'Sljl;le"'

elg* L^€J-"'- r-J*llr Lur,* c9r: rru*Jr-tsJU,

. ",.i r-13

.;Lul ,rs jljtn dE)l .+ r+-r\:a:c+

. .+Jldlo*J+:r ij{:+

Ul ,j-u) <J"ii: 5-5,r agii-. JSI: t !<:+

j a-!jn crr- e]ii.- cjs :t:nf, .d6Yl .J3 ,i4ril* -t cn rJ.fi r5:):l ,t!u ",o:,j"n^ all;llau..Eyr

:(eLr;r v;

.(-lr:.r) ir'- LJ^- a,rti'"Jl LJ.6 ut J^-l L-rJl 4cJ^Jl ;-.p.1;Yg"

.(.tE+;:)A-B-C-D ,lli B-C-D, A-B-D eK lrl'"'i:l .=

;jJl.Jl C- 4-s nL il.n 4l k+-'E <I;i:r;;t.: cj:l:,-i* &lJl riiiJl.l'ru". (uL:.,1lf)

. (uL:;rr)r-lj diil* (JjcJ idi; A,B,C,D .tliJl ;ti A-B-D ., B-C-D ..rjlS lrl

6tJ! 6,SJ.r,t-:

t!, &':uL=r

6ill iJp;r^

iJ.,rtill :str

4:Jill CIS

University of Al-QadisiYaCollage of Education

First Examination2016-2017U2120t7

Time: One hour and half

Dep artm ert ol l4utlr.- ufuQr\ a) Prove or disProve the

1- If (1, +, ') be an idealcommutative.

Qz\ a)b)c)

d)

following statements:

of a ring (R, +, ') such that (R/1, +,

Subject: Ring of Algebra(12 degree)

') commutative ring then (R, +, ')

Z-ln a rtng (2, +, '),-f. Z--Z nontrivial homomorphism then/ equal identity map.

3- Every non zero nilpotent element is zero divisor'

4- The identity element (if there exist) of subring (S, +, ') and ring (R, +, ') is a same'

b) Let (1, +,.) be an ideal of a ring (R, +, '), prove (ann(,Q, +, ') is an ideal' (3 degree)

Let (R, +, ') be a ring with identity and a is nilpotent element, prove l+a is invertible in R'

State without proof theorems of isomorphism ring. (12 degree)

Let (1,+, '; ani (J, *, ') be two ideals of a ring (R, *, '), prove R:1O J iff V x e R can be

written uniquely in the form x:atb , a e I and b e J'

Let X:{1,2,3,,4}, find characteristic of ring (P(X), A,n)'

er\ a) Letf:R-, R'be epirnorphism such that R principal ideal domain, prove that R principal

ideal domain. (9 degree)

b)Letf be a nontrivial homomorphism from simple ring (R, +, ') into ring (R ,* ,' ), prove/

one to one.

c) Prove thal (2,, +,, integral domain iff nGood luck

me number.

Ass.prof. Hassan R.

College of EducationDepartment of Math

EXAM ['t]PART301112017

CalculusTime: 1.5 Hour

Ql: A: Prove or disprove: (Choose Three)

7. lf / is even function then / is injection;

2. Lel a e Z* thenevery number of the tot^ J2a +1is rational;

3. (Q,+,.) Completeness ordered field;

4. lf / and g are odd functions lhen gof is even function.

(9 Mark)

B: What value of x satisfy: (Choose Two)

1. I ' l=r,ls(')l

2. (x2 -2x+l)(x-4)>0;3. sign(x): [r]]

(8 Mark)

Q2: A: Sketch the functions:

(t) f (x): lx' - 3l;

(_t- J,

(2) g(x)= ]r(r,| -rI

Ix+t

(8 Mark)

x>l-l<x<1.x<-l

B: Discuss the composition of the Two functions /(x) : l*l ,

g(x) :.tr +T. (8 Mark)

.\

C: Show lhat Lim ' .. = -cn,-i;1x_z),(7 Mark)

--WITH MY FORTUNE WISHES..

Ass. L. Fieras.load Alysaary

University of Al-Qadisiyah

College of Education

Department of Mathematics

First semester exam 2016 -2017

Subject : Algebra of

Class : second

Time : 90 minutes

Date : 29 lll2017

Groups

Ql / A) Define and give a non-trivial example of the following :

l) finite group 2) odd permutation

3) the order of element in a group 4) semigroup

B) If (H,,r) is asubgroup of the group (G,'r) then prove that either a * H = b * H

or(axH)n(b*H)=@

Q2l A) Prove that every cyclic group is commutative .

B) Write all the subgroups of :

l) the group of symmetries of the triangle . 2) the group (S+,o) of order 4

C)If a,b,c e Z anda= b(modn) then ac = bc(mod n).provethat

ii-,;rJt ,,,ti1

Q3/ Choose the correct answer and prove your answer:

I ) In a group (P(X), A)where X is anon_empty set ,the inverse of .4 is

a)0 b) A" c)A d)x

2) If (G ,*) is a group and x E G then x-2 -_

a) (x-t1-r q ; c) x-L * y-r d) Gx)2

3) In a group (Sr,o) , the number ofnon_cyclic subgroup is

a)1 b)2 c)3 d)4

4) In a group (G,*) , if x,y e G and r-1 _ y then

a)x-y b)x*y-l -e c)x*y-e d)x-1 *y-e

with my best wishes

Examiner : Lect. Dr. Mazen Otnran Karim.

Cyt+Y'----> :2--ts

elJll :, t. rl

,3rJr.!Cl 4ilgj 3;J-lt2011-l-29 '-{1-f)l

d-,Yl cl.aill Or&1 iLi-l2017-2016 sA'lJil PbI

\ -, tr t-r6ll L.l+'t'z l+ or i$d,,l+t1Jll f""!

?V,

Q1) Answer four questionS Only of the following: (5 Marks for each question)

la) Finrl all direct summand submodules of the Z-module Z with proof'

(b) State and prove the Second Isomorphism Theorem for modules'

(c) Let M, N and 11 be finitely generated left X-modules, Show that whether the left R-module M @N @K is

finitely generated or not and whY?

(d) Let M be a left fi-module and A, B,C be left submodules of M. show that whether

.4 + B) n C - GnC) + (B n C) ornotandwhy?

1"1 St oro tttut t hether a submodule of aifrygleft X-module is -module or not and why?

Q2) (r) Give an example of the following with proof:

1- Three left X-homomorphisms ai iNi' M' (i=1,2,3)

2- Short exact sequence of left .R-modules is not split'3- Finitely presented R-module is not free.

(6 Marks)such that a1@ a2@ a3 is an epimorphism'

(b) State an equivalent statement of split monomorphism with proof' (5 Marks)

(c) prove that a cyclic left R-module M =< a > is simple if and only if anna(a) is a maximal left ideal of iR.

d#lrll f {- OL:.J i".ic .r,p.i ;6rlJl gr-.;reP.i OlJ..e OJU .r.t :fl.,,rtll o'+jJ

Qadisiya University

College OfEducation

Department Of Mathematic

Second class

Mathematic

251112017 : time l.5h

Q1,1. If < P, ) is a sequence of parabolas, such that the equation of the

parabora P, is x2 -2ffr*(#)' -+t-ffi-0, which

converges to a parabola P. Find the equation of the parabola P

it's properties with graph. (4marks)

2.Let flx) is a tunction such that f")@):(-1)' /(i, J"*')(o)=o

fla)--l for all neN and a is real number. Find the Taylor series

around 4 ofthis function and the interval of convergence. When a:0determine this function. (5marks)

3. Show that which ofthe following series is absolutely convergent

which ofthem is conditionally convergent or not convergent:

. ..- cos(nn)r- Ln=t--ii-

(3marks)

ii. xff=,L# iii. t;=1C1t*+:! t)* ** + * + * * *,8+ + ** r*.8't + * + t* ** * * ** * *+ * * ** ** **,1. '1.+:i'i{.*,* ** 'l'++:} + + ** + t* * *:t

Q2:1. Convert the periodic number to the rational number: (2marks)

i. 11.171818181818... ii.5.1333333...2. Determine the type and Find the propefties of the following conic

sketch the graph: l6x2 - 9y2 + 64x - 90y = 305. (3marks)

3. Find the value ofa such that the sum ofthe series Xf=, a (!)"equul

to the seventh term of the sequence .; ,; ,+ ,* ,';

With Best Wishes

Alaa Kamel Jaber

(3marks)

g:iJl djLill ; r:tJl

eUll :'; ll

ei.;1 Lt- ; gr;ll

( g>,Ll e+f e ) /trr

'r-:5i-it +-.9.xr r+-9 Ol L" $IL!. s! u(r,y) irill9ill illJl?(r,y) (+lril iilll ul crA;.,l (l.,".ili iiL;l

.4! & 31n,i)l ,G.E ,9 \a) Zs iJ.n ::. OL+-, -.JI ,rilJ.- 6i.j iri- illJ iiB^ l.cl (\.;l.o..1!l 9 iJ"i:lt

Re(27+22) < )!!4 ,tt,t_:: lZ+l + lz3l nls rtr (r14+z4l - llzzl-lz+ll

srilJsj drr., 76 = u(x,y) - iu(x,y) : f(z') = u(x,y) + i?(r,y)'-iK 1! (1

. D djliJl sf iir.E Ah /(z) Ol LA-), . D dlLill

( Ly,Jsl e-.1a ) : siL t- 4+rl !E ','il /\u"

ez -_t+ rJ3 4lJt-lt dir Ct Z #\(rlZ -71 < lZ+11 01s llr!i!1t!t ps(Z') > 0 Ol ,)A-r(r

lezz+i + eiz2l < ezx + e-zrv ,i_,1 oAr (rsinse - 16 Cosa0 - lZ Cos20 + 7,0 + O,* n,l2n ,... ai.!.:Jt uaJi (tsin 0

ez^/1 -zi)i 1i 6;u s 1'(J3 - i)' ('r

6rlrll gr.,1rr

grlrjJl qJe (H .r.r

,SiJillJ Cl+ji.i 3r.+l,l dl+J

s-l!l rt'I qL)l Ol.j,.)l

t.lv-l .\1

i.1urtill i-1-4+_rll lls

,:l:;L"jl r""!

9l.,,"jrri.ll 3 6rU'll

lrlt : iJslls-,lJJl llr.ll &Yl dr.eiJl eJj.'i

l.lv-1.\1

+p,rtilt 3 ir.5e*!t: i$

c.!^i!,rlt : f.!

:A,B,C sr.a+i.ll ..r+l OS tit 3 t..y

- : OE lll B 1,4 t'*0+i'clt 64 L3t;rll -21't!'r r+"'"rl ; fo*

i - 3t - ai ' E =;&trl L-q *.r. qrr: p: .rp ';#rl JrJl ,.,to '{>t r"* : i r'r"X= 50 t+ l0 12

. ,rls:r fi>ErirrU.ll fA f"+! i'L.,Jr"ll a'/l -l-: +si t-0 O.oJl! se

. Jsec crlt .- -.,IJ.I. ;u rr*itr -r . Jsec cxll $e r'+il +',,:3t ic'.p'tl -t

i=3t+3j-21( , E- -t_ 41-*4 ,- d=2t+2j+ftn-r'-r"'.-i.1rt4 ( drE+il ' dxrE+zl ,:..rx+ric (64 0n 1 u-2rri. a-ir-,,.) c*J .E +t .tu:f (qr.('^'i"l : r i x iil ::

. r:"-t-rl f;,', ir,.,*.tr..1*;'::;;lll oia fA *t ' (40 sec )'lt! fi

iJJl gr;'l,r

o".rP O.,..,: -i$ f.e

6biltl ++o.dt+ # e!,.,^,, (i

hh tn.':/'d'r'*lr

=n* sro

Y.TfrspErpGll;,^,+- tfqr? {gstot# rrrir.ro

rr urriq? rEFt

l.(!?n? tdip mJ tnrt#

*mh\Wg#/

rmr;f:; t/ r lt r,t! [ .l-A t .l lPFg: t'fria,

llu'yr:1,r,p

ttlton-l iir.:r: k,,v.u

,.,., 3e<t? !,-Fl

ffi,-,:Jl::*.i:.';rur?rfr

,,? a -- )ilr- " )r:- v s B rrrb r? !?e -r_ {v.t,(2),r,-,.,o

,ro *,'::*. ,. ,r),rv1.1,>e.-:

ll'cng.cf.TF s .

**lr'Y. orfr.,,o v"-i!'f,3,FEc*">

f X r1r

(9'

UniversitY of Al-QadisiYah

College ofEducation

DePartment of Mathematics

c-l'cr" 'hird

Numerical AnalYsis

t-4

Q1: Consider the following set

(4,14), (6,30) bY use divide

aPProximation value of f(2)'

of data (0,-10), (1, 20)'

differences compute the

Q2:Isiterativeformulag(x)=f(x)+xtofindtheroot- t=rl of the functio l9 z!!x -- 1) converge.? if not

n.rain" formula and find the roots for seven iteration

Q3: A : Prove 1- E:ehD' 2' L3Y;Y3Y5'

B : Use Gauss-Jordan elimination method to solve

each sYstem

)y+y-7= -5

x-3y+22+l=0

x-22+Y: -5

best wishes

Dr. KhalidM. MOhammed

2017

,5ll *'Jl ;6.tLll

,Jr!l : iJ.vJlr.ir:gist-;U.jll

&$ d!.!l otsi.l2017- 2016 s+"1.,,.tt1 dr!

i,i-rtiJl ad+L..-,1t1 1+ts

8q.aiJ'll f-6

.i.l*fr &r+ aP ta.Yl : tlr).

d-&eilll fr-Eitl OJ-SJ(rsl b,,br,br*'a'trqr+lJl .tj}iitl rxd.iU,.lt iiLilLJEi^ll '-i* -l lllcv

Xt+X2*2Xr=5,Xr+ Xr= b,

2Xt+X2*3Xr=b,( lii &)E J:ri ) l.l{r. dsl & rt}ci g..if, --

uslll i+.re ii3iall 1l; ql3ft ii3i.a.lt 131 dslJ.ll dll,ll (2) ia.4ill iili.arll 111

; t-i .l,r3,.ll iJLiI LJEj.ll rJ. + -t //2r+X,+2X,-3X, =gzxt+Xr+3X3 =0

X,+5X2-l2Xr=g

.u.arl fu" Jl i.aJi.a. &JS3 f ' cjj e.l, ri uila g crns lll dil i$-; -+

.lel=o' Ol c+l .aq*S J n l€.,,'4-Lqag:1,a^l Os:l-l //3u

ilJi^ll /-rc;,,rrl i.,q-'=fi.aOio>ol oaJE.o{sdli^L1l tn'tr^,,'4'4JA'qi,a,acSl -+

. o=l-' 4l

L3 'l

g ,rs,)l e.;y-)3t3.s.f :6cl4Jl rr,.,a. ((Ct+I+ ei.+X UiUi.3 )) d-.f Olr"r Ojh s.,":,t-alt .11

&lEll; , ;. t1 L";lt+$

rl,arl ;6rUl 2OL6-20L7 flrll dJYl €.,r*Jt Ols:.yl :tt"'i ,:,I+^btcJll f""3!!lttltlltatltllllltttlrlttlltttlltlt!talrllrtrlarrllllllltallllllrrllrlllllt

ep L.Jr g7 ,-P i1.E J,arJ i+.aYl e, sLaill Olsi.yt ,1tl^i (+) JllrJ cit{jll Ols:..Yl Os tit ffii.rull cr4tj d 4J,aill 6Elsi.ylet L...,.r 50s60s 70 s 80 .,Je d,arJ &:l$ll 6.lll,rs,/t+ill OtaS.yl

ft6+.+ r3Ulsi.Yl cr, i*Ell l3a at+J$ g*,$eJl!*.dls (JJt "-rl.L.,3ll JAUi

sllSl €Jl,|S3lt &JJilJ €-rl+lt uil-pjYl3 LHt+ll 'tffi

x 3 7 11 15 19

f 5 4 6 2 3

-:"iYl rjs-lt+ ful- Ll+3o Lte +O 6r"13 ai*l-2.r ailj" (,Jo i.tlqlt ,tLft .rt;"s ip druJ'L..rtl5 ea+

o.td O-r-* .plt-g L,,.g'i.tl dlJrjYl-2

-: UBI c,ru$ LJg.ill s-Lill

Ll"utl

.1l.tr.iYl

.1 iF dS rs.Jl

fuJr. s.3l ffi

4+"S

&/-F lt25-30 30-35 35-40 40-45 45-50 s0-55 55-60

r:rlJlJSill 2 4 5 3 2 L 4

oUlsll o3 cJl.plt-g L,,.,l.tl Lil. -1 iF ds

x 10 30 25 24 15

v 30 25 32 L70 39

''; ;;; ;; ;_';' ;:;;; ;; ; ; ;;;,; ;;; ,;; ;; ;ffiM++"":t i,.4+jlJ*ie 43.,.A oLtr" -,;,nil {* Olps rd

fOtj;.Y oure qi OJS] Ol ,1t-':-t jnt.{3 g osfr Ots l3i-1

t l+ c,r*jl o;*i u* 6l /rirl 9 t-i +:.ir +j+c cils lli-2

t 4* 4j.o OJs Y c.rls !+ oy,i O:E Y Ot cJt-l-tJal-.-3

,:)E dr.JAll {.:.6l l]i. 0.1 dL.:.I+(f) drtr* 0.3 dL.i.!(L) rj+"_r 0.6 dL^l-l+ (W) eS 6JS &.J9 jiffi.P(A) s.313 4i^"Ja freJ dlYl ,* ,x+, &JlljJp gl: A rl.rrJt ;r.at& .r.rr-1 ;oLr".1ty

.P(B) rr.3l3 al.tlri3 e-*tlS &Jsll j39 SAS B r.5rrJl .l,,al& .r.rr-2

.P(C)s.313 ,!Vt 4t' issl3 6-;,o 4i^,r.ia 3136 &.sJl -,y.al& .r.rr-3

,Jl f Cr L!-;* c,,t!-.p u*.r "Jc

B dJJ.all gjir,,;9 ,Jt f r-r" i-i-r. ot!_,p g*:,* n d3ri.a €JL,mi3-,,g aj. .".i-.,U 4+lJ*te 4i..Jh dJri.a -1$il. i

.A &$,.11 ir ,',i- * iSrjt guSi gl cjl*Ll r..JLi -l-,..$ L3,.,*.tt {^!.plt t't gls tit.-1

.l+.r..;p'&-e-,, Ljs..*lt ,.!.,,gll OSSI 6i ,1U^r-t rr -2

i;At-i ob+ cl1..r.p.i ;irt-Jl i-';.Lo Ct+iJls fui. j"lt+ &r.l+X,,.rl# y

o1.r,.,,,1sJ1 iiltJl J.lY1 d,aill 015:')1 1!-'ijB\:Lj.all -/.

6.1113 i.et- ig*;Jl 2O!7 -2A16 g-'l;rll-frJ

iryr6.ll &.1:4#-ill 4$

o$:l+jl fl"!

II

I

L-------

r' :pYl II

. J.-ie 18 ln J,,-L!c lo 5 ;r1+j Jhi"+ (25-1) pG;)l gr =ilJ'ic a'i "i'-" l-if -+

g=(2+3)s-ab(9\* t r[r*' -e

tsl&J O<n+2< 15 cA i.isi..a,.ll ijYl ci..all Fl'ie llJ n€N Ol ''';' 11 i'l3La" tn-,tij e*U* ''i<l :2tl.ifJ , N;.1Ji.,.^11 0.7 J$ll+ cJ:Yl r-i.,all

'-r. j-;j:" qp': X ns:r li;cg ' 3<n+1< !t sA cil3l ' t' tl

.fu X 4+i.,Jl 6;r is,,iLll -;.aLirJl $r i , X a+:^lt Jlj'll SLill {lll ;*,lJS!All L3l-!1 r;5 cr^ Z 4ii4ll

:.L.,,#,Js,-i,i iliiill Je &r.:J lll.c ef-3u 6l command window 4.,,.iLi; r..ji 1lS 4dlEll slil+Yl lij :3cr

1- repmat (4,3,41

2- a(:,21

a

:,-,.lil- i-il L+,.i s! r^ ifu'r.,q P :f.r

Z=1+(x+y)+1+(x+y)2n+ (x+y)3"'.....i."+1+(x+y)n" -i

3- for i=1:5

x-x+1;

disp ('x=',,x);

end

4- x=4,y=6

clea r x

X'

rr;ii .-c.ll o zLr.ill-r .r-r-osll , il-r-r-cl r-c". v v r! li, . ', irlrJ! 4-,-r-;.Lr

rl39 ,9rln c,1"l*, .a,f

i*fJra.!r-tr ZOf Z-ZO\O arr! 4JLyf ;rr-l cj.rYl *,&llil*,,i c'Sl+Jle."i

:(cflz"ll e i3dl) 4Jiuill arr-ll .=Jl e

iJ^ilt ;1rs-i (tcr

Min z:xizxz+X3 s.t.

X2*X3:1

3Xr+X2+3X3> 4

X1,X2,X3 ) 0

i-l*,24 cD-: ;rl :.S .rl-r-ll U a,'iYl rlrsYl eE+ +-ths c,rlS. --

i++-cl e. cJtelJl 4: +i++_l g. dJtcL 4) d+i=+ iJ$ fl gr il'iil L,1.15^ drLou 3 ,J.*; (}JHl ol 't.lu}

'(4tst

qs:I,-!:6rr":..,-6 , i 6-1iill ,J-ll ..,+ jJI u3i+J rJ+Jl u*l!l :rc cJi^: (i:1,...6) X, Ol u--,)I+

.f$ dS,ir JS.. ut cJs!-r 6Jii cJS.d u'lJFll ric

: big-M 4!Jt plrri"'l-r;Yl AiJ'"ill & t+ :2

Min z:80Xr+60X2 s.t.

2X1+3Y, > 1g

xr+x2 > 1

X1,X2 ) 0

-)

'i;r;ii l:.::

$rllrJ..-ryJt^ll

. i:lg

(e

:e-,,11 4!Jt alr:.i'.,L sJYl e 3J,.ill cJ- .+ :3.r

f(X1,X2):Xr+3X2 S.t.

Xr>l

Xr<6

2Xr+ X2 < 20

X1,X2>o

:(\: gi; +SXtillJl MIN I MAX r+ -e

f (xr.,xr) = xl + xr,xz + xg * xq * xs + xZ S.t.

.r+J*ill4qA

Xr*Xz:1

X5 - X6:2

X1,X2>o

S'A+ll Ct;ilI+ rll+-j

crL:Lr Jl .-.,i, ,d rr -J I viJ

dJS ul'lae OjL .r

6JJl g,,Jr^

fiFS 6i'"r.!Er ..