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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/
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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)
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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
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