كتاب (نماذج أسئلة الإمتحان التنافسي/ إعداد علي إبراهيم الموسوي) الجزء الثاني والثلاثون: ماجستير هندسة مدني كلية الهندسة الجامعة المستنصرية ... ماجستير علوم حاسبات كلية علوم الرياضيات والحاسوب جامعة الكوفة ... ماجستير علوم حياة/ أحياء مجهرية كلية التربية للعلوم الصرفة جامعة ديالى ... ماجستير علوم فيزياء كلية العلوم جامعة تكريت ... ماجستير علوم حياة كلية العلوم جامعة تكريت ... ... ماجستير علوم حياة كلية التربية للبنات جامعة تكريت ... ماجستير أدب كلية التربية للبنات قسم اللغة العربية جامعة تكريت ... دكتوراه أدب كلية التربية للبنات قسم اللغة العربية جامعة تكريت ... دكتوراه تأريخ إسلامي قسم التأريخ كلية التربية للعلوم الإنسانية جامعة تكريت ... ماجستير رياضيات كلية علوم الحاسوب والرياضيات جامعة تكريت ... ماجستير قانون/العام كلية الحقوق جامعة تكريت ... دكتوراه قانون/العام التخصص الدقيق كلية الحقوق جامعة تكريت.

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Samples of Competitive Examination
Questions: Part XXXII
Technical Report · June 2016
DOI: 10.13140/RG.2.1.4167.8966/1
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2. ][‫واﻟﺛﻼﺛون‬ ‫اﻟﺣﺎدي‬ ‫اﻟﺟزء‬
‫ﺻﻔﺣﺔ‬0
‫اد‬‫إ‬
‫ي‬‫ا‬‫ا‬‫إ‬
‫راه‬‫د‬
‫ا‬‫ن‬‫ا‬‫أ‬‫ذج‬

3. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬
‫ن‬‫ا‬‫أ‬‫ا‬‫ت‬‫را‬‫ا‬)‫ا‬‫راه‬‫وا‬(
‫ا‬‫و‬ ‫ا‬‫ا‬‫ت‬‫ا‬   ‫د‬‫أن‬‫أر‬  ‫درا‬
‫ا‬‫وا‬‫ا‬.
‫ي‬‫ا‬‫ا‬‫إ‬
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‫ي‬‫ا‬‫ا‬‫إ‬
Ali Ibrahim Al-Mosawi
Ali I.Al-Mosawi

4. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

5. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

6. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

7. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

8. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

9. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

10. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

11. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

12. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

13. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

14. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

15. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

16. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

17. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

18. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

19. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

20. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

21. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

22. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

23. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

24. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

25. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

26. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

27. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

28. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

29. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

30. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

31. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

32. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

33. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

34. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

35. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

36. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

37. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

38. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

39. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

40. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

41. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

42. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

43. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

44. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

45. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

46. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

47. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

48. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

49. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

50. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

51. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

52. ‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

53. ‫الرحيم‬ ‫الرحمن‬ ‫هللا‬ ‫بسم‬
‫رياضيات‬ :‫القسم‬ ‫تكريت‬ ‫جامعة‬
:‫الوقت‬ ‫والرياضيات‬ ‫الحاسوب‬ ‫علوم‬ ‫كلية‬3‫ساعات‬
‫اسئلة‬‫التنافسي‬ ‫االمتحان‬)‫(الماجستير‬ ‫العليا‬ ‫للدراسات‬ ‫المتقدمين‬ ‫للطلبة‬
‫الدراسي‬ ‫للعام‬2015–2016
:‫مالحظات‬
1.: ‫وهي‬ ‫مجاميع‬ ‫اربع‬ ‫من‬ ‫مكونة‬ ‫االسئلة‬-
‫والجزئية‬ ‫االعتيادية‬ ‫التفاضلية‬ ‫المعادالت‬ , ‫والتكامل‬ ‫التفاضل‬ ‫مجموعة‬(20)‫درجة‬
‫العقدي‬ ‫,التحليل‬ ‫الرياضي‬ ‫التحليل‬ , ‫الرياضيات‬ ‫اسس‬ ‫مجموعة‬‫,التبولوجي‬(35)‫درجة‬
‫والحلقات‬ ‫الزمر‬ ‫جبر‬ , ‫الخطي‬ ‫الجبر‬ ‫مجموعة‬(20)‫درجة‬
‫العددي‬ ‫التحليل‬ , ‫الرياضي‬ ‫االحصاء‬ , ‫االحتمالية‬ ‫مجموعة‬(25)‫درجة‬
2.‫لل‬ ‫المحدد‬ ‫الوقت‬( ‫االولى‬ ‫مجموعة‬36( ‫الثانية‬ ‫المجموعة‬ , )‫دقيقة‬63,)‫دقيقة‬
( ‫الثالثة‬ ‫المجموعة‬36, )‫دقيقة‬( ‫الرابعة‬ ‫المجموعة‬45‫د‬.)‫قيقة‬
3.‫لكل‬ ‫يكون‬ ‫وبذلك‬ ‫المجموعة‬ ‫عنوان‬ ‫عليه‬ ‫ويكتب‬ ‫مجموعة‬ ‫لكل‬ ‫خاص‬ ‫دفتر‬ ‫يفرد‬‫مشارك‬
‫التنافسي‬ ‫باالمتحان‬. ‫امتحانية‬ ‫دفاتر‬ ‫اربع‬
4.‫يحق‬‫التنافسي‬ ‫باالمتحان‬ ‫للمشارك‬‫لحساب‬ ‫مجموعة‬ ‫أي‬ ‫في‬ ‫الفائض‬ ‫الوقت‬ ‫يستخدم‬ ‫ان‬
.‫االخرى‬ ‫المجاميع‬
‫ابراهيم‬ ‫حسين‬ ‫أ.م.د.حسن‬
‫التنافسي‬ ‫االمتحان‬ ‫لجنه‬ ‫رئيس‬
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

54. ‫تكريت‬ ‫جامعة‬: ‫المـادة‬‫التفاضل‬,‫والتكامل‬‫التفاضلية‬ ‫المعادالت‬
‫كليـة‬‫علوم‬‫والرياضيات‬ ‫الحاسبات‬‫والجزئية‬ ‫االعتيادية‬
‫الرياضيات‬ ‫قسم‬:‫التـاريخ‬13/8/2015
‫أسئلة‬‫ا‬‫ال‬‫متحان‬‫التنافسي‬)‫(الماجستير‬ ‫العليا‬ ‫للدراسات‬ ‫المتقدمين‬ ‫للطلبة‬
‫الدراسي‬ ‫العام‬2015-2016
‫مالحظة‬:: ‫الزمن‬‫ساعات‬ ‫ثالث‬
ulusalcC
Q1 Mark two of the following by T if it is true or F when it is false
1- If a function f is continuous at x = a, then it has a tangent line at x = a.
2- f(g(x)) = g(f(x)) for any two functions f and g
3- The derivative of f(x) = a x
with respect to x, where a is a constant, is x a x-1
.
(2marks)
Q2 full one of the following blanks with correct answer
1- Let the closed interval [a , b] be the domain of a function f. The domain of f(x - 3)
is given by ---------
(a) (a , b) (b) [a , b] (c) [a - 3 , b - 3] (d) [a + 3 , b + 3]
2- 𝐥𝐢𝐦
𝒙→∞
(
𝒄𝒐𝒔(𝒙)−𝟏
𝒙
)= ---------
(a) 1 (b) 0 (c) -1 (d) ∞
(1marks)
Q3 Answer one of the following
1- For f(x) = ln x, find the first derivative of the composite function defined by
𝑭(𝒙) = 𝒇𝝄𝒇(𝒙)
2- Reverse the order of the double integral ∫ ∫ 𝒅𝒚𝒅𝒙
𝒆 𝒙
𝟏
𝟐
𝟎
.
(2marks)
Ordinary and partial differential equations
Q1 Mark five of the following by T if it is true or F when it is false
1- The partial differential equation contain only one independent variable and only
one dependent variable
2- The differential equation x2
y''+xy' -9y=5x is Euler equation.
3- The general solution for the linear homogeneous differential equation of order n
with constant coefficient when the roots are distinct be in the form
𝒚 = 𝒄 𝟏 𝒆 𝒎 𝟏 𝒙
+𝒄 𝟐 𝒆 𝒎 𝟐 𝒙
+… . +𝒄 𝒏 𝒆 𝒎 𝒏 𝒙
Where c1,c2,….,cn are arbitrary constants.
4- The vector functions u1,u2,…,un are linearly dependent if W(u1,u2,…,un )(t)=0.
5- The integral factor for the differential equation 3
2
dy
xy xy
dx
  is
2xdx
e 
6- The equation
3 3 2
( ) 2 0x y dx xy dy   is homogeneous equation of degree
three
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

55. (5marks)
Q2 Define two of the following
1- Initial condition 2- Ordinary Linear Equation. 3- Rank of Equation. (4marks)
Q3 Solve the following
1- Explain how the initial value problem
𝒚′′′
− 𝟐𝒙 𝟐
𝒚′′
+ 𝟓𝒙𝒚′
+ 𝟑𝒚 = 𝒄𝒐𝒔𝒙
𝒚(−𝟐) = 𝟑
𝒚′(−𝟐) = 𝟏
𝒚′′(−𝟐) = −𝟐
has the unique solution in interval R=(x:- ∞ < 𝒙 < ∞ }.
2- Solve the differential equation 𝒚′′
+ 𝒚 = 𝒔𝒆𝒄(𝒙)
(6marks)
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

56. ‫تكريت‬ ‫جامعة‬: ‫المـادة‬,‫تحليل‬ ,‫رياضيات‬ ‫اسس‬‫تبولوجي‬
‫كليـة‬‫علوم‬‫والرياضيات‬ ‫الحاسبات‬:‫التـاريخ‬13/8/2015
‫الرياضيات‬ ‫قسم‬
‫أسئلة‬‫ا‬‫ال‬‫متحان‬‫التنافسي‬)‫(الماجستير‬ ‫العليا‬ ‫للدراسات‬ ‫المتقدمين‬ ‫للطلبة‬
‫الدراسي‬ ‫العام‬2015-2016
‫مالحظة‬:: ‫الزمن‬‫ساعات‬ ‫ثالث‬
Foundation mathematics
Q1 Mark two of the following by T if it is true or F when it is false
1- ( N , ≤ ) is well ordered
2- If p,n,mN then ((p=nm) ∧ ( n ≠ 1) → ( p < m )
3- Let A,B are countable sets then 𝑨 ∪ 𝑩 is uncountable set.
(2marks)
Q2 full one of the following blanks with correct answer
1- If 𝒇: 𝑿 → 𝒀 is one – to – one mapping and B  Y then ------------
(a) 𝒇 (𝒇−𝟏(𝑩)) = 𝒇(𝑿) ∪ 𝑩 (b) 𝒇 (𝒇−𝟏(𝑩)) = 𝒇(𝑿) ∩ 𝑩 (c) 𝒇 (𝒇−𝟏(𝑩)) = 𝑩
2- Let 𝒇: 𝑿 → 𝒀 be a function and let 𝑨 ⊆ 𝑿 , 𝑩 ⊆ 𝒀 then ------------
(a) A 𝒇−𝟏
(𝒇(𝑨)) (b) 𝒇−𝟏
(𝒇(𝑨)) ⊆ 𝑨 (c) A= 𝒇−𝟏
(𝒇(𝑨))
(1marks)
Q3 Answer one of the following
1- Let 𝒏 ∈ 𝒁+
prove that ≡ 𝒏 is equivalence relation on Z
2- prove that ≤ is a partial order relation on N.
(2marks)
Analysis
Q1 Mark six of the following by T if it is true or F when it is fals
1- If
n
n
n
a 






1
1 , n=1,2,3,... then the sequence  na is converge .
2- The set of rational numbers is closed.
3- If X is metric space then XXf : is continuous function if and only if
𝐥𝐢𝐦
𝒏→∞
{𝒇(𝒙 𝒏)} = 𝒇 (𝐥𝐢𝐦
𝒏→∞
{𝒙 𝒏})
4- If  nf is sequence of differentiable functions that converge
to f then f is differentiable function.
5- √𝟐|𝒛| ≤ |𝑹𝒆𝒁| + |𝑰𝒎𝒁|
6- 𝑨𝒓𝒈𝒁 = −𝑨𝒓𝒈(
𝟏
𝒁
)
7- The function 𝒇(𝒛) = 𝒛 is differentiable every where .
8- The function 𝒇(𝒛) = (𝒛 𝟐
− 𝟐)𝒆−𝒙
𝒆−𝒊𝒚
is not entire
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

57. (6marks)
Q2 Full six of the following blanks with correct answer
1- Every monotonic function is ------------ function.
(a) differentiable (b) Riemann integrable (c) continuous (d) no one of them.
2- The set 𝑺 ⊆ 𝑹 𝒏
is compact set if and only if S is -------------
(a) Closed (b) bounded (c) closed and bounded (d) open and bounded
3- If 𝒂 is positive real number and n is positive integer number then the equation
axn
 has ---------- positive real solution .
(a) One (b) no (c) countable set of (d) no one of them
4- If  nf is ------------ to f then   nn ff limlim .
(a) pointwise converge (b)uniform converge (c) increasing sequence
(d) decreasing sequence
5- The set of irrational numbers is ------------ set .
(a) countable finite (b) countable infinite (c) uncountable (d) negligible
6- The function f is Riemann integrable if and only if the set of
discontinuous points for f is ------------ set.
(a) Finite (b) infinite (c) closed (d) negligible
7- For every complex number z = x+iy we have |𝒔𝒊𝒏𝒉𝒚| ------------- |𝒔𝒊𝒏𝒛|
(a) ≤ (b) ≥ (c) = (d) ≠
(6marks)
Q3 Solve the following
1- Suppose that E and G are sets in a metric space (X,d) , where G is open
set. Prove that if G E   then G E  .
2- prove that 𝒙 𝟐
− 𝒚 𝟐
= 𝟏 can be written as 𝒛 𝟐
+ 𝒛
𝟐
= 𝟐 , where z=x+iy .
(8marks)
Topology
Q1 Mark three of the following by T if it is true or F when it is false
1- In a T2 - space the convergent sequence is convergent to a unique point
2- In T2 – space every finite set has a limited point
3- If 𝒇(𝑬)̅̅̅̅̅̅ ⊆ 𝒇(𝑬̅) then 𝒇: (𝑿, 𝝉) → (𝑿∗
, 𝝉∗
) is closed function such that
(𝑿∗
, 𝝉∗) is a subspace of topological space (𝑿, 𝝉)
4- Every Hilbert space is a topological space
(3marks)
Q2 full three of the following blanks with correct answer
1- Every sequentially compact is ----------
(a) Compact (b) locally compact (c) count ably compact (d) no one of them
2- If E∩d(E) =  we say that E is ---------
(a) Isolated (b) superset (c) perfect (d) ) no one of them
3- Any topological space is connectedness if and only if each if X and  is ----------
(a) Open set (b) closed set (c) perfect (d) clopen
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

58. 4- Every compact Housdorff space is ---------
(a) Normal (b) [CN] (c) Regular (d) no one of them (3marks)
Q3 Prove that A∪ 𝑨′ is closed set where A'=d(A)
(4marks)
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

59. ‫تكريت‬ ‫جامعة‬
‫كليـة‬‫علوم‬‫والرياضيات‬ ‫الحاسبات‬: ‫المـادة‬,‫خطي‬ ‫جبر‬‫وحلقات‬ ‫زمر‬ ‫جبر‬
‫الرياضيات‬ ‫قسم‬:‫التـاريخ‬13/8/2015
‫أسئلة‬‫ا‬‫ال‬‫متحان‬‫التنافسي‬‫المتقدمين‬ ‫للطلبة‬)‫(الماجستير‬ ‫العليا‬ ‫للدراسات‬
‫الدراسي‬ ‫العام‬2015-2016
‫مالحظة‬:: ‫الزمن‬‫ساعات‬ ‫ثالث‬
Linear algebra
Q1 Mark two of the following by T if it is true or F when it is false
1- If A,B,C are matrices such that AB=AC then B=C .
2- If L:V→ 𝑾 is linear transformation and dim(V)=dim(W)=n then L is
one – to – one if and only if L is onto.
3- If A is 𝒏 × 𝒏 matrix then Ax=0 has non zero solution if and only if
|𝑨| ≠ 𝟎, where|𝑨| is the determinate of A.
(2marks)
Q2 full one of the following blanks with correct answer
1- The vectors 𝒙 𝟏 = (𝟒, 𝟐, 𝟔, −𝟖) and 𝒙 𝟐 = (−𝟐, 𝟑, −𝟏, −𝟏)in R4
are ---------- .
(a) orthogonal (b) opposite to each other (c) in the same direction
(d) no one of them .
2- If 𝑳: 𝑹 𝟖
→ 𝑹 𝟔
is linear transformation and dim(rang(L))=5 then
dim(ker(L)) = ----------.
(a) 1 (b) 3 (c) 2 (d) 8 .
(1marks)
Q3 Answer one of the following
1- If A is 𝒏 × 𝒏 matrix prove A=S+K , where 𝑺 𝑻
= 𝑺 , 𝑲 𝑻
= −𝑲.
2- Find basis for W={(
𝒂
𝒃
𝒄
) : 𝒃 = 𝒂 + 𝒄, 𝒂, 𝒃, 𝒄 ∈ 𝑹}.
(2marks)
Groups and rings
Q1 Mark five of the following by T if it is true or F when it is false
1- Let G be a group and 𝒂 ∈ 𝑮 . if 𝒂 𝒎
= 𝒆 then O(a) divides m.
2- Let p be a prime number and n , m be positive integers s.t p divides nm, then p
divides n and m .
3- If G=(a) is a group such that O(G)=n , then for each positive integer k divides n ,
the group G has one subgroup of order k .
4- Let A be a finite ring and 𝒂, 𝒃 ∈ 𝑨 s.t ab=1 then ba=1
5- The ideal generated by 2 is maximal in 𝒁 𝒏 for any odd positive integer 𝒏 ≥ 𝟐
6- The Centre of ring is an ideal
(5 marks)
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

60. Q2 Full four of the following blanks with correct answer
1- Let H be a subgroup of a group G , then H is normal iff …………… for each g
in G
2- Let a be an element in a group G s.t O(a) =20 then O(𝒂 𝟔
)=…………..
3- Let a, b be two elements in a group G s.t O(a)=n and O(b)=m and gcd(n,m)=1 ,
then 𝑯 = (𝒂) ∩ (𝒃) = ……………
4- The all zero devisor elements of the ring 𝒁 𝟏𝟐 are ………….
5- Let 𝒇: 𝑹 𝟏 → 𝑹 𝟐 be a homomorphism of rings, then 𝒌𝒆𝒓 𝒇 = {𝟎} if and only if f is
………….
(4 marks)
Q3 Solve the following
1- If G is a group such that (𝒂𝒃) 𝟐
= (𝒂 𝟐
𝒃 𝟐) ∀ 𝒂, 𝒃 ∈ 𝑮. Prove that G is commutative .
2- Let I,J be two distinct maximal ideal of a commutative ring A with 1 , prove that
𝑰𝑱 = 𝑰⋂𝑱
(6 marks)
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

61. ‫ةةت‬‫ة‬‫تكري‬ ‫ةةة‬‫ة‬‫جامع‬: ‫ةةـادة‬‫ة‬‫الم‬‫ةةيح‬‫ة‬‫رياض‬ ‫ةةاي‬‫ة‬ ‫اح‬ ‫ةةةح‬‫ة‬‫احتمالي‬
‫كليـة‬‫علوم‬‫والرياضيات‬ ‫الحاسبات‬‫عددي‬ ‫تحليل‬
‫الرياضيات‬ ‫قسم‬:‫التـاريخ‬13/8/2015
‫أسئلة‬‫ا‬‫ال‬‫متحان‬‫التنافسي‬)‫(الماجستير‬ ‫العليا‬ ‫للدراسات‬ ‫المتقدمين‬ ‫للطلبة‬
‫الدراسي‬ ‫العام‬2015-2016
‫مالحظة‬:: ‫الزمن‬‫ساعات‬ ‫ثالث‬
Probability
Q1 Mark two of the following by T if it is true or F when it is false
1- If 𝑨 anb 𝑩 are two Events then 𝑷(𝑨 ∪ 𝑩) = 𝑷(𝑨) + 𝑷(𝑩)
2- If 𝑿 and 𝒀are two independent random variables then 𝑷(𝑿𝒀) = 𝑷(𝑿)
3- If 𝑨 anb 𝑩 are two Events where 𝑷(𝑨) = 𝟎. 𝟑 anb 𝑷(𝑩) = 𝟎. 𝟒
then 𝑷(𝑨 ∪ 𝑩) = 𝟎. 𝟏𝟐
(2marks)
Q2 full one of the following blanks with correct answer
1- If 𝑿 ∼ 𝑮𝒆𝒐𝒎𝒆𝒕𝒓𝒊𝒄 (𝒑) then 𝑬(𝑿) = -----------
2- The moment generated function of r.v. 𝑿 is 𝑴 𝑿(𝒕) =----------
(1marks)
Q3 Answer one of the following
1- If 𝑷(𝑨) = 𝒂 anb 𝑷(𝑩) = 𝒃 find 𝑷(𝑨̅⋂𝑩̅)
2- If 𝑿 ∼ 𝒃𝒊𝒏𝒐𝒎𝒊𝒂𝒍 (n,p) find the probability distributed of r.v. 𝒀 = 𝒏 − 𝑿.
(2marks)
Mathematical statistic
Q1 Mark three of the following by T if it is true or F when it is false
1- The Poisson Distribution is continuous.
2- The distribution function of r.v. X is equal's to ∫ 𝒇(𝒙)𝒅𝒙
∞
−∞
3- The characteristic function of r.v. X is equal's to 𝑬(𝒆𝒕𝒙)
4- If 𝑿 ∼ 𝒃𝒊𝒏(𝒏, 𝒑) if 𝑿𝒀 =
𝑿−𝒏𝒑
𝒏𝒑(𝟏−𝒑)
∼ 𝑵(𝟎, 𝟏) where p is larger and n is smaller
(3 marks)
Q2 full three of the following blanks with correct answer
1- The r.v.s 𝑿 𝒏 Converges in Distribution to r.v 𝑿 if ----------
2- Two r.v.s are Equivalent if ------------
3- Let 𝑨 ⊆ 𝛀 and function 𝑰 𝑨: 𝛀 → {𝟎, 𝟏} which is define by---------- is called
characteristic function.
4- The density function of r.v 𝑿~ Uniform (a,b) is equal's ------------ (3 marks)
Q3 Prove that if 𝑿 ∼ 𝑵(𝟎, 𝟏) then 𝑿 𝟐
∼ 𝝌 𝟐
(𝟏). (4 marks)
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

62. Numerical analysis
Q1 Mark three of the following by T if it is true or F when it is false
1- The Inherent Error is the error product by substitute the infinite equation by
finite equation.
2- The Secant method depended on the different of signal.
3- The convergence of Newton-Raphson method is global.
4- The Partial Pivot is substitute the row between them.
(3 marks)
Q2 full three of the following blanks with correct answer
1- A formula of Newton- Raphson method to find the sequare root is
a-
2
1
2
i
i
i
x A
x
x


 b-
2
1
2
i
i
i
x A
x
x



c- 1
1
( )
2
i i
i
A
x x
x
   d- 1
1
(2 )
2
i i
i
A
x x
x
  
2- The enough condition to convergent the fixed point method is
a- ( ) , 0g x L L  b- ( ) , 1g x L L 
c-
( ) , 1g x L L 
d-
( ) , 0g x L L 
3- If we have the following data
x 3.1 3.2
( )f x 1.1311 1.1632
Then (3.16)f =
a- 1.1334 b-1.1552 c-1.1505 d-1.1515
4- If you have the equation ln( ) 1 0x x   with root in the interval [1,2] ,
then if you use the bisection method then the interval which contain the root
in the second iteration is
a- [1.2,2] b- [1.5,2] c- [1.75,2] d-[1,1.75]
(3 marks)
Q3 Solve one of the following
1- What is the number of iteration which you need to find the approximation root
with any  by use Bisection method.
2- Use the Newton-Raphson Method to find the general term of r
a .
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

63. (4 marks)
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

64. ‫الرحيم‬ ‫الرحمن‬ ‫هللا‬ ‫بسم‬
‫رياضيات‬ :‫القسم‬ ‫تكريت‬ ‫جامعة‬
:‫الوقت‬ ‫والرياضيات‬ ‫الحاسوب‬ ‫علوم‬ ‫كلية‬3‫ساعات‬
‫اسئلة‬‫التنافسي‬ ‫االمتحان‬)‫(الماجستير‬ ‫العليا‬ ‫للدراسات‬ ‫المتقدمين‬ ‫للطلبة‬
‫الدراسي‬ ‫للعام‬2015–2016
:‫مالحظات‬
1.: ‫وهي‬ ‫مجاميع‬ ‫اربع‬ ‫من‬ ‫مكونة‬ ‫االسئلة‬-
‫والجزئية‬ ‫االعتيادية‬ ‫التفاضلية‬ ‫المعادالت‬ , ‫والتكامل‬ ‫التفاضل‬ ‫مجموعة‬(20)‫درجة‬
, ‫الرياضيات‬ ‫اسس‬ ‫مجموعة‬‫التحليل‬‫,التبولوجي‬(35)‫درجة‬
‫مجموعة‬‫والحلقات‬ ‫الزمر‬ ‫جبر‬ , ‫الخطي‬ ‫الجبر‬(20)‫درجة‬
‫العددي‬ ‫التحليل‬ , ‫الرياضي‬ ‫االحصاء‬ , ‫االحتمالية‬ ‫مجموعة‬(25)‫درجة‬
2.‫لل‬ ‫المحدد‬ ‫الوقت‬( ‫االولى‬ ‫مجموعة‬36( ‫الثانية‬ ‫المجموعة‬ , )‫دقيقة‬63,)‫دقيقة‬
( ‫الثالثة‬ ‫المجموعة‬36, )‫دقيقة‬( ‫الرابعة‬ ‫المجموعة‬45.)‫دقيقة‬
3.‫لكل‬ ‫يكون‬ ‫وبذلك‬ ‫المجموعة‬ ‫عنوان‬ ‫عليه‬ ‫ويكتب‬ ‫مجموعة‬ ‫لكل‬ ‫خاص‬ ‫دفتر‬ ‫يفرد‬‫مشارك‬
‫التنافسي‬ ‫باالمتحان‬. ‫امتحانية‬ ‫دفاتر‬ ‫اربع‬
4.‫يحق‬‫التنافسي‬ ‫باالمتحان‬ ‫للمشارك‬‫لحساب‬ ‫مجموعة‬ ‫أي‬ ‫في‬ ‫الفائض‬ ‫الوقت‬ ‫يستخدم‬ ‫ان‬
.‫االخرى‬ ‫المجاميع‬
‫ابراهيم‬ ‫حسين‬ ‫أ.م.د.حسن‬
‫التنافسي‬ ‫االمتحان‬ ‫لجنه‬ ‫رئيس‬
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

65. ‫تكريت‬ ‫جامعة‬: ‫المـادة‬‫التفاضل‬,‫والتكامل‬‫التفاضلية‬ ‫المعادالت‬
‫كليـة‬‫علوم‬‫والرياضيات‬ ‫الحاسبات‬‫والجزئية‬ ‫االعتيادية‬
‫الرياضيات‬ ‫قسم‬:‫التـاريخ‬2/9/2015
‫أسئلة‬‫ا‬‫ال‬‫متحان‬‫التنافسي‬)‫(الماجستير‬ ‫العليا‬ ‫للدراسات‬ ‫المتقدمين‬ ‫للطلبة‬
‫الدراسي‬ ‫العام‬2015-2016
‫مالحظة‬:: ‫الزمن‬‫ساعات‬ ‫ثالث‬
ulusalcC
Q1 Mark two of the following by T if it is true or F when it is false
1- If f ' is the derivative of f, then the derivative of the inverse of
f is the inverse of f '.
2- 𝒄𝒐𝒔(𝒎𝒙) 𝒄𝒐𝒔(𝒏𝒙) = 𝟏/𝟐[(𝒄𝒐𝒔( 𝒎 + 𝒏) 𝒙 − 𝒄𝒐𝒔( 𝒎 − 𝒏) 𝒙]
3- To find the linear approximation to a function at x = a you need to know the
first derivative of that function.
(2marks)
Q2 full one of the following blanks with correct answer
1- lim [e x
-1] / x as x approaches 0 is equal to -----------
(a) 1 (b) 0 (c) is of the form 0 / 0 and cannot be calculated.
2- A critical number c of a function f is a number in the domain of f such that
(a) 𝒇′(𝒄) = 𝟎 (b) 𝒇′(𝒄) is undefined (c) (a) or (b) (d) no one of them
(1marks)
Q3 Answer one of the following
1- Let 𝒈(𝒙) =
𝒙
𝒙−𝟏
and 𝒇(𝒙) =
𝟏
𝒙+𝟐
are functions define on R{1},R{-2}
respectively. Find the domain of g𝝄𝒇
2-Find the volume of the solid in the first quarter bounded by 𝒙 = 𝟒 − 𝒚 𝟐
𝒂𝒏𝒅 𝒕𝒉𝒆 𝒑𝒍𝒂𝒏𝒆 𝒛 = 𝒚 𝒂𝒏𝒅 𝒙 = 𝟎 , 𝒚 = 𝟎
(2marks)
Ordinary and partial differential equations
Q1 Mark five of the following by T if it is true or F when it is false
1- Every linear combination for solutions of linear homogeneous differential equation
is not a solution of this equation.
2- [
𝒄𝒐𝒔𝒕
−𝒔𝒊𝒏𝒕
] is not a solution of the equation y'=Ay where A= [
𝟎 𝟏
−𝟏 𝟎
]
3- The system 𝒚 𝟏(𝒕) = 𝒆𝒕
, 𝒚 𝟐(𝒕) = 𝟏 +
𝒆 𝟐𝒕
𝟐
is solution for initial value problem:
𝒚′ 𝟏 = 𝒚 𝟏 , 𝒚 𝟐′ = 𝒚 𝟏
𝟐
𝒚 𝟏(0)=1 , 𝒚 𝟐(0)=3/2
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

66. 4-
2xdx
e  is integral factor for the differential equation
3
2
dy
xy xy
dx
 
5- The general solution for the equation 4 3 8 x
y y y xe    is
3x x
cy Ae Be
 
6- Let the initial value problem
𝒅𝒚
𝒅𝒕
= 𝒇(𝒕, 𝒚) ; 𝒚(𝒕 𝟎) = 𝒚 𝟎
If 𝒇 and
𝝏𝒇
𝝏𝒚
are continuous , then the initial value problem has a unique solution.
(5marks)
Q2 Define two of the following
1- Partial differential equation
2- Linear differential equation
3- initial Value Problem.
(4marks)
Q3 Solve the following
1- Find the solution of the equation
2
2 1 0x y xy   
2- Determine whether the functions sin(t) and cos(t-𝝅/𝟐) are linearly
independent or not on any arbitrary interval
(6marks)
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

67. ‫تكريت‬ ‫جامعة‬: ‫المـادة‬,‫تحليل‬ ,‫رياضيات‬ ‫اسس‬‫تبولوجي‬
‫كليـة‬‫علوم‬‫والرياضيات‬ ‫الحاسبات‬:‫التـاريخ‬2/9/2015
‫الرياضيات‬ ‫قسم‬
‫أسئلة‬‫ا‬‫ال‬‫متحان‬‫التنافسي‬‫للدراسات‬ ‫المتقدمين‬ ‫للطلبة‬)‫(الماجستير‬ ‫العليا‬
‫الدراسي‬ ‫العام‬2015-2016
‫مالحظة‬:: ‫الزمن‬‫ساعات‬ ‫ثالث‬
Foundation mathematics
Q1 Mark two of the following by T if it is true or F when it is false
1- If X,Y,Z,W are sets and 𝒇: 𝑿 → 𝒀 , 𝒈: 𝒀 → 𝒁 , 𝒉: 𝒁 → 𝑾 are functions then
(𝒉𝝄𝒈)𝝄𝒇 = 𝒉𝝄(𝒈𝝄𝒇)
2- If {𝑨𝒊}𝒊 is family of subsets of X then 𝒇(⋂ 𝑨𝒊𝒊 ) = ⋂ 𝒇𝒊 (𝑨𝒊)
3- let n > 0 be any natural number. On Z , define the relation R by xRy if and only if n
divide x-y , then R is an equivalence relation
(2marks)
Q2 full one of the following blanks with correct answer
1- If 𝒇𝝄𝒈 is one – to – one then 𝒈 is ------------------
(a) one – to – one (b) onto (c) bounded (d) no one of them
2- let R be a relation on the natural numbers N define by xRy if and only if x divide y in
N then R is -------------
(a) symmetric (b) reflexive (c) antisymmetric (d) no one of them
(1marks)
Q3 Answer one of the following
1- Show that ⋂ (−
𝟏
𝒏
, 𝟏 +
𝟏
𝒏
)∞
𝒏=𝟏 = [𝟎, 𝟏]
2- Let 𝒈 and 𝒇 are functions such that 𝒈𝝄𝒇 is onto prove g is onto.
(2marks)
Analysis
Q1 Mark six of the following by T if it is true or F when it is false
1- The set






 ,...,,
3
1
2
1
1S is negligible set.
2- If 𝝁∗( 𝑺) = 𝟎 then S is countable set, where 𝝁∗( 𝑺) is the outer measure of
the set S.
3- the set of irrational number is closed set in R
4- Every Cauchy sequence in R is converge.
5- If 𝒇(𝒛) and 𝑓(𝑧) are entire function on the domain D then 𝒇 is constant function
6- 𝐜𝐨𝐬(𝒊𝒛) = 𝒄𝒐𝒔𝒚 𝒄𝒐𝒔𝒉𝒙 + 𝒊𝒔𝒊𝒏𝒚 𝒔𝒊𝒏𝒉𝒙.
7- the function 𝒇(𝒛) = (𝒛 𝟐
− 𝟐)𝒆−𝒙
𝒆−𝒊𝒚
is not entire
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

68. 8- 𝒕𝒂𝒏−𝟏
𝒛 =(i/2)log((i+z)/(i-z))
(6marks)
Q2 Full six of the following blanks with correct answer
1- If f is Riemann integrable function then f is ------------ function .
(a) continuous (b) not continuous (c) Riemann integrable
(d) not Riemann integrable
2- If 𝒇: [𝒂, 𝒃] → 𝑹 is continuous function then f is --------------
(a) differentiable (b) not differentiable (c) bounded (d) unbounded
3- Every increasing function is ------------ function.
(a) differentiable (b) Riemann integrable (c) continuous
(d) non of the above .
4- The equation 82 x has ---------------
(a)One positive rational root (b) one positive real root (c) two rational roots
(d) no one of them.
5- If 𝒆 𝒛
= 𝟏 + √ 𝟑𝒊 𝒕𝒉𝒆𝒏 𝒛 = --------------------
(a) 2i (b) i (c) 2 (d) -i
6- Log(-i) = ----------------
(a) -𝝅i (b) 𝝅i (c) -𝝅i/2 (d) 𝝅i/2
7- (−𝒊)𝒊
= ---------------- .
(a) 𝝅i/2 ( b) 𝒆 𝝅𝒊/𝟐
(c) –𝝅i/2 (d) 𝒆−𝝅𝒊/𝟐
(6marks)
Q3 Solve the following
1- Suppose that ),( dX is metric space where X is the set of rational numbers
and Xyxyxyxd  ,),( . Prove that the set  32: 2
 xXxE is closed
subset of X .
2- prove that
𝟏
𝟐𝝅𝒊
∮
𝒆 𝜶𝒛
𝒛 𝟐+𝟏
𝒅𝒛𝒄
= 𝒔𝒊𝒏𝜶 where 𝒄: |𝒛| = 𝟑
(8marks)
Topology
Q1 Mark three of the following by T if it is true or F when it is false
1- Every metric space is T2 – space
2- Hilbert space is locally compact space
3- Every compact set in T2 – space is closed
4- Every locally connectedness is connected
(3marks)
Q2 full three of the following blanks with correct answer
1- Let E be a subset of a topological space (𝑿, 𝝉). If E=d(E) then E is called --------
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

69. (a) Scattered set (b) dense (c) perfect (d) clopen
2- Let X={𝒂, 𝒃, 𝒄} and let 𝝉 = {∅, 𝑿, {𝒂}, {𝒃}, {𝒂, 𝒃}} then (X, 𝝉) is ---------
(a) Normally space (b) regular space (c) [CN] (d) [CR]
3- Any topological space is connectedness if and only if each of X and  is ----------
(a) Open set (b) closed set (c) perfect (d) clopen
4- Every metric space is -----------
(a) [CN] (b) [CR] (c) [N] (d) [R]
(3marks)
Q3 Prove that if E ⊆(X, 𝝉) then 𝑬 𝚶
= 𝑬 𝒄̅̅̅ 𝒄
(4marks)
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

70. ‫تكريت‬ ‫جامعة‬
‫كليـة‬‫علوم‬‫والرياضيات‬ ‫الحاسبات‬: ‫المـادة‬,‫خطي‬ ‫جبر‬‫وحلقات‬ ‫زمر‬ ‫جبر‬
‫الرياضيات‬ ‫قسم‬:‫التـاريخ‬2/9/2015
‫أسئلة‬‫ا‬‫ال‬‫متحان‬‫التنافسي‬)‫(الماجستير‬ ‫العليا‬ ‫للدراسات‬ ‫المتقدمين‬ ‫للطلبة‬
‫الدراسي‬ ‫العام‬2015-2016
‫مالحظة‬:: ‫الزمن‬‫ساعات‬ ‫ثالث‬
algebraLinear
Q1 Mark two of the following by T if it is true or F when it is false
1- If V is n dimensional vector space and 𝑺 = { 𝒙 𝟏, 𝒙 𝟐, … , 𝒙 𝒏}are linearly
independent vectors in V then S is basis for V.
2- Let V be the space of all continuous functions on the interval (−∞, ∞) and
𝑾 = { 𝒇 ∈ 𝑽: 𝒇( 𝟎) = 𝟓}then W is subspace of V.
3- If A ,B are two matrices such that AB=0 then either A=0 or B=0 .
(2marks)
Q2 full one of the following blanks with correct answer
1- If A is 𝒏 × 𝒏 matrix has distinct eigenvalues then A is similar to ----------
matrix.
(a) non singular (b) singular (c) diagonal (d) no one of them.
2- If 𝑳: 𝑽 → 𝑹 𝟓
is onto linear transformation and dim(ker(L)) = 2
then dim(V) =---------.
(a) 7 (b) 5 (c) 3 (d) 1 .
(1marks)
Q3 Answer one of the following
1- If A is diagonalizable matrix prove 𝑨 𝒌
is diagonalizable matrix for any
positive integer number k.
2- If { 𝑿 𝟏, 𝑿 𝟐, 𝑿 𝟑} are linearly independent prove { 𝒀 𝟏, 𝒀 𝟐, 𝒀 𝟑} are linearly
independent where 𝒀 𝟏 = 𝑿 𝟏 + 𝑿 𝟐 + 𝑿 𝟑 , 𝒀 𝟐 = 𝑿 𝟐 + 𝑿 𝟑 , 𝒀 𝟑 = 𝑿 𝟑 .
(2marks)
Groups and rings
Q1 Mark five of the following by T if it is true or F when it is false
1- If G=(a) s.t O(G)=n , then for each positive integer k divides n , the group G has one
subgroup of order k .
2- Let G be a finite group and let H be a subgroup of G , then O(G) divides O(H) .
3- For any elements a, b in G and any integer n , then (𝒂−𝟏
𝒃𝒂) 𝒏
= 𝒂−𝒏
𝒃 𝒏
𝒂 𝒏
.
4- The ideal generated by 2 is maximal in 𝒁 𝒏 for any odd positive integer 𝒏 ≥ 𝟐
5- A ring 𝒁 𝟓 is an integral domain
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

71. 6- Let 𝒇: 𝑹 𝟏 → 𝑹 𝟐 is a homomorphism of rings, if I is an ideal of 𝑹 𝟐 then f −𝟏
(I) is an
ideal 𝑹 𝟏 .
(5 marks)
Q2 Full four of the following blanks with correct answer
1- Let a be an element in a group then the subgroup (𝒂 𝟏𝟐) ∩ (𝒂 𝟏𝟖) = …………….
(a) (𝒂 𝟑𝟔
) (b) (𝒂 𝟏𝟐) (c) (𝒂 𝟏𝟖) (d) (𝒂 𝟔)
2- Let H and K be two subgroups of a group G s.t 𝑫 = 𝑯 ∩ 𝑲 ≠ {𝒆} if O(H)=14 and
O(K)=35 then O(D)=………………….
(a) 49 (b) 14 (c) 35 (d) 7
3- Intersection of two subrings is ………
a- not subring b- subring c- ideal d – center of subring
4- Let 𝒇: 𝑹 → 𝑹́ is a homomorphism of rings, if I is an ideal of R and f is ……….
function, then f (I) is an ideal of 𝑹́ .
a- one-one b- onto c- identity d- inverse
5- The all zero devisor elements of the ring 𝐙 𝟏𝟐 are …………..
a-{2,3,4,6,8} b-{4,6,8,9,10} c-{2,3,8,9,10} d-{2,3,4,6,8,9,10}
(4 marks)
Q3 Solve the following
1- Let G be a group with exactly 4 elements . prove that G is abelian .
2- If U is an ideal of a ring R; let r(U) = {x ∈ R : xu = 0 for all u ∈ U}.Prove that r(U) is
an ideal of R
(6 marks)
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

72. ‫ةةت‬‫ة‬‫تكري‬ ‫ةةة‬‫ة‬‫جامع‬: ‫ةةـادة‬‫ة‬‫الم‬‫ةةيح‬‫ة‬‫رياض‬ ‫ةةاي‬‫ة‬ ‫اح‬ ‫ةةةح‬‫ة‬‫احتمالي‬
‫كليـة‬‫علوم‬‫والرياضيات‬ ‫الحاسبات‬‫عددي‬ ‫تحليل‬
‫الرياضيات‬ ‫قسم‬:‫التـاريخ‬2/9/2015
‫أسئلة‬‫ا‬‫ال‬‫متحان‬‫التنافسي‬)‫(الماجستير‬ ‫العليا‬ ‫للدراسات‬ ‫المتقدمين‬ ‫للطلبة‬
‫الدراسي‬ ‫العام‬2015-2016
‫مالحظة‬:: ‫الزمن‬‫ساعات‬ ‫ثالث‬
Probability
Q1 Mark two of the following by T if it is true or F when it is false
1- If 𝑨 anb 𝑩 are two Events then 𝑷𝒓(𝑨⋂𝑩) = 𝑷𝒓(𝑨) + 𝑷𝒓(𝑩)
2- If 𝑷𝒓(𝑿) = 𝟏 then 𝑷𝒓(𝑿 𝒄) =0.5.
3- 𝐏𝐫(𝒂 ≤ 𝑿 ≤ 𝒃) = 𝐏𝐫(𝑿 > 𝒂) + 𝐏𝐫(𝑿 > 𝒃)
(2marks)
Q2 full one of the following blanks with correct answer
1- If 𝑿 ∼ 𝑵 (𝝁, 𝝈 𝟐) then Var(𝑿) =-----------
2- If 𝑿 ∼ 𝝌 𝟐(𝜶) then 𝑴 𝑿(𝒕) =------------
(1marks)
Q3 Answer one of the following
1- If 𝑨 and 𝑩 are two events prove that 𝐏𝐫(𝑨⋃𝑩) = 𝑷𝒓(𝑨) + 𝐏𝐫(𝑩) − 𝐏𝐫(𝑨⋂𝑩).
2- If 𝑿𝒊 ; 𝒊 = 𝟏, 𝟐, … , 𝒏 are random sample with size 𝒏
where 𝑿𝒊 = 𝒊 𝒇𝒐𝒓 𝒆𝒂𝒄𝒉 𝒊 = 𝟏, 𝟐, … , 𝒏 prove that 𝑬(𝑿) =
𝒏+𝟏
𝟏
.
(2marks)
Mathematical statistic
Q1 Mark three of the following by T if it is true or F when it is false
1- The Normal Distribution is continuous Distribution .
2- The distribution function of discrete r. v. 𝑿 is equal's 𝐏𝐫(𝑿 ≤ 𝒙) = ∫ 𝒇(𝒙)𝒅𝒙
∞
−∞
.
3- If 𝑿 ∼ 𝑷𝒐𝒊𝒔𝒔𝒐𝒏(𝒑) then 𝑬(𝑿) = 𝒑
4- If 𝑿 ∼ 𝑩𝒆𝒓𝒏𝒐𝒖𝒍𝒍𝒊(𝒑) then 𝑬(𝑿) = 𝟏 − 𝒑
(3 marks)
Q2 full three of the following blanks with correct answer
1- If 𝑿 ∼ 𝑵 (𝝁, 𝝈 𝟐) then Var(𝑿) = -----------
2- If 𝑿 ∼ 𝝌 𝟐
then 𝒀 = √ 𝑿 ∼ --------------
3- If 𝑿 ∼ Geometric(p) then 𝒇(𝒙) = ------------
4- Let 𝑿𝒊 are independent r.vs. and 𝑿𝒊 ∼ 𝑩𝒆𝒓𝒏𝒐𝒖𝒍𝒍𝒊 (p) 𝒊 = 𝟏, 𝟐, … , 𝒏 ,
then 𝒀 = ∑ 𝑿𝒊 ∼𝒏
𝒊=𝟏 ----------------
(3 marks)
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

73. Q3 Solve the following
1- If 𝑿𝒊; 𝒊 = 𝟏, 𝟐, … , 𝒏 are independent r. vs. such that 𝑿𝒊~ 𝑬𝒙𝒑 (𝜷) for each 𝒊
prove that 𝒀~𝚪(𝒏, 𝜷) where 𝒀 = ∑ 𝑿𝒊
𝒏
𝒊=𝟏 .
(4 marks)
Numerical analysis
Q1 Mark three of the following by T if it is true or F when it is false
1- The convergence speed of False Position method is Linear.
2- In the interpolation by lagrange method then the different between points equals.
3- The Bessel method Used when value be in the first table.
4- The Bool method in numerical integral used when the number of point is
seven.
(3 marks)
Q2 full three of the following blanks with correct answer
1- A formula of Jaccobi method to solve the system of equations is
a-
1
[ ]/
n
i i ij ij jj
j
x b a x a

   b-
1
[ ]/
j i
n
i i ij ij jj
j
x b a x a


  
c-
1
[ ]/
j i
n
i i ij ij ii
j
x b a x a


   d-
1
[ ]/
j i
n
i i ij j ii
j
x b a x a


  
2- In the Romberg method 𝑹(𝒌, 𝒊) = -----------, k=2,3,…,n
a-
2
2
1 1
1
1 1
[ ( 1,1) ( ( ) ]
2 2
k
k k
i
R k h f a i h

 

   
b-
2
2
4 ( , 1) ( 1, 1)
4 1
i
i
R k i R K i

   

c-
2
2
1 1
1
1 1
[ ( 1,1) ( ( ) ]
2 2
k
k k
i
R k h f a i h

 

   
d-
1
1
4 ( , 1) ( 1, 1)
4 1
i
i
R k i R K i

   

3-
3
if 
a- 3 1 1 3
2 2 2 2
3 3
i i i i
f f f f
   
   b- 3 1 1 3
2 2 2 2
3 3
i i i i
f f f f
   
  
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

74. c- 3 1 1 3
2 2 2 2
3 3
i i i i
f f f f
   
   d- 3 1 1 3
2 2 2 2
3 3
i i i i
f f f f
   
  
4- If you have the following data
x 0 0.5 1 1.5 2
( )f x -0.5 0.5 1 1.3 1.5
and you want to find (1.4)f by use Bessel formula, then the value of
0x is
a- 0 b- 1 c- 2 d-0.5
(3 marks)
Q3 Solve one of the following
1- What is the convergence conditions for the Fixed point iterative method to solve
system of nonlinear equations.
2- Derive the formula of Modified Euler method for solve ordinary Differential
Equations.
(4 marks)
‫ذج‬‫أ‬‫ن‬‫ا‬‫ا‬====))))((((()====‫ا‬‫إ‬‫ي‬‫ا‬

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89. (FLrJl dlJt &itiJl ; i.rt.Jl c+JS. ir.5
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90. frt't);rst
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91. ell.:
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92. .t-
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93. .t},
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