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الاجوبة النموذجية لاسئلة الاختبار النهائي - الدور الاول

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الكلية كلية العلوم     القسم قسم الكيمياء     المرحلة 1
أستاذ المادة فؤاد حمزة عبد الشريفي       23/02/2017 14:24:41
Ministry of Higher Education Primary - Studies
and Scientific Research First class stage
Babylon University Allowed time is 3 Hours
College of Science Total marks is 100 marks
Chemistry Department Examine of Mathematics

Final Examination Questions of first course for Scholar year 2016 – 2017

Q1: A. Find the domain for the function f(x)=?(x^2-6x+8 ) ?10 marks?

B.Find the derivative for y=3x cos^(-1)?3x-?(1-?9x?^(2 ) ) ?10 marks?

Q2:( Do only 2) A.What is the molarity of [H^+ ] when pH = 7.4
B. The radioactive element strontium 90 has a half-life of approximately 28 years. If y_0 is the initial amount of the element, then the amount y remaining after t years is given by
y(t)=y_0 (0.5)^(t?28). In how many years will 60g remain? if y_0=100g.
C. Solve for x: 4 cosh?x-3 sinh??x ?=4
Q3: Evaluate the limits( Do only 2) ?20 marks?
A. lim?(x??)??(x+4)^6/(x^2+3x+5)^3 ? B. ?lim ??(x? -1) (1+cos?x)/(x^2+2x+1) C. lim?(x?0) ( ln?(1+x)-x)/(cos?x-1)

Q4: Evaluate the following integrals ?20 marks?
A. ??(3x-7)/(x^2-2x-15) dx B.??x^3 e^3x dx

Q5:A.Evaluate ?_1^3?dx/(x^2-2x+5) ?10 marks?

B. Estimate ?_1^2??(1+x^3 ) dx by using Simpson^ s rule for n=6. ?10 marks?



Dr. Ahmed Abed Ass.Prof. Dr. Abbas Jassim Attiah Ass. Lecturer Fouad Hamza Ass.Prof.
Instructor Instructor Head of department
??? With our best wishes ???

الاجوبة النموذجية
Q1: A. Find the domain for the function f(x)=?(x^2-6x+8 ) ?10 marks?
x^2-6x+8?0
x^2-6x+8=0 ? (x-2)(x-4)=0 ? x=2 ,4
? (-??,2] ,[2,4] and [4,? ?)?
0?? (-??,2] ? 8?0 True ?
3?[2,4] ?9-6×3+8?0? -1?0 False ?
5?[4,? ?)? ? 25-6×5+8?0 ? 3?0 True ?
Thus the domain is D=? (-??,2] ?[4,? ?)?

B.Find the derivative for y=3x cos^(-1)?3x-?(1-?9x?^(2 ) ) ?10 marks?
dy/dx=3x×(-1)/?(1-(3x)^(2 ) )×3+3 cos^(-1)?3x-(-18x)/(2?(1-?9x?^(2 ) ))
=(-9x)/?(1-?9x?^(2 ) )+3 cos^(-1)?3x+9x/?(1-?9x?^(2 ) )=3 cos^(-1)?3x
Q2: A.What is the molarity of [H^+ ] when pH = 7.4
pH=-log?[H^+ ] ? 7.4=-log?[H^+ ]
log?[H^+ ]=- 7.4 ? [H^+ ]=?10?^(- 7.4)
[H^+ ]=?10?^0.6×?10?^(- 8)=3.98×?10?^(- 8) M
B. The radioactive element strontium 90 has a half-life of approximately 28 years. If y_0 is the initial amount of the element, then the amount y remaining after t years is given by
y(t)=y_0 (0.5)^(t?28). In how many years will 60g remain? if y_0=100g.
60=100×(0.5)^(t?28) ? (0.5)^(t?28)=0.6
ln??(0.5)^(( t )/28) ?=ln??(0.6) ? ? ( t )/( 28) ln?(0.5)=ln??(0.6) ?

( t )/( 28)=ln??(0.6) ?/ln?(0.5) ? ( t )/( 28)=(-0.511)/(-0.693) ? t=28×0.737=20.636
C. Solve for x: 4 cosh?x-3 sinh??x ?=4
4((e^x+e^(-x))/2)-3((e^x-e^(-x))/2)=4 ? (4e^x+4e^(-x)-3e^x+3e^(-x))/2=4
e^x+7e^(-x)=8 ? e^2x-8e^x+7=0 ? (e^x-7)(e^x-1)=0
e^x=7 ? x=ln?7 ? or e^x=1 ? x=ln?1=0
Q3: Evaluate the limits( Do only 2) ?20 marks?
A. lim?(x??)??(x+4)^6/(x^2+3x+5)^3 ?=(lim?(x??)??(x+4)^2/(x^2+3x+5)? )^3=(lim?(x??)??(x^2+8x+16)/(x^2+3x+5)? )^3
=(lim?(x??)??(1+(8?x)+(16?x^2 ))/(1+(3?x)+(5?x^2 ) )? )^3=1^3=1
B. ?lim ??(x? -1) (1+cos?x)/(x^2+2x+1)=?lim ??(x? -1) (-? sin??x)/(2x+2)=?lim ??(x? -1) (-?^2 cos?x)/2= ? ??^2/2
C. lim?(x?0) ( ln?(1+x)-x)/(cos?x-1)= lim?(x?0) ( 1/((1+x) )-1)/(-sin?x )= lim?(x?0) ((1-1-x)/((1+x) ))/(-sin?x )= lim?(x?0) (x/((1+x) ))/sin?x
= lim?(x?0) x/((1+x) sin?x )=lim?(x?0) 1/((1+x) cos?x+sin?x )=1

Q4: Evaluate the following integrals ?20 marks?
A. ??(3x-7)/(x^2-2x-15) dx
(3x-7)/(x^2-2x-15)=(3x-7)/(x-5)(x+3) =A/((x-5) )+B/((x+3) )
x=5 ? A=(3x-7)/((x+3) )=1 and x=-3 ? B=(3x-7)/((x-5) )=2
??(3x-7)/(x^2-2x-15) dx=??(1/((x-5) )+2/((x+3) )) dx=ln?(x-5)+2 ln?(x+3)+c
B.??x^3 e^3x dx
x^3 and it^ s D. e^3x and it^ s I.
x^3 e^3x
?3x?^2 (1?3) e^3x
6x (1?9) e^3x
6 (1?27) e^3x
0 (1?81) e^3x
???x^3 e^(-3x) ? dx=( x^3 )/3 e^3x -( 3x^2 )/9 e^3x+( 6x )/27 e^3x-( 6 )/81 e^3x+c
=( 1 )/27 e^3x (9x^3-3x^2+6x-2)+c


Q5:A.Evaluate ?_1^3?dx/(x^2-2x+5) ?10 marks?
?_1^3?dx/(x^2-2x+5)=?_1^3?dx/(x^2-2x+1+4)=?_1^3?dx/((x-1)^2+4)
=( 1 )/( 2 ) ? tan^(-1)??((x-1))/2? ?|_1^3=( 1 )/( 2 ) (tan^(-1)??((3-1))/2?-tan^(-1)??((1-1))/2? )
=( 1 )/( 2 ) (tan^(-1)?1-tan^(-1)?0 )=( ? )/( 8 )

B. Estimate ?_1^2??(1+x^3 ) dx by using Simpson^ s rule for n=6. ?10 marks?
?_a^b?f(x)dx?( h )/3 [?(_^)y_0 +4y_1+2y_2+4y_3+2y_4+4y_5+y_6?_^ ]
h=(b-a)/n=(2-1)/6=( 1 )/( 6 )
x_n y_n Factors Product
1 y_0=1.4142 1 1.4142
1+1?6 y_1=1.6087 4 6.4348
1+2?6 y_2=1.8359 2 3.6718
1+3?6 y_3=2.0917 4 8.3668
1+4?6 y_4=2.3727 2 4.7454
1+5?6 y_5=2.6762 4 10.7048
2 y_6=3 1 3
Sum 38.3378

?_1^2??(1+x^3 ) dx?(1?6)/3×38.3378=38.3378/18=2.1299


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