:ناــحتملال مدقتلا طباوضو تمايلعت...Be sure to show all your work and...
Transcript of :ناــحتملال مدقتلا طباوضو تمايلعت...Be sure to show all your work and...
تعليامت وضوابط التقدم لالمتحــان:
الحضور إىل اللجنة قبل عرش دقائق من بدء االمتحان لألهمية. –إبراز البطاقة الشخصية ملراقب اللجنــة. –
مينع كتابة رقم الجلوس أو االسم أو أي بيانات أخرى تدل عىل –شخصية املمتحن يف دفرت االمتحان، وإال ألغي امتحانه.
يحظر عىل املمتحنني أن يصطحبوا معهم مبركز االمتحان كتبا دراسية –أو كراسات أو مذكرات أو هواتف محمولة أو أجهزة النداء اآليل أو
أي يشء له عالقة باالمتحان كام ال يجوز إدخال آالت حادة أو أسلحة من أي نوع كانت أو حقائب يدوية أو آالت حاسبة ذات
صفة تخزينية.يجب أن يتقيد املتقدمون بالزي الرسمي )الدشداشة البيضاء واملرص –أو الكمة للطالب والدارسني والزي املدريس للطالبات واللباس العامين
للدارسات ( ومينع النقاب داخل املركز ولجان االمتحان.ال يسمح للمتقدم املتأخر عن موعد بداية االمتحان بالدخول إال –
إذا كان التأخري بعذر قاهر يقبله رئيس املركز ويف حدود عرشدقائق فقط.
يتم االلتزام باإلجراءات الواردة يف دليل الطالب ألداء امتحان شهادة –دبلوم التعليم العام.
يقوم املتقدم باإلجابة عن أسئلة االمتحان املقالية بقلم الحرب )األزرق –أو األسود(.
يقوم املتقدم باإلجابة عن أسئلة االختيار من متعدد بتظليل – ( وفق النموذج اآليت: الشكل )
عاصمــة سلطنة عمــــان هي: س – الدوحة القاهرة أبوظبي مسقط
( باستخدام القلم الرصاص وعند يتم تظليل الشكل ) مالحظة: الخطأ، امسح بعناية إلجراء التغيري.
غري صحيح صحيح
امتحان دبلوم التعليم العام للمدارس الخاصة )ثنائية اللغة(للعام الدرايس 1436/1435 هـ - 2014 / 2015 م
الدور األول - الفصل الدرايس األول
• املادة: الرياضيات البحتة . تنبيه • األسئلة يف ) 14 ( صفحة.
• زمن اإلجابة: ثالث ساعات.• اإلجابة يف الورقة نفسها.
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Diploma, Bilingual Private Schools, Pure Mathematics First Session – First Semester Academic Year: 2014/2015
دة، ال يتم تصحيحها مسو
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Diploma, Bilingual Private Schools, Pure Mathematics First Session – First Semester Academic Year: 2014/2015
Instructions:
1. Programmable calculators are not allowed. 2. A list of formulae is provided on the final pages.
Question 1 (28 marks)
There are 14 multiple-choice items worth two marks each. Shade in the correct answer for each of the following items .
1) If 3x + 1( x + 2 )( x + 1)
A( x + 2 )
+ B( x + 1 )
, then B equals:
– 2 4
5 7
2) limh 0 f ( x + h ) – f ( x )
3h =
3f ′ ( x ) 13
f ′ ( x )
f ′ ( 3 ) f ′ ( 13
)
3) If y = ax2 has a gradient of 12 at point ( 2, 12 ), then a equals:
16
13
3 6
4) If f ′ ( x ) = 6x2 – 4 x3, then the number of inflexion point(s) of f ( x ) equals:
0 1
2 3
2
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Diploma, Bilingual Private Schools, Pure Mathematics First Session – First Semester Academic Year: 2014/2015
Question 1 continued
5) The period of f ( x ) = 6 cosec 2( x – π ) is:
π 2π
4π 6π
6) If sin θ = 35
where 0 < θ < π2
, then sec θ =
34
45
54
43
7) If 3 tan ( x ) – 1 = 0 where 0 < x < π2
, then the value of x is:
π6
π4
π3
π2
8) If sec θ – tan θ ≠ 0, then 1sec θ – tan θ
=
sec θ + tan θ sec θ – tan θ
cot θ + cos θ cos θ – cot θ
9) 7x6 dx =
x6 + c x7 + c
7x7 + c 42 x5 + c
3
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Diploma, Bilingual Private Schools, Pure Mathematics First Session – First Semester Academic Year: 2014/2015
Question 1 continued
10) From the figure, if area A = 12 square units and area B = 9 square units,
then 9
0
f (x)dx equals:
– 21 – 3
3 21
11) If f ′ ( x ) = 6x2 , f ( 0 ) = 1, then f ( 2 ) =
25 24
17 16
12) If 2
0
( 2 x – m )dx = – 4, then m =
– 8 – 4
4 8
13) The probability that there will be at least one ‘tails’ in tossing a coin four times is:
1516
78
12
1
16
14) A basket contains 10 green apples and 15 red apples. Ahmed chooses one apple at random and eats it, followed by another one. The probability that Ahmed eats one green and one red apple is:
625
12
14
2325
y
xA
0 6 9
B
4
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Diploma, Bilingual Private Schools, Pure Mathematics First Session – First Semester Academic Year: 2014/2015
Extended Questions
Write your answer for each of the three questions in the space provided. Be sure to show all your work and the correct units where applicable.
Question 2 (16 marks)
15) Express 5x2
( x2 + 2x + 1)( x – 2 ) as a sum of partial fractions. [3 marks]
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Diploma, Bilingual Private Schools, Pure Mathematics First Session – First Semester Academic Year: 2014/2015
Question 2 continued
16) Find d2ydx2 if y = x
32 + x + 3 [4 marks]
6
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Question 2 continued
17) a. Find the equation of the tangent to the curve y = 16 – x2 for x ≥ 0 at the point of intersection of curve y with the x-axis. [3 marks]
b. Given that y = 13
x3 – x2, find the range of values x for which y is
an increasing function [3 marks]
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Diploma, Bilingual Private Schools, Pure Mathematics First Session – First Semester Academic Year: 2014/2015
Question 2 continued
18) A closed rectangular box is to be constructed with a base whose length is twice its width x. If the total surface area is 108 cm2, calculate the value for x which maximises the box’s volume. [3 marks]
y
x2x
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Diploma, Bilingual Private Schools, Pure Mathematics First Session – First Semester Academic Year: 2014/2015
Question 3 (14 marks)
19) If 4x3 + 10x + 4x ( 2x + 1)
( 2x – 1) + Ax
+ B( 2 x + 1 )
, calculate B. [3 marks]
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Diploma, Bilingual Private Schools, Pure Mathematics First Session – First Semester Academic Year: 2014/2015
Question 3 continued
20) Without using a calculator, calculate the value of:
sin 105° cos 75° + cos 105° sin 75° [3 marks]
21) a. Given that cot A = 14
and cot ( A – B ) = 8, calculate the value of tan B.
[3 marks]
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Diploma, Bilingual Private Schools, Pure Mathematics First Session – First Semester Academic Year: 2014/2015
Question 3 continued
21) b. Express 5
3 cos θ –
53
sin θ in the form R cos ( θ + α ) where R > 0.
[3 marks]
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Diploma, Bilingual Private Schools, Pure Mathematics First Session – First Semester Academic Year: 2014/2015
Question 3 continued
22) Prove that:
( cos x2
– sin x2
)2 + sin x = 1 [2 marks]
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Diploma, Bilingual Private Schools, Pure Mathematics First Session – First Semester Academic Year: 2014/2015
Question 4 (12 marks)
23) Calculate: 1
0
( 4 x3 – x + 1) dx [4 marks]
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Question 4 continued
24) Calculate the shaded area of the region bounded by these graphs of
f ( x ) = x3 – x2 – x and g( x ) = x2 + 2 x [4 marks]
1
0
–1
2
3
x
y
–3 –2
(–1, –1)
–1 0 1 2
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Diploma, Bilingual Private Schools, Pure Mathematics First Session – First Semester Academic Year: 2014/2015
Question 4 continued
25) A and B are two events such that P ( A ) = 0.3, P ( B ) = 0.25 and P ( A B ) = 0.05 .
Draw a Venn diagram to represent the events A and B, and the sample space S. [2 marks]
26) A and B are two events such that P ( A ) = 0.75, P ( B ) = 0.5 and P ( A | B ) = 0.4 .
Calculate ( A B′ ). [2 marks]
[ End of Examination ]
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Diploma, Bilingual Private Schools, Pure Mathematics First Session – First Semester Academic Year: 2014/2015
List of Formulae (2 sheets)
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Diploma, Bilingual Private Schools, Pure Mathematics First Session – First Semester Academic Year: 2014/2015
End of Formulae
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دة مسو
Diploma, Bilingual Private Schools, Pure Mathematics First Session – First Semester Academic Year: 2014/2015
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دة مسو
Diploma, Bilingual Private Schools, Pure Mathematics First Session – First Semester Academic Year: 2014/2015
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دة مسو
Diploma, Bilingual Private Schools, Pure Mathematics First Session – First Semester Academic Year: 2014/2015
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دة مسو
Diploma, Bilingual Private Schools, Pure Mathematics First Session – First Semester Academic Year: 2014/2015
زءلج
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حارض
• املادة: الرياضيات )ثنائية اللغة(. • اإلسئلة يف ) 15 ( صفحة.
• زمن االجابة: ثالث ساعات• اإلجابة يف الورقة نفسها
غائب
التكتب يف هذا الجزء
امتحان شهادة دبلوم التعليم العام للمدارس الخاصة )ثنائية اللغة(للعام الدرايس 1٤٣٥/1٤٣٤هـ - 201٤/201٣م
الدور األول - الفصل الدرايس األول
تعليامت وضوابط التقدم لالمتحان:- الحضور إىل اللجنة قبل عرش دقائق من بدء االمتحان لألهمية.
- إبراز البطاقة الشخصية ملراقب اللجنة.- مينع كتابة رقم الجلوس أو االسم أو أي بيانات أخرى تدل عىل
شخصية املمتحن يف دفرت االمتحان ، وإال ألغي امتحانه.- يحظر عىل املمتحنني أن يصطحبوا معهم مبركز االمتحان كتبا دراسية أو كراسات أو مذكرات أو هواتف محمولة أو أجهزة النداء اآليل أو أي
يشء له عالقة باالمتحان كام اليجوز إدخال آالت حادة أو أسلحة من أي نوع كانت أو حقائب يدوية أو آالت حاسبة ذات صفة تخزينية.
- يجب أن يتقيد املتقدمون بالزي الرسمي)الدشداشة البيضاء واملرص أو الكمة للطالب والدارسني والزي املدريس للطالبات واللباس العامين
للدارسات( ومينع النقاب داخل املركز ولجان االمتحان.- ال يسمح للمتقدم املتأخر عن موعد بداية االمتحان بالدخول إال إذا
كان التأخري بعذر قاهر يقبله رئيس املركز ويف حدود عرش دقائق فقط.
- يتم االلتزام باإلجراءات الواردة يف دليل الطالب ألداء امتحان شهادة دبلوم التعليم العام.
- يقوم املتقدم باإلجابة عن أسئلة االمتحان املقالية بقلم الحرب )األزرق أو األسود(.
- يقوم املتقدم باإلجابة عن أسئلة االختيار من متعدد بتظليل الشكل ( وفق النموذج اآليت: (
س – عاصمة سلطنة عامن هي: القاهرة. الدوحة.
مسقط. أبو ظبي.
( باستعامل القلم الرصاص وعند مالحظة: يتم تظليل الشكل )الخطأ، امسح بعناية إلجراء التغيري.
غري صحيح صحيح
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Diploma, Semester One - First Session –Bilingual Private Schools, Mathematics 2013/2014
دة، اليتم تصحيحها مسو
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1
Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
Formulae sheets on pages 16-17
Question One : (28 marks)
There are 14 multiple-choice items worth two marks each. Shade in the ( ) next to the correct answer for each of the following items.
1) If y = 7x +9, then =
2) The equation of the tangent to curve f (x)=3x 2-2x +4 at x = 0 is :
3) If = 12 x 2 - 4 and the stationary points of y are at x = 0, 1
and -1, then the maximum point(s) at x =
4) If = + , then 4B =
dydx
12
7x
7
y = -2x +4
y = x +4
-1, 0
1
-1
0
0, 1
1
x +9
9
y = x -2
y = 6x-2
-1 6
14
d 2 yd x 2
1
x 2 + 2x -3
A
( x +3)
B
( x -1)
-14
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5) If two fair spinners each with four faces numbered 1 to 4 are thrown together and the product of numbers indicated on each spinner is recorded, what is the probability of the spinners indicating a product which is square and even ?
6) In a group of 20 men and 12 women, half of the men have black eyes and a third of the women have black eyes. If a person is chosen randomly, what is the probability that the person is a women or black eyes?
7) If y =1 + 0.4 sec θ, and θ = π, then the value of y equals:
8) If 3 tan2x = 1, then the values of x (where 90° < x < 270° ) are :
18
2632
1432
2232
1232
316
14
34
-1.4
0.6
150°, 210°
150°, 240°
120°, 240°
120°, 210°
-0.6
1.4
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3
Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
(3(
9) If sin 2 0 , then 22cos
sin 2
cos 2ec equals:
tan 2 2tan
cot 2
2cot
10) If 82cos sin
cosA A
ecA , then sin 2 A
11) ∫ √
√ c √ c
√ √
12) If ( ) ∫ , ( ) , then ( )
Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
(3(
9) If sin 2 0 , then 22cos
sin 2
cos 2ec equals:
tan 2 2tan
cot 2
2cot
10) If 82cos sin
cosA A
ecA , then sin 2 A
11) ∫ √
√ c √ c
√ √
12) If ( ) ∫ , ( ) , then ( )
Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
(3(
9) If sin 2 0 , then 22cos
sin 2
cos 2ec equals:
tan 2 2tan
cot 2
2cot
10) If 82cos sin
cosA A
ecA , then sin 2 A
11) ∫ √
√ c √ c
√ √
12) If ( ) ∫ , ( ) , then ( )
Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
(3(
9) If sin 2 0 , then 22cos
sin 2
cos 2ec equals:
tan 2 2tan
cot 2
2cot
10) If 82cos sin
cosA A
ecA , then sin 2 A
11) ∫ √
√ c √ c
√ √
12) If ( ) ∫ , ( ) , then ( )
Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
(3(
9) If sin 2 0 , then 22cos
sin 2
cos 2ec equals:
tan 2 2tan
cot 2
2cot
10) If 82cos sin
cosA A
ecA , then sin 2 A
11) ∫ √
√ c √ c
√ √
12) If ( ) ∫ , ( ) , then ( )
9) If sin 2θ ≠0, then - cosec 2θ equals:
10) If cos A sin 2 A = , then sin 2A =
12) If f (x)= ∫ x-2dx, f (1) = 1, then f (x) =
11) ∫ =
tan 2θ
cot 2θ
tan2θ
cot2θ
4
10
8
16
2 cos2 θ sin 2θ
8 cosecA
- -2
- +1
- -1
- +2
1x
1x
1x
1x
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
(4(
13) Consider the sketch below. If A=4, B=5, and C=2 are three areas, then ∫ ( ( ) ( ))
-1
1
6
11
14) ∫
13) Consider the sketch below. If A = 4, B = 5, and C = 2 are three areas, then ∫ ( f (x) - g(x)) dx =
b
a
1
0
-1
1
6
11
14) ∫ dx =x 3-x 2 + x-1 x-1
-
112
13
1112
43
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
(5(
EXTENDED QUESTIONS
Question Two (14 marks)
a) i. If 2
2 1
x
x
2 1
BAx
, compute A B . (3 marks)
Write your answers for each of the three questions in the space provided. Be sure to show all your work and the correct units where applicable .
EXTENDED QUESTIONS
Write your answers for each of the three questions in the space provided. Be sure to show all your work and the correct units where applicable.
Question Two : (14 marks)
a) i. If = A + , compute A + B (3 marks) x 2
x2-1 B x2-1
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
(6(
ii. Express 35
( 1)( 5)x
x x
in partial fractions. (3 marks)
ii. Express in partial fractions. (3 marks) x-35
(x-1)(x+5)
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
(7(
b) Differentiate 2( ) ( 3)f x x 2 1x with respect to . (3 marks)
c) Express sin 3 cos in the form ( ) and calculate its minimum value. (5 marks)
Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
(7(
b) Differentiate 2( ) ( 3)f x x 2 1x with respect to . (3 marks)
c) Express sin 3 cos in the form ( ) and calculate its minimum value. (5 marks)
b) Differentiate f (x) = (x+3)2 + 2x -1 with respect to x. (3 marks)
c) Express sinθ - cosθ in the form R sin (θ + ) and calculate its minimum value. (5 marks)
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
(8(
QUESTION THREE (14 marks)
a) i. Given that( ) ( ) 2lim 2 1
0f x h f x x
h h
, calculate (5)f .
(2 marks) ii. Prove that is the equation for the normal to curve
3( ) 2 1f x x at point (1,1) . (2 marks)
Question Three : (14 marks)
a) i. Given that = 2x 2-1, calculate f '(5). (2 marks)
ii. Prove that 6y + x - 7 = 0 is the equation for the normal to curve f (x) = 2x 3 -1 at the point (1,1). (2 marks)
Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
(8(
QUESTION THREE (14 marks)
a) i. Given that( ) ( ) 2lim 2 1
0f x h f x x
h h
, calculate (5)f .
(2 marks) ii. Prove that is the equation for the normal to curve
3( ) 2 1f x x at point (1,1) . (2 marks)
f (x) -
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
(9(
b) i. Determine the range of values of x for which y is decreasing, if 2 312 4y x x . (3 marks)
b) i. Determine the range of values of x for which y is decreasing, if y'= 12x 2- 4x 3. (3 marks)
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
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ii. A rectangular resort is to be constructed with one side open to the sea. A
security fence is required along the remaining 3 sides of the resort. What is
the maximum area that can be enclosed with 800 m of fencing?
(3 marks)
sea
resort
ii. A rectangular resort is to be constructed with one side open to the sea. A security fence is required along the remaining 3 sides of the resort. What is the maximum area that can be enclosed with 800 m of fencing? (3 marks)
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
c) i. Calculate ∫ x 0 .01 dx. (2 marks)
ii. Given that f "(x )=6 x , f ' (0) = 0 and f (1) = 0, Calculate f (x).
(2 marks)
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
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Question Four : (14 marks)
a) i. If sin A = , cos A = , sin B = , cos B = ,
Calculate tan ( A - B ). (3 marks)
1213
513
810
610
ii. Prove that (1- ) 2 + cos2 x = 2 - 2sin x
(3 marks)
1cosec x
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
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b) i. Calculate an approximation to the area bounded by axes, , and √ . Use the Trapezium Rule with one strip. (2 marks)
Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
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b) i. Calculate an approximation to the area bounded by axes, , and √ . Use the Trapezium Rule with one strip. (2 marks)
b) i. Calculate an approximation to the area bounded by axes: x = 8 and y = + 1. Use the Trapezium Rule with one strip. (2 marks)
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
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ii. From the diagram below, calculate the shaded area. (2 marks)
ii. From the diagram below, calculate the shaded area. (2 marks)
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
C) i. A coin is flipped first and then a six-sided die is thrown. The results are recorded. Draw a tree diagram to represent this information. (1 mark)
ii. In a set of exam results, the percentage of success in Physics is 70% and in Maths 80% and in both subjects is 65%. If a student is chosen randomly, what is the probability that he will fail in Maths if he fails in Physics ? (3 marks)
END OF THE EXAMINATION
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
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Formulae sheet for semester 1 Differentiation:
1. ,ny x
R )1( nnxdxdy n
2. ( ) ( )lim( )0
f x h f xf xh h
3. ( 1 ), Rn ndyy kx knx ndx
4. ( ) ( ), ( ) ( )dyy f x g x f x g xdx
5. ( ), ( )dyy k f x k f xdx
6. Area and Volume of a cuboid with length, width and height as hwl and ,, respectively.
hwlVolumelhwhlwArea 222 7. Area and Volume of a cylinder with radius r and height h .
2 22 r h 2 r Area Volume r h 8. Area and Volume of a sphere with radius r .
2 344 r 3
Area V olume r
Trigonometry: Pythagorean Formulas Double Angle Formulas: 1. 1cossin 22 AA 2. AA 22 tan1sec 3. AAec 22 cot1cos
1. AAA cossin22sin 2. 2 2cos 2 cos sinA A A 2cos 2 2 cos 1A A 2cos 2 1 2sinA A 3.
AAA 2tan1
tan22tan
Compound Angle Formulas: Half Angle Formulas: 1. BABABA sincoscossin)sin( 1. )cos1(
21
21sin 2 AA
2. BABABA sincoscossin)sin( 2. )cos1(21
21cos2 AA
3. BABABA sinsincoscos)cos( 3. )2cos1(21sin 2 AA
4. BABABA sinsincoscos)cos( 4. )2cos1(21cos 2 AA
5. BABABA
tantan1tantan)tan(
6. BABABA
tantan1tantan)tan(
The form sincos ba : sincos ba can be expressed in the form )cos( R or )sin( R where 2 2R , tan tanb aa b and or
a b
n
n (n-1)
Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
(16(
Formulae sheet for semester 1 Differentiation:
1. ,ny x
R )1( nnxdxdy n
2. ( ) ( )lim( )0
f x h f xf xh h
3. ( 1 ), Rn ndyy kx knx ndx
4. ( ) ( ), ( ) ( )dyy f x g x f x g xdx
5. ( ), ( )dyy k f x k f xdx
6. Area and Volume of a cuboid with length, width and height as hwl and ,, respectively.
hwlVolumelhwhlwArea 222 7. Area and Volume of a cylinder with radius r and height h .
2 22 r h 2 r Area Volume r h 8. Area and Volume of a sphere with radius r .
2 344 r 3
Area V olume r
Trigonometry: Pythagorean Formulas Double Angle Formulas: 1. 1cossin 22 AA 2. AA 22 tan1sec 3. AAec 22 cot1cos
1. AAA cossin22sin 2. 2 2cos 2 cos sinA A A 2cos 2 2 cos 1A A 2cos 2 1 2sinA A 3.
AAA 2tan1
tan22tan
Compound Angle Formulas: Half Angle Formulas: 1. BABABA sincoscossin)sin( 1. )cos1(
21
21sin 2 AA
2. BABABA sincoscossin)sin( 2. )cos1(21
21cos2 AA
3. BABABA sinsincoscos)cos( 3. )2cos1(21sin 2 AA
4. BABABA sinsincoscos)cos( 4. )2cos1(21cos 2 AA
5. BABABA
tantan1tantan)tan(
6. BABABA
tantan1tantan)tan(
The form sincos ba : sincos ba can be expressed in the form )cos( R or )sin( R where 2 2R , tan tanb aa b and or
a b
Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
(16(
Formulae sheet for semester 1 Differentiation:
1. ,ny x
R )1( nnxdxdy n
2. ( ) ( )lim( )0
f x h f xf xh h
3. ( 1 ), Rn ndyy kx knx ndx
4. ( ) ( ), ( ) ( )dyy f x g x f x g xdx
5. ( ), ( )dyy k f x k f xdx
6. Area and Volume of a cuboid with length, width and height as hwl and ,, respectively.
hwlVolumelhwhlwArea 222 7. Area and Volume of a cylinder with radius r and height h .
2 22 r h 2 r Area Volume r h 8. Area and Volume of a sphere with radius r .
2 344 r 3
Area V olume r
Trigonometry: Pythagorean Formulas Double Angle Formulas: 1. 1cossin 22 AA 2. AA 22 tan1sec 3. AAec 22 cot1cos
1. AAA cossin22sin 2. 2 2cos 2 cos sinA A A 2cos 2 2 cos 1A A 2cos 2 1 2sinA A 3.
AAA 2tan1
tan22tan
Compound Angle Formulas: Half Angle Formulas: 1. BABABA sincoscossin)sin( 1. )cos1(
21
21sin 2 AA
2. BABABA sincoscossin)sin( 2. )cos1(21
21cos2 AA
3. BABABA sinsincoscos)cos( 3. )2cos1(21sin 2 AA
4. BABABA sinsincoscos)cos( 4. )2cos1(21cos 2 AA
5. BABABA
tantan1tantan)tan(
6. BABABA
tantan1tantan)tan(
The form sincos ba : sincos ba can be expressed in the form )cos( R or )sin( R where 2 2R , tan tanb aa b and or
a b
2 2
32
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Integration:
1) ( 1)
, 11
nn xx dx c n
n
2) [ ( ) ( )] ( ) ( ) f x g x dx f x dx g x dx
3) ( ) ( ) kf x dx k f x dx
4) Area and volume of solids of revolution
( ) b
a
Area f x dx
2( ( )) b
a
V olume f x dx
5) Trapezium rule
0 1 2 1( ) [ 2( ......... )2
b
n na
hf x dx y y y y y
Probability: 1) Addition Rule:
( ) ( ) ( ) ( )P A B P A P B P A B 2) Conditional Probability:
( )( ) ( | )( )
P A BP A given B P A BP B
3) Multiplication Rule: ( ) ( | ) ( ) ( | ) ( )P A B P A B P B or P B A P A
4) Independent Rule: A and B are independent if:
( | ) ( ) ( | ) ( ) or P(A B) = ( ) ( )P A B P A or P B A P B P A P B 5) Mutually Exclusive Rule:
A and B are Mutually Exclusive if: P(A B)=0
6) )(1
)(1)(
)(AP
ABPAP
ABP
7) ( ) ( ) ( )P B A P A P A B
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
دة مسو
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
دة مسو
زءلج
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Diploma, First Semester - First Session –Bilingual Private Schools, Mathematics 2013/2014
رقم الورقة
رقم املغلف
تعليامت وضوابط التقدم لالمتحــان:
الحضور إىل اللجنة قبل عرش دقائق من بدء االمتحان لألهمية. –إبراز البطاقة الشخصية ملراقب اللجنــة. –
مينع كتابة رقم الجلوس أو االسم أو أي بيانات أخرى تدل عىل –شخصية املمتحن يف دفرت االمتحان، وإال ألغي امتحانه.
يحظر عىل املمتحنني أن يصطحبوا معهم مبركز االمتحان كتبا دراسية –أو كراسات أو مذكرات أو هواتف محمولة أو أجهزة النداء اآليل أو
أي يشء له عالقة باالمتحان كام ال يجوز إدخال آالت حادة أو أسلحة من أي نوع كانت أو حقائب يدوية أو آالت حاسبة ذات
صفة تخزينية.يجب أن يتقيد املتقدمون بالزي الرسمي )الدشداشة البيضاء واملرص –أو الكمة للطالب والدارسني والزي املدريس للطالبات واللباس العامين
للدارسات ( ومينع النقاب داخل املركز ولجان االمتحان.ال يسمح للمتقدم املتأخر عن موعد بداية االمتحان بالدخول إال –
إذا كان التأخري بعذر قاهر يقبله رئيس املركز ويف حدود عرشدقائق فقط.
يتم االلتزام باإلجراءات الواردة يف دليل الطالب ألداء امتحان شهادة –دبلوم التعليم العام.
يقوم املتقدم باإلجابة عن أسئلة االمتحان املقالية بقلم الحرب )األزرق –أو األسود(.
يقوم املتقدم باإلجابة عن أسئلة االختيار من متعدد بتظليل – ( وفق النموذج اآليت: الشكل )
عاصمــة سلطنة عمــــان هي: س – الدوحة القاهرة أبوظبي مسقط
( باستخدام القلم الرصاص وعند يتم تظليل الشكل ) مالحظة: الخطأ، امسح بعناية إلجراء التغيري.
غري صحيح صحيح
امتحان دبلوم التعليم العام للمدارس الخاصة )ثنائية اللغة(للعام الدرايس 1435/1434 هـ - 2013 / 2014 م
الدور الثاين - الفصل الدرايس األول
• املادة: رياضيات. تنبيه: • األسئلة يف ) 14 ( صفحة.
• زمن اإلجابة: ثالث ساعات.• اإلجابة يف الورقة نفسها.
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Diploma, Second Session – First Semester Bilingual Private Schools, Maths Academic Year: 2013/2014
Question 1 (28 marks)
There are 14 multiple-choice items worth two marks each.Shade in the correct answer for each of the following items .
1) If y = x5, then dydx
=
x4 x5
5x4 5x5
2) The equation of the normal to curve f ( x) = x2 + 3x at x = 0 is y =
– 12 x + 3 – 1
3 x
13 x 2x + 3
3) If f ' ( x ) = 3x2 – 12x + 9, and the stationary point y of occurs at x = 3, 1 , then the minimum point at x =
0 1
3 4
4) If 5x – 1( x + 1)( x – 2 )
= A( x + 1)
+ B( x – 1)
, then B =
–1 1
2 3
5) A six sided die is thrown twice and the numbers landing face up are recorded. What is the probability of not same numbers landing face up?
16
712
512
56
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Diploma, Second Session – First Semester Bilingual Private Schools, Maths Academic Year: 2013/2014
6) A student made two applications to two colleges. The probability of acceptance in the colleges of Engineering and Medicine are 0.5 and 0.3 respectively. If the probability of being rejected by both colleges is 0.85,what is the probability of acceptance in one of the two colleges?
0.15 0.2
0.65 0.8
7) If y = – 4+ cos ecθ , and θ = π2
, then the value of y equals:
–5 –3
3 5
8) If y = 6cos x + 3 = 0 , then the value of x (where 0˚ < x < 360˚ ) are:
60˚, 240˚ 60˚, 300˚
120˚, 240˚ 120˚, 300˚
9) (sec2θ – 1)
sec2θ =
tan2θ sin2θ
cos2θ cot2θ
10) sin 12 A × cos 1
2 A =
14
sin2A 14
cos2A
12 sin A 1
2 cos A
11) xdx=
43 x
34 + C 3
4 x
43 + C
34 x
34 + C 4
3 x
43 + C
3
3
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Diploma, Second Session – First Semester Bilingual Private Schools, Maths Academic Year: 2013/2014
12) If f ( t ) = ( 2 – 6t )dt, and f (1) = 0, then f ( t ):
2t – 3t2 – 1 2t – 3t2 – 2
2t – 3t2 + 1 2t – 3t2 + 2
13) Consider the sketch below.
If A, B and C are three areas, then a
0
( f (x) – g( x ) )dx =
A + B + C A + B – C
A + B B + C
14) 1
0
x – 4 x – 2
dx =
– 83
– 43
43
83
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Diploma, Second Session – First Semester Bilingual Private Schools, Maths Academic Year: 2013/2014
Extended Questions
Write your answer for each of the three questions in the space provided. Be sure to show all your work and the correct units where applicable.
Question Two: (14 marks)
15) If 4x3 – 5x2 – 2x – 4
x3 – 1 = A
Bx2 + C
( x3 – 1) compute A, B and C. [3 marks]
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Diploma, Second Session – First Semester Bilingual Private Schools, Maths Academic Year: 2013/2014
16) Express 6x + 1
( x2 – 1) in partial fractions. [3 marks]
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Diploma, Second Session – First Semester Bilingual Private Schools, Maths Academic Year: 2013/2014
17) Differentiate f ( x ) = x4 + 2x
x2 with respect to x. [3 marks]
18) Express 2 2 cosθ + 2 2 sinθ in the form R cos ( θ + α) and calculate its maximum value. [5 marks]
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Diploma, Second Session – First Semester Bilingual Private Schools, Maths Academic Year: 2013/2014
Question Three: (14 marks)
19) Given that limh 0 f ( x + h ) – f ( x )
h = x + 1, calculate the gradient of the
curve f (x) when x = 2. [2 marks]
20) If y = 3a x – 16 is the equation of the tangent of f ( x )= 5x3 – 3x + c at x = 1, then calculate the value of a . [2 marks]
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Diploma, Second Session – First Semester Bilingual Private Schools, Maths Academic Year: 2013/2014
21) Determine the range of values of x for which is increasing, if y′ = x2 + 6x + 5 [3 marks]
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Diploma, Second Session – First Semester Bilingual Private Schools, Maths Academic Year: 2013/2014
22) A cylindrical can has radius ( r ), and height ( h ) = 16 r2 . The material of can costs three rials
per square metre. Calculate the radius ( r ) and height ( h ) of the can with the lowest cost. [3 marks]
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Diploma, Second Session – First Semester Bilingual Private Schools, Maths Academic Year: 2013/2014
23) Calculate x – 1 dx [2 marks]
24) What is the equation of the curve whose gradient at ( x, y ) is given by 5x4 – 3 and which passes through the point (1, 0 )? [2 marks]
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Diploma, Second Session – First Semester Bilingual Private Schools, Maths Academic Year: 2013/2014
Question Four: (14 marks)
25) If sin A = 513
, cos A = 1213
, sin B = 45 , cos B = 3
5 , calculate sin ( A + B ).
[3 marks]
26) Prove that ( 1 – sec2α )( 1 – cos2α ) = 1 – tan2α – cos2α. [3 marks]
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Diploma, Second Session – First Semester Bilingual Private Schools, Maths Academic Year: 2013/2014
27) Calculate an approximation to the area bounded by the axis, x = 2 and y = x3 + 1. Use the Trapezium Rule with one strip. [2 marks]
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Diploma, Second Session – First Semester Bilingual Private Schools, Maths Academic Year: 2013/2014
28) From the diagram below, calculate the shaded area. [2 marks]
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Diploma, Second Session – First Semester Bilingual Private Schools, Maths Academic Year: 2013/2014
29) Two coins are flipped and the results are recorded. Draw a tree diagram to represent this information. [1 mark]
30) The probability that a person is training to drive a car is 0.8 and the probability that he will pass the driving test if he trains is 0.6. What is the probability that he will train and not pass the driving test? [3 marks]
[End of Examination]
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Diploma, Second Session – First Semester Bilingual Private Schools, Maths Academic Year: 2013/2014
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دة مسو
تعلي�ت وضوابط التقدم لالمتحــان:
الحضور إىل اللجنة قبل عرش دقائق من بدء االمتحان لألهمية. –إبراز البطاقة الشخصية ملراقب اللجنــة. –
�نع كتابة رقم الجلوس أو االسم أو أي بيانات أخرى تدل عىل –شخصية املمتحن يف دفرت االمتحان، وإال ألغي امتحانه.
يحظر عىل املمتحن© أن يصطحبوا معهم ¥ركز االمتحان كتبا دراسية –أو كراسات أو مذكرات أو هواتف محمولة أو أجهزة النداء اآليل أو
أي يشء له عالقة باالمتحان ك´ ال يجوز إدخال آالت حادة أو أسلحة من أي نوع كانت أو حقائب يدوية أو آالت حاسبة ذات
صفة تخزينية.يجب أن يتقيد املتقدمون بالزي الرسمي (الدشداشة البيضاء واملرص –أو الكمة للطالب والدارس© والزي املدريس للطالبات واللباس الع´¿
للدارسات ) و�نع النقاب داخل املركز ولجان االمتحان.ال يسمح للمتقدم املتأخر عن موعد بداية االمتحان بالدخول إال –
إذا كان التأخÉ بعذر قاهر يقبله رئيس املركز ويف حدود عرشدقائق فقط.
يتم االلتزام باإلجراءات الواردة يف دليل الطالب ألداء امتحان شهادة –دبلوم التعليم العام.
يقوم املتقدم باإلجابة عن أسئلة االمتحان املقالية بقلم الحرب (األزرق –أو األسود).
يقوم املتقدم باإلجابة عن أسئلة االختيار من متعدد بتظليل –:Ñوفق النموذج اآل ( الشكل (
عاصمــة سلطنة عمــــان هي: س – الدوحة القاهرة أبوظبي مسقط
) باستخدام القلم الرصاص وعند يتم تظليل الشكل ( مالحظة: .Éالخطأ، امسح بعناية إلجراء التغي
غ� صحيح صحيح
رقم الورقة
رقم املغلف
امتحان شهادة دبلوم التعليم العام للمدارس الخاصة )ثنائية اللغة(
للعام الدرايس 1434/1433 هـ - 2012 / 2013 م
الدور األول - الفصل الدرايس األول
• الرياضيات. تنبيه:
• األسئلة يف ) 16 ( صفحات.
• زمن اإلجابة: ثالث ساعات.
• اإلجابة يف الورقة نفسها.
Diploma, Semester First – First Session, Bilingual Private Schools, Mathematics 2012/2013
1
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Question One (28 marks)
There are 14 multiple–choice items worth two marks each.
Shade the correct answer for each of the following items.
1. lim h"0
ƒ(–2) –ƒ(–2+h)
h =
ƒ’(2) – ƒ’(2)
ƒ’(–2) – ƒ’(–2)
2. If y = a x2+5 and d2y
dx2 = 6 at x = –1, then a=
–3 –1
1 3
3. The coordinates of the stationary point of the curve y = 2x – x2 is:
(1, – 2) (0, 0)
(1, 1) (2, 0)
4. If 6x
(x – 1)(x + 2) = A
(x – 1) + B
(x + 2) , then the value of 2A – B is:
0 2
4 6
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5. Which of trigonometric functions are both odd?
cosθ, cosecθ cosecθ, cotθ
secθ, cotθ cosθ, secθ
6. If cotθ = 4 3
and θ is reflex, then secθ=
-5 3
-5 4
5 4
5 3
7. If cot(3θ –30°) = 1√3
, 0° < θ < 90°, then θ=
10° 20°
30° 60°
8. If t = cosθ , then t2 – 12 =
cos2θ 12 cos2θ
cos θ 2
12 cos θ
2
9. (3π2 – 3) dt =
π3t – 3πt + c 3π2t – 3t + c
3π3 – 3π + c π3 – 3π + c
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10. x3 – 8
x – 2 dx =
x3
3 + x2 + 4x + c x3
3 – x2 + 4x + c
x3
3 + 2x2 + 4x + c x3 – 2x2+ 4x + c
11. Consider the sketch,
If A1 ,A2 are two areas, then b
a
f (x)dx =
8
5.4
– 5.4
– 8A1=6.7
a
x
y
f(x)
A2=1.3
b
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12. Consider the sketch. It's symmetric around y-axis. If the sum of ordinates's values
( y1 , y2 , ... , yn-1) is 5 and 3
0
f(x)dx=3, then the width of each interval for the shaded area is:
1 3
3 5
2 3
6 5
13. If E1 and E2 are two mutually exclusive events, P(E1) = 0.05 , P(E2' )= 0.07 ,
then P(E1 ∪ E2) =
0.02 0.12
0.35 0.98
14. On an experiment of throwing a fair die (has each face number 1 to 6) and tossing a coin, the results were recorded on each of them. If A is "the event of observing tail", B is "the event of observing 3" , then P(A ∪ B) =
1 12
1 3
7 12
2 3
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f(x)
y
x3–3
(3,4)
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Extended Questions
Write your answer for each of the three questions in the constructed response section in the space provided. Be sure to show all your work and correct units where applicable.
Question Two: [14 marks]
a) i. If 5x+7
(x – 5)(x2 + 7) = A(x2+7) + Bx(x – 5) + c (x – 5)
(x – 5)(x2 + 7) , find A. (3 marks)
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ii. Express x3 + 4x2
(x + 1)(x + 3) in partial fractions (3marks)
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b) Find the equation of the tangent to y = x2 + x at x = 1 (3marks)
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c) Without using a calculator: Find the value of sin120° + tan75° (5marks)
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Question Three: [14 marks]
a) i. If f(x) = x14 , find f " (x) (2 marks)
ii. Given that y = 2x3 + x has gradient equal 7 at the point (a , b), find possible values for a and b. (2 marks)
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b) i. Find the range of values of x for which y is decreasing, given that y = 4 3 x3 – 16x + 9.
(3 marks)
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ii. A container in the shape of a right circular cylinder with no top. It has surface area 3π square metres. What height (h) and base radius (r) which makes the volume of the container as maximum as possible? (3 marks).
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c) i. Find ( 3√y 5 – 8 )dy (2 marks)
ii. Find the equation of the curve which its gradient is given by 3x2 – 2x and f(2) = 7 (2 marks)
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Question Four: [14 marks]
a) i. Find the value of R and tan α in this identity: 4sin θ + 2cos θ= R cos (θ – α) (3 marks)
ii. Prove the identity 2cot2(90° – θ) – 6 + 6 sec2θ
– 2 + 2cosec2θ = ± 2 tan2θ.� (3 marks)
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b) i. If f(6) = 13 and f(9) = 17, find 9
6
f ' (x) dx (2 marks)
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ii. Consider the sketch. Find the shaded area. (2 marks)
f1(x) = x2 – 8
f2(x) = – x2
(2, –4)(– 2, –4)
y
x
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c) If A and B are defined in the sample space, P(A∪B) = 3 4 , P(A) =
2 3 and P(A∩B) =
1 4 ,
find:
i. P(A'). (1 mark)
ii. P(A⎟B). (3 marks)
[ End of Examination ]
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دة، ال يتم تصحيحها مسو
تعلي�ت وضوابط التقدم لالمتحــان:
الحضور إىل اللجنة قبل عرش دقائق من بدء االمتحان لألهمية. –إبراز البطاقة الشخصية ملراقب اللجنــة. –
�نع كتابة رقم الجلوس أو االسم أو أي بيانات أخرى تدل عىل –شخصية املمتحن يف دفرت االمتحان، وإال ألغي امتحانه.
يحظر عىل املمتحن© أن يصطحبوا معهم ¥ركز االمتحان كتبا دراسية –أو كراسات أو مذكرات أو هواتف محمولة أو أجهزة النداء اآليل أو
أي يشء له عالقة باالمتحان ك´ ال يجوز إدخال آالت حادة أو أسلحة من أي نوع كانت أو حقائب يدوية أو آالت حاسبة ذات
صفة تخزينية.يجب أن يتقيد املتقدمون بالزي الرسمي (الدشداشة البيضاء واملرص –أو الكمة للطالب والدارس© والزي املدريس للطالبات واللباس الع´¿
للدارسات ) و�نع النقاب داخل املركز ولجان االمتحان.ال يسمح للمتقدم املتأخر عن موعد بداية االمتحان بالدخول إال –
إذا كان التأخÉ بعذر قاهر يقبله رئيس املركز ويف حدود عرشدقائق فقط.
يتم االلتزام باإلجراءات الواردة يف دليل الطالب ألداء امتحان شهادة –دبلوم التعليم العام.
يقوم املتقدم باإلجابة عن أسئلة االمتحان املقالية بقلم الحرب (األزرق –أو األسود).
يقوم املتقدم باإلجابة عن أسئلة االختيار من متعدد بتظليل –:Ñوفق النموذج اآل ( الشكل (
عاصمــة سلطنة عمــــان هي: س – الدوحة القاهرة أبوظبي مسقط
) باستخدام القلم الرصاص وعند يتم تظليل الشكل ( مالحظة: .Éالخطأ، امسح بعناية إلجراء التغي
غ� صحيح صحيح
رقم الورقة
رقم املغلف
امتحان شهادة دبلوم التعليم العام للمدارس الخاصة )ثنائية اللغة(
للعام الدرايس 1434/1433 هـ - 2012 / 2013 م
الدور الثاين - الفصل الدرايس األول
• الرياضيات. تنبيه:
• األسئلة يف ) 15 ( صفحة.
• زمن اإلجابة: ثالث ساعات.
• اإلجابة يف الورقة نفسها.
Diploma, Semester One – Second Session, Bilingual Private Schools, Mathematics 2012/2013
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Question One (28 marks)
There are 14 multiple–choice items worth two marks each.
Shade the best correct answer for each of the following items.
1. limh 0 f (7) – f ( 7 + h )
2 h =
– 12
f ′ ( 7 ) 12
f ′ ( 7 )
f ′ ( 7 ) – f ′ ( 7 )
2. If f ( x ) = a x2 + 3 x and f ′ ( 2 )= –5, then the value of a is:
–2 – 12
12
2
3. If f ( x ) = x3
3 +
3x3
3 + 9, then the x - coordinates of the stationary points are:
0, 92
0, –92
0, 3 0, –3
4. If 1( x + 3 ) ( x + 2 )
= A
( x + 3 ) + B
( x + 2 ) , then 4 A + 6 B =
0 1
2 4
5. Which of trigonometric functions are both have domain θ ∈ R, θ ≠ ( 2 n + 1 ) 90º:
cot θ, sec θ sec θ, tan θ
cosec θ, tan θ cot θ, cosec θ
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6. If cosec θ = 54 and θ is obtuse, then tan θ equals:
– 43
– 45
– 34
– 35
7. The cosec e 12
θ – 5 o = √2 , 0º < θ < 180º, then θ =
20º 25º
80º 100º
8. If t = sin θ, then 1 – 2 t2 =
12
cos θ 12
cos 2 θ
cos 2 θ 2 cos θ
9. dtπ –5 =
π 6
6 + c 5π 4+ c
π 5 t + c π 6
6 t + c
10. x2 - 9x + 3
dx =
x2
2 + 3x + c
x2
2 – 3x + c
x 2 – 3x + c x 2 + 3x + c
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11. Consider the sketch, if b
a
f ( x ) dx = – 4 and
x
y
a b c
c
b
f ( x ) dx = 3.7, then the area bounded by the
curve of f ( x ) and the x – axis from x = a to x = c is:
0.3 0.7
3.7 7.7
12. If 3
a
–2x dx = 16, then a =
± 1 ± √17
± √7 ± 5
13. If A and B are independent events, P( A ) = 14
and P( B ) = 13
, then P( A′ B) =
112
16
14
1112
14. If E1, E2 and E3 are three mutually exclusive events, P ( E1 ) = 2 P ( E2 ) = 0.32
and P ( E′1 E′2 E′3) = 0.21, then P ( E3 ) =
0.15 0.31
0.52 0.69
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Extended Questions
Write your answer for each of the three questions in the constructed responsesection in the space provided.
Be sure to show all your work, including the correct units where applicable.
Question Two: (14 marks)
A) i) If 6 – 9x( x + 4 ) ( x2 + 5)
= A ( x2 + 5 )+ Bx ( x + 4 )+ c ( x + 4)( x + 4 ) ( x2 + 5)
, find A. (3 marks)
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ii) Express x4 – 4x2+ 1x ( x – 2 ) ( x + 2)
in partial fractions. (3 marks)
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B) Find the equation of the normal to y = x2 at the point where x = 1. (3 marks)
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C) Without using calculator: Find the value of cos 15º – tan 135º. (5 marks)
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Question Three: [14 marks]
A) i) Find, from the first principles, the derivative of 3x. (2 marks)
ii) Show that there is only one point on the curve y = 83
x3 + 4x2 + 2x, where the tangent
is parallel to y = 4. (3 marks)
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B) i) Find the range of values of x, for wich y is increasing, y = – x3 + 27x. (3 mark)
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ii) The sum of two natural numbers equals 120, in wich the multiplication of the square of one of them by the other number is to be as maximum as possible, Find the two numbers. (3 marks)
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C) i) Find √x45 dx. (2 marks)
ii) For all x if the second derivative of a curve equals 4, and it’s gradient at (4,1) is 12, find the equation of the curve. (2 marks)
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Question Four: (14 marks)
A) i) Find the value of R and tan in the identity: 5sin θ + 3cos θ = Rcos ( θ – ) ( 3 marks)
ii) Prove the following identity: ( 3 marks)
c 4sin2 ( 90º – θ )
cosec2 θ – cot2 θ + 8cos2 ( 90º – θ )
2sec2 θ – 2tan2 θ m tan2 θ = 8tan θ
2 – sec2 θ
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B) i) Consider the sketch. Find an approximation to the shaded area by using the Trapezium Rule. ( 2 marks)
4
2
2
3
3 4 5
1
10x
y
(2, 2.27) (4, 2.27)
(3, 2)
(5, 4)(1, 4)
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ii) Consider the sketch. Find the shaded area. (2 marks)
x
y
(0, 1)
(2, 5)
–2 –1 1 2
2
3
4
5
30
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C) A and B are defined in sample space such that, P ( A )= 0.5, P ( B ) = 0.4, P ( A B ) = 0.8 . Find:
i) P ( A' ) (1 marks)
ii) P ( A⎟ B) (3 marks)
END OF EXAMINATION AND GOOD LUCK!
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Mathematics first semester, Second session 2012/2013
(1(
Formulae Sheet For First Semester Differentiation:
1. ( )
2. Area and volume of a cuboid with length, width and height as l, w, and h respectively.
3. Area and volume of a cylinder with radius, r, and height, h.
4. Area and volume of a sphere with radius, r.
Trigonometry: Pythagorean Formulas:
1.
2.
3.
4. ( )
Compound Angle Formulas:
1. ( ) 2. ( )
3. ( ) 4. ( )
5. ( )
6. ( )
Double Angle Formulas:
1.
2.
3.
Half Angle Formulas:
1. ( )
2.
( )
The form acos bsin :
2 2 R= ,acos bsin can be expressed in the form Rcos( ) or Rsin( ) where a b
barctan .a
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Mathematics first semester, Second session 2012/2013
(2(
Integration:
∫ ( )
2. Area and volume of solids of revolution
∫ ( )
∫ ( ( ))
3. Trapezium rule
[ ( ) ]
Probability: 1. Addition Rule:
( ) ( ) ( ) ( ) 2. Conditional Probability:
( ) ( ) ( ) ( )
3. Multiplication Rule: ( ) ( ) ( ) ( ) ( )
4. Independent Rule: A and B are independent if :
( ) ( ) ( ) ( ) ( ) ( ) ( )
5. Mutually Exclusive Rule: A and B are Mutually Exclusive if :
P(A B) 0
Diploma, Semester One – Second Session, Bilingual Private Schools, Mathematics 2012/2013
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دة مسو
Diploma, Semester One – Second Session, Bilingual Private Schools, Mathematics 2012/2013
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دة مسو