امتحان شهادة دبلوم التعليم العام - المدارس الخاصة - ثناي ية اللغة للعام الدراسي ١٤٣٣/١٤٣٢ ه - ٢٠١١ ٢٠١٢ / م الدور الا ول - الفصل الدراسي الا ول
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1 رقم الورقة رقم المغلف تنبيه: المادة: الرياضيات - ثناي ية اللغة. الا سي لة في ) ١٤ ( صفحة. امتحان شهادة دبلوم التعليم العام - المدارس الخاصة - ثناي ية اللغة للعام الدراسي ١٤٣٣/١٤٣٢ ه - ٢٠١١ ٢٠١٢ / م الدور الا ول - الفصل الدراسي الا ول زمن الا جابة: ثلاث ساعات. الا جابة في الورقة نفسها. تعليامت وضوابط التقدم للامتحان: الحضور إلى اللجنة قبل عشر دقاي ق من بدء الامتحان للا همية. إبراز البطاقة الشخصية لمراقب اللجنة. مينع كتابة رقم الجلوس أو الاسم أو أي بيانات أخرى تدل على شخصية الممتحن في دفتر الامتحان وإلا ألغي امتحانه. يحظر على الممتحنني أن يصطحبوا معهم في لجان الامتحان كتبا دراسية أو كراسات أو مذكرات أو هواتف محمولة أو أجهزة النداء الا لي أو أي شيء له علاقة بالامتحان كام لا يجوز إدخال آلات حادة أو أسلحة من أي نوع كانت أو حقاي ب يدوية أو آلات حاسبة ذات صفة تخزينية. يجب أن يتقيد المتقدمون بالزي الرسمي (الدشداشة البيضاء والمصر أو الكمة للطلاب والدارسني والزي المدرسي للطالبات واللباس العامين مع العباءة للدارسات ( ومينع النقاب داخل المركز ولجان الامتحان. لا يسمح للمتقدم المتا خر عن موعد بداية الامتحان بالدخول إلا إذا كان التا خري بعذر قاهر يقبله ري يس المركز وفي حدود عشر دقاي ق فقط. يتم الالتزام بالا جراءات الواردة في دليل الطالب لا داء امتحان شهادة دبلوم التعليم العام. يقوم المتقدم بالا جابة عن أسي لة الامتحان المقالية بقلم الحبر (الا زرق أو الا سود). يقوم المتقدم بالا جابة عن أسي لة الاختيار من متعدد بتظليل الشكل ) ( وفق النموذج الا يت: س عاصمة سلطنة عمان هي: القاهرة الدوحة مسقط أبوظبي ملاحظة: يتم تظليل الشكل ) ( باستخدام القلم الرصاص وعند الخطا امسح بعناية لا جراء التغيري. صحيح غري صحيح
2 QUESTION ONE (28 marks) There are 14 multiple-choice questions worth two marks each. Shade in the bubble next to the BEST answer for each question. 1. lim h 0 f ( 3 + h ) f ( 3 ) h f ' ( 3 ) f ' ( 3 ) = 1 f ' ( 3 ) 1 f ' ( 3 ) 2. If f (x) = 3x 2 + 5, then the f ' (x) equals 5x + 5 6x + 5 6x 5x 3. If y = 1 2 x2 4x, then it will have a stationary point at: ( 0, 4 ) ( 4, 0 ) ( 4, 8 ) ( 4, 8 ) 4. If 4x + 1 ( x )( x 1) = Ax + B ( x ) + C ( x 1 ), then A + C = The function that is sketched ( for π < x < π ) in the diagram is: y = cos ec x y = cos x y = s ec x y = sin x 1
3 6. If sin θ = and tan θ > 0, then sec θ = 7. The period of the function y = 3 sin 2 ( x π 4 ) is 2 3 π 2π 3π 4π If sin θ + cos θ = 4 3, then sin 2 θ = x 2 + 2x x dx = x dx + 2x dx x dx + 2 dx x 2 dx + 2x dx x 2 dx + 2 dx 10. ( 1 3x )( 1 + 3x + 9x 2 ) dx = x 27 4 x4 + c x 27 2 x2 + c x x2 + c x x4 + c 11. If b a ( x 2 10x + 25 ) dx = 39, then 1 3 b a ( x 5) 2 dx = ( 2x + a ) dx = 7, then the value of a is
4 13. Consider the Venn diagram. If P ( E 1 ) = 0.14, P ( E 2 ) = 0.04, P ( E 3 ) = P ( E 4 ) = 0.19, P ( E 5 ) = 0.07, P ( E 6 ) = 0.3 and P ( E 7 ) = 0.07 then P ( A' B ) = A E 2 E 1 E 3 E 4 E 7 E 6 E 5 B The table below shows the results of a survey about students in college and the cost of attending college, the probability that the cost of attending college is too low, given that the student in college. Cost too much Cost just right Cost too low Student in college Student not in college
5 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. Express 3x ( x 2 ) ( 2x + 1 ) ( x + 1 ) as a sum of partial fraction: (6 marks) 4
6 B. If y = 4 3 x3 2 x 2 x, find d 2 y dx 2 (3 marks) 5
7 C. Solve the following equation: (5 marks) tan 2 ( x + π 6 ) = 7 for 0 x π 3 6
8 QUESTION THREE (14 marks) A. Find the points on y = x + 3 x where the gradient is 1 4 (4 marks) 7
9 B. Find the radius and height of the right circular cylinder of the largest volume that can be inscribed in a right circular cone with radius 6 inches and height 10 inches. (6 marks) r 10 in h 6 in 8
10 C. The gradient of a curve is given by dy dx = k x2 + 6, where k is a constant, given that the curve passes through ( 1, 6 ) and ( 1, 2 ). Find the value of y in terms of x. (4 marks) 9
11 QUESTION THREE (14 marks) A. i. Express ( 12cos θ + 5 sin θ ) in the form: R cos ( θ α ) where R > 0 and 0 < α < 90 (3 marks) 10
12 ii. Prove the following identity: (3 marks) cos 4θ + cos 2θ sin 2θ sin 4θ = cot θ 11
13 B. i. Find an approximation to the area bounded by the axes y = x 2, x = 3 and x = 8 using the trapezium rule with five intervals. (2 marks) 12
14 ii. Find the area between the curves y 2 = x and 3y 2 = 4 x (2 marks) 3y 2 = 4 x y 2 = x 13
15 C. i. Draw a Venn diagram to represent this information: (1 mark) Observing at least one tail in the experiment of tossing two balanced coins. ii. Consider the experiment of tossing a fair die that has each face numbered 1 to 6 and let: A: {observe an even number} B: {observe a number less than or equal to 4}. Show that A and B are independent (3 marks) [ End of the Examination ] 14
16 Formulae sheet for semester 1 Differentiation 1. y = x n dy n 1) = nx( dx 2. Area and volume of a cuboid with length, width and height as l, w and h respectively: Area = 2lw + 2wh + 2lh Volume = l w h 3. Area and volume of a cylinder with radius, r, and height, h: Area = 2πrh + 2πr 2 Volume = πr 2 h 4. Area and volume of a sphere with radius, r : Area = 4πr 2 Volume = 4 3 πr3 Trigonometry Pythagorean Formulae 1. sin 2 A + cos 2 A = 1 2. sec 2 A tan 2 A 3. cosec 2 A = 1 + cot 2 A Compound Angle Formulae 1. sin ( A + B) = sin A cos B + cos A sin B 2. sin ( A B) = sin A cos B cos A sin B Double Angle Formulae 1. sin 2A = 2 sin A cos A 2. cos 2A = cos 2 A sin 2 A cos 2A = 2cos 2 A 1 cos 2A = 1 2sin 2 A 3. tan 2A = 2 tan A 1 tan 2 A 3. cos ( A + B) = cos A cos B sin A sin B 4. cos ( A B) = cos A cos B + sin A sin B 5. tan ( A + B) = 6. tan ( A B) = tan A + tan B 1 tan A tan B tan A tan B 1 + tan A tan B Double Angle Formulae 1. sin A = 1 2 ( 1 cos A ) 2. cos A = 1 2 ( 1 + cos A ) The form a cos θ + b sin θ: a cos θ + b sin θ can be expressed in the form R cos ( θ ± α ) or R sin ( θ ± α ) where R = a 2 + b 2, α = arctan b a
17 Integration 1. x n dx = x ( n + 1) n c, n 1 2. Area and volume of solids of revolution: Area = b a f ( x ) dx Volume = Area = π b a ( f ( x ) ) 2 dx 3. Trapezium rule: Area = 1 2 h [ y ( y 1 + y y n 1 ) + y n ] Probability 1. Addition Rule: P ( A B ) = P ( A ) + P ( B ) P ( A B ) 2. Conditional Probability: P ( A given B ) = P ( A B ) = P ( A B ) P ( 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: P ( A \ B ) = P ( A ) or P ( B \ A ) = P ( B ) or P ( A B ) = P ( A ) P ( B ) 5. Mutually Exclusive Rule: A and B are mutually exclusive if: P ( A B ) = 0
18 م س ود ة لا يتم تصحيحها 17
س الدوحة القاهرة أبوظبي
رقم الورقة رقم المغلف تنبيه: المادة: رياضيات. الا سي لة في ) ١٤ ( صفحة. امتحان دبلوم التعليم العام للمدارس الخاصة (ثناي ية اللغة) للعام الدراسي ١٤٣٥/١٤٣٤ ه - ٢٠١٣ ٢٠١٤ / م الدور الثاين - الفصل الدراسي
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