14 ح ر وجع ومطابق لألصل اليدوى وي طبع على مسئولية اللجنة الفنية. a b x a x b c. a b c

Transcript

1 ر وجع ومطابق لألصل اليدوى وي طبع على مسئولية اللجنة الفنية ا االسم التوقيع التاريخ االسم التوقيع التاريخ 4 ح ث.ع.ج / أول ARAB REPUBLIC OF EGYPT Ministry of Education General Secondary Education Certificate Examination, 04 { First Session New System } The Pure Mathematics { Algebra and Solid Geometry } {70} Time: Hours 4 ح تابع {70} ث.ع.ج / أول الرياضيات البحتة { الجبر والهندسة الفراغية } باللغة اإلنجليزية } األسئلة فى صفحتين { تنبيه مهم : يسلم الطالب ورقة امتحانيه باللغة العربية مع الورقة المترجمة. Remark: Calculators are allowed. Note that :{,, } are the cubic roots of unity and i = - I- Algebra Answer the following questions: First Question: (6 marks) a) Complete the following statements: - The solution set of the equation x = is - If the number of terms of the expansion of n x y is terms, then the value of n = - When solving the two equations: a x + b y = c and a x + b y = c, if x = 7, y =, then c=, c= b) If n = 70, n p r = 0, find the value of : n C + n C r r Second Question: (6 marks) a) In the expansion of 4x x 5, find : ( i ) The value of term free of x. ( ii ) The value of x which makes the two middle terms of the expansion equal. b) Prove that: = 9, where is one of the cubic roots of the unity. 7 Third Question: (6 marks) a) Put the number z = + i in the trigonometric form, then find its square roots in the exponential form. b) Without expanding the determinant, prove that : a bx ax b c a a bx ax b c = ( x b c ) a b c a b x a x b c a b c II- Solid Geometry Fourth Question: (6 marks) a) Complete the following statements: - If the measure of the angle between two skew lines equals 90, then the two lines are... - If each of two intersecting planes is perpendicular to a third plane, then the line of intersection of the two planes is... - If the dimensions of a cuboid are x, y and z and xy = 48 cm, xz = 44 cm and yz = 9 cm, then its volume = cm and the length of its diagonal =... cm. b) X, Y and Z are three parallel planes. The line L intersects them in the points A, B and C respectively and the line M intersects them in the points D, E and F respectively. If AF intersects the plane Y in N and AB : BC = :, prove that: CF + AD = 4(BN + NE). Fifth Question: (6 marks) MABC is a triangular pyramid in which MA plane ABC, AB = AC =0 cm, BC = cm, MA = 8 cm and D is the mid-point of BC. ( i ) Calculate the length of AD and prove that MD BC (ii) Find m ( M- BC - A). (iii) Prove that the two planes MAD and MBC are perpendicular. انتهت األسئلة بقية األسئلة فى الصفحة الثانية

2 ) ) ) جمهورية مصر العربية وزارة التربية والتعليم امتحان شهادة إتمام الدراسة الثانوية العامة لعام 04 م نموذج إجابة الرياضيات البحتة [ الجبر والهندسة الفراغية - باإلنجليزية الدرجة العظمى ( 0 الدرجة الصغرى ( - عدد الصفحات ( 5 ] 70 [ الدور األول ( نظام حديث ) First: Algebra Answer to the first question: (6 Marks) (A) ( marks) - (B) ( marks) A) - {, } C = 6, C = 5 B) ي n = 70 = 6 n ى = 6 n ى 7 r P ي, 7 = 0 F P r 7 = P ى = r ى = 5 r ى n + C r n + + C r n + = C r 8 = C 5 8 = C = 56

3 تابع 7 ث.ع / أول / ح نموذج إجابة [ الرياضيات البحتة الجبر والهندسة الفراغية ( باللغة اإلنجليزية ) ] نظام حديث 04 Answer to the second question: (6 Marks) (A) ( marks) - (B) ( marks) 5 A) [ i ] T r+ = C r ( x )r (4x ) 5 r 5 = C r ( )r (4) 5 r x 0 r To find the term free of x, put 0 r = zero = 0 r ى T ى is the term free of x, T = 5 C 0 ( )0 (4) 5 = 00 [ ii ] The two middle terms are T 8, T 9 T 9 ى T 8 = x = = 8 8 x = F x ى = x ى B) L. H. S. = ( 5 ω 5ω 7ω ω 7 ) 4 = ( 5ω ω 5ω = [ ω (5ω ) 5ω ω 7ω ω 7 ) 4 ω (ω 7) ω 7 ] 4 = (ω ω) 4 = (± i) 4 = 9 i 4 = 9 = R. H. S.

4 تابع 7 ث.ع / أول / ح نموذج إجابة [ الرياضيات البحتة الجبر والهندسة الفراغية ( باللغة اإلنجليزية ) ] نظام حديث 04 Answer to the third question: (6 Marks) (A) ( marks) - (B) ( marks) A) ي Z = + i = + = Z ى, cos θ = θ = π ى, sin θ = ] Z = [cos π + i sin π ى, The square roots of the number Z = [cos π + i sin π ] = [cos π + n π + i sin π + n π ], where n = 0, at n = 0 F one root = [cos π 6 + i sin π 6 ] = eπ 6 i at n = F the other root = [cos 7π 6 + i sin 7π 6 ] = e7π 6 i B) L. H. S. = a + b x a x + b C a + b x a x + b C C x C a + b x a x + b C = a a x a x + b C a a x a x + b C a a x a x + b C = a ( x ) a x + b C a ( x ) a x + b C a ( x ) a x + b C taking ( x ) as a common Factor from the st col. = ( x ) a a x + b C a a x + b C a a x + b C, C x C = ( x ) a b C a b C a b C = R. H. S.

5 تابع 7 ث.ع / أول / ح نموذج إجابة [ الرياضيات البحتة الجبر والهندسة الفراغية ( باللغة اإلنجليزية ) ] نظام حديث 04 Second: Solid Geometry Answer to the fourth question: (6 Marks) (A) ( marks) - (B) ( marks) A) - Skew orthogonal lines - Perpendicular to this plane - 5, 6 B) ي The plane Y // the plane Z and the plane ACF intersects them D M L in BN, CF respectively X A BN // CF ى BN ى = AB CF AC BN ى CF = 4 = AN AF = 4 (). BN CF = 4 ى Y E N B Similarly: plane X // plane Y, C plane FAD intersects them Z F ى NE = FN AD FA = 4 for drawing the model (). NE AD = 4 ى adding (), () NE) CF + AD = 4 (BN + ى 4

6 تابع 7 ث.ع / أول / ح نموذج إجابة [ الرياضيات البحتة الجبر والهندسة الفراغية ( باللغة اإلنجليزية ) ] نظام حديث 04 Answer to the fifth question: (6 Marks) [ i ] In DABC: AB = AC, D is mid-point of BC ي M AD BC ى 8 cm (CD) (AD) = (AC) ى = 00 6 B 0 cm A = 64 (AD) ى AD = 8 cm ى 6 cm D 6 cm 0 cm MA plane ABC, MD is ي, inclined to the plane, its projection AD BC MD BC ى [ii] ي Each of MD, AD is perpendicular to BC C for drawing the model MDA is a plane angle for the dihedral angle M BC A ى ( MDA) m ( M BC A) = m ى MA = AD = 8 cm ي = 45 ( MDA) m ( M BC A) = m ى [iii] ي BC both MD, AD BC plane MAD ى Plane BC plane MBC ي Plane MBC plane MAD ى انتهى نموذج اإلجابة 5