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〔 E.N / 15〕 ح/ أ/ ع. ج 〔ث/ 54〕 ARAB REPUBLIC OF EGYPT Ministry of Education General Secondary Education Certificate Examination, 2015 〔 New System – First Session 〕 Time: 2 hours Pure Mathematics 〔 Algebra & Solid Geometry 〕 الرياضيات البحتة 〕 الجبر والهندسة الفراغية 〔باللغة اإلنجليزية . ـ يسلم الطالب ورقة امتحانية باللغة العربية مع الورقة المترجمة1 : تنبيه مهم . ـ اإلجابات المتكررة عن أسئلة االختيار من متعدد لن تقدر ويتم تقدير اإلجابة األولى فقط2 Remark: 1- Calculators are allowed. 2- { 1, , 2 } are the cubic roots of unity and i2 = -1 〔 〕 األسئلة فى صفحتين First: Answer only One Remark: 1. Calculators are question allowed of the following: First Question: Complete the following statements: (6 marks) a) If a five different digit odd number is formed from the digits 1,2,3,4,5,6 and 8, then the number of numbers that can be formed equals ………… . b) The numerical value of the expression: (2 + 5 + 2 2 )6 equals ………… . c) If two coplanar lines intersect and are parallel to another two intersecting coplanar lines, then ………… d) If ABCDA'B'C'D', is a cuboid, then the measure of the angle between AB , B'C' equals ………… ْ . e) In the cube ABCDA'B'C'D' ,if the area of إBA'C' is 50 3 cm 2 , then its volume equals ………… cm 3 . f) In the opposite figure: M MXYZ is a triangular pyramid. Its base is the isosceles right-angled triangle XYZ at Y. If MX ‘ plane XYZ, then the measure Second Question: Choose the correct answer from those given: (6 marks) a) The numerical value of the expression e bi ــe ــbi Z X of the dihedral angle Y ــMX ــZ equals …..… ْ . Y = ………… [ ‒2 or 0 or 1 or 2] 1 1 1 b) 1 x + 1 1 = ………… [ 0 or x + y or xy or ــxy] 1 1 y +1 c) Two lines which are perpendicular to the same plane are ………… [parallel or intersecting or skew or perpendicular] d) In a regular triangular pyramid, if its edge length is L and its height is h, 2 then h2 = ………… L [ 3 2 or 2 or 3 3 or 2 〔 〕 بقية األسئلة فى الصفحة الثانية ، ُروجع ومطابق لألصل اليدوى ويطبع على مسئولية اللجنة الفنية التوقيع االسم التوقيع االسم 2 3 ] 〔 E.N / 15〕 ح/ أ/ ع. ج 〔ث/ 54〕 تابع 〔 2〕 e) If !AB is inclined to a plane at angle of measure 60 ْ , and if the length of the projection of !AB in the plane is 10 cm, then AB = ……… cm. [10 3 or 20 or 20 3 or 5 ] f) If the lateral height of a regular triangular pyramid is 10 3 cm , then the sum of the areas of its faces equals ………… cm 2 [ 400 3 or 80 3 or 120 3 or 800 3 ] Second: Answer the following Questions: Third Question: (8 marks) a) ( i ) If m+n P2 = 182, n ــ1 C3 = 84, find the value of each of m , n. 3 ( ii ) If the two middle terms in the expansion of ( 2 + x )9 are equal, x 8 find the value of x. x 1 1 b) Without expanding the determinant, prove that: 1 1 x = ( x 1)2 x 1 x Fourth Question: (8marks) a) If z = 16 , write z in the trigonometric form, then find the cubic roots 1 3 i of z in the exponential form. b) In إABC, if: c cos A + a cos C = b, b cos A + a cos B = c, c cos B + b cos C = a, 2 2 2 use Cramer’s rule to prove that: cos A = b + c a 2bc Fifth Question: (8 marks) a) MXYZ is a triangular pyramid. If L is the mid-point of !XY and a plane is drawn passing by point L parallel to both !MX and !YZ and intersecting !MY , !MZ and !XZ at N,P and Q respectively, show that LNPQ is a parallelogram with perimeter equals to MX + YZ. b) MABCD is a right quadrilateral pyramid, where its base ABCD is a square of side length 48 cm and its diagonals intersect at N. If E is the mid-point of !AB , MN = 24 cm: ( i ) Prove that the plane MAC ‘ the plane ABCD. ( ii ) Prove that !AB ‘ the plane MNE. ( iii ) Find m ( M ــAB ــD ) 〔 〕 انتـهـت األســئـلـة ، ُروجع ومطابق لألصل اليدوى ويطبع على مسئولية اللجنة الفنية التوقيع االسم التوقيع االسم