Download REPÚBLICA ÁRABE DE EGIPTO

〔 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