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CCS Mathematics March 2014
Class of G8 Exam of 𝟐 𝒏𝒅
semester Duration: 2hs
Name :…………………………………..
:‫مالحظة‬‫يناسب‬ ‫الذي‬ ‫بالترتيب‬ ‫اإلجابة‬ ‫المرشح‬ ‫يستطيع‬ ‫البيانات‬ ‫لرسم‬ ‫أو‬ ‫المعلومات‬ ‫الختزان‬ ‫أو‬ ‫للبرمجة‬ ‫قابلة‬ ‫غير‬ ‫حاسبة‬ ‫آلة‬ ‫باستعمال‬ ‫يسمح‬‫بترتيب‬ ‫االلتزام‬ ‫(دون‬ ‫ه‬
.)‫المسابقة‬ ‫في‬ ‫الوارد‬ ‫المسائل‬
I. ( 3 points)
1) Given :
𝐴 = √
5
6
× √
8
7
× √
21
5
; 𝐵 =
1
√2
× √
2
3
×
√3
2
and 𝐶 =
2,8×1019+0.77×10²⁰
46×1018−0.031×10²¹
a) Prove that
𝐴
𝐵
= 4.
b) Calculate 𝐴 × 𝐵 then choose the correct answer :
i- A and B are opposites.
ii- A and B are inverses.
c) By writing all the step of calculation, prove that C is an integer.
II. ( 2 points)
Consider the following numbers : 𝐴 = −
1
2
√1.34̅ × 10 and 𝐵 =
2+
1
3
1−
1
8
÷
4
3
−
1
4
×
2
3
1) Write A and B in the form of irreducible fractions.
2) What can you say about A and B ? Justify.
III. ( 4 points)
Given the following polynomials :
𝐴(𝑥) = (3𝑥 − 2)2
− (2𝑥 + 1)2
𝑎𝑛𝑑 𝐵(𝑥) = (𝑥 − 3)2
− (9 − 𝑥2) + (𝑥 + 1)(2𝑥 − 6)
1) Develop and reduce 𝐴(𝑥) and 𝐵(𝑥).
2) Factorize 𝐴(𝑥) and 𝐵(𝑥).
3) Solve :
a) 𝐴(𝑥) = 𝐵(𝑥).
b) 𝐴(𝑥) = 5𝑥².
4) Consider the fraction 𝐹(𝑥) =
𝐴(𝑥)
𝐵(𝑥)
a) For what values of 𝑥, 𝐹(𝑥) is defined ?
b) Simplify 𝐹(𝑥).
c) Solve 𝐹(𝑥) = 0 and 𝐹(𝑥) =
4
5
.
IV.( 1 point)
Given a fixed circle C( O, R) of fixed diameter [AB].
Let E be a variable point on this circle. Find the locus of M the
midpoint of the chord [BE].

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V.( 3 points)
In the adjacent figure, the points M, N, P and Q are the
respective midpoints of the segments [AB], [BD], [DC]
and [AC].
1) Prove that (MQ) is parallel at (NP).
2) Prove that MQ=NP.
3) Deduce the nature of the quadrilateral MNPQ.
VI.(2 points)
In the adjacent figure, ABCD is a rectangle and DEFG is
a square.
1) Express the areas of ABCD and DEFG, in function
of 𝑥.
2) Determine 𝑥 such that the area of the hachured part is
equal to 26 cm².
3) Determine the dimensions of ABCD.
VII.( 4 points)
Draw a trapezoid ABCD right at A and D such that AB= 6 cm and DC= 8 cm. let H be the foot of the
perpendicular passing from B at (DC).
1) What is the nature of ABHD ? Justify.
2) Let E be the midpoint of [AD], J is the midpoint of [HC] and I the midpoint of [BC].
a) Prove that (IJ) and (AD) are parallel.
b) Deduce the nature of the quadrilateral AEJI., justify.
c) (EI) cuts (BH) in L, prove that L is the midpoint of [BH].
GOOD WORK.