This document is the exam for the Class of G8 in CCS Mathematics from March 2014. It consists of 7 questions testing various math skills. Question I involves simplifying fractions and expressions. Question II requires writing fractions in irreducible form and determining properties. Question III has students developing, factorizing, and solving polynomials. Question IV asks about the locus of a midpoint on a circle. Question V proves properties of midpoints and determines a quadrilateral. Question VI expresses areas in terms of x and solves for x given an area condition. Question VII draws and analyzes a trapezoid, proving properties about its midpoints and segments.

1. Page 1 of 2
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].

2. Page 2 of 2
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.