The document is a math exam consisting of 8 questions covering topics in statistics, geometry, algebra, trigonometry, and coordinate geometry. Some key details: - Question I analyzes a pie chart about the number of books students read and involves calculating relative frequencies, determining cumulative frequencies, and constructing graphs. - Question II involves constructing a point from vector additions and proving properties about the resulting quadrilateral. - Question III expands algebraic expressions and solves equations involving square roots. - Question IV proves proportionality and finds the linear equation from a table. - Question V factors polynomials and solves equations equal to zero and each other, and analyzes the ratio of two polynomials. - Questions VI-VII apply properties of

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CCS Mathematics June 2015
Class of G8 Exam of 𝟑 𝒕𝒉 semester Duration : 2 h
Name :…………………………………..
:‫مالحظة‬‫ينا‬ ‫الذي‬ ‫بالترتيب‬ ‫اإلجابة‬ ‫المرشح‬ ‫يستطيع‬ ‫البيانات‬ ‫لرسم‬ ‫أو‬ ‫المعلومات‬ ‫الختزان‬ ‫أو‬ ‫للبرمجة‬ ‫قابلة‬ ‫غير‬ ‫حاسبة‬ ‫آلة‬ ‫باستعمال‬ ‫يسمح‬‫االلتزام‬ ‫(دون‬ ‫سبه‬
‫بترت‬.)‫المسابقة‬ ‫في‬ ‫الوارد‬ ‫المسائل‬ ‫يب‬
I. (4 points)
A statistical survey made on the numbers of books read by 200 students of the class of G8, is showing in
the following pie chart.
1) Complete the following table :
Number of books read 1 2 3 4 5 6 total
Relative frequency in %
frequency
Cumulative frequency
2) Determine the number of students read at least 3 books.
3) What is the percentage of the students that read at least 5 books ?
4) Indicate, with justification, the most frequently number of books which the students have read.
5) Draw the bar graph of the numbers of the students.
6) Draw the polygon of the cumulative frequencies.
II. (3 points)
O, A and B are three distinct non-collinear points.
1) Construct the point C such that 𝑂𝐶⃗⃗⃗⃗⃗ = 𝑂𝐴⃗⃗⃗⃗⃗ + 𝑂𝐵⃗⃗⃗⃗⃗ .
2) What is the nature of the quadrilateral OACB ? Justify.
3) Let D be the midpoint of [AB]. E is the symmetry of D with respect to O and F is the symmetry
of D with respect to B. Prove that (EF) and (OB) are parallel. Deduce that 𝐸𝐹⃗⃗⃗⃗⃗ = 2𝑂𝐵⃗⃗⃗⃗⃗ .
III. (3 points)
Consider the following real numbers :
𝐴 = √2(1 + √6) and 𝐵 = 2 − √6
1) Write 𝐴 × 𝐵 in the form of 𝑎√2 + 𝑏√3, where a and b are two integers.
2) Calculate 𝐴² and 𝐵². Deduce that: 𝐴² + 𝐵² = 24.
1 book
40%
2 books
20%
3 books
16%
4 books
10%
5 books
8%
6 books
6%

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IV. (3 points)
Consider the adjacent table :
1) Prove that this table is proportional.
2) Calculate the coefficient of proportionality
and deduce its linear equation .
V. (4 points)
Given the polynomials : 𝑃( 𝑥) = (2𝑥 − 3)( 𝑥 + 4) − 𝑥² + 16 and 𝑄( 𝑥) = ( 𝑥 + 5)2 − 1
1) Factorize 𝑃(𝑥) and 𝑄(𝑥).
2) Solve the following equations :
a) 𝑃( 𝑥) = 0.
b) 𝑃( 𝑥) = 𝑄( 𝑥).
c) 𝑄( 𝑥) = 24.
3) Let : 𝐹( 𝑥) =
𝑃(𝑥)
𝑄(𝑥)
a) For what values o x, 𝐹(𝑥) is defined?
b) Simplify 𝐹(𝑥) then solve 𝐹( 𝑥) =
1
2
.
VI. (2 points)
1) Prove that for all the nonzero numbers, this inequality 𝑥² + 𝑦² ≥ 2𝑥𝑦 is always true.
2) Deduce that 𝑥 𝑎𝑛𝑑 𝑦 are two nonzero numbers with same sign then
𝑥
𝑦
+
𝑦
𝑥
≥ 2.
VII. (6 points)
1) In an orthonormal system of axis x’Ox, y’Oy, place the points A(1 ;3), B( 5 ;8) and C(5 ;3).
2) Calculate the measure of [AB],[AC] et [BC], the deduce the nature of the triangle ABC.
3) Determine the coordinates of the point D, such that 𝐶𝐴⃗⃗⃗⃗⃗ = 𝐵𝐷⃗⃗⃗⃗⃗⃗
4) What is the nature of the quadrilateral ACBD ? Justify.
5) Determine the coordinates of the point I , the midpoint of [AC] and of the point J that of [AB].
6) Deduce the length of segment IJ.
7) Prove that OACB is a parallelogram.
VIII. (5 points)
In the adjacent figure, (C ) is a circle of center O
and radius 3 cm, BP= AB and 𝐵𝐴̂ 𝑁 = 30°.
1) Calculate AP,BN and AN.
2) Calculate AQ and BQ.
3) Verify that M is the midpoint of [AP].
4) Verify that (OM) is perpendicular at (AB).
5) Find the locus of P when N varies on ( C).
GOOD WORK
0.16 × 10² 1.3̅
3 3
4
×
25
100
÷
3
4