This document contains a mathematics exam with 7 sections covering various topics: 1) Choosing the correct answers with justifications for questions about equations, inequalities, and trigonometric functions. 2) Verifying relationships between algebraic expressions. 3) Calculating relative frequencies, means, and number of buildings needed based on floor distributions. 4) Solving a system of equations and word problems about prices and discounts. 5) Working with polynomials including determining coefficients, evaluating, and solving equations. 6) Analyzing points, lines, triangles, and circles in a coordinate plane. 7) Proving geometric relationships involving circles, lines, and triangles.

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CCS Mathematics March 2014
Class of G9 Exam of 𝟐 𝒏𝒅
semester Duration : 120 min
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. (1.5 point)
In the table below one answer only is correct to each question. Choose the correct answer with justification.
𝑵° Questions Answers
a b c
1)
𝐴 =
1
3
−
4
3
÷
3
8
− (
1
3
−
1
6
)
2
𝐴 = −
13
4
𝐴 =
1
9
𝐴 = −
5
3
2) The solution of :
(𝑥 − 1)2
≤ (𝑥 − 3)2
𝑥 > 2 𝑥 ≤ 2 𝑥 ≥ 1
3) If 𝑎 > 0, cos 𝛼 = 𝑎 +
1
2
𝑎𝑛𝑑
sin 𝛼 = 𝑎 −
1
2
𝑡ℎ𝑒𝑛 𝑎 =
2 1
2
3
4) If 𝐴 = 3√3 and 𝐵 = 2 + 2√2 then 𝐴 = 𝐵 𝐴 < 𝐵 𝐴 > 𝐵
II. (1.5 point)
Given the following numbers :
𝐴 = 1 +
√76
√19
+
√135
√15
; 𝐵 =
40×10‾5−34×10‾⁴
50×10‾⁵
and 𝐶 = √(
1
4
−
3
16
) − (
1
2
−
1
3
)
2
−
1
9
÷ 2
1) Verify that A and B are opposite.
2) Verify that A and C are inverse.
III. (2.5 points)
In a certain city the numbers of floors in each building are distributed as follows:
Number of floors 2 3 4 5 6 7 8
Number of buildings 28 56 60 64 80 68 44
Relative frequencies in percentage
1) - What is the number of buildings in this city?
- Complete the above table showing your calculations.
2) Find the mean of this distribution.
3) How many building must the government build of 10 floors each such that the average number of floors
in this city becomes 6.005?
IV. (2.5 points)
1) Solve the following system : {
3𝑥 + 2𝑦 = 130
24𝑥 + 14𝑦 = 1000
2) Rami went into a store and bought three jackets and two shirts. He paid 130$. After a sold of 20% on
the price of the jacket and 30% on the price of the shirt, Salim bought from the same store the same 3
jackets and 2 shirts as Rami, but he paid 100 $. Find the original price of the jacket and shirt.

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V. ( 3 points)
1) Consider the polynomials :
𝑃(𝑥) = (𝑎 − 1)𝑥² + (2𝑏 + 1)𝑥 + 24 and 𝑄(𝑥) = (𝑚 + 4)𝑥² + (𝑛2
+ 8)𝑥 + (8𝑝 − 3).
a) Can you determine a and b such that P(𝑥) is identical to zero? Justify.
b) Can you say that 𝑄(𝑥) is identical to zero ? justify.
2) Consider the polynomial : 𝐴(𝑥) = (4𝑥 − 1)2
+ (𝑥 + 2)(𝑥 + 6)
a) Show that 𝐴(𝑥) = 17𝑥² + 13.
b) Does the equation 𝐴(𝑥) = 0 admit any solutions ? Justify.
c) Solve :
i) 𝐴(𝑥) = 166
ii)
𝐴(𝑥)
3𝑥²+4.2
= 5
3) Calculate 𝐴(√2 − 1).
VI. (4 points)
In an orthogonal system of axis x’Ox and y’Oy, consider the points A(4 ;1) ; B(5 ;4) and C(1 ;2).
1) Determine the equation of the line (AB) and that of the line (AC).
What can you say about these lines ? Justify.
2) Calculate AB, AC and BC. What is the nature of the triangle ABC ? Justify.
3) Calculate sin 𝐴𝐵̂ 𝐶 and tan 𝐴𝐵̂ 𝐶.
4) Determine the coordinates of the center M and the radius of the circumscribed circle about the triangle
ABC.
5) Does the origin O belong to this circle ? why ?
6) Draw the line (D) of equation 𝑦 = 2 which cuts (AB) in P. Calculate the coordinates of P.
7) Let D be the translate of A by translation of vector 𝐶𝐵⃗⃗⃗⃗⃗ . Find the coordinates of D.
VII. ( 5 points)
In the adjacent figure we have :
 (C ) is a circle of center O.
 [AB] is a fixed diameter of (C ) such that AB= 6 cm.
 M is a variable point on (C ) and N is the point diametrically
opposite to M.
 E is the symmetric of A with respect to M.
1) Reproduce the figure.
2) a- Prove that (OM) and (BE) are parallel; deduce the length of BE.
b- Prove that (BM) the perpendicular bisector of [AE].
c- Prove that when M varies on (C ), the point E varies on a fixed
circle which its center and radius will be determined.
3) Let I be the point of intersection of the lines (EN) and (AB).
a- Prove that the triangles ION and IBE are similar and deduce that :
𝐼𝐵 = 2 × 𝐼𝑂.
b- Calculate IO and IB.
c- What does that the point I represent for the triangle MBN ? Justify.
d- (EN) cuts (MB) at F. Prove that (OF) is perpendicular to (MB).
GOOD WORK.