This document is a mathematics exam consisting of 6 sections with multiple questions in each section. Some of the questions involve solving systems of equations, analyzing geometric shapes and properties, calculating lengths and angles, plotting lines on a coordinate plane, and determining loci of points. The exam covers a wide range of mathematics topics including algebra, geometry, trigonometry, and coordinate geometry. It tests the student's ability to set up, solve, and justify various mathematical problems.

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CCS Mathematics 23 Dec. 2013
Class of G9 Exam of 𝟏 𝒔𝒕
semester Duration : 120 min
Nom :…………………………………..
:‫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)
Consider the four numbers A, B , C and D.
𝐴 = (√5− 1)
2
+ (√5+ 1)
2
, 𝐵 =
3
4
+
5
4
×
7
15
, 𝐶 =
(10√3)
4
25×103×3+103×6×37.5
and
𝐷 =
3.4×10−3×(102)³
4×10⁻³
In the following questions all the steps of calculations must be shown :
1) Prove that A is a natural numbers.
2) Write B in the form of fraction in simplest form.
3) Prove that C is a decimal number.
4) Give the scientific form of D.
II. (2 points)
A bookshop offers a 10 % discount on its articles.
The sum of original prices of a pen and an agenda is three times the original price of the pen.
The sum of prices of the pen and the agenda after discount is 54 000 LL.
1) Model the previous information into a system of two equations with two unknowns.
2) Solve this system and find the original price of a pen and that of an agenda.
III. (2 points)
A restaurant proposes to its customers two offers for lunch:
Offer A: Monthly subscription of 200,000 LL.
Offer B: 11,000 LL per meal.
1) Yara takes 20 meals per month. What is the offer that most advantageous for Yara? Justify.
2) Youssef takes 10 meals per month. What is the offer that most advantageous for Youssef?
Justify .
3) Let x be the number of meals per month. Find in the terms of x, the price with both offers.
4) By solving an inequality, indicate the number of meals which offer B is the most advantageous.
IV. (3 points)
Given a rectangle ABCD such that AB= 5 cm and AD= 4 cm, Let M be a point of [AB] such
that BM= 2cm.
1) Calculate DM.
2) Deduce that the triangle DMC is isosceles.
3) Let I be the midpoint of [MC]. Prove that 4 points A, M, I and D belong to the same circle
which its center is to be determined and calculate its radius.
4) Calculate MC , MI.

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V. (5 points)
In an orthonormal system of axes, x'Ox and y'Oy, consider the line (D) with equation y = -2x-3
and the two points A (-2 ; 1) and B (6 ; 5).
1) Verify that (D) passes through A.
2) Plot A and B and draw (D).
3) Determine the equation of (AB) and deduce that (D) is perpendicular to (AB).
4) The line (D) intersects y'Oy at C. Find the coordinates of C.
5) Let (S) be the circle circumscribed about the triangle ABC.
I is the center of this circle . Calculate the coordinates of I.
6) The line (D') is the parallel through C to (AB) . (D') intersects the circle (S) at another point E.
a. What is the nature of the quadrilateral ABEC? Justify.
b. Calculate the coordinates of the point E.
c. (d) is the tangent at A to (S). Find the equation of (d).
VI. (5 points)
In the adjacent figure we have :
 (C ) is a circle of center O and diameter
[AB].
 OA= OB= 3 cm.
 P is a point of [AB) such that OP= 5 cm.
 E is a point of ( C) such that EP= 4 cm.
 (D) is tangent in A at (C ).
 (PE) cuts (D) in J.
1) Reproduce the figure.
2) Prove that (PE) is tangent at (C ) in E. Deduce that
JE= JA.
3) Let JE= JA= 𝑥 and JP= 𝑥 + 4 where 𝑥 is a unit of
cm.
a. Apply Pythagoras theorem in the triangle APJ
for calculating 𝑥.
b. Deduce that ABJ is a right isosceles triangle.
4) (JB) cuts ( C) in another point F. Prove that F is the midpoint of [BJ] and (FO) is the
perpendicular bisector of [AB].
5) Let N be the midpoint of [MB]. Find the locus N where M varies on (C).
GOOD WORK.