This document appears to be an exam for a mathematics class covering several topics: 1) Identifying the correct question and answer from a multiple choice list and justifying the selection. 2) Performing calculations such as simplifying fractions, writing numbers in scientific notation, and evaluating expressions. 3) Solving geometry problems involving properties of circles, triangles, quadrilaterals, and finding lengths, midpoints, and areas. 4) Simplifying and solving polynomial equations. 5) Proving geometric properties about lines and figures. The exam consists of 6 sections involving various math problems and calculations to show work for full credit. Correct solutions require identifying and applying relevant math concepts and formulas.

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CCS Mathematics Dec. 2014
Class ofG9 Exam of 𝟏 𝒕𝒉 semester Duration : 120 min
Name:…………………………………..
:‫مالحظة‬‫(د‬ ‫يناسبه‬ ‫الذي‬ ‫بالترتيب‬ ‫اإلجابة‬ ‫المرشح‬ ‫يستطيع‬ ‫البيانات‬ ‫لرسم‬ ‫أو‬ ‫المعلومات‬ ‫الختزان‬ ‫أو‬ ‫للبرمجة‬ ‫قابلة‬ ‫غير‬ ‫حاسبة‬ ‫آلة‬ ‫باستعمال‬ ‫يسمح‬‫االلتزام‬ ‫ون‬
.)‫المسابقة‬ ‫في‬ ‫الوارد‬ ‫المسائل‬ ‫بترتيب‬
I. (2 points)
In the following table, only one of the proposed question is correct. Write the number of each question
and its corresponding answer. Justify your choice.
No Questions Answers
a b c
1 216 + 213
212 − 210
227 24 27
2
(4 −
5
2
)
2
(4 +
5
2
)
2
(1 −
5
2
)
2
(
5
2
− 4)
2
3 If 𝑚2 + 𝑛2 = 20 and 𝑚𝑛 = 8, then( 𝑚 − 𝑛)2 = −6 4 2
4
If 𝐴 = √(√2 − 2)
2
− √(2 − √3)
2
− √(√2− √3)
2 −2√3 − 4 0 2√2 − 4
II. (3 points)
Given the following numbers:
𝐴 =
7
18
×
2
7
− (
5
3
− 1)
2
; 𝐵 =
0.3×10‾³×0.006×10⁶
0.9×(10²)4
𝐶 = 2√5 + 2√125 − √45 ; 𝐷 = √(3 − 2√2)
32
× √(3+ 2√2)
32
All the steps of calculation must be shown:
1) Write A in the form of irreducible fraction.
2) Write the scientific notation of B.
3) Write C in the form of 𝑎√5; a is a natural number.
4) Prove that D is a natural number.
III.(3 points)
1) In the following figure, ABCD is a quadrilateral such that :
AD= 5 𝑐𝑚, 𝐷𝐶 = 2√5, AB= 3 cm, BC= 6 cm, and AC=3√5
Verify that A,B, C and D belong to the same circle which its
center and diameter will be determined.
2) ABC is a right triangle at A such that AB=3 + √5. Calculate
AC if the area of this triangle is equal to 2 cm2 and give the
approximation of that area to nearest 0.001.

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IV. (2.5 points)
Given that 2
P(x) 4x 9 (x 2)(2x 3)     and Q(x) (2x 3)(x 1).  
1) Prove that P(x) (2x 3)(3x 5).  
2) Solve the equation .0)x(Q 
3) Let
P(x)
F(x) .
Q(x)

a- For what values of x, is F(x) defined ?
b- Simplify F(x), then solve the equation 2)x(F  , and write the solution in the form
c
2ba 
where a, b and c are integers.
V. (3 points)
Consider a semi-circle (C) of center O, radius R and diameter [AB]. Let M be a point on (C)
distinct from A and B. The tangent at M to (C) cuts the tangent at A in point N and the tangent
at B in point P. (OP) cuts [MB] in D and (ON) cuts [AM] in E.
1) Draw a figure.
2) Prove that D is the midpoint of [MB] and that E is the midpoint of [MA].
3) Calculate ED in terms of R.
4) Prove that ODME is a rectangle.
5) Let J be the midpoint of [DE]. Prove that, when M moves on (C), J moves on a semi-circle
whose center and radius are to be determined.
VI. (6 ½ points)
Consider, in an orthonormal system of axes Oxx and Oyy where the unit of length is the
centimeter, the points A(0 ; – 4) , E(0 ; 1) , F(4 ; – 1) and the straight line (d) of equation
.1x
2
1
y 
1) Plot the points A, E and F.
2) Verify by calculation, that E and F are two points of (d), then draw (d).
3) Prove that I(2 ; 0) is the midpoint of [EF].
4) We know that .52EF 
a- Calculate AE and AF. Deduce that triangle AEF is isosceles of principal vertex A.
b- Is the straight line (AI) perpendicular to (EF)? Justify.
5) Let B be the symmetric of A with respect to I.
a- Prove that AFBE is a rhombus.
b- Calculate the coordinates of B.
6) Let (d') be the straight line passing through B and parallel to (d). Determine the equation of
(d').
7) (AE) and (AF) intersect (d') in M and N respectively. Prove that EMNF is an isosceles
trapezoid and calculate its area.
BON TRAVAIL.