G9 exam of first semster

1. Page 1 of 2
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.