CCS Mathematics Dec. 2014 Class of G9 Exam of 1th semester Duration : 120 min

Name:…………………………………..اإلجابة مالحظة: المرشح يستطيع البيانات لرسم أو المعلومات الختزان أو للبرمجة قابلة غير حاسبة آلة باستعمال يسمح

) المسابقة ) في الوارد المسائل بترتيب االلتزام دون يناسبه الذي .بالترتيب

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 Answersa b c

1 216+213

212−210

227 24 27

2 (4−52 )

2

(4+ 52 )

2

(1−52 )

2

( 52−4 )

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= 718

×27−( 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√5Verify 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 and

1( Prove that

2( Solve the equation Q( x ) = 0 .

3( Let a- For what values of x, is F)x( defined ?

b- Simplify F)x(, then solve the equation F ( x )= √2 , and write the solution in the form a + b√2

c 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 x' Ox and y

' Oy 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

y =− 12

x + 1 .

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 EF = 2√5 .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.

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BON TRAVAIL.

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