7002 - 7002 التجربة الثالثة لعام الشهادة المتوسطة
: الرقم : اإلسم ساعتين: المدّة مسابقة في الرياضيات االنكليزي
. يسمح ابستخدام آلة حاسبة غري قابلة للربجمة او اختزان املعلومات او رسم البياانت- :ارشادات عامة
.يستطيع املرشح االجابة ابلرتتيب الذي يناسبه دون االلتزام برتتيب املسائل الواردة يف املسابقة -
. ومجيعها إلزامية6 عدد املسائل-
33. العالمة القصوى-
1st exercise: (4 ½ pts)
1) Given the following two numbers:
2
3 1 2
A ; B 3 5 2 25 45
8 8
By writing all the steps of calculations:
2
a- Calculate A and show that A . ( ¾ pt)
1
8
b- Calculate B and show that B 8 2 . ( ¾ pt)
c- Deduce that A B 1 . ( ½ pt)
2) Let x b e a real number defined by: x 8 2 7 8 2 7 .
a- Compare 8 2 7 and 8 2 7 . Deduce the sign of x. (1pt)
b- Calculate x 2 and show that x 2 4 . Deduce the value of x. (1 ½ pts)
3cm
2nd exercise: (2 ½ pts)
Consider a square whose side measures x cm. the side is
increased by 3cm.
1) Determine, among the 3 algebraic expressions given
below, the expression that expresses the increase in area x cm
of the square:
x 3 2 x 2 ; x 2 x 3 2 ; x 3 2 x 2 . (1pt)
2 2
2) Show that the obtained expression can be written as: 3cm
32x 3 .
Deduce the value of x when the increase in area is
33cm2.(1 ½ pts)
3rd exercise: (3 ½ pts)
Consider the two polynomials: f ( x ) 4 x 2 1 2x 12 2 4 x x 2 ; g ( x ) 3 x 12 x 2 2
1) Develop and reduce f(x). ( ¾ pt)
2) Show that f(x) = 2(2x-1) (3x+2). ( ¾ pt)
3) Write g(x) in the form of a product of two factors of the first degree. ( ¾ pt)
f (x )
4) Let h(x)=
g (x )
a- For what values of x, h(x) is defined. ( ½ pt)
b- Simplify h(x), then solve the equation h(x) = 2. ( ¾ pt)
2/1 الصف التاسع – رياضيات انكليزي- امتحان التجربة الثالثة- نيسان- 2002
4th exercise: (7pts)
A video-club proposes two options to its clients:
Option A: pay a sum of 40$ each month and 2$ for each videotape rented.
Option B: pay 12$ for each videotape rented.
1) Let:
x be the number of videos rented,
Y A be the sum paid in option A,
Y B be the sum paid in option B.
a- Express in each of the two options, the sum paid in terms of x. (1pt)
b- Recopy and complete the following table: (1pt)
x 3 4 5
YA
YB
2) Represent graphically these two options in the same system.
(1cm on the x-axis represents 1 video tape; 1 cm on the y-axis represents 10$) (1 ½ pts)
3) What is the advantageous option if the client wants to rent in a month:
a. 3 videotapes? ( ½ pt)
b. 5 videotapes? ( ½ pt)
4) Determine the value of x, so that option A is more advantageous than option B. (1pt)
5) A person chooses option B and paid 156$. How many videotapes did he rent? Is he right for choosing
option B? Justify. (1 ½ pts)
5th exercise: (5 ½ pts)
ABC is an isosceles triangle at vertex A. The altitude issued from A cuts [BC] at H. (see figure below)
Given BC = 6cm and AH = 4cm. Q
Let M be a point of [BH] such that BM = x. The parallel to (AH)
through M cuts (AB) at P and (AC) at Q.
1) a- Calculate BH and give an encirclement for x. (1pt) A
MP x
b- Show that: . Deduce MP in terms of x. (1pt)
AH 3 P
2) a- Express MC in terms of x. ( ½ pt)
4
b- Show that MQ = (6 x ) . (1pt)
3
c- Find the value of x so that MQ = 3MP. (1pt)
d- In this case, precise the position of P on [AB]. (1pt) B x M H C
6th exercise: (7pts)
In an orthonormal system x’ox, y’oy, consider the two straight-lines (d) and (d’) of respective equations
1
(d): y = -2x + 5 and (d’): y = x and the two points A(-1;7) and B(-2;-1)
2
1) Draw (d) and (d’) then plot A and B. (1 ½ pts)
2) Show that the point I(2,1) is the point of intersection of (d) and (d’). ( ¾ pt)
3) Verify by calculation that A belongs to (d) and that (d’) passes through B. (1pt)
4) Given: AI = 45 ; BI = 20 ; AB = 65 .
a- Deduce that the triangle ABI is right at I. ( ¾ pt)
b- Let J be the center of the circle circumscribed about the right triangle ABI. Calculate the radius of
the circle and determine the coordinates of J. (1pt)
5) a- Write the equation of the straight-line (Δ) passing through J and parallel to (d’). (1pt)
b- (Δ) cuts (d) at L. Find the coordinates of L. (1pt)
2/2 الصف التاسع – رياضيات انكليزي- امتحان التجربة الثالثة- نيسان- 2002
