This document contains a multi-part math exam with the following questions: 1) Identifying the correct statement about solutions to an equation. 2) Simplifying and graphing expressions involving square roots. 3) Comparing costs of purchasing shirts for a volleyball team under two different offers when varying the number of shirts. 4) Plotting points, calculating lengths and angles of triangles, finding equations of lines, and proving points belong to lines. 5) Drawing and analyzing a circle with tangent lines and finding measures of angles and proving triangles are congruent.

1. Page 1 of 2
I. ( 3 points)
One answer of the following statements is correct. Find the correct one with justification.
Nₒ Statements Answers
A B C
1 The equation 3𝑥² − 21 = 0 has a solution 𝑥 = √21
or 𝑥 =
−√21
𝑥 = √7 and
𝑥 = −√7
𝑥 = −√7 or 𝑥 =
√7
2 𝑥 2 + √3
√3(2 − √3) √3
If this table is proportional , then :
1 √3 2 + √3
2 − √3
3 Consider the line (𝐷 𝑚)of equation
𝑦 = (2𝑚 − 1) 𝑥 + 𝑚 − 4 where m is a real
number and the point A(2 ; -1). If A belongs
to (𝐷 𝑚) then m=
2 -5 1
4 ABCD is a rhombus of side :
AB=√3(2 − √3)² + 7(3 − √3) .ABCD is a
square if AC =
√2 9√2 7-4√3
5 During the holiday the price of a jacket has
increased by 25% and after the holiday its
price has decreased by 20% and became 60$,
then its original price was :
60$ 64$ 80$
II. (3 points)
Consider the two numbers :
𝐴 = −2√48 + 6√3 × √
25
36
+ √4√144 and B= (2 − √3)² + (√5 − 1)(3√5+ 2) − √400 +
√48
1) Simplify A and B.
2) Rationalize the expression
𝐴+𝐵
𝐴−𝐵
3) Deduce that
𝐴+𝐵
𝐴−𝐵
=−
1
2
( 𝐴 + 𝐵)².
4) MNOP is a square of side = 𝐴 − 𝐵. Calculate MO.
Name:…………………
Classof G9
Subject: Mathematics
LaCitéCulturelleScientifique
(Mraige)
Exam of 𝟏 𝒇𝒊𝒓𝒔𝒕
semester
Date:19/12/2016
Duration:2h

2. Page 2 of 2
III. ( 3 points)
Before the opening of the season,a team of volleyball decides to change the shirts. They want to
print a number on each shirt. The team receives two offering:
Offer A: A non-printed shirt costs $ 125.
Its price increases 12% in order to print a number.
Offer B: A non-printed shirt costs $ 90.
The price for printing the number on all the shirts is $ 500.
1) How much does a printed shirt cost by the offer A ?
2) Designate by x the number of shirts purchased, and 𝑦1the final cost of a shirt x by the
offer A, while 𝑦2 designates the final cost of a shirt x by the offer B.
a. Prove that 𝑦1 = 140𝑥 and 𝑦2 = 90𝑥 + 500.
b. Represent 𝑦1 and 𝑦2 𝑔𝑟𝑎𝑝ℎ𝑖𝑐𝑎𝑙𝑙𝑦 𝑖𝑛 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑥, in the same orthogonal system.
(Unit : 1 cm for each shirt on the abscissas axis, and 1 cm for each 100 $ on the
ordinates axis).
3) On the graph, determine the price of 5 shirts by the offer A, the by the offer B.
4) Find graphically the number of shirt after which the offer B becomes the best choice?
IV. (5 points)
1) In an orthogonal system x’Ox, y’Oy plot the points A (2 ; 7), B( 8 ;1) and C(-2 ; -3).
2) Calculate AB, AC and BC. What is the nature of the triangle ABC?
3) I is the midpoint of [AB].
a. Calculate the coordinates of I.
b. Calculate the coordinates of M center of the circumscribed circle about the triangle BIC.
c. Find the equation of the line (D) tangent to (C) at B.
4) Prove that P( 5 ;0) belongs to the perpendicular bisector [AC].
5) Find the equation of the line (𝐷1) parallel to ( BC) and passing through A.
6) Prove graphically that (𝐷1) is perpendicular at (D) then verify the result by calculation.
V. (5 points)
Consider the circle C(0 ; 2.5 cm). E is a point such that OE= 5 cm. (d) and (d’) are two tangents
at
( C) in P and Q such that (d) is parallel to (d’).
1) Draw the figure.
2) Prove that P,O and Q collinear.
3) Construct the circle ( C’) of diameter [OE]. (C’) cuts (C ) at A and B prove that A𝑂̂B=
120°.
4) Find the measure of the angles E𝐴̂O and E𝐵̂O.
5) (EA) cuts (BO) at M and EB cuts (AO) at N. Prove that the two triangles MEB and NEA are
congruent, and determine their homologous elements.
6) Prove that (EO) is perpendicular to [MN]. What does (MN) for ( C).
GOOD WORK.