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1st-Trial 15-16

This document contains a mathematics exam for the 9th grade in English with 4 exercises: 1) The first exercise asks students to identify the correct answer from multiple choice questions and justify their responses. 2) The second exercise involves calculating properties of a quadrilateral given side lengths and determining if expressions are rational or irrational. 3) The third exercise asks students to calculate values that make algebraic expressions identical, factorize an expression, and find integer values. 4) The fourth exercise is word problem involving properties of circles, triangles, perpendiculars, and tangents to circles. Students are asked to reproduce diagrams, calculate lengths and areas, identify collinear points, and determine loci.

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422 views2 pages

1st-Trial 15-16

This document contains a mathematics exam for the 9th grade in English with 4 exercises: 1) The first exercise asks students to identify the correct answer from multiple choice questions and justify their responses. 2) The second exercise involves calculating properties of a quadrilateral given side lengths and determining if expressions are rational or irrational. 3) The third exercise asks students to calculate values that make algebraic expressions identical, factorize an expression, and find integer values. 4) The fourth exercise is word problem involving properties of circles, triangles, perpendiculars, and tangents to circles. Students are asked to reproduce diagrams, calculate lengths and areas, identify collinear points, and determine loci.

Uploaded by

api-253679034
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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1st exercise : (4 pts)


In the following table, just one of the proposed answers is correct. Indicate the number of the question
and its corresponding answer and justify.
No

Answers

Questions

1
, then
12

3 n

4
9

43
9

1
1
1
, b 25 and c 5 and if x
,
10
b a c 1
2
2

25

2 5

2 5 1

65
28

13
8

Isosceles

Right isosceles

equilateral

If a is a real number such that : a 4.694

1.

a-1 =

If

2.

then x = ...
If C

3.

Let

252
45 5

, then C
112
245 4
ABC

be

triangle

such

4 3 5
15 3
5
3
, AB
,

and AC
3
3
3 3
then the triangle ABC is

4.

BC =

that:

2nd exercise: (4 pts)


Let ABCD be a quadrilateral such that: (The unit of length is the cm).
2

10
3
64
61 5
1 1 1
AB 2.25
; BC 12.25
; CD 7.3 and AD 2 12 .
21 2
9
12 3
2 3 4
1) a) Reduce, then say if each of the given numbers is either rational or irrational, and justify your
answer. (2 pts)
b) Deduce the nature of ABCD. ( pt)
2) Let E be the foot of the height issued from A to [DC].

h is the length of [AE] (AE = h), such that :

2 4

2 1

2 2 .

a) Show that h is an integer. (1 pt)


b) Calculate the area of the quadrilateral ABCD. ( pt)

2/1

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3rd exercise : (6 pts)


Consider the following algebraic expressions:
E x m 1 x 3 3x 2 m 3 x 2m 1 ,

G x 4x 2 4x 1 6x 3 x 2 5 20x 2 , and

F x (a x ) 2 x 2 1 b 2x a x 3 a c x 1 1 .

(a, b, c and m are four real numbers and m is a parameter)


1) Calculate m so that 1 a root of E (x) . (1 pt)
For the rest of the exercise, consider : E x 2x 3 3x 2 4x 5 .
2) Show that E

2 is an integer. (1pt)

3) a) Show that : F x a 2 x 3 1 b x 2 2a 2 c x b c 1. (1pts)


b) Calculate a, b and c so E (x) and F (x) are identical. (1pt)
4) Factorize G(x) then deduce its roots. (1 pts)
4th exercise: (10 pts)
In the following figure we have:
* (C) is a circle of center O, of radius R and of diameter [AB].
2
* C is a point on [AO] such that: AC R .
3
* (d) is the perpendicular to (AB) at B.
* D is a point on (d) such that: DO B 60 .
o

1) Reproduce the figure. ( pt)


4
2) a) Show that : BC = R . ( pt)
3
b) Determine the nature of the triangle DOB,
and then calculate DB in terms of R. ( pt)
c) Calculate, in terms of R, the area of the triangle COD. ( pt)
3) Let M be the orthogonal projection of B on [CD]. The perpendicular bisector of [AB] cuts [CM] at N.
Show that M, N, O and B belong to the same circle of center I and of diameter to be precised. ( pt)
4) The tangent to (C) issued from D other than (d), touches the circle at E, (E is the point of tangency),
and it cuts the line (AB) at P.

B ? Justify. ( pt)
a) What does [DO) represent for the angle ED
b) Using the triangle BDP, show that: DP = 2 BD. ( pt)
c) Deduce that E is the midpoint of [DP]. ( pt)
5) F is the symmetric of E with respect to (AB).
a) Show that AEOF is a rhombus. ( pt)
b) Deduce that the points F, O and D are collinear. ( pt)
c) Show that (PF) is tangent to (C) at F. ( pt)
6) In this part, we suppose that D varies on the straight line (d).
a) Indicate the fixed points among O, M, N, D and B. ( pt)
b) What is the locus of I, the midpoint of [BN]? Justify. ( pt)
c) Let K be the intersection point of [DO] and [EB].
On which line does K vary? Justify. ( pt)
2/2

- - 5102

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