mid-year exam of math (Al taybe high school)

1. ‫الرسمية‬ ‫الطيبة‬ ‫ثانوية‬(‫العاشر‬ :‫الصف‬A-B-C(:‫الثانوية‬ ‫رقم‬1436
‫النبطية‬ :‫المحافظة‬‫االول‬ ‫الفصل‬ :‫االمتحان‬‫رياضيات‬ :‫المادة‬
‫مرجعيون‬ :‫القضاء‬: ‫المدة‬150‫دقيقة‬:‫تاريخ‬24-1-2019
I- 2 ½ points
In the table below only one answer is correct for each question .Choose the correct
answer and justify :
No Questions
Answers
A B C
1
A={2 ; x3+1 ; 5} and B={9 ; 5 ; 2}.
If A=B then x =
16 4 2
2
In a class of 34 students, 18 practice
football, 17 practice swimming, and 8
practice both activities. How many students
practice neither football nor swimming?
9 7 27
3
Let x be the measure of an arc such that
x ∈]-𝜋; 0[ then √1 − 𝑐𝑜𝑠2( 𝑥) =
Sinx 1-cosx -sinx
4 √(𝑥 − 1)33
+ √𝑥44
− | 𝑥| − 𝑥 = 1 1 − | 𝑥| + 𝑥 -1
II- 2 ½ pts
Given the following sets:
E = {1 , 2 , 3 , 4 , 5 , 6 , 7 , 8}, A = {1 , 3 , 5 , 7}, B = {1 , 2 , 4 , 5} and C = {3 , 5 , 4}.
1) Write, in extension: A  B, B  C and BA  .
2) Answer by TRUE or FALSE:
7 ∈ A ; {4} ∈ B ; Card P(C) = 9 ; [1;9[ ∩ ℕ = E .
3) Find a subset X of E such that C ∪ X = A ∪ B and Card (X) = 3 .
III- 1 ½ pts
A and B are two subsets of a set E such that:
A  B = {1 , 2}, Card (A) = 3, Card (B) = 4, Card(E) = 7 .
A = {3 , 4 , 5 , 9}, and B = {7 , 5 , 9}
Write the sets E , A and B in extension.
IV- 3 pts
1) Calculate A = (√5
3
− √3
3
)(√25
3
+ √15
3
+ √9
3
) .
2) Let D = √3 + 2√2 × (1 − √2) .
a) Study the sign of D .
b) Verify that D2 = 1 then deduce D .
3) Simplify: B =
√𝑥65
× √𝑥34
√𝑥710
× √ 𝑥4
; (x>0) .
4) Verify that
𝐵
𝑥
= A + D .

2. V- 2 ½ pts
A. Given | 𝑥 − 2| < 1 , 2 < y < 5 and -1 < z < 2 .
Bound x , y + 2z , z-y and xy .
B. Write in increasing order :
𝑠𝑖𝑛𝑥; 𝑠𝑖𝑛2 ( 𝑥) ; √ 𝑠𝑖𝑛𝑥 𝑎𝑛𝑑
1
𝑠𝑖𝑛𝑥
where x ∈]0; 𝜋[ .
VI- 4 pts
A. 𝐿𝑒𝑡 𝛼 =
16𝜋
3
𝑟𝑎𝑑 𝑎𝑛𝑑 𝛽 = 450°
1) Find the principal values of 𝛼 𝑎𝑛𝑑 𝛽
2) Express 𝛼 𝑖𝑛 𝑑𝑒𝑔𝑟𝑒𝑒 𝑎𝑛𝑑 𝛽 𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛
B. Given sinx =
3
5
where x ∈ ]
𝜋
2
; 𝜋[ .
1) Calculate cosx and tanx .
2) Calculate E =cos(11𝜋 + 𝑥) − cos(
𝜋
2
+ 𝑥) × tan(12𝜋 − 𝑥) .
C. Verify the following equality : (
1+tan2 𝑥
1+cot2 𝑥
) + 1 =
1
cos2 𝑥
.
D. Let x be the measure of an arc such that x ∈] 0 ;
𝜋
2
[ and |𝑐𝑜𝑠𝑥 +
1
2
| = 1 .
Find x .
VII- 4 pts
ABC is an isosceles triangle and I is the midpoint of [BC], AB=AC=5 et BC=8
We define the point D such that 𝐴𝐷⃗⃗⃗⃗⃗ = 𝐴𝐵⃗⃗⃗⃗⃗ + 𝐴𝐶⃗⃗⃗⃗⃗ and 𝐴𝐸⃗⃗⃗⃗⃗ =
3
2
𝐴𝐵⃗⃗⃗⃗⃗ 𝑎𝑛𝑑 𝐴𝐹⃗⃗⃗⃗⃗ =
3
4
𝐴𝐶⃗⃗⃗⃗⃗ .
1) Plot D , E and F .
2) Write 𝐸𝐹⃗⃗⃗⃗⃗ 𝑎𝑠 𝑙𝑖𝑛𝑒𝑎𝑟 𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝐴𝐵⃗⃗⃗⃗⃗ 𝑎𝑛𝑑 𝐴𝐶⃗⃗⃗⃗⃗ .
3) Show that 𝐸𝐼⃗⃗⃗⃗ = −𝐴𝐵⃗⃗⃗⃗⃗⃗⃗⃗⃗ +
1
2
𝐴𝐶⃗⃗⃗⃗⃗
4) Deduce that E, I and F are collinear
5) Let 𝑉⃗ = 6𝐴𝑀⃗⃗⃗⃗⃗⃗ − 3𝐵𝑀⃗⃗⃗⃗⃗⃗ − 3𝐶𝑀⃗⃗⃗⃗⃗⃗
show that 𝑉⃗ = 3𝐴𝐷⃗⃗⃗⃗⃗ . calculate ‖𝐴𝐷⃗⃗⃗⃗⃗⃗⃗ ‖ then deduce ‖𝑉⃗⃗⃗ ‖.
Good Work