CHU VAN AN SECONDARY SCHOOL TEST NUMBER 02 GIFTED STUDENTS INVESTIGATION TEST SUBJECT MATH IN ENGLISH GRADE 8 Duration 120 minutes PART I MULTIPLE – CHOICE (100 marks) Question 1 Caculate 2 275 45 240[.]
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TEST NUMBER 02
GIFTED STUDENTS INVESTIGATION TEST SUBJECT: MATH IN ENGLISH - GRADE 8
Duration: 120 minutes
PART I: MULTIPLE – CHOICE (100 marks)
Question 1: Caculate
75 45
240 ?
A 10 ; B 12 ; C 15 ; D 18 ; E 30
Question 2: When the number N 1 2 3 9 is writen as a dicimal number, how many zeros 1 .2 3 9 does it end in ?
A ;3 B ;4 C ;5 D ;6 E None of the above
Question 3: CE and BD are angle bisectors of ABC which intersect at point F If
o
BFC 110 , find the measure of BAC
A 30 ; o B 35 ; o C 40 ; o D 45 ; o E 70 o
Question 4: If y x y x
x z z y for three positive numbers ,x y and z , all different, then
x
y ?
A 1
;
3
;
2
;
5
;
Question 5: The polynomials x2 3 7x2 ax 6 and x3 8x2 2a 1 x 16 leaves the same
remainder when divided by x 2 Find the value of a?
A 10 ; B 11 ; C 12 ; D 13 ; E None of the above
Question 6: There are twenty people in a room, with a men and b women Each pair of men
shakes hands, and each pair of women shakes hands, but there are no handshakes between a man and a woman The total number of handshakes is 106 Determine the value of a b
A 72 ; B 75 ; C 80 ; D 84 ; E None of the above
Question 7: Rectangles FENR and HGMS are inscribed in triangle ABC as shown The area of
triangle ABC is 60 and its base, BC , has length 10 If EN GM 3 then the sum of the areas
of the two rectangles is:
A 30 ; B 32 5 , ; C 36 ; D 36 5 , ; E 37 5 ,
Question 8: For how many integers n is n n
2 2
5 4 an integer ?
A 1 ; B 2 ; C 3 ; D 4 ; E 5
Question 9: For how many n in 1 2 3; ; ; ;100 is the tens digit of n2
odd ?
A 10 ; B 20 ; C 30 ; D 40 ; E 50
S
R M
F E
A
G
Trang 2Question 10: In the figure, ABCD is an isosceles trapezoid with side lengths
,
AD BC 5 AB 4 and DC 10 The point C is on DF and B is the midpoint of hypotenuse DE in the right triangle DEF Then CF ?
A 3 25, ; B 3 5, ; C 3 75, ; D 4; E 4 25,
PART II: COMPOSE (200 marks)
Problem 1 Prove that if n is a perfect cube then n2 3n 3 cannot be a perfect cube
Problem 2 Let , ,a b c and d be real numbers such that a2 b2 c2 d2 1 and ac bd 0
Determine the value of ab cd
Problem 3 Let CH be the altitude of triangle ABC with ACB 90 The bisector of BAC 0 intersects CH , CB at P M respectively The bisector of , ABC intersects CH , CA at Q N ,
respectively Prove that the line passing through the midpoints of PM and QN is parallel to line
AB
Problem 4 Let ,a b and c be positive real numbers such that abc 1 Find the minimum value
P
-THE END -
F
E
B
A
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TEST NUMBER 02
GIFTED STUDENTS INVESTIGATION TEST SUBJECT: MATH IN ENGLISH - GRADE 8
Duration: 120 minutes
PART I: MULTIPLE – CHOICE (100 marks)
ANSWERS AND MARKS
Marks 10,0 10,0 10,0 10,0 10,0 10,0 10,0 10,0 10,0 10,0
PART II: COMPOSE (200 marks)
1
(50 marks)
Suppose by way of contradiction that n2 3n 3 is a cube
Note that n n2 3n 3 n3 3n2 3n 1 1 n 1 3 1 20
and since n 13 1 is not a cube, we obtain a contradiction 10
2
(50 marks) Since a2 b2 1, a and b are not both 0
We may assume that a 0 From ac bd c bd
a
Substituting into c2 d2 1, we have b d d a b d d
2
It follows that a2 d2 Hence ab cd ab bd a d b
2
3
(50 marks)
Let ,E F be the midpoints of QN PM, respectively
Let X Y, be the intersection of CE CF, with AB respectively 10
Now CMP 90 CAM 90o BAM APH CPM
So CM CP then CF AF
15
P
Q F E
N
N
Y
C
Trang 4Since AF bisects CAY
Hence CAF YAF A S A so CF FY
Similarly CE EX
15
By the midpoint theorem, we have EF parallel to line XY , which is the
4
P
P
15
P
P
1
20
P
2 1
MinP 2 when a b c 1
15