![CCS Mathematics Oct. 2014
G9 Exam 1 Duration : 1h
Name:…………………………..
Page 1 of 2
I. (1 point)
Solve the following inequality: 푥 −
3
2
≤ −
5
2
푥 + 1.
II. ( 2 points)
Consider the following numbers :
퐴 = √12 − 2√27 + 4√75 ; 퐵 = 14
45
× 27
49
÷ 3
5
푎푛푑 퐶 = 14 ×10⁵
0.7×10²
1) Write A in the form 푎√3 , where a is an integer.
2) Write B in the form of irreducible fraction.
3) Write C in the form 푏 × 10ⁿ, where b is an integer.
III. ( 3 points)
BAL is a right triangle at A. O is the midpoint of [BL].The circle of diameter [BO], cuts [BA]
in another point I.
1) Construct the figure.
2) Prove that I is the midpoint of the segment [AB].
IV. ( 6 points)
Consider two circles C (O ; 4 cm) and C’(O’ ; 2 cm) that are tangent externally at T. To every
point M of (C ) we associate appoint N of (C’) such that M푇̂푁 = 90°. (MN) cuts (OO’) at I.
The common tangent at T cuts [MN] at T’.
1) Compare the angles
푂̂
푇 푎푛푑 푀푇̂
푇′ .
2) Compare the angles
N푂′ ̂ 푇 푎푛푑 푁푇̂
푇′.
3) Deduce that (OM) and
(O’N) are parallel.
4) Assume that :
IN =4√2 .
Calculate the length IO’
and IO.
Deduce that I remains
fixed as M varies on (C ).
5) through P, the midpoint of [MN], draw the parallel to [OM) that cuts [OO’] at A.
a. Verify that as M varies on ( C), A remains fixed and the length AP remains constant.
b. What is the locus of the point P, as M varies on (C ).](https://image.slidesharecdn.com/g9exam1-141109014826-conversion-gate01/85/g9-exam-1-1-320.jpg)
![Page 2 of 2
V. ( 4 points)
Given the triangle ABC right at A and [AH] is the height relatively to the hypotenuse [BC]. The
circle of diameter [AH], cuts [AC] at D and [AB] at E ; the tangents at D and E to the circle , cut
[BC] respectively in M and N.
1) Draw the figure.
2) What is the nature of the quadrilateral ADHE ? Justify.
3) Prove that BC = 2MN.
GOOD WORK.](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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G9 exam 1
- 1. CCS Mathematics Oct. 2014 G9 Exam 1 Duration : 1h Name:………………………….. Page 1 of 2 I. (1 point) Solve the following inequality: 푥 − 3 2 ≤ − 5 2 푥 + 1. II. ( 2 points) Consider the following numbers : 퐴 = √12 − 2√27 + 4√75 ; 퐵 = 14 45 × 27 49 ÷ 3 5 푎푛푑 퐶 = 14 ×10⁵ 0.7×10² 1) Write A in the form 푎√3 , where a is an integer. 2) Write B in the form of irreducible fraction. 3) Write C in the form 푏 × 10ⁿ, where b is an integer. III. ( 3 points) BAL is a right triangle at A. O is the midpoint of [BL].The circle of diameter [BO], cuts [BA] in another point I. 1) Construct the figure. 2) Prove that I is the midpoint of the segment [AB]. IV. ( 6 points) Consider two circles C (O ; 4 cm) and C’(O’ ; 2 cm) that are tangent externally at T. To every point M of (C ) we associate appoint N of (C’) such that M푇̂푁 = 90°. (MN) cuts (OO’) at I. The common tangent at T cuts [MN] at T’. 1) Compare the angles 푂̂ 푇 푎푛푑 푀푇̂ 푇′ . 2) Compare the angles N푂′ ̂ 푇 푎푛푑 푁푇̂ 푇′. 3) Deduce that (OM) and (O’N) are parallel. 4) Assume that : IN =4√2 . Calculate the length IO’ and IO. Deduce that I remains fixed as M varies on (C ). 5) through P, the midpoint of [MN], draw the parallel to [OM) that cuts [OO’] at A. a. Verify that as M varies on ( C), A remains fixed and the length AP remains constant. b. What is the locus of the point P, as M varies on (C ).
- 2. Page 2 of 2 V. ( 4 points) Given the triangle ABC right at A and [AH] is the height relatively to the hypotenuse [BC]. The circle of diameter [AH], cuts [AC] at D and [AB] at E ; the tangents at D and E to the circle , cut [BC] respectively in M and N. 1) Draw the figure. 2) What is the nature of the quadrilateral ADHE ? Justify. 3) Prove that BC = 2MN. GOOD WORK.