PHNG GIO DC &O TO THI HC SINH GII CP HUYN

3

THI HC SINH GII CP TRNG NM HC 2012 2013 Mn: Ton hc Lp 9 (Thi gian: 120 pht khng k thi gian giao )

Cu 1 (4,0 im): Cho biu thc

P =

a) Tm iu kin P c ngha?

b) Khi P c ngha, chng t P khng ph thuc vo x

Cu 2 (4,0 im):

1- So snh: v

2- Gii bt phng trnh:

Cu 3 (4,5 im):

1- Cho a + b = 1. Tm gi tr nh nht ca biu thc Q = a3 + b3 + ab

2- Bit ax + by + cz = 0. Hy tnh gi tr ca biu thc

Cu 4 (4 im):

Cho tam gic ABC, ly im C . Qua A v ng thng song song vi CC ct BC ti A. Qua B v ng thng song song vi CC ct AC ti B. Chng minh rng:

Cu 5 (3,5 im):

Mt hc sinh vit dy s sau: 49,4489,444889, 44448889,.. (S ng sau c vit 48 vo gia s ng trc). Chng minh rng tt c cc s vit theo quy lut trn u l s chnh phng.

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. P N MN TON 9 HC SINH GII NM HC 2012 2013Cu 1 (4,0 im): Cho biu thc

P =

a) iu kin P c ngha

b) Khi

Khng ph thuc vo xCu 2 (4,0 im):

1- So snh: v

Ta c:

Do : =

2- Gii bt phng trnh: Lp bng xt du v tri:x 1 2

2x 3 - - 0 + +

x 1 - 0 + + +

x 2 - - - 0 +

V tri - || + 0 - || +

Vy tp nghim ca bt phng trnh l: 1 b = 1 - a

Biu thc Q = a3 + b3 + ab = (a + b)3 3ab(a + b) + ab

Q = 1 3ab + ab = 1 2ab = 1 2a(1- a)

Q = 1 2a + 2a2 = 2(a2 2.a. + ) + 1 -

Q = 2(a-)2 +

Du = xy ra (

Vy gi tr nh nht ca Q l khi v ch khi

2- Bit ax + by + cz = 0

hay 0= (ax + by + cz)2 = a2x2 + b2y2 +c2z2 + (2abxy + 2bcyz + 2caxz)

=> - (2abxy + 2bcyz + 2caxz) = a2x2 + b2y2 +c2z2 (1)Hy tnh gi tr ca biu thc

Cu 4 (4,0 im) A

C

A B a

BGi ng thng a//CC//a;

a ct BC ti A; a ct AC ti B

Suy ra: AA//CC//BB

p dng h qu nh l ta lt ta c:

Thay (2) vo (1) ta c:

Cu 5 (3,5 im):

A = = 9 + 8.10 + 8.102 ++ 8.10n + 4.10n+1 4.10n+2+4.102n+1Ta vit 9 = 1+4+4 va 8 = 4+4 ta c:

A=1+4+4+(4+4).10+(4+4).102++(4+4).10n+4.10n+1+4.10n+2++4.102n+1 = 1+(4+4.10+4.102++4.10n)+(4+4.10+4.102++4.102n+1)

= 1+4.(1+10+102++10n)+4.(1+10+102++102n+1)

= 1+4.+4.

=

=

=

Ta c: 2.10n+1+13 (C tng cc ch s chia ht cho 3) Nn s trong ngoc to thnh mt s chnh phng. Suy ra A la s chnh phng_1416757513.unknown

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