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WEBSITE VNMATH.COM THI TH TUYN SINH VO LP 10 NM 2012Mn: TON

CHNH THC Thi gian lm bi: 150 pht; khng k thi gian pht

Bi 1 (2,5 im)

a) Rt gn biu thc A =1

a 1

1

a + 1

+ 1.

b) Khng s dng my tnh cm tay, chng minh1

2 + 1 +

1

3 +

2 +1

4 +

3 l s t nhin.

c) Bit phng trnh x2 3x + 1 = 0 c hai nghim dng x1, x2. Khng gii phng trnh, hytnh tng

x1 +

x2.

Bi 2 (2,5 im) Cho hm s y = mx2 c th (P) i qua im A(1; 1).

a) Xc nh m v v th (P).

b) Gi B l im trn (P) c honh bng 2. Tnh h s gc ca ng thng AB.

c) Tm im C nm trn trc tung sao cho ba im A,B,C thng hng.

Bi 3 (1,0 im) Mt tm tn hnh ch nht c chu vi l 48 cm. Ngi ta ct b mi gc mt hnhvung c cnh 2 cm ri gp ln thnh mt hnh hp ch nht khng c np c th tch l 96 cm3. Tnhchiu di v chiu rng ca tm tn hnh ch nht.

Bi 4 (1,0 im) Cho tam gic vung ABC c AB = 3, BC = 4, AC= 5. Hnh vung XYZWnmtrong tam gic c X, Y nm trn on AC, W nm trn on AB v Z nm trn BC. Tnh dichiu cao xut pht t B ca tam gic ABC v chu vi hnh vung XYZW.

Bi 5 (2,0 im) Cho hnh vung ABCD c din tch bng a2 v E l im di ng trn on CD(E khc D). ng thng AE ct BC ti F, ng thng qua A vung gc vi AE ct CD ti K. GiI l trung im ca FK.

a) Chng minh tam gic ADK bng tam gic ABF. Suy ra tam gic AFK vung cn.

b) Khi E di ng trn CD th I di ng trn ng c nh no?

c) Chng minh bn im A,B,F,Icng nm trn mt ng trn.

Bi 6 (1,0 im) Mt chu nc hnh bn cu bng thy tinh c bn knh R = 10 cm, t trong mtkhung hnh hp ch nht (hnh 1). Trong chu c cha sn mt khi nc hnh chm cu c chiu caoh = 4 cm. Ngi ta b vo chu mt vin bi hnh cu bng kim loi th mt nc dng ln va phkn vin bi (hnh 2). Tnh bn knh ca vin bi (kt qu lm trn n 2 ch s l thp phn). Cho bitcng thc tnh th tch khi chm cu ca hnh cu tm I bn knh R, c chiu cao h l:

V = h2(Rh

3).

Hnh 1 Hnh 2

HtTh sinh khng c s dng ti liu. Cn b coi thi khng gii thch g thm.

H v tn th sinh:.....................................................; S bo danh:....................................

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VNMATH.COM P N THI TH TUYN SINH VO LP 10 NM 2012Mn: Ton

CHNH THC p n thang im gm 2 trang.

P N - THANG IM

Cu p n im

I a.(1 im)

(2,0 im) iu kin: a 0 v a = 1 0,25Kt qu rt gn a+1

a1. 0,75

b.(1 im)

Trc cn mu s 0,25

Kt qu bng 1. 0,75

c.(0.5 im)

Vit ra nh l Viete cho phng trnh x2 3x + 1 = 0. 0,25(

x1 +

x2)2 = 5 do

x1 +

x2 =

5. 0,25

II 1.(1 im)

(2,0 im) (P) i qua (1, 1) nn m = 1. 0,25

V th (P) 0,75

2.(1 im)

Ch ra B(2, 4). 0,25

Vit phng trnh ng thng AB : y = 3x

2. 0,5

H s gc ca AB bng 3. 0,25

3.(0.5 im)

A , B, C thng hng khi C thuc ng thng AB. 0,25

C thuc trc tung nn C(0,2). 0,25III Gi chiu rng ca tm tn l x, khi chiu di ca n l 24x. iu kin 0 < x < 12. 0,25

(1,0 im) Ct b 4 gc th chiu rng, chiu di v chiu cao ca hnh hp l x4, 20x v 2. 0,25Suy ra (x 4)(20 x)2 = 96 0,25Gii ra x = 16 (loi) v x = 8, sau kt lun. 0,25

IV Tam gic ABC vung ti B. 0,25(1,0 im) Tnh BH = 12

5. 0,25

S dng nh l Ta-let tm W Z. 0,25

Chu vi hnh vung bng 4W Z =240

37. 0,25

V 1.(0.75 im)

(2,0 im) Tam gic ADK bng tam gic ABF (g.c.g). 0,5

Suy ra AF K vung cn. 0,25

2.(0.5 im)

Chng minh IA = IC. 0,25I di ng trn trung trc ca AC. 0,25

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3.(0.75 im)

AF K vung cn nn AI va l trung tuyn va l ng cao nn gc AIF bng 900 0,5

ABF + AIF = 1800 nn A , B, F , I nm trn mt ng trn. 0,25

VI Gi x l bn knh hnh cu. iu kin 0 < 2x < 10. Th tch khi nc khi cha b

vin bi vo V1 = 416

3 .

0,25

(1,0 im) Th tch khi nc khi cha b vin bi vo V2 =4x2(30 2x)

3. 0,25

Ta c V2 V1 = 4x3

3. Phng trnh ny tng ng vi 3x3 30x2 + 104 = 0. 0,25

Gii ra 3 nghim, loi 2 nghim do iu kin ban u. Kt qu l 2, 09 cm. 0,25

. . . . . . . . . . . . . . . . . . . . . . . .Ht . . . . . . . . . . . . . . . . . . . . . . . .

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