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s 3

N TP HC K 2 Nm hc Mn TON Lp 10

Thi gian lm bi 90 pht

Cu 1:

a) Cho a, b, c > 0. Chng minh rng: a b c

b c a

1 1 1 8

b) Gii bt phng trnh: x x x x2 2

2 5

5 4 7 10

Cu 2: Cho phng trnh: x m x m m2 22( 1) 8 15 0

a) Chng minh phng trnh lun c nghim vi mi m . b) Tm m phng trnh c hai nghim tri du .

Cu 3: Trong mt phng Oxy, cho ABC vi A(1; 2), B(2; 3), C(3; 5). a) Vit phng trnh tng qut ca ng cao k t A. b) Vit phng trnh ng trn tm B v tip xc vi ng thng AC.

c) Vit phng trnh ng thng vung gc vi AB v to vi 2 trc to mt tam gic c din tch bng 10.

Cu 5:

a) Chng minh: k k2 33

cos sin1 cot cot cot , .

sin

b) Rt gn biu thc: A2

tan2 cot 2

1 cot 2

. Sau tnh gi tr ca biu thc khi

8

.

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H v tn th sinh: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SBD :. . . . . . . . . .

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s 3

P N N TP HC K 2 Nm hc Mn TON Lp 10

Thi gian lm bi 90 pht

Cu 1:

a) Do a, b, c > 0 nn a a b a c c

b b c b a a

1 2 , 1 2 , 1 2

2

Nhn cc bt ng thc trn, v theo v, ta c: a b c a b c

b c a b c a

1 1 1 8 8

b) Gii bt phng trnh: x x x x x x x x2 2 2 2

2 5 2 50

5 4 7 10 5 4 7 10

x x x x x x

x x x x x x x x

2 22( 7 10) 5( 5 4) (3 11)

0 0

( 1)( 4)( 2)( 5) ( 1)( 2)( 4)( 5)

x11

( ;0) (1;2) ;4 (5; )

3

Cu 2: Cho phng trnh: x m x m m2 22( 1) 8 15 0 x m x m m2 22( 1) 8 15 0

a) m m m m m m m R2 2 2 21 23

( 1) 8 15 2 6 16 (2 3) 0,

2 2

Vy phng trnh bc hai cho c hai nghim phn bit vi mi m. b) Tm m phng trnh c hai nghim tri du .

PT c hai nghim tri du ac < 0

m m m m m2 21(( 8 15) 0 8 15 0 ( ;3) 5; Cu 3: Trong mt phng Oxy, cho ABC vi A(1; 2), B(2; 3), C(3; 5). a) Vit phng trnh tng qut ca ng cao k t A.

A VTPT BC(1;2), : (1;8)

PT ng cao k t A l x y x y1 8( 2) 0 8 17 0

b) Vit phng trnh ng trn tm B v tip xc vi ng thng AC.

Tm B(2; 3), Phng trnh AC: x y

x y1 2

3 2 1 0

2 3

,

Bn knh R d B AC3.2 2.( 3) 1

( , ) 13

9 4

Vy phng trnh ng trn l x y2 2( 2) ( 3) 13

c) Vit phng trnh ng thng vung gc vi AB v to vi 2 trc to mt tam gic c din tch bng 10.

Gi s Ox M m Oy N n( ;0), (0; ) . AB (1; 5) uur

, MN m n( ; ) uuur

.

Phng trnh MN: x y

nx my mn

m n

1 0 .

Din tch tam gic MON l: ABC

S m n mn1. 10 20

2 (1)

Mt khc MN AB MN AB m n m n. 0 5 0 5

(2)

T (1) v (2) m

n

10

2

hoc

m

n

10

2

Phng trnh l: x y5 10 0 hoc x y5 10 0

Cu 5:

a) cos 2 2

3 2 2

cos sin 1 1. cot .(1 cot ) 1 cot

sinsin sin sin

2 31 cot cot cot (pcm)

3

b) A 22

tan2 cot 2 1.sin 2 tan2

sin2 .cos21 cot 2

Khi 8

th A tan2. tan 1

8 4

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