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BỘ GIÁO DỤC VÀ ĐÀO TẠO

⎯⎯⎯⎯⎯⎯⎯⎯

ĐỀ CHÍNH THỨC

ĐÁP ÁN – THANG ĐIỂM

ĐỀ THI TUYỂN SINH ĐẠI HỌC NĂM 2012

Môn: TOÁN; Khối A và khối A1

(Đáp án – thang điểm gồm 04 trang)

Câu Đáp án Điểm

a) (1,0 điểm)

Khi m = 0, ta có: 4 2

y x = − 2 . x

• Tập xác định: D = \.

• Sự biến thiên:

− Chiều biến thiên: 3 y x ' 4 = − 4x; y ' = 0 ⇔ x = 0 hoặc x = ±1.

0,25

Các khoảng nghịch biến: (−∞;−1) và (0; 1); các khoảng đồng biến: (−1; 0) và (1;+ ∞).

− Cực trị: Hàm số đạt cực tiểu tại x = ±1, yCT = −1; đạt cực đại tại x = 0, yCĐ =0.

− Giới hạn: lim lim . x x

y y →−∞ →+∞

= = +∞

0,25

− Bảng biến thiên:

0,25

• Đồ thị:

0,25

Trang 1/4

b) (1,0 điểm)

Ta có 3 2 y x ' 4 = − 4(m +1)x = 4x(x − m −1).

Đồ thị hàm số có 3 điểm cực trị khi và chỉ khi m +1 > 0 ⇔ m > −1 (*).

0,25

Các điểm cực trị của đồ thị là 2 A(0; m ), B m ( 1 − + −; 2m −1) và C m ( 1 + −; 2m −1).

Suy ra: 2 AB = −( 1 m + ; − (m +1) JJJG ) và 2 AC = + ( 1 m ; − (m +1) ). JJJG 0,25

Ta có AB = AC nên tam giác ABC vuông khi và chỉ khi AB.AC = 0 JJJG JJJG 0,25

1

(2,0 điểm)

⇔ . Kết hợp (*), ta được giá trị m cần tìm là 4 ( 1 m m + − ) ( +1) = 0 m = 0. 0,25

+∞

y

y ' – 0 + 0 – 0 +

x −∞ –1 0 1 +∞

–1

0

–1

+ ∞

O

2

1

– 1

–1

–2

8

x

y

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Page 2 of 4

Câu Đáp án Điểm

Phương trình đã cho tương đương với ( 3 sin x x + cos − = 1)cos x 0. 0,25

π

cos 0 π ( ) 2 • =x x ⇔ = + k k ∈] . 0,25

• + 3 sin x x cos −1= 0 ( ) π π

cos cos

3 3

⇔ −x = 0,25

2

(1,0 điểm)

⇔ x = k2π hoặc

2π 2π ( ) 3

x k = + k ∈] .

Vậy nghiệm của phương trình đã cho là π

π, 2

x = + k x = k2π và 2π 2π (). 3

x k = + k ∈]

0,25

Hệ đã cho tương đương với: ( ) ( )

3 3

2 2

( 1) 12( 1) ( 1) 12( 1) (1)

1 1 1. (2) 2 2

x x y y

x y

⎧ − − − = + − + ⎪

⎪ − + + = ⎩

0,25

Từ (2), suy ra

1 1 1

2

− ≤ x − ≤ và 1 1 1

2 − ≤ +y ≤ ⇔

3 1 1

2 2

− ≤ −x ≤ và 1 3 1 .

2 2

− ≤ +y ≤

Xét hàm số 3 f ( )t t = −12t trên 3 3

;

2 2

⎡−

⎢⎣ ⎤

⎥⎦

, ta có 2 f t '( ) 3 = (t − 4) < 0 , suy ra f(t) nghịch biến.

0,25

Do đó (1) ⇔ x – 1 = y + 1 ⇔ y = x – 2 (3).

Thay vào (2), ta được ( ) ( ) 2 2 1 3 1

2 2

x x − + − = ⇔ 2 4 8 x x − + =3 0 ⇔

1

2

x = hoặc

3

.

2

x = 0,25

3

(1,0 điểm)

Thay vào (3), ta được nghiệm của hệ là ( ) 1 3 ( ; ) ;

2 2 x y = − hoặc ( ) 3 1 ( ; ) ; . 2 2 x y = − 0,25

Đặt u = +1 ln(x +1) và 2

d d , suy ra

d d

1

x

u

x = +

và 1

v . x

v

x = x = − 0,25

3 3

1 1

1 ln( 1)

( 1)

x dx I

x x x

+ + = − +

+ ∫ 0,25

( ) 3

1

2 ln 2 1 1

3 1

dx

x x

+

= + − + ∫

3

1

2 ln 2 ln

3 1

x

x

+

= +

+ 0,25

4

(1,0 điểm)

2 2 ln3 ln 2. 3 3

= + − 0,25

Ta có SCH n là góc giữa SC và (ABC), suy ra n o SCH = 60 .

Gọi D là trung điểm của cạnh AB. Ta có: ,

6

a HD= 3

, 2

a CD=

2 2 7 , 3

a HC = + HD CD = o 21

.tan60 . 3

a SH = = HC

0,25

2 3

.

1 1 21 3 . . . . 7

3 3 3 4 12 S ABC ABC

a a a V SH S = = ∆ = . 0,25

Kẻ Ax//BC. Gọi N và K lần lượt là hình chiếu vuông góc

của H trên Ax và SN. Ta có BC//(SAN) và 3

2

BA = HA nên

3 ( , ) ( ,( )) ( ,( )). 2

d SA BC = = d B SAN d H SAN

Ta cũng có Ax ⊥ (SHN) nên Ax H ⊥ K. Do đó

HK ⊥ (SAN). Suy ra d H( ,( )) .

Trang 2/4

SAN = HK

0,25

5

(1,0 điểm)

o

2 2

2 3 . 42

12 , sin 60 , . 3 3

a a SH HN a AH HN AH HK

SH HN

= = = = =

+

Vậy

S

B

C

H

x

N

K

D

A

42 ( , ) . 8

a d SA BC = 0,25

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Page 3 of 4

Câu Đáp án Điểm

Ta chứng minh 3 1, (*). t ≥ +t ∀t ≥ 0

Xét hàm ( ) 3 1 t f t = − t − , có '( ) 3 ln 3 1 0, 0 t f t = − > ∀t ≥ và f (0) = 0 , suy ra (*) đúng.

Áp dụng (*), ta có | | | | | | 3 3 3 3 | | | | | x y y z z x x y y z z x|. − − − + + ≥ + − + − + −

0,25

Áp dụng bất đẳng thức | a b | + ≥ | | | a + b | , ta có:

2 2 2 2 (| x− + y y | | −z|+|z−x|) =| x− y| +| y−z| + |z−x| +|x− y|(| y−z|+|z−x|)+| y−z|(|z−x|+| x− y|)

( ) 2 2 2 + − |z x|(|x y − |+| y−z|) ≥ 2 | x y − | | + y−z| | + z−x| .

0,25

Do đó ( ) ( ) 2 2 2 2 2 2 2 | x− + y y | | −z|+|z−x| ≥ 2 | x− y| | + y−z| | + z−x| = 6x +6y +6z −2 x+ y+z .

Mà x y + +z=0, suy ra 2 2 2 | | x− + y y| −z|+|z−x| ≥ 6x +6y +6z .

0,25

6

(1,0 điểm)

Suy ra | | | | | | 2 2 2 3 3 3 6 6 6 x y y z z x P x − − − = + + − + y + z ≥3.

Khi x = y = z = 0 thì dấu bằng xảy ra. Vậy giá trị nhỏ nhất của P bằng 3.

0,25

Gọi H là giao điểm của AN và BD. Kẻ đường thẳng qua H

và song song với AB, cắt AD và BC lần lượt tại P và Q.

Đặt HP = x. Suy ra PD = x, AP = 3x và HQ = 3x.

Ta có QC = x, nên MQ = x. Do đó ∆AHP = ∆HMQ, suy ra

AH ⊥ HM.

0,25

Trang 3/4

Hơn nữa, ta cũng có AH H= M .

Do đó AM = 2 2 MH d = = (M ,(AN)) 3 10 . 2

0,25

A∈AN, suy ra A(t; 2t – 3).

3 10

2

MA = ⇔ ( ) ( ) 2 2 11 7 45 2

2 2

t t − + − = 2

0,25

7.a

(1,0 điểm)

⇔ t t 2 5 4 0

A B

D C

N

M

H

P Q

− + = ⇔ t =1 hoặc t = 4.

Vậy: A(1;−1) hoặc A (4;5). 0,25

Véc tơ chỉ phương của d là a = (1; 2; 1). Gọi H là trung điểm của AB, suy ra IH ⊥ AB. JJG

Ta có H∈d nên tọa độ H có dạng H t( −1; 2t;t+ ⇒2) IH =(t−1; 2t;t −1). JJJG 0,25

IH ⊥ AB ⇔ .0 IH a = ⇔ ⇔ JJJG JJG t t − +1 4 +t −1= 0 1

3

t = ( ) 2 2 2

; ; . 3 3 3

⇒ = IH − − JJJG 0,25

Tam giác IAH vuông cân tại H, suy ra bán kính mặt cầu (S) là 2 6 2 .

3

R I = = A IH = 0,25

8.a

(1,0 điểm)

Do đó phương trình mặt cầu cần tìm là 2 2 2 8 ( ): ( 3) .

3

S x + y + −z = 0,25

1 5 n C C n n

− = 3 ⇔ ( 1)( 2) 5

6

n n n

n − − = 0,25

⇔ n = 7 (vì n nguyên dương). 0,25

Khi đó ( )

7 7 2 2 7 7 2

7 14 3

7 7

0 0

1 1 1 ( 1) . 14 2 2 2

n k k k k

k k

k

k k

nx x x C C x

x x x

− = =

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ − ⎜ ⎟ − =−= ⎜ ⎟ ⎜ ⎟ − = ⎝ ⎠ ⎝ ⎠ ∑ ∑ ⎝ ⎠ 0,25

9.a

(1,0 điểm)

Số hạng chứa 5x tương ứng với 14 − 3k = 5 ⇔ k = 3 .

Do đó số hạng cần tìm là

3 3

7 5 5

4

( 1) . 35 . 0,25

2 16

C

x x − = −

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