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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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