A-2011

B GIAO DC VA AO TO
THI TUYN SINH I HC NM 2011

Mon: TOAN; Khi: A
CHINH THC
Thi gian lam bai: 180 phut, khong k thi gian phat


PHN CHUNG CHO TT C THI SINH (7,0 im)
-x + 1
Cau I (2,0 im) Cho ham s y =
.
2x - 1
1. Kho sat s bin thien va v th (C) ca ham s a cho.
2. Chng minh rng vi mi m ng thng y = x + m luon ct th (C) ti hai im phan bit A va
B. Gi k1, k2 ln lt la h s goc ca cac tip tuyn vi (C) ti A va B. Tim m tng k + k t
1
2
gia tr ln nht.
Cau II (2,0 im)
1+ sin 2x + cos 2x
1. Gii phng trinh
= 2 sin xsin 2 .
x
2
1+ cot x
2
2
3
5

x y - 4xy + 3y - 2(x + y) = 0
2. Gii h phng trinh
(x, y ).
2
2
2

xy(x + y ) + 2 = (x + y)

4 xsin x + (x +1)cos x
Cau III (1,0 im) Tinh tich phan I =
d .
x


x sin x + cos x
0
Cau IV (1,0 im) Cho hinh chop S.ABC co ay ABC la tam giac vuong can ti B, AB = BC = 2a;
hai mt phng (SAB) va (SAC) cung vuong goc vi mt phng (ABC). Gi M la trung im ca AB;
mt phng qua SM va song song vi BC, ct AC ti N. Bit goc gia hai mt phng (SBC) va (ABC)
bng 60o. Tinh th tich khi chop S.BCNM va khong cach gia hai ng thng AB va SN theo a.
Cau V (1,0 im) Cho x, y, z la ba s thc thuc on [1; 4] va x y, x z. Tim gia tr nh nht ca
x
y
z
biu thc P =
+
+
.
2x + 3y
y + z
z + x
PHN RIENG (3,0 im): Thi sinh ch c lam mt trong hai phn (phn A hoc B)
A. Theo chng trinh Chun
Cau VI.a (2,0 im)
1. Trong mt phng to Oxy, cho ng thng : x + y + 2 = 0 va ng tron
2
2
(C) : x + y - 4x - 2y = 0. Gi I la tam ca (C), M la im thuc . Qua M k cac tip tuyn
MA va MB n (C) (A va B la cac tip im). Tim ta im M, bit t giac MAIB co din tich
bng 10.
2. Trong khong gian vi h ta Oxyz, cho hai im A(2; 0; 1), B(0; -2; 3) va mt phng
(P) : 2x - y - z + 4 = 0. Tim ta im M thuc (P) sao cho MA = MB = 3.
Cau VII.a (1,0 im) Tim tt c cac s phc z, bit:
2
2
z = z + z.
B. Theo chng trinh Nang cao
Cau VI.b (2,0 im)
2
2
x
y
1. Trong mt phng ta Oxy, cho elip (E):
+
= 1. Tim ta cac im A va B thuc
4
1
(E), co hoanh dng sao cho tam giac OAB can ti O va co din tich ln nht.
2. Trong khong gian vi h ta Oxyz, cho mt cu
2
2
2
(S) : x + y + z - 4x - 4y - 4z = 0 va im
(4
A ; 4; 0) . Vit phng trinh mt phng (OAB), bit im B thuc (S) va tam giac OAB u.
Cau VII.b (1,0 im) Tinh moun ca s phc z, bit: (2z - 1)(1 + i) + (z + 1)(1 - i) = 2 - 2i .
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Thi sinh khong c s dng tai liu. Can b coi thi khong gii thich gi them.
H va ten thi sinh:.............................................; S bao danh:................................

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