TRƯỜNG THPT CHUYÊN NGUYÊN TẤT THÀNH (Đề có 01 trang) ĐỀ THI THỬ ĐẠI HỌC LẦN THỨ 2 NĂM 2014 Môn: TOÁN Thời gian làm bài: 180 phút, không kể giao đề. I. PHẦN CHUNG CHO TẤT CẢ THÍ SINH (7 điểm) Câu 1. (2 điểm) Cho hàm số x 2 y x 1    . (1) a) Khảo sát sự biến thiên và vẽ đồ thị hàm số (1). b) Tìm m để đường thẳng d: y = -x + m cắt đồ thị hàm số (1) tại 2 điểm A, B tạo thành tam giác OAB thỏa mãn 1 11  OB OA với O là gốc tọa độ. Câu 2. (1 điểm). Giải phương trình:   3 3 x x x x 2 2 sin cos cos 2 sin x cos 2 2 2 2 4                  Câu 3. (1 điểm). Giải phương trình: 32 3 3x 5 x 1 8x 5 0      . Câu 4. (1 điểm). Tính tích phân: 2 0 cos2x I sin x sin x dx 1 3cosx            . Câu 5. (1 điểm). Cho hình trụ có trục OO’ bằng bán kính đáy và bằng a. Gọi A là điểm thuộc đường tròn tâm O, A’ là điểm thuộc đường tròn tâm O’ sao cho AA’ = 2a. Tính thể tích của tứ diện OAA’O’. Câu 6. (1 điểm). Cho 3 số dương x, y, z thỏa mãn x + y + z = 2. Tìm giá trị lớn nhất của biểu thức: . S x y y z z x       II. PHẦN RIÊNG (3 điểm). Thí sinh chỉ làm 1 trong 2 phần A hoặc B. A. Theo chương trình chuẩn. Câu 7a. (1 điểm). Trong mặt phẳng với hệ tọa độ Oxy, cho tam giác ABC có phương trình cạnh AB: 2x + y – 1 = 0, phương trình cạnh AC: 3x + 4y + 6 = 0. Tìm tọa độ trọng tâm G của tam giác ABC biết M(1; -3) nằm trên cạnh BC thỏa mãn: 3MB = 2 MC . Câu 8a. (1 điểm). Trong không gian Oxyz, cho 3 điểm M(3; 1; 1), N(4; 8; -3), P(2, 9, -7) và mặt phẳng (Q): x + 2y – z – 6 = 0. Tìm trên (Q) điểm A sao cho AM AN AP      nhỏ nhất. Câu 9.a (1 điểm). Giải phương trình sau trên tập số phức: 2 2 2 (z i) (z i) 5z 5 0      B. Theo chương trình nâng cao. Câu 7.b. (1 điểm). Trong mặt phẳng tọa độ Oxy, cho đường tròn (C):     2 2 x - 1 + y - 2 = 4 và điểm N(2; 1). Tìm trên đường thẳng d: x + y + 2 = 0 điểm M sao cho từ M kẻ được hai tiếp tuyến MA, MB đến (C) (với A, B là 2 tiếp điểm) và đường thẳng AB đi qua N. Câu 8b. (1 điểm). Trong không gian Oxyz, cho các điểm A(2; 3; 0), B(0; 2 ; 0) và đường thẳng d có phương trình: x t y 0 z 2 t          . Tìm điểm C trên đường thẳng d sao cho tam giác ABC có chu vi nhỏ nhất. Câu 9b. (1 điểm). Giải hệ phương trình:   2 2 5 5 2y x y x 1 log x 3y 1 log y 2x 4y 1 2 2 2                 Hết www.DeThiThuDaiHoc.com www.MATHVN.com . 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