Tài liệu đề thi toán kèm lời giải quốc gia 2016 (2)

TRU€JNG DAI HOC KHOA HOC TQ NHIEN DE THI THE THPT QUOC GIA 2016 LAN 1 TRU©NG THPT CHUYEN KHTN Dé thi gom 01 trang Mfin thi: TOAN Théi gran. 180 phiit tKhong ké théi gran phét dé; Cau 1 (1,0 diém). Khao sat su bién thién va ve do thi cua ham Cau 2 (1,0 diém). Viét phuong trinh tiép tuyén ciia do thi ham -“' biét tiép tuyén cat hai truc Ox, Oy lan lupt tai cac diem A, B phan bi(t thfia man diéu ki(n OB = 3 OA. Cau 3 (1,0 diém) z thoa man: l× l _ a)Tim phan thtrc vii phan :io cua b)Giai phuong trinh trén t a,p so thpc (3 — Cau 4 (1,0 diém). Tinh tich phan os2.r )² -I— (3 -I— + 2iz + = 0. )² = 2²+'. dx. Cau 5 (1,0 diém). Trong khfing gian v‹ i he toa do vufing goc Oxyz, cho mat phang (P): x + 2y + z — 4 - 0 z+2 - —Tim toa do giao diem A cua ducing thang d va mat phang (P) va viét phuong 2 trinh duéng thang A nam trong mat phang (P), dong thcri cat va vuong goc vcii duéng thang d. Cau 6 (1,0 diém). a) Giai phuong trinh luong giac sins — . sin2x —— . cosx + cos2x. b) Xét mot da giac déu 12 canh, hfii co bao nhiéu tain giac khfing can co ba dinh la cac dinh ciia mot da giac déu da cho. Cau 7 (1,0 diém). Cho hinh chop S.ABC co day ABC la tain giac can tai A trong dfi AB —— AC —— a, BAC — 120' ; m)at bén SAB la tain giac déu va nam trong m a)t phang vufing goc vcii day. Tinh theo a the tich khoi chop S.ABC va ban kinh m)at cau ngoai tiép khoi chop S.ABC. Cau 8 (1,0 diém). Trong ma)t phang véri h( truc tpa do vufing goc Oxy, cho tain gi:ic ABC co A(4;6), truc tain H(4;4), trung diem M cua canh BC thuoc duéng thang 6: x — 2y — 1 - 0. Gpi E, F lan lupt la chiin duéng cao ha tii cac dinh B,C cua tain giac. Tim tpa do cac dinh B, C biét duéng thang EF song song voi duéng thang d . x — 3y + 5 - 0. 7× Cau 9 (1,0 diém). Giai h( phuong trinh trén tap so thuc 2x' 2 5x = —2 -I- 5 Cau 10 (1,0 diém) Xét cac so thuc duong x,y, z thfia man diéu kien x' + y' + z’ ½ x’ + y 4 + 5 chfmg minh rang >> Truy ca(p trang http://tuvensinh247.com/ de hpc Toan — Ly — Hfia — Sinh — Van — Anh tot nhat! l x' + y' + z' 3. >> Truy ca(p trang http://tuvensinh247.com/ de hpc Toan — Ly — Hfia — Sinh — Van — Anh tot nhat! l TRACING DAI H©C KHOA H©C TJ NHIEN TRANG THPT CHUYEN KHTN DAP AN DE THI THE THPT QUOC GIA 2016 LAN 1 Mfin thi: TOAN 1.TXD: D = R 2.Siy bién thién y'- 4x ³ — 4x. x —— 0 BBT: Ham so dat circ tiéu tai x = +1; xCT ——— ³ - 3.Do th) Giao ciia do th] véri Oy: (0;0) Do th| nhan truc tung lam truc doi xiing. >> Truy ca(p trang http://tuvensinh247.com/ de hpc Toan — Ly — Hfia — Sinh — Van — Anh tot nhat! 2 Ta co y' = — . Gpi xc 1:i hoanh do tiép diém, vi OB = 3OA nén he tiép tuyén la (×-1)² OA (x o 1)' Co hai tiép tuyén: THI: = - 0 => y ——2. Ta co phuong trinh tiép tuyén la: y = y’(x ). (x — x ) + y =a Ta co: ' OB bi, a, b c R; Khi do phuong trinh tr‹ thanh: ""-1- 2iz + '(“"- on z + 2‹z -t- (z -t- i)(1 -1- i) —l — 1 13 — 5 13 I. Vay phan thuc va phan :io tuong ung la: — 1 5 b) Chia ca 2 ve ctia phuong trinh cho 2² ta dupe: Phuong trinh da cho tuong ring vcii (' “)² + (' “)² = 2. 9- Ta 2 2 4 = 1. Nén 2 Thi ta --2 W /'— 2/ + 1= 0 tim dupe t = 1 (Tin), Vay, phuong trinh da cho co 1 nghiem la x = 0. Ta phan tich: dx = cos’ x cos' x — siii'x 1 d(tnrix) cos ² x cos ² x = 1 cos ² x d(tune) (cos²x — ton 2 x). (tan x — tan’ 5 = (tan —— 4 tan’— 0 4 ) — (tan 0 —tan’ 5 5 4 5 (P): x + 2y + z — 4 = 0; d: “'- -“' 0 2 1 3 Toa do giao diém A ciia d va (P) la nghiem ctia he phuong trinh z ———2 + 3/ Tpa do giao diém A(1;1; 1). Do d cat (P) tai A ma A lai cat d nén ta co A di qua A(1;l; 1). Do A nam trong ma)t phang (P) va vuong goc vcri d nén vtcp A vuong goc v‹ i vtpt rig va vtcp up. Ta co: rig(1; 2; 1); up (2; 1; 3); Vector chi phuong ciia duéng thang A la u = [np; d]'(5; — 1; —3) nén A: a) sins — . sin2x =.cosx + cos2x O sins — Phuong trinh da cho tuong duong v‹ i: . cosx —— . sin2x + cos2x. —sinx — — cosx 2 x — — - rr — 2x 3 2 6 6 2 2 k 2ri Giai ta tim duoc cac hp nghi(m x - — — + k 2 ii,x 7r — 2r +h — , k, b) Gpi da giac déu da cho la A A . . .A ,. Vi A A7 la tru.c doi xting cua da giac nén so tain giac can dinh A la 5 tain giac, trong dfi co mot tain giac déu la AjA 5Ag. gTuong In co 4 tain giac can (khfing déu) dinh A„ A... Suy ra so tain giac can ma khong la tain giac déu bang 12.4 = 48, so tain giac déu la 4. Do do, so tain giac can la 48 4 = 52. B A Gpi H la trung diém cua AB thi H la chan duéng cao ha tit dinh S ciia hinh chop. Ta cfi ² >ABC - 2 - on.sfnl20’= —. Gpi D 1:i diém doi xung cua Aqua BC thi D la tain duéing tron ngoai tiép tain giac ABC. Ta co tain gi:ic DAB déu va do do, DH ± AB. Suy ra DH ± (SAB). Tit D, dung duéng thang A song song vcri duéng thang SH thi A la truc cua ducing trfin ngoai tiép day. Gpi I la tain tain giac déu SAB v:i trong ma)t phang (SHD), dung duéng thang d di qua I va song song voi DH thi d la trpc cua ducrng tron ngoai tiép ma)t (SAB). Gpi O = a n d thi O la tain m)at cau ngoai tiép hinh chop S.ABC. Ta + -’ 6 ’ Cau 8. Gpi I, M lan lupt la trung diém ciia AH, BC. De thay cac diém A, H, E, F cung thuoc du6ng trfin duéing kinh AH, co tain la I; cfin cac diém B, C, E, F cung thuoc duéng tron duémg kinh BC, co tain la M. Vi EF la day cung chung cua hai duong tron nfii trén nén IM ± EF, kéo theo IM ± d. Tit do, viét dupc phuong trinh duéng thang IM: 3s + y — 17 = 0. Do M nén suy ra M(5;2). Duéng thang BC vuong goc v‹ AHdi qua M nén BC: y — 2 = 0. Tu do, gpi tpa do diém B(b; 2) thi tpa do C(10 — b; 2). Vi BH ± AC nén AC. HB —— 0, suy ra (6 — b).(b — 4) + $—4).$—2) = 0, tu do tim dupc b = 2 ho)ac b = 8. Suy ra B(2;2), C(8;2) hoa)c B(8;2), C(2;2). Cau 9. Gia su cac can thuc 1:i co nghia. Xét hai truéng hop. THI. Néu y = 0 thi tu phuong trinh thu nhat ta co ngay x = 0, khfing thfia man. TH2. Néu y >0, thi chia ca hai ve ciia phuong trinh thu nhat cho , ta dupe Ve trai ciia phuong trinh 1:i ham dong bién theo an t =, ve phiii 1:i hann nghich bién theo t nén phuong trinh theo an t - co nghiém duy nhat, va tit dfi ta phiii co - 1, kéo theo x = y > 0. Chia ca hai ve cho x > 0, rfii da)t t - x +-q, dna ve phuong trinh t 2 — 5 - 5 — t, tim dupe t = 3. Tii do, x = 1, x = 2. x = 1 =› y = 1 (thoa man) x = 2 =› y = 2 (thfia man) Vay, h( da cho co hai nghiem (1; 1), (2;2) Cau 10: _0 Ta se chung minh cac bat dang thuc sau — 1 fi 3(x ³ y ³— 1 fi 3(y’— y ³)(2) z'— 1 ½ 3(z5 — z )(3) >> Truy ca(p trang http://tuvensinh247.com/ de hpc Toan — Ly — Hfia — Sinh — Van — Anh tot nhat! 8 Th(at vay, cac bat dang thtrc (1), (2), (3) lan lupt tuong duong vcri cac bat dang thuc diing ducii day (x — 1)'(2x + 1) 10, (y — 1)'(3y' + 2y + 1) 0, (z — 1)²(3z³ + 2z + 1) 0. Cong ttmg ve cac bat dang thuc (1), (2), (3) va sfi dqng giii thiét dupc viét lai, ta co ngay diéu phai chtmg minh.