Nối tiếp nội dung phần 1 tài liệu Tuyển chọn 39 đề thử sức học kì môn Toán 12 nâng cao, phần 2 giới thiệu tới người đọc phần đáp án và hướng dẫn giải chi tiết các đề thi thử. Mời các bạn cùng tham khảo nội dung hi tiết.
Tuygn chpn 39 dg Cty TNHH MTV D W H Khang Vi$l t h i i siic hqc ki mOn ToAn I6p 12 NSng cap - Ph?m Trpng Tha " i n h the tich ciia hinh chop S.ABCD va the tich chop S.BHKC Ohii'ng minh diem S, A, H, E vii K ciing nam tren mot mat cau Tinh the tich c a khoi cau ngoai tiep cua hinh chop S.AHEK Go! M la hinh chieu cua H trcn canh SA Tinh the tich cua hinh chop M.AHEK X" 02 HU6NG D A N - D A P S O Caul Klu'io scU si( hien thien vu ve thj (C) ciia ham vr^fdoc gia tiT giiii) Viet phUcfnii trinh tiep tityen ciia (C), hiet tiep tuyen (6dp so y = 24x - 43) Xcic (linh ni de phU(/nf^ trinh PhifcJng trinh da cho c6 ba nghiem phan biel YCBT X + > < - x >() x +3>0 -X x + = (x + 3)- < x>-3 2x- + 7x + = () Vay phu'dng trinh dii cho c6 nghiem x = -3, - ~ 2 Gidi phUifns^ trinh Dieu kien: < x < va x 182 X = - • X = - < X l o g , ( ' ' - l ) ( l + l o g ( ' ' - ) ) = (1) + Iog2(2'' - l ) l o g ( ' ' ' ^ ' - ) = (*) trinh Trong ASAH viiong tai A c6 SA - AE.tan6()° = — t a n " - — • Dat t = log2(2'' - ) + S^,^=iAE.BC = 73a a= S-d- PhiTdng trinh ( I ) v i c l lai t ( l + t) = o 3a V3a3 + (dvlt) tdni /, IHIH kiiilt va tiiili dien tick vat can iiiioai Xcic Jjiili ticp lunh chop + + Goi O la tarn ciia tam giac A B C ihi O la tam difdng Iron ngoai t i c p K h i t = t h i l o g ( ^ - l ) - < ^ ' = < : > x = log.,5 K h i t = - thi l o g , ( " - ) = - o " - + - - X = l o g , - = l o g , - 8 " S.Ai theo a t " + t - = o t = hoSc t := - V a y phiTdng trinh da cho c6 nghicm la x = log2 5, x = l o g j - lain giac A B C Goi d la true ciia difring tron ngoai ticp tam giac A B C t h i d l ( A B C ) \ ; i Giiii hat trinh 2^ + 2"""^' - < {*) phiOrni; Ta c6: {*) « 2" + 2.2"" - < ( I ) d//SA.Trong m p ( d ; S A ) k c difiJng trung t r i f c A c i i a SA (P lii trung d i c m ciia SA) D i l l t - 2"^ > 0, ( I ) v i c l lai r - 31 ^ < o > ho3c < t < dirring thang A cat true d tai I , ta c6: I e d => l A = IS; I e A => l A = IB = IC Khi > Ihi 2"^ > 2 x > N c n I lii l i i m m i l l ciiii ngoai t i c p hnih chop S.ABC c6 ban kinh R = l A + Tu" giac A O I P lii hnih chiJ nhiit c6 l A lii difdng chco K h i < l < thi < " < I c> X < Vay lap n g h i c m bat phi/cmg lrinhS = (-00; ( ) ) u ( l ; + o o ) R = IA = N/OA- + AP- - J -SA -AE Cau V a Tim i^ici tri hhi nhat va :^ia tri nho nini'l ciia ham so 12 Ti\p xac dinh I ) = x + Tinh the tick ciia khoi chop A.BCNM + Hiim so' dii ' h.; irc't th;iiih g(t) = r - Vit + theo a T ' l CO- ^ s A M N _ SA S M SN _ S M SN ^^.PMC + D a o h a m g\'.i Tinh gia i r i g Trong tam giac S A B c6 SA^ = SM.SB TiTcJng tiT tam giac SAC ta c6: Ncn V.S.AMN Vs.ABC ^ V 190 ^A.BCNM S M SB =V i i - ^ , g'(t) = t = ^ e [-1; ]] " SA • SB • SC ~ SB • SC SB + D a l I = s i n x , I e ( - l ; 1| D i c n lich mat can: S^.^^ = t R ' : = i - ^ ^ ^ = Y^7ta'^(dvdt) ^S.ABC SN_ SC - V SN SC 13 81 81 169 169 ^S.AMN I- SA2 + A B 'S.ABC 13 V D o d o max y = xeK = g ( l ) = - N / , g ( - l ) = + V3 / max g(t) = + ^3 o x = - - + k t , k e Z tel-1; 1] x.= - + miny= 88 >/3a^ 81 169 SB^ 169 llS'd^ 169 xeK (dvtl) m i n g(i) = — < z > le|-l; Ij >:\ k2Tt , keZ x= — «j ill + k27t 191 Tuygn chon 39 di \hil sCfc hpc ki mfln Toan Iflp 12 Nang cao - Pham Trpng Thu Cty TIMHH MTV DWH Khang Viet Dodo Cfiu I V b Chiinii ininh ( - ' ' + + ") = -1 + (2" -1)2 _l +i(2"+2-'') = 1+ mm y = mm g(l) = X6|0;7i| I6|(); l | " 1-2' 2" DE s o l + - ( " - - " ) = | + ^ l + i ( 2 " - + 2-2") +, , i ( 2 " + + ") = V4 ( " + ) ^ + 2^ I l +i.(2"+2-") = 2" 2" n x =— 37t x = — ,x+l (do x < ( ) n c n " rf/^'//i; 4.3 H a m so da cho trcl lhanh g(t) = -^"^ " l Thi sink chi didic Idm mot troitg hai phdii (phdn A hoac phdn li) A T h e o chi/(/n}i t r i n h C h u a n D a o h a m g'(l) = 4l2 - , g'(i) = o i ^}_^ 1= Tinh gia I r i g 192 CdulVii (2,0 diem) G i a i phi/dng i r i n h l o g ( x - ) +1 = 21og4(6x - 10) ^ , g ( i ) - - | g(0)-o € 0;1 ' ' Giai bal phiTOng I r i n h " < " " " + ^&u\d (1,0 diem) ' T i m gia I r j Idn nha't va gia t r i nho nhii't ciia ham so y = 2x"' - x _ i x +10 trendoan [0; 31 Tuyg'n chon 39 de iliii '.uc hoc ki mbn ToAn k3p 12 Niang cao - Ph^m Trpng ThU I IJ Theo chi//3 = ^ , The tich khoi chop S.ABC: V = ^S^,j(_ SA = ^ - y - N / i = ^ ( c m ^ (cm^) Tinh tarn D vd ban kinh ciia mat cau ntfoai tiep hinh chop S.ABC Giaiphir(1ngtrinh2^^ ' ' - ^ ^ ^ ^ ^ = C&uVh (1,0 diem) Hai goc SAC va SBC lii hai g()c vuong nen tam D cua mat cau ngoai ijcp hinh chop S.ABC la trung diem canh SC Tim gia tri Idn nhal va giti tri nho nhat cua ham so y = V2x - Ban kinh mat cau R = - S C = - \ / s A - + AC" = — - - 2'"'"- cauiva H U d N G D A N - D A P SO Giai pludfnii cau I trinh Khdo sat sU hien thien va ve thi (Q ^uu ham w (doc gia lif giai) Viet phU(fng trinh ciia ^/wvV/i.i; thani^ d son}; son;; iY>( (dap so y = -3x -1) , f x - - 3>() r Dicu kien V3 6x~l()>() cau I I V(' 27t.a- = 7t.OA.2a => OA = a Khi ni = la c6 l o g j X - 3k)g2X - = logoX = Trong ASOA vuong lai O, la c6: SO = VsA" - O A " = 7(2a)- - a" = -ASThe lich ciia hlnh non can lim la V = ^Tt.OA'.SO = Z L I L ^ (jvii) cau ivb GidiphU(fnii trinh (I) m = Khi m = 3, la c6 4^ - 4.2'' + - 1=1 Dal I = 2^, I > Phu-dng Irlnh Iren ltd ihiinh r - 4l + o = 3" • Vdi I = ihi 2" = « X = • Vdi t = Ihl 2'' = t h i (r(cos(p + isincp ) ) " - r " ( c o s n q ) + isinncp) a = b o iy = y'' a + b = (x±x';y±y')297 296 Tuyg'n chQn 39 6i thCl sijtc hqc ki mfln Toan I6p 12 Nang cao - Phgm Trpng TtiU • k i i = ( k x ; k y ) vi'li k e :huy • a C l i n g phiTdng b (h ^ ()) 3k e • a b = x x ' + Cty TNHH MTV DVVH Khang Vi?t = kx' ^ x y ' - x ' y = y = ky X • ri.ii = Ne'u A //A' ihi A' c6 dang ax + by + c' ( v d i c' ^c) • Ne'u A _L A' thi A' c6 dang bx - ay + c' = (hoSc - bx + ay + c' = 0) yy' N e u du'iJng lhang di qua M(x^,: y,,) va nhan li = (a; b) lam V T P T Ihi di/tlng th^ng tren c6 phU'dng trinh a(x - x^,) + b(y - y^,) = a.b • cos( li, b ) = • + Cac tru'dng hdp dac b i e l : XX + y y ^ x - +y-.^/x'- + y ' Chu y a b o a.b = xx' + yy' = Ne'u du'cJng lhang A song song hoac trung v d i Oy thi A c6 dang ax + c = • Ne'u du'(:(ng lhang A song song hoac trung v d i Ox thl A c6 dang by + c = • Ne'u du'itng lhang A di qua goc loa ihi A c6 dang ax + by = • DuTtng thang di qua hai d i e m A(a; 0), B(0; b) (vdi a.b II DUdNG T H A N G /) Cdc daiiii phiMiig trinh didiiifi tlmiifi • ChodircJng i h a n g A qua M(x^,; y , , ) va c6 V T C P li = ( U j i u-, ).Ta c o : + Phifting irinli ihani so ci'ia di/rtng ihdng A la X=X, +11,1 T ii+y^l a b (1) Di/dng lhang di qua hai d i e m A { x ^ ; y ^ ) , BCx^; y ^ ) c6 phu'ctng trinh lii ( x - x ^ ) ( y g - y ^ ) - ( y - y ^ ) ( X g - x ^ ) K h i x,^ - x^ ?t va y ^ - y ^ "^hi (2) vie't l a i X - X, + Pliif(t diem den mot diilfng thang Cho du'cing lhang A : ax + by + c = va M(x^^; A Ta c6: / • Fhifdng Irinh diTifng thang A qua M(x^,; y^,) va c6 he so goc k la d ( M ; A) = y - y , , =k(x-x^,) Chiiy A l A ' o k ^ ^ k y = - A/ZA'c^k,^ = k^- • Phu-ctng trinh long qiuil difctng t h i n g A c6 dang ax + by + c = ( a " + h~ ^ 0) Trong d o : • li = (a; b) la n i o l vectcf phap luyen cua A • li = ( - b; a) ( hay li = (b; - a)) la mot vcclcJ chi phu^dng cua A k = — la he s o g c ( b ^ ( ) ) b 298 Vi tri tiidng doi cua hai diiitiig thang Cho hai di/clng lhang c6 phUdng trinh A , : ajX + b,y + C[ = A-,: ajX + b2y + Cj = So' giao d i e m ciia A j va A j chinh la so' nghiem cua he phu'dng trinh: aix + b , y - - c , ajX + b2y = - C j 299 Cfy TIMHII M[V DVVM Kh,ni,j Vn^ Tuyin chpn 39 6i thil siic hqc kl m6n ToAn Idp 12 Nang cap - Ph?m Trpng Thi/ + Ta linh: D = a, b, aT b-, -c^ ^ a ^ x + b T y + CT a | X + b | y + C| = a[b2 - a ^ b j b, ^aj^ + = -c,b-, +C2b, ^a; bj^ -r)/ + b ; 6) Vf tri ciia hai diem doi v('fi dUifng thdiig C h o diTiJng l h a n g A : a x + b y + c - 0; a, -c, = - a | C T + a^Cj a-, - L s hai d i e m A ( x ^ ; y ^ ^ ) , ^ ( x , , ; y ^ j l p h a n b i c l Ta c6: + B i c n liuln: • A , va A-, c a t n h a i i D A , / / A T A, s A , o o • A va IJ n a m l i a i phi'a d o i v(' R c > M i'< n g o a i ( C ) + I M = R M l u i m i r e n ( C ) 1- I M < R M irong (C) 301 Tuygn chpn 39 6i thil sOc hpc ki m6n ToAn I6p 12 Nang qao - Ph?m Trpng Thu Cty TNHH MIV 5} Vi tri ciia ditifng thang doi vi'ii dififng trbn Cho diTclng Iron (C ) c6 tarn I , ban kinh R va difdng thang A + d(I;A)>RAkh6ngcrit(C) > > , 2 u2 c2 = b - a2 L i e n he a, b, c c Tieu diem F,(-e;0),F.(e;0) F,(0; - c ) , F ( ; e ) D i n h tren true Idn A , ( - a ; ) , A3(a;()) B,((); - b ) , B ( ; b ) Dinh tren true nho B,((); - b ) , B , ( ; b ) A , ( - a ; ( ) ) , A^Ca: 0) Tarn sai e.^ l a + Fgoi la lieu diem ciia papabol (P) ^>1 b + DifcJng lhang A goi la diTiing chuan + MF goi la ban kinh qua lieu diem cua M + Khoang each tCr F den A goi la tham so lieu K i hieu p 2) PhiMng trinh chinh tdc cuaparabol va cdc ye'u to'cua parabol -) Di/cJng chuan Ticm can a a" A, ^ : x = ± - = ± — e c b y=± A, , y ^ ^ ^ y c —X Ban kinh qua lieu MF, = ex^ +a MF, = cy^^+b ( V d i M ( x ^ ; y^^)e(H)) MF2 MF2 = c y ^ , - b eXj^ y'=2px ^b y = ± —X a a = c - a Tieu diem Di/dng chuan PhiTdng trinh hlnh chff X = ±a, y = ±b X = ±a, y = ±b nhat cd set PT tiepluyenvi'n(H) taiM(x,,; y^) (bd sun^) -; x y^ =-2px r P — x2=-2py p P X = x-=2py — P ''1 Dicu kicn (P) tiep xiic '^'^0 a2 yyo_, b^ Ax + By + C = b2 a^ " • (bd B-p = 2AC B2p = -2AC A p = 2BC A p = -2BC Miiii^) Dieu kicn (H) tiep xuc Ax + By + C = () AV-BV=C2 3V-AV=C2 (hd sunjLf) 304 3fi^ Cty TNHH MTV D W H Khang Vi?t lyfi'n chqn ii thif sire hoc kl mOn Toan Idp 12 Nang cao - P h ? m Trpnq T h J ' a + b = (x ± x'; y ± y'; / ± z ' ) /Hdf28 PHaONG PHAP TOA OO TRONG KHONG GIAN kii = ( k x ; k y ; k / ) v»'ti k e a b = x x ' + y y ' + zz' VECTO VA TOA O O =^x Toa cua m o t vccut: a = (x; y ; / ) o a = x i + y j + / k Toa cua m o t d i e m : M ( x ; y; /.) O M = x i + y j + /.k + y +z • cos( a b ) = ii.li _ x x ' + y y ' + zz' yj\ +•/ .\]\ +•/ Cho A ( x ^ ; y ^ ; z ^ ) , B ( X g ; y ^ ; z ^ ) Chii y a b o a.b = () x x ' + y y ' + zz' = + AB = ( X a - x ^ ; y a - y ^ ; z B - / A ) - AB • Tich c() hiohiii ciui luii vecUt | a, b ] = I • DieII ki('n ddni; + Toa irung d i e m I cua doan A B : yA+yb ZA y ' v: ' + Toa d i e m M chia doan A B ihco ti so k ^ 1: y i = x y X' y Tlic tich hiiih Iwp ABCD A'B'C'D': V= B A , BC C A , CB AB A D AB, A D AA' 1-k yA-^^yB rhC' tich hinh chop A BCD • V = - A B A C A D l - k z - k zB ' x z' x ' AB, AC Dii'ii ticli li'inh Ivnh lutnli ABCD : S^^^j - X, - k x , / z pluliii^ : a , b, c dong phang | a, b |c = • Dii'ii tich tain f^iac ABC : SA B C + ^ - U y II P H U O N G T R i N H C U A M A T P H A N G l - k /) Phif(tii}i trinh tdiif> qiidt ciia mat phang • TrongkgOxyzchotamgiac ABCco A ( x ^ ; y ^ ^ z ^ ) , B(Xy; yg; z^) ^A V C(Xj,; y ^ ; /-c^- ^ ^""""S '-"a A A B C thi yc- '^B yA+yB+yc Z^+Z^+Zc • PhiTdng irinh A x + By + Cz + D = v d i A " + B " + C " > la phiTdng trinh tdng qiiiit ciia mat phfing c6 V T P T ri = ( A ; B; C ) • N c i i mat phiing ( a ) q i i a d i e m M(x^,; y^^; z ^ J va c6 V T P T ri = ( A : B; C ) t h i phiTdng trinh tdng quat ciia ( a ) la A ( x - x ^ , ) + B(y - y^,) + C ( z - z ^ ^ ) ® Cdc trUifiig lufp dac bi^t: X e t mat phang ( a ) : A x + By + Cz + D = "H^P TOAN + A x + By + Cz = o ( a ) di qua gdc toa C h o a - ( x ; y ; z ) va b = (x'; y'; z') X=x a = b o y = y' z = z' + By + Cz + D = ( D ?t 0) ( a ) song song \(Vi true Ox A x + Cz + D = ( D ;t 0) ( a ) song song vi'li true O y A x + By + D = ( D 0) o ( a ) song song v d i iriic Oz 307 Tuy6'n chqn 39 dg thir sCfc hpc ki m6n Toan I6p 12 Nang cao - Phgm Trpng Thu + By + C/ = o ( a ) cliiJa true Ox Cty TNHH MTV DWH Khang Vi^t 5) Khodng each tii mot diem den mat phdng Ax + C/ = c:> ( a ) clu'ra liiic Oy Cho mat phang ( a ) : A x + By + Cz + D = 0, IVKx^^; y^,; z^,) g (rx) Ax + By = o ( a ) clu'ra true Oz + Cz + D = ( D ?t 0) ( a ) song song v d i (Oxy) KhoangciichtirMdcn A x + D = ( D ^ 0) U V = () b) He tren v6 nghiem d // (a) 5) Vi tri tiiifiig doi ciia dUifiig thang va mat phang Cho mat ph;1ng ( a ) ; Ax + By + C / + D = c6 VTPTri= (A; B; C), Ihiing d di qua M(\^,; y^,; a) d tilt (a) «> li.ri ) va t o VTCP li - (a; b; t ) Aa Bb + Ct ^ li.ri = b) dl(a)) d//((X) M g(a) Aa+ Bb + Ct = {) dirCtng c) He tren vo so nghiem o d e (u) 6) Gtic ciia hai diding thang Cho hai difCJng thang d va d' Ian Iiritl t o VTCP li = (a; b; t ) , v = (a'; b'; t ' ) Got (p giijfa hai diA'Jng thang d vad'diCitt xat dinh b('R (a) khong cat (S) H d(I,(u)) = R (a) tiep xuc (S) FT (u) cilt (S) Iheo du'cing lion (C): d(l,(a))