#VLJUBCOIFSIBLLTBLMESWF\":%*/:\":*/-\"3*OBBJUUJSTBZMZBTBOOIÐLÐNMFSJOF HËSFLJUBCOEÐ[FOJ NFUOJ TPSVWFõFLJMMFSJLTNFOEFPMTBIJ¿CJSõFLJMEFBMOQZBZNMBOB- NB[ GPUPLPQJZBEBCBõLBCJSUFLOJLMF¿PóBMUMBNB[ :BZO4PSVNMVTV $BO5&,÷/&- :BZO&EJUÌSÑ %J[HJ–(SBGJL5BTBSN &TSB:·,4&-)BLBO\"ó$\" *4#//P :BZOD4FSUJGJLB/P \"ZEO:BZOMBS%J[HJ#JSJNJ #BTN:FSJ ÷MFUJöJN &SUFN#BTN:BZO-UEõUJr \":%*/:\":*/-\"3* JOGP!BZEJOZBZJOMBSJDPNUS 5FMr 'BLT 0533 051 86 17 aydinyayinlari aydinyayinlari * www.aydinyayinlari.com.tr %¸O¾P.DSDáñ11.SINIF %FOLMFNWF&öJUTJ[MJL4JTUFNMFSJ KARMA TEST - 1 Karma Testler 11. SINIF 4. MODÜL 1. x2 +Z2 = 26 5. _ x4 - 4x2 i x $ 0 Modülün sonunda YpZ= 5 x2 - 2x tüm alt bölümleri L©HUHQNDUPDWHVWOHU EFOLMFNTJTUFNJOJTBôMBZBOGBSLMYHFSÀFLTB- FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO \\HUDOñU ZMBSOOUPQMBNLBÀUS IBOHJTJEJS Alt bölümlerin A) - # - 3 C) 0 D) 5 E) 10 A) [ -2, 3 # -2, 3 ) EDĜOñNODUñQñL©HULU C) [ 2, 3 ) D) [ -2, 2 ] E) [ -2, 3 ) - { 0, 2 } DENKLEM VE EŞİTSİZLİK 2. #JS EJLEÌSUHFOJO LÌöFHFO V[VOMVôV 2 5 CS 6. 4x – 2 · _ x2 - 16 i # 0 SİSTEMLERİ EJLEÌSUHFOJOBMBOJTFCS2PMEVôVOBHÌSF CV _ x - 4 i3 · _ x + 3 i EJLEÌSUHFOJO ÀFWSFTJ BöBôEBLJMFSEFO IBOHJTJ- FöJUTJ[MJôJOJ TBôMBZBO Y UBN TBZ EFôFSMFSJ LBÀ UBOFEJS EJS ³ İkinci Dereceden İki Bilinmeyenli Denklem Sistemleri t 2 \" # $ % & \" # $ % & ³ İkinci Dereceden Bir Bilinmeyenli Eşitsizlikler - I t 10 ³ İkinci Dereceden Bir Bilinmeyenli Eşitsizlikler - II t 17 ³ İkinci Dereceden Bir Bilinmeyenli Eşitsizlikler - III t 23 6ñQñIð©LðĜOH\\LĜ 3. x2 - 8x + 7 < 0 7. 3 - x2 + 2x - 15 2 0 <D]ñOñ6RUXODUñ _ x - 2 i2 www.aydinyayinlari.com.tr YAZILI SORULARI FöJUTJ[MJôJOJTBôMBZBO%YFUOBNLMFTNBZWFMB&SöOJUOTJU[PMJQLMB4NJTUFNMFSJ FöJUTJ[MJôJOJTBôMBZBOYUBNTBZEFôFSMFSJUPQMB- 2NXO\\D]ñOñVñQDYODUñQGD ³ İkinci Dereceden Bir Bilinmeyenli Eşitsizlikler - IV t 30 ©ñNDELOHFHNVRUXODUñL©HULU 11. SINIF 4. MODÜL %&/,-&.7&&õ÷54÷;-÷,4÷45&.-&3÷ LBÀUS 1. ( a - 2) x2 - 2ax - 1 < 0 NLBÀUS 4.C) -2 \" # $ % & A) - # -4 D) 0 E) 3 y ³ Karma Testler ÷,t÷/4$1÷%&3&$&%&/÷,÷#÷-÷/.&:&/-÷%&/,-&.4÷45&.-&3÷ FöJUTJ[MJôJYJOIFSHFSÀFLTBZEFôFSJJÀJOTBôMB- OZPSTBBOOCVMVOEVôVFOHFOJöBSBMôCVMV- ÷MJöLJMJ,B[BONMBS OV[ –4 3 x O ³ Yazılı Soruları t 4511.4.1.1 : öLJODJEFSFDFEFOJLJCJMJONFZFOMJEFOLMFNTJTUFNMFSJOJO¿Ë[ÐNLÐNFTJOJCVMVS %XE¸O¾PGHNL¸UQHN VRUXODUñQ©¸]¾POHULQH rx !3JÀJOFöJUTJ[MJLTBôMBOZPSJTFÔ<WFB- 2 < 0 y = f(x) DNñOOñWDKWDX\\JXODPDVñQGDQ Nesil Sorular t 47TANIM ÖRNEK 3 PMNBM B C D E F G`3WFB C DHFS¿FLTBZMBSOEBO x2 -Z2 = 9 ³ Yeni B2 -p B- 2 ) ( -1 ) <0 jB2 +B- 2 < B-2 < 0 ôFLJMEFZ=G Y QBSBCPMÐOÐOHSBGJóJWFSJMNJõUJS x -Z= 9 B[JLJTJTGSEBOGBSLMPMNBLÐ[FSF 4. x2 + ax - 18 $ 0 a –2 1 2 8. x+4· x-2 #VOBHÌSF JLJODJEFSFDFEFOJLJCJMJONFZFOMJEFOLMFNTJTUFNJOJO #0 ^ x + 2 h f^ x h ax2 +CYZ+DZ2 +EY+FZ+G= 0 ÀÌ[ÑNLÑNFTJOFEJS x+b a2 + a – 2 + – + + #0 x2 + 12x - 13 x2 -Z2 = (>x - 2y) . (>x + 2y) = 9 CJ¿JNJOEFLJJGBEFMFSF JLJODJEFSFDFEFOJLJCJ- 91 FöJUTJ[MJôJOÀÌ[ÑNLÑNFTJ[ -9, -3 ) , [2, 3 PM- FöJUTJ[MJôJOJTBôMBZBOOFHBUJGUxB2N–T1B6ZEFôFSMFSJ- MJONFZFOMJ EFOLMFN EFOJS öLJ CJMJONFZFOMJ FO B[JLJEFOLMFNEFOPMVõBOTJTUFNJOEFOLMFNMF- x -Z= 9 a–2 – – – + OJOUPQMBNLBÀUS FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ SJOEFOFOB[CJSJJLJODJEFSFDFEFOJTFCVTJTUF- + x +Z= 1 EVôVOBHÌSF B-CGBSLLBÀUS NF JLJODJ EFSFDFEFO JLJ CJMJONFZFOMJ EFOL- MFNTJTUFNJEFOJS A) - # - 2 C)B-!1( -2, 1D) ) 1 E) 4 A) - # -13 C) -12 D) -11 E) -10 x+2=0 x2 - 16 = 0 G -4 ) =G = 0 x = -2 x = 4, x = -4 %FOLMFNMFSJO PSUBL ¿Ë[ÐN LÐNFTJ EFOLMFN 2x = 10 j x = Z= -2 XODĜDELOLUVLQL] 1. $ 22. .$ x32. E· ^ 3 –4. xEh5 # 0 41 x –4 –2 3 4 TJTUFNJOJO¿Ë[ÐNLÐNFTJEJS ÇK = {(5, -2)} ^ 5 – x h3 5. E 6. \" 7. $ 8. E 1 (x + 2) f(x) + + – + – ÖRNEK 1 x2 – 16 FöJUTJ[MJôJOJTBôMBZBOYUBNTBZMBSOOLÑNFTJ- ÇK = [ -2, 3] , ( 4, 3 ) OJZB[O[ 3x2 -Z2 = 11 ÖRNEK 4 x2 +Z2= 9 Z2 - 2x2 = 23 x 035 JLJODJEFSFDFEFOJLJCJMJONFZFOMJEFOLMFNTJTUFNJOJO x +Z= 6 x2 (3 – x)5 + + – + <HQL1HVLO6RUXODU ÀÌ[ÑNLÑNFTJOFEJS (5 – x)3 0RG¾O¾QJHQHOLQGH\\RUXP 3x2 -Z2 = 11 JLJODJ EFSFDFEFO EFOLMFN TJTUFNJOJO ÀÌ[ÑNLÑNF- 5. x2 – 4x > 0 \\DSPDDQDOL]HWPHYE + x2 +Z2 = 9 TJOFEJS x+4 EHFHULOHUL¸O©HQNXUJXOX $OW%¸O¾P7HVWOHUL <(1m1(6m/6258/$5YUBNTBZMBSOOLÑNFTJ{0, 3, 4} %FOLMFNWF&öJUTJ[MJL4JTUFNMFSJ VRUXODUD\\HUYHULOPLĜWLU x2 – 16 3. #JS PUPNPCJM NBSLBT USBGJL LB[BMBSO ËOMFNFL $\\UñFDPRG¾OVRQXQGD 4x2 = 20 j x2 = 5 jZ2 = 4 Z= 6 - x, (6 - x)2 - 2x2 = 23 >0 x2= 5 j x = 36 - 12x + x2 - 2x2 = 23 j x2 + 12x - 13 = 0 TEST - 2 1. /FTJCF\"ZEO:ME[MBS-JTFTJhOJOZBQxU2S–E2óxC–JS8SFL- 5 j x = - ÷L5JODJ%FSFDFEFO÷LJ#JMJONFZFOMJ%FOLMFN4JTUFNMFSJ MBNUBCFMBTOBZEOMBUNBLJ¿JOLVSFVöMBJUOTJE[ÐMJ[LFTOJFTLUUFFN JOJOÀÌ[ÑNBNLÑBDNZFMTBJHOFJMCJõVUJMSVEOJóVJ[CJSTJTUFNEFBSB¿ ËOÐOEFLJCJS Z2 = 4 jZ= 2 jZ= -2 (x + 13).(x - 1) = 0 x = -13 jZ= 19 , x = 1 jZ= 5 SFLMBNUBCFMBTOOZFSFEFóEJóJOPLUBJMFBZOIJ[B- DJTNJBMHMBEóOEBPUPNBUJLPMBSBLGSFOZBQNBLUB- Ç .K = % a 5, 2 k, a 5, - 12 .k, a - 5x, 22 k+, aZ-2 5=, -62 k / ÇK = {(-13, 19), (1, 5)}5. x2 +YZ+Z2 = 7 3. ( a - 5 ) x2 - 4ax + a - 3E=BL0J\"OPLUBTOBLPOVMBOõLLBZOBóUBCFMBOOFO ES 'SFO õJEEFUJOJ ZPMVO TÐSUÐONF LBUTBZTOB WF Z2 - 2x = 3 ÐTUUFLJ$OPLUBTOBZEOMBUNBLJ¿JOx2ZF-S4JxMF=F0OGB[MBx+ 4 = 0 BSxB2D-O1D6JT=N0JBMHMBEóBOEBLJI[OBHËSFEFóJõUJS- 2x +Z= 5 EFOLMFNJOJO LÌLMFSJOJO UFSTJöBB¿SFZUBMJQNPMBBLCUBJMENSFTJ x = 4, x = 0 x = -4 NFxL=UF4E, JSx = -4 JÀJOBOOÀÌ[ÑNLÑNFTJOJCVMVOV[ x2C- 2x - 8= 0 ÖRNEK 2 ÷LJODJ EFSFDFEFO EFOLMFN TJTUFNJOJO HFSÀFL JLJODJ EFSFDFEFO EFOLMFN TJTUFNJOJ TBôMBZBO Her alt bölümün FOLÑÀÑLYEFôFSJWFFOCÑZÑLZEFôFSJJÀJO VRQXQGDRE¸O¾POHLOJLOL x +Z= 4 TBZMBSEBLJÀÌ[ÑNLÑNFTJLBÀFMFNBOMES x -ZGBSLLBÀUS WHVWOHU\\HUDOñU a–3 x = 4, x = -2 x –4 –2 0 4 Z= x2 - 6x + 10 \" # $ %Ö RNEK &5 5FSTJöBSFUMJJLJLÌLJÀJOY1 · x2 < 0 <0 a– 5 x2 +Z2 = 19 A) - # $ % & Ô>PMNBMB2 - B- B- 3 ) > 0 JLJODJ EFSFDFEFO EFOLMFN TJTUFNJOJO ÀÌ[ÑN LÑNF- x2 – 4x – + + – + A B x+4 OFHBUJGPMEVôVCJMJOJZPS \"SBDODJTNJBMHMBEóBOEBLJI[7 TÐSUÐONFLBU- TJOFEJS x +Z= 5 a x2 – 16 A Bx2 – 2x – 8 a 3 5 + – + TB+ZTL+PMNBLÐ[FSFBSB¿JMFZBZBBSBTOEBLJV[BL- a–3 + –+ MóWFSFOGPOLTJZPO Z= 4 - x, 4 - x = x2- 6x + 10 ,x02 =-(xYZ-+3)Z(x2 - 2) JLJODJEFSFDFEFOEFOLMFNTJTUFNJOFHÌSF YZÀBSQ- a–5 x= 3 jZ 2. = 17 NLBÀUS B! ( 3, 5 ) G 7 = L- 72 - L- 7+ 1 | |#$ = ( x2 - NWFöLLBZOBôÇKZM=B(U-B2C,F0MB) ,B(S4B,-3 ) x = 2 jZ Z2 -YZ= 8 (x +Z 2 = 52 TOEBLJNFTBGF Y- NPMEVôVOBHÌSFY2JGB- PMBSBLWFSJMJZPS ÇK = {(3, 1), (2, 2)} 6. YWFZHFSÀFLTBZMBSPMNBLÑ[FSF \"SBDO \" OPLUBTOEBZLFO BMHMBEô # OPLUB- EFTJOJO BMBCJMFDFôJ LBÀ GBSLM UBN TBZ EFôFSJ ÷LJODJEFSFDFEFOEFOLMFNTJTUFNJOxJ2T+BôMYBZZ+BOZ2Y= 25 j YZ x2+jZY2Z- 4x WFZEFôFSJJÀJOY-ZJGBEFTJOJOQP[JUJGEFôFSJ 5. (–T2,O0E)BbC V MßV OBOZBZBZB7OJOUÑNEFôFSMFSJJÀJO +Z+ 13 = 0 1. (–2, 1) 2. {0, 3, 4}WBSE3.S(3, 5) 45 4. <m >b ß ÀBSQNBEô CJMJOEJôJOF HÌSF \"# ZPMVOEBLJ TÑS- $ UÑONF LBUTBZTOO BMBCJMFDFôJ FO CÑZÑL UBN LBÀUS PMEVôVOBHÌSF YZÀBSQNLBÀUS \" # % & 1. % a 5, 2 k, a 5, - 2 k, a - 5, 2 k, a -\" 5,- 2 k / #2 .{(3 , 1), (2,$2 )} 2 % 3&. {(5, –2)} 4. {(–13A,)1-9), (1, 5)} # 5.-33 C) -2 D) 3 E) 6 TBZEFôFSJLBÀUS \" # $ % & 3. YWFZHFSÀFLTBZMBSPMNBLÑ[FSF 2. 4. )FSCJSLBUOHFOJõMJóJNFUSFPMBOLBUMCJSCJOB- WDPDPñ\\HQLQHVLOVRUXODUGDQ x2 -Z= 0 f(x) = x2 + (k +1)x + k + 16 OO Eõ ZÐ[FZJOEFO CJS SFLPS EFOFNFTJ J¿JO USNB- ROXĜDQWHVWOHUEXOXQXU OBO ËSÐNDFL BEBN MBLBQM öTNBJM IFS EBLJLBEB Y Z- | x - 6 | = 0 7. YWFZHFSÀFLTBZMBSPMNBLÑ[FSF NFUSFJMFSMFNFLUFEJS x2 -YZ= 43 EFOLMFNTJTUFNJOJTBôMBZBOYEFôFSMFSJOJOÀBS- QNLBÀUS Z2- 21 =ZY A) - # -6 C) 0 D) 6 E) 8 )BNEJ ²ôSFUNFO TOGUBO TFÀUJôJ JLJ ÌôSFODJTJ \"OM WF #BOV JMF BöBôEBLJ HJCJ CJS PZVO PZOB- PMEVôVOBHÌSF Z-YJGBEFTJOJOEFôFSJBöBôEB- NBLUBES LJMFSEFOIBOHJTJPMBCJMJS O #BOVUBIUBEBZB[BOG Y GPOLTJZPOEBYZFSJOF A) - # -6 C) 4 D) 6 E) 16 IFSIBOHJCJSHFS¿FLTBZZB[BS 4. x2 -Z2 = 15 8. 2x2 +Z2 -Z+ 2 = 0 O :B[EóYEFóFSJJ¿JOGPOLTJZPOEBYFLTFOJÐ[F- U EBLJLB TPOVOEB öTNBJMhJO LBU HF¿UJóJ WF CJOB x2 +Z+ 4 = 0 SJOEFCJSOPLUBCVMVSJTFZBSõNBZ#BOVLB[B- Ð[FSJOEFCJSOPLUBEBPMEVóVCJMJONFLUFEJS YZ= -4 OBDBLUS ÷LJODJ EFSFDFEFO EFOLMFN TJTUFNJOJO HFSÀFL YCJSUBNTBZWFYJMFUBSBTOEBY=U-CBôO- ÷LJODJ EFSFDFEFO EFOLMFN TJTUFNJOJO ÀÌ[ÑN TBZMBSEBLJÀÌ[ÑNLÑNFTJLBÀFMFNBOMES O \"LTJUBLEJSEFPZVOV\"OMLB[BOBDBLUS UTCVMVOEVôVOBHÌSF YLBÀUS LÑNFTJBöBôEBLJMFSEFOIBOHJTJEJS A) { (4, 1 ) } #VPZVOV\"OMhOLB[BONBPMBTMôPMEVôVOB # { ( 4, -1 ), ( -4, 1 ) } HÌSF L ZFSJOF ZB[MBCJMFDFL LBÀ GBSLM UBN TBZ C) { ( -4, 1 ) } EFôFSJWBSES D) { ( -4, 1 ), ( 4, - J J -J -J } E) { ( -4, 1 ), ( 4, - J -J -J J } \" # $ % & \" # $ % & \" # $ % & 1. \" 2. $ 48 3. D 4. \" 1. $ 2. D 3. # 4. D 7 5. $ 6. \" 7. \" 8. \"
www.aydinyayinlari.com.tr 11. SINIF 11. SINIF 4. MODÜL DENKLEM VE EŞİTSİZLİK SİSTEMLERİ ³ İkinci Dereceden İki Bilinmeyenli Denklem Sistemleri t 2 ³ İkinci Dereceden Bir Bilinmeyenli Eşitsizlikler - I t 10 ³ İkinci Dereceden Bir Bilinmeyenli Eşitsizlikler - II t 17 ³ İkinci Dereceden Bir Bilinmeyenli Eşitsizlikler - III t 23 ³ İkinci Dereceden Bir Bilinmeyenli Eşitsizlikler - IV t 30 ³ Karma Testler t 41 ³ Yazılı Soruları t 45 ³ Yeni Nesil Sorular t 47 1
11. SINIF 4. MODÜL %&/,-&.7&&õ÷54÷;-÷,4÷45&.-&3÷ www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/÷,÷#÷-÷/.&:&/-÷%&/,-&.4÷45&.-&3÷ ÷MJöLJMJ,B[BONMBS 11.4.1.1 : öLJODJEFSFDFEFOJLJCJMJONFZFOMJEFOLMFNTJTUFNMFSJOJO¿Ë[ÐNLÐNFTJOJCVMVS TANIM ÖRNEK 3 B C D E F G`3WFB C DHFS¿FLTBZMBSOEBO x2 -Z2 = 9 FOB[JLJTJTGSEBOGBSLMPMNBLÐ[FSF ax2 +CYZ+DZ2 +EY+FZ+G= 0 x -Z= 9 CJ¿JNJOEFLJJGBEFMFSFJLJODJEFSFDFEFOJLJCJ- JLJODJEFSFDFEFOJLJCJMJONFZFOMJEFOLMFNTJTUFNJOJO MJONFZFOMJEFOLMFNEFOJS ÀÌ[ÑNLÑNFTJOJCVMVOV[ ö¿JOEF JLJODJ EFSFDFEFO JLJ CJMJONFZFOMJ FO B[ x2 -Z2 = (>x - 2y) . (>x + 2y) = 9 CJSEFOLMFNCVMVOBOWFCJSEFOGB[MBEFOLMFN- EFOPMVõBOTJTUFNFJLJODJEFSFDFEFOJLJCJMJO- 91 NFZFOMJEFOLMFNTJTUFNJEFOJS x -Z= 9 %FOLMFNMFSJO PSUBL ¿Ë[ÐN LÐNFTJ EFOLMFN + x +Z= 1 TJTUFNJOJO¿Ë[ÐNLÐNFTJEJS 2x = 10 j x = Z= -2 ÇK = {(5, -2)} ÖRNEK 1 ÖRNEK 4 3x2 -Z2 = 11 Z2 - 2x2 = 23 x2 +Z2= 9 x +Z= 6 JLJODJEFSFDFEFOJLJCJMJONFZFOMJEFOLMFNTJTUFNJOJO ÀÌ[ÑNLÑNFTJOJCVMVOV[ JLJODJ EFSFDFEFO EFOLMFN TJTUFNJOJO ÀÌ[ÑNLÑNF- TJOJCVMVOV[ 3x2 -Z2 = 11 + x2 +Z2 = 9 Z= 6 - x, (6 - x)2 - 2x2 = 23 4x2 = 20 j x2 = 5 jZ2 = 4 36 - 12x + x2 - 2x2 = 23 j x2 + 12x - 13 = 0 x2= 5 j x = 5 j x = - 5 Z2 = 4 jZ= 2 jZ= -2 (x + 13).(x - 1) = 0 x = -13 jZ= 19 , x = 1 jZ= 5 Ç .K = % a 5, 2 k, a 5, - 2 k, a - 5, 2 k, a - 5, - 2 k / ÇK = {(-13, 19), (1, 5)} ÖRNEK 2 ÖRNEK 5 x +Z= 4 x2 +Z2 = 19 Z= x2 - 6x + 10 x +Z= 5 JLJODJEFSFDFEFOEFOLMFNTJTUFNJOFHÌSF YZÀBSQ- JLJODJ EFSFDFEFO EFOLMFN TJTUFNJOJO ÀÌ[ÑN LÑNF- NLBÀUS TJOJCVMVOV[ (x +Z 2 = 52 Z= 4 - x, 4 - x = x2- 6x + 10 , 0 = (x - 3) (x - 2) x2 +YZ+Z2 = 25 j YZ jYZ x = 3 jZ x = 2 jZ ÇK = {(3, 1), (2, 2)} 1. % a 5, 2 k, a 5, - 2 k, a - 5, 2 k, a - 5, - 2 k / 2. {(3, 1), (2, 2)} 2 3. {(5, –2)} 4. {(–13, 19), (1, 5)} 5. 3
www.aydinyayinlari.com.tr %&/,-&.7&&õ÷54÷;-÷,4÷45&.-&3÷ 4. MODÜL 11. SINIF ÖRNEK 6 ÖRNEK 9 x2 +Z2 = 20 x2 -Z2 = 5 YZ= 8 JLJODJ EFSFDFEFO EFOLMFN TJTUFNJOJO ÀÌ[ÑN LÑNF- YZ= 6 TJOJCVMVOV[ JLJODJ EFSFDFEFO EFOLMFN TJTUFNJOJO ÀÌ[ÑNLÑNF- TJOJCVMVOV[ 8 2 8 2 2 64 y= 6 & x2 -d 6 2 = 5 & x2 - 36 =5 y= x x x x x n x n 2 j + d = 20 & + x2 = 20 x4 - 5x2 - 36 = 0 x x4 + 64 20x2 j x4 - 20x2 + 64 = 0 (x2 - 9).(x2 + 4) = 0 = x2 x2 x = 3, x = -3, x =J Y= -J (x2 - 16).(x2- 4) = 0 j (x - 4).(x + 4).(x - 2).(x + 2) = 0 Z= Z= - Z= -J Z=J x = Z= 2; x = - Z= -2 ; x = Z= 4 ; x = - Z= -4 ÇK = { ( 3, 2 ), (-3, - J -J -J J } ÇK = {(4, 2), (-4, -2), (2, 4), (-2, -4)} ÖRNEK 7 ÖRNEK 10 x2 +Z2 = 8 x2 +YZ+Z2 = 7 x +Z2 = 6 x +Z= 3 JLJODJ EFSFDFEFO EFOLMFN TJTUFNJOJO ÀÌ[ÑN LÑNF- TJOJCVMVOV[ JLJODJEFSFDFEFOEFOLMFNTJTUFNJOJOHFSÀFLTBZMBS- EBLJÀÌ[ÑNLÑNFTJOJCVMVOV[ Z2 = 6 - x j x2 - x - 2 = 0 (x - 2) (x + 1) = 0 j x = 2, x = -1 Z= 3 - x j x2 + x (3 - x) + (3 - x)2 = 7 x = 2 jZ= Z= -2 j x = -1 j y = 7 , y = - 7 ÇK = {(2, 2), (2, -2), (-1, 7 ), (-1, - 7 )} 22 2 x + 3x - x + 9 - 6x + x = 7 x2 - 3x + 2 = 0 (x - 2) (x - 1) = 0 x = Z=WY= Z= 2 ÇK = {(2, 1), (1, 2)} ÖRNEK 8 ÖRNEK 11 x2 -YZ= 2 YWFZHFSÀFLTBZMBSPMNBLÑ[FSF Z2 -YZ= 7 x2+Z2 - 10x +Z+ 41 = 0 EFOLMFNTJTUFNJOJTBôMBZBOYWFZEFôFSMFSJJÀJO EFOLMFNJOJTBôMBZBOYWFZEFôFSMFSJJÀJOY+ZUPQ- x -ZOJOQP[JUJGEFôFSJLBÀUS MBNLBÀUS x2 -YZ= 2 x2 - 10x + 25 + Z2 +Z+ 16) = 0 Z2 -YZ= 7 (x - 5)2 + Z+ 4)2 = 0 + x = Z= -4 x2 -YZ+Z2 = 9 j (x -Z 2 = 9 j x -Z= 3 2x +Z= 2.5 - 4 = 6 6. {(4, 2), (–4, –2), (2, 4), (–2, –4)} 3 9. \\ m m J mJ mJ J ^ 10. {(2, 1), (1, 2)} 11. 6 7. {(2, 2), (2, –2), (–1, 7 ), (–1, – 7 )} 8. 3
11. SINIF 4. MODÜL %&/,-&.7&&õ÷54÷;-÷,4÷45&.-&3÷ www.aydinyayinlari.com.tr ÖRNEK 12 ÖRNEK 15 x2 +Z2 +Z- 9 = 0 ¥FWSFTJ DN PMBO EJLEËSUHFO õFLMJOEFLJ CJS MFWIBOO x2 +Z+ 1 = 0 LËõFMFSJOEFO CJS LFOBS V[VOMVóV DN PMBO LBSFMFS LF- TJMJQBUMZPS,BMBOQBS¿BMBSLBUMBOBSBLÐTUÐB¿LCJSEJL- JLJODJEFSFDFEFOEFOLMFNTJTUFNJOJOHFSÀFLTBZMBS- EËSUHFOMFSQSJ[NBTPMVõUVSVMVZPS EBLJÀÌ[ÑNLÑNFTJOJCVMVOV[ 0MVöBOQSJ[NBOOUBCBOBMBODNPMEVôVOBHÌSF x2 +Z2 +Z- 9 = 0 EJLEÌSUHFOMFWIBOOLFOBSV[VOMVLMBSLBÀDNEJS – x2 +Z+ 1 = 0 Z2 +Z- 10 = 0 2 y–4 2 2(x +Z = 40 x x–4 Taban 2 Z+ Z- 2) = 0 jZ= Z= -5 jY Z x2 + 5 = 0, x2 = -5, Ç = q 2 2 x2 - 9 = 0 , x =WY= -3 j ÇK = {(3, -5), (-3, -5)} y 2 0MVöBOQSJ[NBOOUB- CBO BZSUMBS Y - 4) WF Z- PMVS (x - Z- 4) = 32 x +Z= 20 jZ= 20 - x (x - 4) (16 - x) = 32 j -x2 + 20x - 64 = 32 x2 - 20x + 96 = 0 ÖRNEK 13 (x - 8) . (x - 12) = 0, j x = 8, x = 12 \"TMJMF#BOVhOVOCPZMBSUPQMBNOO\"TMhOOCPZVOBPSB- Z= Z= 8 O \"TMhOOCPZVOVO#BOVhOVOCPZVOBPSBOOBFõJUUJS #VOBHÌSF \"TMhOOCPZVOVO#BOVhOVOCPZVOBPSB- OLBÀUS \"TMhOOCPZV= x ÖRNEK 16 #BOVhOVOCPZV=ZPMTVO 5BõZD FóSJTJ Z = x2 FóSJTJ PMBO ZPM Ð[FSJOEF IBSFLFU x+y x 2 FEFO \" BSBD JMF UBõZD EPóSVTV Z = 2x + EPóSVTV 22 2 Ð[FSJOEFIBSFLFUFEFO#BSB¿MBSIBSFLFUMFSJCPZVODB x = y & x = xy + y & x - xy - y = 0 LF[LBSõMBõNõUS d x 2 x -1=0 d x = t olsun. n y y = x2 y = 2x + 8 y y y n- \"BSBD U2 -U- 1 = 0 j D 1± 5 x x 1+ 5 t= , y > 0 PMEVôVOEBO y = 2 2 1, 2 ÖRNEK 14 x #JSEJLEËSUHFOJOCPZVJMFFOJBSBTOEBLJGBSLDNEJS O #VEJLEÌSUHFOJOBMBODN2JTFEJLEÌSUHFOJOÀFW- SFTJLBÀDNEJS #BSBD %JLEÌSUHFOJOFOJY CPZVZPMTVO ÷LJLBSöMBöNBBSBTV[BLMLLBÀCSEJS Z- x = 3 jYZ Z= x + 3 , x.(x + 3) = 88 j x2 + 3x - 88 = 0 Z= x2JMFZ= 2x +CJSCJSJOFFöJUMFOJS (x + 11) . (x - 8) = 0, x = -11, x = 8 x > 0 jY JÀJOZ x2 = 2x + 8 j x2 - 2x - 8 = 0 (x - 4) . (x + 2) = 0 ¦FWSF= 2.(8 + 11) = 38 x =JÀJOZ= 16 x = 4 x = -2 x= -JÀJOZ= JMF - JÀJ ^ 4 + 2 h2 + ^ 16 - 4 h2 = 6 5 12. { ( 3, –5 ), ( –3, – 5 ) } 13. ^ 1 + 5 h / 2 14. 38 4 15. WF12 16. 6 5
www.aydinyayinlari.com.tr %&/,-&.7&&õ÷54÷;-÷,4÷45&.-&3÷ 4. MODÜL 11. SINIF ÖRNEK 17 ÖRNEK 20 Z= x2 + x - 1 öLJODJEFSFDFEFOEFOLMFNTJTUFNJOJO¿Ë[ÐNÐJ¿JO¿J[JMFO Z= -2x2 + 7x - 4 HSBGJLBõBóEBLJõFLJMEFWFSJMNJõUJS JLJODJ EFSFDFEFO EFOLMFN TJTUFNJOJO ÀÌ[ÑN LÑNF- y TJOJCVMVOV[ A O B x x2+ x - 1 = -2x2 + 7x - 4 –2 –2 5 3x2 - 6x - 3 = 0 3.(x - 1)2 = 0, x =JÀJOZ= 1 j ÇK = {(1, 1)} C –10 ÖRNEK 18 %FOLMFN TJTUFNJOJO ÀÌ[ÑN LÑNFTJ \" WF $ OPLUBMB- SPMEVôVOBHÌSF $OPLUBTOOLPPSEJOBUMBSUPQMBN Z= x2 - 4x + a LBÀUS Z= 2x - 3 EFOLMFNTJTUFNJOJOHFS¿FLTBZMBSEBLJ¿Ë[ÐNLÐNFTJJMF 1BSBCPMEFOLMFNJZ= (x + 2) (x - 5) JMHJMJ Z= x2- 3x - 10 %PôSVEFOLMFNJZ= -x - 2 * B`C-RF JTFTJTUFNJO¿Ë[ÐNLÐNFTJJLJFMFNBO- x2 - 3x - 10 = -x - 2 j x2 - 2x - 8 = 0 MES (x - 4) . (x + 2) = 0, x = 4, x = -2 \" - WF$ -6), 4 + (-6) = -2 ** B` ( 6, R JTFTJTUFNJO¿Ë[ÐNLÐNFTJTPOTV[FMF- NBOMES ÖRNEK 21 *** B=JTFTJTUFNJO¿Ë[ÐNLÐNFTJ{( 3, 3 )}UÐS BWFCSFFMTBZMBSPMNBLÐ[FSF JGBEFMFSJOEFOIBOHJMFSJEPôSVEVS Z= 2x2+ x + a Z= x2+ 5x +C x2 - 4x +B= 2x - 3 j x2 - 6x +B+ 3 = 0 EFOLMFN TJTUFNJOJO HFSÀFL TBZMBSEBLJ ÀÌ[ÑN LÑ- D = 36 - B+ 3) = 24 -B NFTJZMFJMHJMJ *B<JTFD > ÀÌ[ÑNLÑNFTJJLJFMFNBOMES * B<CJTF¿Ë[ÐNLÐNFTJJLJFMFNBOMES **B>JTFD < ÀÌ[ÑNLÑNFTJCPöLÑNFEJS ** C+ 4 <BJTF¿Ë[ÐNLÐNFTJCPõLÐNFEJS ***B=JTFD = 0 (x2 - 6x - 9) = 0 j (x - 3)2 = 0 *** B=CJTF¿Ë[ÐNLÐNFTJUFLFMFNBOMES x =JTFZ= 3 j ÇK = {(3, 3)} $FWBQ *WF*** JGBEFMFSJOEFOIBOHJMFSJEPôSVEVS ÖRNEK 19 2x2+ x +B= x2+ 5x +C x2- 4x + B-C = 0 a `3PMNBLÐ[FSF D = 16 - B-C Z= x2 - 5x + 5 * B<CJTF>PMEVôVOEBOÀÌ[ÑNLÑNFTJFMF- Z= -x2 + x + a EFOLMFN TJTUFNJOJO HFSÀFL TBZMBSEBLJ ÀÌ[ÑN LÑ- NBOMES NFTJCJSFMFNBOMPMEVôVOBHÌSF BLBÀUS ** C+ 4 <BJTFD <PMEVôVOEBOÀÌ[ÑNLÑNFTJ x2 - 5x + 5 = x2 + x +B CPöLÑNFEJS 2x2 - 6x + 5 -B= 0 *** B=Cj D =PMBDBôOEBOÀÌ[ÑNLÑNFTJJLJFMF- 1 NBOMES D = 0 j 36 - 4.2.(5 -B = 0 j a = 2 17. {(1, 1)} 1 5 20. –2 21. *WF** 18. *WF***19. 2
TEST - 1 ÷LJODJ%FSFDFEFO÷LJ#JMJONFZFOMJ%FOLMFN4JTUFNMFSJ 1. 2x2 +Z2 = 17 5. Z- x = 3 x2 -Z2 = -5 Z2 - x2 = 7 EFOLMFNTJTUFNJOJTBôMBZBOYEFôFSMFSJOJOÀÌ- JLJODJEFSFDFEFOEFOLMFNTJTUFNJOJTBôMBZBOZ [ÑNLÑNFTJBöBôEBLJMFSEFOIBOHJTJEJS EFôFSMFSJOJOUPQMBNLBÀUS A) { -^ # \\-2 } C) { -3, -2 } A) - # -6 C) 6 D) 10 E) 12 D) { 2 } E) { -2, 2 } 2. x +Z= 3 6. x2 +Z2 = 26 Z= x2 + x - 12 x -Z= 6 JLJODJEFSFDFEFOEFOLMFNTJTUFNJOJTBôMBZBOY EFOLMFN TJTUFNJOJ TBôMBZBO Y WF Z EFôFSMFSJ WFZEFôFSMFSJJÀJOYZJGBEFTJOJOBMBCJMFDFôJFO JÀJOYZÀBSQNLBÀUS LÑÀÑLEFôFSLBÀUS A) - # -3 C) 3 D) 5 E) 10 A) - # -40 C) -24 D) -15 E) -10 7. x2 -YZ= 72 YZ+Z2 = 7 3. x +Z= 7 JLJODJEFSFDFEFOEFOLMFNTJTUFNJOJTBôMBZBOY WFZEFôFSMFSJJÀJOY-ZOJOEFôFSJBöBôEBLJ- x2 -Z2 = 21 MFSEFOIBOHJTJPMBCJMJS JLJODJEFSFDFEFOEFOLMFNTJTUFNJOJTBôMBZBOY A) - # -8 C) 0 D) 6 E)8 WFZEFôFSMFSJJÀJOY-ZLBÀUS \" # $ % & 8. x2 +Z2 = 10 4. 4x2 -Z2 = 91 YZ= 3 2x -Z= 7 EFOLMFN TJTUFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJ- MFSEFOIBOHJTJEJS EFOLMFN TJTUFNJOJ TBôMBZBO Y WF Z EFôFSMFSJ A) { ( 3, 1 )} # { ( -3, -1 ), ( 3, 1 ) } JÀJO x PSBOLBÀUS C) { ( -3, -17 } D) { ( 3, 1 ), ( 3, -1 ) } y A) - # - 1 C) -1 1 E) 5 E) { ( 3, 1 ), ( -3, -1 ), ( 1, 3 ), ( -1, -3 ) } 5 D) 5 1. E 2. # 3. # 4. E 6 5. # 6. \" 7. \" 8. E
÷LJODJ%FSFDFEFO÷LJ#JMJONFZFOMJ%FOLMFN4JTUFNMFSJ TEST - 2 1. x2 +Z2 = 6 5. x2 +YZ+Z2 = 7 Z2 - 2x = 3 2x +Z= 5 ÷LJODJ EFSFDFEFO EFOLMFN TJTUFNJOJO HFSÀFL JLJODJ EFSFDFEFO EFOLMFN TJTUFNJOJ TBôMBZBO TBZMBSEBLJÀÌ[ÑNLÑNFTJLBÀFMFNBOMES FOLÑÀÑLYEFôFSJWFFOCÑZÑLZEFôFSJJÀJO x -ZGBSLLBÀUS \" # $ % & A) - # $ % & 2. x2 -YZ+Z2 = 17 6. YWFZHFSÀFLTBZMBSPMNBLÑ[FSF Z2 -YZ= 8 x2 +Z2 - 4x +Z+ 13 = 0 ÷LJODJEFSFDFEFOEFOLMFNTJTUFNJOJTBôMBZBOY PMEVôVOBHÌSF YZÀBSQNLBÀUS WFZEFôFSJJÀJOY-ZJGBEFTJOJOQP[JUJGEFôFSJ A) - # -3 C) -2 D) 3 LBÀUS \" # $ % & E) 6 3. YWFZHFSÀFLTBZMBSPMNBLÑ[FSF x2 -Z= 0 Z- | x - 6 | = 0 7. YWFZHFSÀFLTBZMBSPMNBLÑ[FSF EFOLMFNTJTUFNJOJTBôMBZBOYEFôFSMFSJOJOÀBS- x2 -YZ= 43 QNLBÀUS Z2- 21 =ZY A) - # -6 C) 0 D) 6 E) 8 PMEVôVOBHÌSF Z-YJGBEFTJOJOEFôFSJBöBôEB- LJMFSEFOIBOHJTJPMBCJMJS A) - # -6 C) 4 D) 6 E) 16 4. x2 -Z2 = 15 8. 2x2 +Z2 -Z+ 2 = 0 YZ= -4 x2 +Z+ 4 = 0 ÷LJODJ EFSFDFEFO EFOLMFN TJTUFNJOJO ÀÌ[ÑN ÷LJODJ EFSFDFEFO EFOLMFN TJTUFNJOJO HFSÀFL LÑNFTJBöBôEBLJMFSEFOIBOHJTJEJS TBZMBSEBLJÀÌ[ÑNLÑNFTJLBÀFMFNBOMES A) { (4, 1 ) } \" # $ % & # { ( 4, -1 ), ( -4, 1 ) } C) { ( -4, 1 ) } D) { ( -4, 1 ), ( 4, - J J -J -J } E) { ( -4, 1 ), ( 4, - J -J -J J } 1. $ 2. D 3. # 4. D 7 5. $ 6. \" 7. \" 8. \"
TEST - 3 ÷LJODJ%FSFDFEFO÷LJ#JMJONFZFOMJ%FOLMFN4JTUFNMFSJ 1. 2a2 +BC-C2 = 28 3. Z= x2 - x + 5 2a -C= 4 Z= 3x + a EFOLMFN TJTUFNJOJ TBôMBZBO B WF C EFôFSMFSJ ÷LJODJ EFSFDFEFO EFOLMFN TJTUFNJOJO ÀÌ[ÑN JÀJOB+CUPQMBNLBÀUS LÑNFTJCJSFMFNBOMPMEVôVOBHÌSF BLBÀUS \" # $ % & A) - # $ % & 2. Z= x2 - x - 6 4. Z= x2 + 2x - 3 Z= x - 2 Z= 4x +D %FOLMFN TJTUFNJOJO ÀÌ[ÑN LÑNFTJOJ CVMNBL ÷LJODJ EFSFDFEFO EFOLMFN TJTUFNJOJO HFSÀFL TBZMBSEBLJÀÌ[ÑNLÑNFTJFMFNBOMPMEVôVOB JTUFZFOCJSÌôSFODJBöBôEBLJHSBGJLMFSJOIBOHJ- HÌSF DOJOBMBCJMFDFôJFOLÑÀÑLUBNTBZEFôF- SJLBÀUS TJOJLVMMBONBMES A) - # -4 C) -3 D) -2 E) -1 A) y B) y 5 –2 O 3x O 2 x –1 –3 –6 –6 C) y D) y O 2 O x –2 3 23 x –3 –2 –2 5. Z= x2 +NY+ 1 –6 –6 Z= 4x - 8 E) y O 2 x ÷LJODJ EFSFDFEFO EFOLMFN TJTUFNJOJO HFSÀFL –2 3 TBZMBSEBLJÀÌ[ÑNLÑNFTJCPöLÑNFPMEVôVOB HÌSF N OJO CVMVOEVôV BSBML BöBôEBLJMFSEFO IBOHJTJEJS –6 A) ( - # -1, 11 ) C) ( 2, 10 ) D) ( -10, 2 ) E) ( -2, 10 ) 1. # 2. $ 8 3. $ 4. $ 5. E
÷LJODJ%FSFDFEFO÷LJ#JMJONFZFOMJ%FOLMFN4JTUFNMFSJ TEST - 4 1. \"OBMJUJLEÑ[MFNEF 4. \"õBóEBLJõFLJMEFYFLTFOJOJ\" - WF# y = x2 WFZ2 = x2 OPLUBMBSOEBLFTFOQBSBCPMWFEEPóSVTVOVOHSBGJ- 3 óJWFSJMNJõUJS EFOLMFN TJTUFNJOJ TBôMBZBO Y Z JLJMJMFSJOJO y 8 CFMJSUUJôJ OPLUBMBS LÌöF LBCVM FEFO ÑÀHFOJO BMBOLBÀCS2EJS C 4 \" # $ % & A B x –1 O 4 d: y + 2x – 8 = 0 1BSBCPMJMFEPôSVOVOLFTJNOPLUBMBS#WF$PM- EVôVOBHÌSF A^ & hLBÀCS2EJS ABC \" # $ % & 2. \"OBMJUJLEÑ[MFNEF Z= x +EPóSVTVJMFZ= x2 - 5x + 8 QBSBCPMÑOÑOLFTJNOPLUBMBSBSBTOEBLJV[BL- 5. Z= -x2 + 2x MLLBÀCSEJS Z= x2 - 4x - 8 A) 2 2 # $ 4 2 QBSBCPMMFSJOJO LFTJN OPLUBMBSOO PSEJOBUMBS D) 4 5 E) 8 UPQMBNLBÀUS A) - # -11 C) -8 D) -3 E) 0 3. 6. a `3PMNBLÐ[FSF y Z= x2 - 3x + a 5 Z= x + 1 A x JLJODJEFSFDFEFOEFOLMFNTJTUFNJZMFJMHJMJ 5/2 * B JTFEFOLMFNJO¿Ë[ÐNLÐNFTJJLJFMFNBO- MES O ** B JTFEFOLMFNJO¿Ë[ÐNLÐNFTJUFLFMFNBO- MES Z måBY2 y = –2x + 5 *** a =JTFQBSBCPMJMFEPóSV OPLUBTOEB \"OBMJUJLEÐ[MFNEFZ= 4 - ax2QBSBCPMÐJMF UFóFUUJS Z= -2x +EPóSVTVOVOHSBGJóJ¿J[JMNJõUJS %PôSVJMFQBSBCPM\"OPLUBTOEBUFôFUPMEVôVOB JGBEFMFSJOEFOIBOHJMFSJEPôSVEVS HÌSF BEFôFSJLBÀUS A) 1 # 2 \" :BMO[* # :BMO[** $ *WF** 3 3 C) 1 D) 2 E) 3 % *WF*** & * **WF*** 1. # 2. $ 3. $ 9 4. # 5. # 6. \"
11. SINIF 4. MODÜL %&/,-&.7&&õ÷54÷;-÷,4÷45&.-&3÷ www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/#÷3#÷-÷/.&:&/-÷&õ÷54÷;-÷,-&3* ÷MJöLJMJ,B[BONMBS 11.4.2.1 : öLJODJEFSFDFEFOCJSCJMJONFZFOMJFõJUTJ[MJLMFSJO¿Ë[ÐNLÐNFTJOJCVMVS ÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ ÖRNEK 3 &öJUTJ[MJLMFSJO¦Ì[ÑN,ÑNFTJ x2 - x > 0 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJJöBSFUUBC- TANIM MPTVPMVöUVSBSBLCVMVOV[ B C D`3WFBáPMNBLÐ[FSF x2 - x = 0 j x.(x - ax2 +CY+D> 0 , ax2+CY+D< 0 x = 0 , x = B> 0 ax2 +CY+D$ 0 , ax2 +CY+D# 0 x –ß 0 1 +ß õFLMJOEFLJ JGBEFMFSF JLJODJ EFSFDFEFO CJS CJ- MJONFZFOMJFöJUTJ[MJLEFOJS x2 – x + – + #V FõJUTJ[MJLMFSJO ¿Ë[ÐN LÐNFTJ CVMVOVSLFO ÇK= mß b ß EFOLMFNJOJõBSFUUBCMPTVPMVõUVSVMVS %m/*m ÖRNEK 4 öLJODJEFSFDFEFOCJSCJMJONFZFOMJEFOLMFNMFSEF -x2 + x + 6 # 0 D >JTFGBSLMJLJHFS¿FLLËLWBSES FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ (x1 < x2 FõJUTJ[MJóJOJOJõBSFUUBCMPTVBõBóEBLJ HJCJPMVõUVSVMVS x x1 x2 BY2 + bx + c BJMF BJMF BJMF -x2 + x + 6 = 0 j -(x2 - x - 6) = 0, - (x - 3).(x + 2) = 0 BZO UFST BZO x = 3, x = - B< 0 JõBSFUMJ JõBSFUMJ JõBSFUMJ x –ß –2 3 +ß ÖRNEK 1 –x2 + x + 6 – + – x2 - 4x + 3JGBEFTJOJOJöBSFUUBCMPTVOVPMVöUVSVOV[ ÇK= (-ß -2] b [ ß x2 - 4x + 3 = 0 (x - 3) . (x - 1) = 0 x = 3 , x = B> 0 x 13 x2 – 4x + 3 + –+ ÖRNEK 2 %m/*m -x2 + 7x - 12 JGBEFTJOJO JöBSFU UBCMPTVOV PMVöUV- öLJODJEFSFDFEFOCJSCJMJONFZFOMJEFOLMFNMFSEF SVOV[ D =JTFEFOLMFNJOFõJUJLJHFS¿FLLËLÐWBSES -x2 + 7x - 12 = -(x - 4) . (x - 3) &õJUTJ[MJóJOJõBSFUUBCMPTVBõBóEBLJHJCJPMVõUV- x = 4 , x = B< 0 SVMVS x 34 x x1 = x2 BY2 + bx + c BJMF BJMF BZO BZO JõBSFUMJ JõBSFUMJ –x2 + 7x – 12 – +– 10 3. mß b ß 4. mß m>b< ß
www.aydinyayinlari.com.tr %&/,-&.7&&õ÷54÷;-÷,4÷45&.-&3÷ 4. MODÜL 11. SINIF ÖRNEK 5 ÖRNEK 8 x2 - 4x + 4 x2 - x + 4 # 0 JGBEFTJOJOJöBSFUUBCMPTVOVPMVöUVSVOV[ FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ x2 - 4x + 4 = 0, (x - 2)2 = 0, x1 = x2= B> 0 x2 - x + 4 = 0 , D = 1 - 16 = -15 x2 x mß ß x2 – x + 4 + + + + + + + + ÇK = q x2 – 4x + 4 + + ÖRNEK 6 ÖRNEK 9 4x2 - 12x + 9 > 0 x2 +NY+ 9 > 0 FöJUTJ[MJôJrx `3JÀJOTBôMBOEôOB FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ HÌSF NEFôFSMFSJOJOLÑNFTJOJCVMVOV[ rx`3JÀJOEPôSVJTFD < N2 - 36 < 0 N- N+ 6) < 0 4x2 - 12x + 9 = 0, (2x - 3)2 = 0, x1 = x2= 3 m –6 6 N` (-6, 6) B> 0 m2 – 36 + – + 2 x 3/2 4x2 – 12x + 9 + + ÇK =3- ( 3 2 = d - 3 , 3 n , d 3 , 3 n ÖRNEK 10 2 22 ,BSFTJ LFOEJTJOJO LBUOO GB[MBTOEBO LÑÀÑL PMBOUBNTBZMBSOUPQMBNLBÀUS %m/*m x2 < 3x + 10 (x - 5) . (x + 2) < 0 x2 - 3x - 10 < 0 –2 5 öLJODJEFSFDFEFOCJSCJMJONFZFOMJEFOLMFNMFSEF D <JTFHFS¿FLLËLZPLUVS x +– + &õJUTJ[MJóJOJõBSFUUBCMPTVBõBóEBLJHJCJPMVõUV- x2 – 3x – 10 SVMVS -1 + (0) + (1) + 2 + 3 + 4 = 9 x mÞ Þ BY2 + bx + c BOOJõBSFUJJMFBZO ÖRNEK 11 x 2 –1 x–7 f 2 p >f 9 p 34 ÖRNEK 7 FöJUTJ[MJôJOJTBôMBZBOYUBNTBZEFôFSMFSJLBÀUBOFEJS x2 - 3x + 5 3 1– x 2 2x – 14 n d >d 3 n 1 - x2 > 2x - 14 , JGBEFTJOJOJöBSFUUBCMPTVOVPMVöUVSVOV[ 22 x2 - 3x + 5 = 0 , D = 9 - 4.5 = -11 < B> 0 0 > x2 + 2x - 15 , 0 > (x + 5) (x - 3) x mß ß x –5 3 +– + x2 – 3x + 5 + + + + + + + + {–4, –3, ..., 2} , 7 tane 6. R - ( 3 2 11 8. q 9. (–6, 6) 10. 9 11. 7 2
11. SINIF 4. MODÜL %&/,-&.7&&õ÷54÷;-÷,4÷45&.-&3÷ www.aydinyayinlari.com.tr ÖRNEK 12 ÖRNEK 15 a `3PMNBLÐ[FSF BY2 - ( a + 2 ) x + 2 = 0 .BMJZFUGJZBU\"5-PMBOCJSÐSÐOÐO 4PMBOTBUõGJZBUJ¿JO ËOFSJMFOJLJEVSVNBõBóEBLJHJCJEJS EFOLMFNJOJOJLJGBSLMHFSÀFLLÌLÑPMEVôVOBHÌSF B *EVSVN4= A2 OOFOHFOJöÀÌ[ÑNLÑNFTJOJCVMVOV[ **EVSVN4= 13A - 22 D > B+ 2)2 -B> B2 -B+ 4 > 0 ** EVSVNVO * EVSVNEBO EBIB L»SM PMEVôV CJMJOEJ- B- 2)2 > Bâ ôJOF HÌSF \" ZFSJOF HFMFCJMFDFL UBN TBZMBSO UPQMB- NLBÀUS a2 3- {0, 2} ++ \"- 22 -\">\"2 -\"PMNBMES \"2 -\"+ 22 < \"- \"- 11) < 0 A 2 11 +– + 3 + 4 + 5 + ... + 10 = 52 ÖRNEK 13 ÖRNEK 16 G Y CJSGPOLTJZPOPMNBLÐ[FSF %JLEËSUHFO CJ¿JNJOEF CJS NBTB ZBQNBL JTUFZFO ,B[N 6TUBhOOFMJOEFNBTBOOÐTULTNJ¿JOHFSFLMJUBIUBMFW- f (x) = - x2 + 2x + 15 IBEBOCS2EFOB[CVMVONBLUBES GPOLTJZPOVOVOFOHFOJöUBONLÑNFTJOJCVMVOV[ ·TULTN 2n g^ x hUBONLÑNFTJH Y $ TBôMBZBOEFôFSMFSEJS -x2 + 2x + 15 $ 0, -(x - 5) . (x + 3) $ 0 x –3 5 –x2 + 2x + 15 – + – &OHFOJöUBONLÑNFTJ[-3, 5]EJS ÖRNEK 14 :BQNBTHFSFLFONBTBOOCJSLFOBSV[VOMVóV B- CS JLFOEJóFSLFOBSCVLFOBSEBOCSEBIBV[VOEVS #VOB HÌSF ZBQMBDBL NBTBOO ÀFWSF V[VOMVôVOVO BMBCJMFDFôJNBLTJNVNUBNTBZEFôFSJLBÀUS G Y CJSGPOLTJZPOPMNBLÐ[FSF B- B+ 1) < B- 3 > B> 3 B2 -B- 3 < B+ 1 > B> -1 f^ x h = 7 B- B+ 6) < 0 8 2x2 - ax + 2 a –6 + GPOLTJZPOVOVOFOHFOJöUBONLÑNFTJHFSÀFLTBZ- MBSPMEVôVOBHÌSF BOOEFôFSBSBMôOCVMVOV[ a2 – 2a – 48 + – 2x2 -BY+â D <PMNBMES B- 3 >PMEVôVOEBO<B< 8 B2- 16 < B- B+ 4) < 0 .BTBOOÀFWSFTJ[ B+ 3) + B+ 1)] =B- 4 a –4 4 4 ( 3 <B< 8) j 12 <B< 32 a2 – 16 + – + 8 <B- 4 < 28 B- 4 jÀFWSFV[VOMVôVCSEJS B` (-4, 4) 12. 3m\\ ^13. <m > 14. (–4, 4) 12 15. 52 18. 27
÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ&öJUTJ[MJLMFSJO¦Ì[ÑN,ÑNFTJ TEST - 5 1. x2 - 5x - 24 < 0 5. x2 - 7x + 10 $ 0 FöJUTJ[MJôJOJTBôMBZBOYUBNTBZMBSOOUPQMBN FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO LBÀUS IBOHJTJEJS \" # $ % & A) ( -Þ b Þ # -Þ b Þ C) R - [ 2, 5 ] D) (-Þ ] b [ Þ E) [ 2, 5 ] 2. x2 - 2x - 15 # 0 FöJUTJ[MJôJOJTBôMBZBOFOCÑZÑLUBNTBZEFôFSJ LBÀUS \" # $ % & 6. x2 - 9x + 18 # 0 FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO IBOHJTJEJS A) R - # [ 3, 6 ] C) ( 2, 9 ) D) R - [ 3, 6 ] E) ( 3, 6 ) 3. 2x2 - 3x + 2 < 0 FöJUTJ[MJôJOJTBôMBZBOYEFôFSMFSJOJOÀÌ[ÑNLÑ- NFTJLBÀFMFNBOMES \" # $ % & 4. 2x2 - 6x + 9 # 0 7. -x2 + x + 12 # 0 2 FöJUTJ[MJôJOJ TBôMBZBO JLJ OFHBUJG UBN TBZOO FöJUTJ[MJôJOJTBôMBZBOYEFôFSMFSJOJOÀÌ[ÑNLÑ- UPQMBNFOÀPLLBÀUS A) - # -7 C) -8 D) -9 E) -10 NFTJLBÀFMFNBOMES \" # $ % & 1. $ 2. \" 3. \" 4. # 13 5. D 6. # 7. #
TEST - 6 ÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ&öJUTJ[MJLMFSJO¦Ì[ÑN,ÑNFTJ 1. ( x - Y> x - 1 4. 3x2 + 5x < f 1 3 9 p FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO IBOHJTJEJS A) R- $ 3+ IBOHJTJEJS # Þ A) ( - # -3, 2 ) C) ( -3, -2 ) D) R - { 1 } E) R D) ( -6, 5 ) E) ( -1, 5 ) 2. ,BSFTJOJOLBUOOGB[MBTLFOEJTJOJOLBUO- 5. x2 - N+ 3 ) x +N# 0 EBOLÑÀÑLPMBOUBNTBZMBSOUPQMBNLBÀUS FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOEFUBOFUBNTB- ZPMEVôVOBHÌSF NOJOBMBCJMFDFôJUBNTBZEF- \" # $ % & ôFSMFSJOJOUPQMBNLBÀUS A) - # $ % & 3. 6. x2 + 2x +N- 3 > 0 x –6 2 + – + ÇK = (–6, 2) :VLBSEB ÀÌ[ÑN LÑNFTJ WFSJMFO JLJODJ EFSFDF- FöJUTJ[MJôJr x `3JÀJOTBôMBOZPSTBNOJOEFôFS EFOFöJUTJ[MJLBöBôEBLJMFSEFOIBOHJTJPMBCJMJS BSBMôBöBôEBLJMFSEFOIBOHJTJEJS A) x2 - 4x + 12 > # Y2 - 4x + 12 < 0 \" N> # N< 4 C) -2 <N< 4 C) x2 + 4x - 12 < 0 D) x2 - 4x - 12 > 0 D) -4 <N< & N> 5 E) x2 - 4x - 12 < 0 1. D 2. E 3. $ 14 4. $ 5. D 6. \"
÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ&öJUTJ[MJLMFSJO¦Ì[ÑN,ÑNFTJ TEST - 7 1. x2 - 2ax + a > 0 4. G Y CJSGPOLTJZPOPMNBLÐ[FSF FöJUTJ[MJôJr x `3JÀJOTBôMBOEôOBHÌSF BOO f_ x i = - x2 + 13x - 30 EFôFSBSBMôBöBôEBLJMFSEFOIBOHJTJEJS GPOLTJZPOVOVOFOHFOJöUBONLÑNFTJOEFLBÀ A) ( -Þ # -Þ $ UBNTBZWBSES % Þ & -2, 3 ) \" # $ % & 2. ax2 + ( 4 - 2a ) x + a + 5 < 8 5. G Y CJSGPOLTJZPOPMNBLÐ[FSF f^ x h = 2x + 5 FöJUTJ[MJôJr x `3JÀJOTBôMBOEôOBHÌSF BOO x2 - ax + 2x + 1 EFôFSBSBMôBöBôEBLJMFSEFOIBOHJTJEJS GPOLTJZPOVr x `3JÀJOUBONMPMEVôVOBHÌSF A) ( -Þ # Þ $ -Þ BOOBMBCJMFDFôJUBNTBZEFôFSMFSJUPQMBNLBÀ- % Þ & q US \" # $ % & 6. G Y CJSGPOLTJZPOPMNBLÐ[FSF f^ x h = x-3 3. f 5 2 6 x+2 x2 + ^ a - 1 hx + a + 2 –x p p #f 65 GPOLTJZPOVYJOJLJGBSLMEFôFSJJÀJOUBONT[PM- FöJUTJ[MJôJOJTBôMBZBOYUBNTBZMBSOOUPQMBN EVôVOBHÌSF BOOFOLÑÀÑLJLJQP[JUJGUBNTBZ EFôFSJOJOUPQMBNLBÀUS LBÀUS A) - # -1 C) 2 D) 3 E) 5 \" # $ % & 1. $ 2. E 3. $ 15 4. $ 5. D 6. D
TEST - 8 ÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ&öJUTJ[MJLMFSJO¦Ì[ÑN,ÑNFTJ 1. NCJSSFFMTBZPMNBLÑ[FSF 4. ax2 + 2ax - 2x2 - 5x + a > 0 -x2 + 3x +N FöJUTJ[MJôJ rx `3 JÀJO TBôMBOEôOB HÌSF B OO BMBCJMFDFôJFOLÑÀÑLUBNTBZEFôFSJLBÀUS JGBEFTJr x `3JÀJOUFOLÑÀÑLPMEVôVOBHÌ- SF NOJOÀÌ[ÑNBSBMôBöBôEBLJMFSEFOIBOHJ- A) - # $ % & TJEJS \" N< # m < 3 $ N> 0 4 E) m > 2 % N> 3 3 2. .FSU Y+ 5-LBSõMóOEBBMEóCJSNBM 5. NCJSSFFMTBZPMNBLÐ[FSF ( x2 - 3x + 5-ZFTBUZPS Z= 2x2 - N- 2 ) x + 21 #V BMöWFSJöUFO .FSUhJO L»S ZBQUô CJMJOEJôJOF QBSBCPMÑOÑO HSBGJôJ Z = EPôSVTVOVO EBJNB HÌSF YZFSJOFZB[MBCJMFDFLFOLÑÀÑLJLJQP[JUJG ÑTUÑOEFPMEVôVOBHÌSF NOJOFOCÑZÑLUBNTB- UBNTBZOOUPQMBNLBÀUS ZEFôFSJLBÀUS \" # $ % & \" # $ % & 3. (ÐOEFBUBOFÐSFUJMFOCJSÐSÐOÐOUBOFTJOJONBMJZFUJ 6. #JSTOGUBCVMVOBOFSLFLËóSFODJTBZTL[ËóSFODJ ( 3a + 5-WFUBOFTJOJOTBUõGJZBU B2 + 5- TBZTOEBOGB[MBES#JSËóSFUNFOTOGUBLJIFSCJS L[ËóSFODJZFFSLFLËóSFODJTBZTOOGB[MBTLB- EJS EBS IFSCJSFSLFLËóSFODJZFJTFL[ËóSFODJTBZT- OOLBULBEBSLBMFNEBóUZPS #VTBUöUBOL»SFMEFFEJMFCJMNFTJJÀJOFOB[LBÀ UBOFÑSÑOÑSFUJMNFTJHFSFLJS &SLFL ÌôSFODJMFSJO BMEô UPQMBN LBMFN TBZT L[ÌôSFODJMFSJOBMEôUPQMBNLBMFNTBZTOEBO \" # $ % & GB[MBPMEVôVOBHÌSF EBôUMBOUPQMBNLBMFNTB- ZTFOB[LBÀUS \" # $ % & 1. # 2. # 3. # 16 4. D 5. # 6. D
www.aydinyayinlari.com.tr %&/,-&.7&&õ÷54÷;-÷,4÷45&.-&3÷ 4. MODÜL 11. SINIF ÷,÷/$÷%&3&$&%&/#÷3#÷-÷/.&:&/-÷&õ÷54÷;-÷,-&3** ÷MJöLJMJ,B[BONMBS 11.4.2.1 : öLJODJEFSFDFEFOCJSCJMJONFZFOMJFõJUTJ[MJLMFSJO¿Ë[ÐNLÐNFTJOJCVMVS Çarpım Şeklindeki Eşitsizliklerin ÖRNEK 3 Çözüm Kümesi ( 2x - - x ) < 0 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ ÖRNEK 1 2x - 6 = 0 5-x=0 x=3 x=5 ( 2x - Y+ 2 ) $ 0 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ x 35 (2x–6).(5–x) –+– ¦Ì[ÑN 2x - 5 = 0 x+2=0 Ç.K = mß b ß x= 5 x = -2 2 –2 5/2 ÖRNEK 4 x +–+ ( x - Y2 - 4x - 5 ) # 0 (2x – 5) . (x + 2) FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ Ç .K = ^ - 3, - 2@ , > 5 , 3 p 2 x-7=0 x2 - 4x - 5 = 0 x=7 (x - 5) (x + 1) = 0 x = 5 x = -1 x –1 5 7 (x–7) . (x2–4x–5) – +– + ÖRNEK 2 Ç.K = mß m] b [5, 7> (x2 - Y- 1) # 0 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ ¦Ì[ÑN 2x - 1 = 0 ÖRNEK 5 x2 - 1 = 0 x= 1 2 ( x - 2 )2 Y+ 3 ) > 0 (x - 1) (x + 1) = 0 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ x = 1 x = -1 (x - 2)2 = 0 x = ¦JGULBUM x –1 1/2 1 x+3=0 x = -3 (x2–1) . (2x – 1) – +– + x –3 2 (x – 2)2 . (x + 3) –+ + Ç .K = ^ - 3, - 1@ , > 1 , 1H Ç.K = m ß] – { 2 } 2 17 3. mß b ß 4. mß m>b< >5. m ß m\\^
11. SINIF 4. MODÜL ÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ&öJUTJ[MJLMFSJO¦Ì[ÑN,ÑNFTJ www.aydinyayinlari.com.tr ÖRNEK 6 ÖRNEK 9 ( x - x2 Y2 - 3x ) # 0 5'' TÐQFSMJH NB¿MBSOEB LVMMBOMBO ¿J[HJ UFLOPMPKJTJ J¿JO HFMJõUJSJMFOCJSCJMHJTBZBSQSPHSBNOEBUPQVOLBMF¿J[HJTJ- FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ OFPMBOV[BLMóIFTBQMBONBLUBES1SPHSBNEBUPQTBIB J¿JOEFCJSOPLUBEBJTFLBMF¿J[HJTJOFPMBOV[BLMóIFTBQ- x - x2 = 0 x=0,x=1 MBOS WF CV V[BLMó FLSBOB QP[JUJG PMBSBL ZBOTUMS 5PQ x2- 3x = 0 TBIBEõCJSOPLUBEBJTFLBMF¿J[HJTJOFPMBOV[BLMóIF- x=0,x=3 TBQMBOSWFFLSBOBOFHBUJGPMBSBLZBOTUMS5''4ÐQFS- x 01 3 MJHhEFPZOBOBO'FOFSCBI¿Fm#FõJLUBõNB¿OEB¿J[HJUFL- OPMPKJTJOFCBõWVSBOIBLFNFLSBOEB (x–x2) . (x2–3x) – –+– ( 2x - 5) (x - 6 ) Ç.K = (-ß, 1> b [ ß JGBEFTJOJHËSÐQHPMLBSBSWFSNJõUJS ÖRNEK 7 #VOB HÌSF Y JO BMBCJMFDFôJ UBN TBZ EFôFSMFSJOJO a <C< 0 <DPMNBLÐ[FSF UPQMBN LBÀUS (Gol için topun çizgiyi tamamen geç- BY CY - D < FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJOJ CVMV- mesi gerekmektedir.) OV[ 5PQ UBNBNFO ÀJ[HJZJ HFÀNJö WF HPM LBSBS WFSJMEJôJOF HÌSF FLSBOEBZB[BOJGBEFOFHBUJGPMNBMES BY CY-D < 0 (2x - 5) (x - 6) < 0 6 c a 5/2 x=0 x= <0 2x – 5 – ++ b x–6 &OCÑZÑLEFSFDFMJUFSJNJOLBUTBZTBC> 0 c x b0 ax.(bx–c) + – – + – + ÇK = d 5 , 6 n + –+ Ç.K = d c , 0 n 2 3 + 4 + 5 = 12 b ÖRNEK 8 ÖRNEK 10 ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS y y 4 3 –2 x –5 –2 O 4 x O3 y = f(x) y = f(x) #VOBHÌSF Y+ G Y >FöJUTJ[MJôJOJOÀÌ[ÑNLÑ- #VOBHÌSF Y2- G Y $FöJUTJ[MJôJOJOÀÌ[ÑN NFTJOJCVMVOV[ LÑNFTJOJCVMVOV[ x + 4 = 0 j x = -4 x2 - 9 = 0 j x = 3, x = -3 G Y = 0 j x = -2, x = 3 G Y = 0 j x = -5, x = -2, x = 4 x –4 –2 3 x –5 –3 –2 3 –4 (x + 4) . f(x) + – + – (x2mæ G Y – + – – + – Ç.K = (-ß, -4) b (-2, 3) Ç.K = [-5, -3] b {-2} b [3, 4] c 18 9. 12 10. <m m>b {-2} b< > 6. mß >b< ß 7. d , 0 n 8. mß, –4) b (–2, 3) b
¦BSQNõFLMJOEFLJ&öJUTJ[MJLMFSJO¦Ì[ÑN,ÑNFTJ TEST - 9 1. Y Y- 7 ) # 0 5. ( x + Y- - x ) < 0 FöJUTJ[MJôJOJTBôMBZBOLBÀUBOFYUBNTBZTWBS- FöJUTJ[MJôJOJTBôMBZBOFOCÑZÑLOFHBUJGUBNTBZ ES JMFFOLÑÀÑLQP[JUJGUBNTBZOOUPQMBNLBÀUS \" # $ % & A) - # -3 C) 0 D) 3 E) 5 6. B C`3WF< a <CPMNBLÐ[FSF ( ax +C CY+ a ) # 0 FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO IBOHJTJEJS 2. ( x + -2x + 5 ) > 0 A) f - 3, - b H # <- a , 3 n a b FöJUTJ[MJôJOJO TBôMBZBO FO CÑZÑL Y UBN TBZT LBÀUS C) >- a , - b H D) >- b , - a H ba ab A) - # -1 C) 0 D) 1 E) 2 E) > a , b H ba 7. B C D` R , a <C< 0 <DWF ( ax -C DY-B C-DY > 0 3. ( -x - -x - 5 ) # 0 PMEVôVOBHÌSF BöBôEBLJMFSEFOIBOHJTJFöJUTJ[- MJôJOÀÌ[ÑNLÑNFTJOJOCJSBMULÑNFTJEJS FöJUTJ[MJôJOJTBôMBZBOYUBNTBZMBSOOUPQMBN LBÀUS A) f a , b p # f b , b p A) - # -8 C) -7 D) -4 E) -3 ca ca C) f b , - a p D) d - 3 , a n cc c E) f b , 3 p a 4. B C D` R, 8. x –2 2 6 a <C<DWF Y+B Y+C Y+D > 0 + – + – ÇK = (–Þ, –2] b [2, 6] PMEVôVOB HÌSF, BöBôEBLJMFSEFO IBOHJTJ FöJU- :VLBSEBJöBSFUUBCMPTVWFSJMFOFöJUTJ[MJLBöBô- TJ[MJôJOÀÌ[ÑNLÑNFTJOJOCJSBMULÑNFTJEJS EBLJMFSEFOIBOHJTJPMBCJMJS A) ( x2 - Y- 6 ) $ 0 \" C D # B C $ -C -D # ( x2 - 8x - Y+ 2 ) $ 0 C) ( x2 - 4x - - x ) $ 0 D) ( -a, -C & -D -C D) ( x - Y2 - 8x + 12 ) $ 0 E) ( x2 - 4x - Y+ 2 ) $ x 1. $ 2. E 3. \" 4. E 19 5. $ 6. D 7. E 8. $
TEST - 10 ¦BSQNõFLMJOEFLJ&öJUTJ[MJLMFSJO¦Ì[ÑN,ÑNFTJ 1. ( x - 3 )2 Y2 - 16 ) < 0 5. x3 + 3x2 - 4x - 12 < 0 FöJUTJ[MJôJOJTBôMBZBOJLJGBSLMUBNTBZOOUPQ- FöJUTJ[MJôJOJTBôMBZBOJLJGBSLMOFHBUJGUBNTBZ- MBNFOÀPLLBÀUS OOUPQMBNFOÀPLLBÀUS A) - # -9 C) -1 D) 1 E) 3 A) - # -4 C) -5 D) -7 E) -9 2. ( x2 - -x2 - 6 ) $ 0 FöJUTJ[MJôJOJTBôMBZBOLBÀGBSLMQP[JUJGYUBNTB- 6. x3 + 4x2 - 16x - 64 < 0 ZEFôFSJWBSES FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO \" # $ % & IBOHJTJEJS A) ( -ß > # -4, 4 ) C) ( 4, ß) D) ( -ß, 4 ) E) ( -ß, 4 ) - { -4 } 3. ( x + Y2 - 5x + 6 ) $ 0 7. B CWFDCJSFSHFSÀFLTBZWF FöJUTJ[MJôJOJ TBôMBZBO GBSLM QP[JUJG UBN TBZ- ( 4 -Y Y4 + ax3 -CY2 +D # 0 OOUPQMBNFOB[LBÀUS \" # $ % & FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJ[ ß] b {0}PMEV- ôVOBHÌSF B+C+DUPQMBNLBÀUS A) - # -8 C) -6 D) 2 E) 10 4. ( x2 - 3x + Y2 - x - 2 ) # 0 8. ( x + Y2 - 6 ) < x2 - 2x - 8 FöJUTJ[MJôJOJTBôMBZBOLBÀGBSLMUBNTBZEFôFSJ FöJUTJ[MJôJOJ TBôMBZBO WF QP[JUJG PMNBZBO Y EF- WBSES ôFSMFSJOJOÀBSQNLBÀUS \" # $ % & A) - # $ % & 1. E 2. $ 3. \" 4. $ 20 5. $ 6. E 7. \" 8. #
¦BSQNõFLMJOEFLJ&öJUTJ[MJLMFSJO¦Ì[ÑN,ÑNFTJ 4. TEST - 11 1. y y 4 y = f(x) 3 O1 x x –3 O y = f(x) :VLBSEBHSBGJôJWFSJMFOZ=G Y GPOLTJZPOVOB :VLBSEBHSBGJôJWFSJMFOZ=G Y GPOLTJZPOVOB HÌSF ( x - G Y $ 0 FöJUTJ[MJôJOJTBôMBZBOY HÌSF ( x2 - G Y < 0FöJUTJ[MJôJOJOÀÌ[ÑNLÑ- UBNTBZMBSOOÀÌ[ÑNLÑNFTJLBÀFMFNBOMES NFTJBöBôEBLJMFSEFOIBOHJTJEJS \" # $ % & \" Þ # mÞ $ - Þ D) ( -3, 3 ) E) ( -Þ - { -3 } 2. y 5. y 6 9x –4 O y = f(x) –2 x O 3 y = f(x) :VLBSEBHSBGJôJWFSJMFOZ=G Y GPOLTJZPOVOB :VLBSEB HSBGJôJ WFSJMFO Z = G Y GPOLTJZPOVOB HÌSF ( x - G Y $ 0FöJUTJ[MJôJOJTBôMBZBOWF HÌSF ( x6 + G Y > 0FöJUTJ[MJôJOJTBôMBZBO OFHBUJGPMNBZBOLBÀGBSLMUBNTBZEFôFSJWBS- ES LBÀGBSLMUBNTBZWBSES \" # $ % & \" # $ % & 6. y 3. y y = f(x) y = f(x) O –3 O 2 4 x 3 –3 x –9 :VLBSEBHSBGJôJWFSJMFOZ=G Y GPOLTJZPOVOB HÌSF ( x2 + G Y < 0FöJUTJ[MJôJOJOÀÌ[ÑNLÑ- NFTJBöBôEBLJMFSEFOIBOHJTJEJS :VLBSEBHSBGJôJWFSJMFOZ=G Y GPOLTJZPOVOB A) ( -Þ # -3, 2 ) C) ( 2, 4 ) HÌSF ( 2x + G Y $ 0FöJUTJ[MJôJOJTBôMBZBO GBSLMYUBNTBZTOOUPQMBNFOB[LBÀUS D) ( - & Þ \" # $ % & 1. D 2. # 3. $ 21 4. E 5. E 6. $
TEST - 12 ¦BSQNõFLMJOEFLJ&öJUTJ[MJLMFSJO¦Ì[ÑN,ÑNFTJ 1. y 4. y y = f(x) 3 5 4 3x O –1 6 x O y = g(x) y = f(x) :VLBSEB HSBGJôJ WFSJMFO Z = G Y GPOLTJZPOVOB :VLBSEBHSBGJLMFSJWFSJMFOZ=G Y WFZ=H Y HÌSF G Y G Y- 4 ) < 0FöJUTJ[MJôJOJTBôMBZBOY GPOLTJZPOMBSOBHÌSF G Y H Y > 0FöJUTJ[MJôJ- SFFMTBZMBSJÀJOx2 - 8x + 19JGBEFTJOJOBMBCJ- OJTBôMBZBOYUBNTBZMBSOOUPQMBNLBÀUS MFDFôJFOCÑZÑLWFFOLÑÀÑLUBNTBZMBSOUPQ- MBNLBÀUS \" # $ % & \" # $ % & 2. y y = g(x) 1 5. y –3 –1 y = f(x) O 3x –5 O x y = f(x) –2 6 –2 :VLBSEBHSBGJLMFSJWFSJMFOZ=G Y WFZ=H Y :VLBSEBWFSJMFOZ=G Y GPOLTJZPOVOVOHSBGJ- GPOLTJZPOMBSOBHÌSF ôJOFHÌSF G Y+ G Y- 3 ) < 0FöJUTJ[MJôJOJTBô- G Y H2 Y -x2 + 5x - 9 ) # 0 FöJUTJ[MJôJOJ TBôMBZBO LBÀ GBSLM Y UBN TBZT MBZBOYUBNTBZMBSLBÀUBOFEJS WBSES \" # $ % & \" # $ % & 3. y y = f(x) 6. y 3 y = f(x) –5 O x –2 6 –2 –1 O –2 1 4 x :VLBSEBWFSJMFOZ=G Y GPOLTJZPOVOVOHSBGJ- y = g(x) ôJOFHÌSF :VLBSEBHSBGJLMFSJWFSJMFOZ=G Y WFZ=H Y fd x - 1 n.f_ 3 - x i $ 0 GPOLTJZPOMBSOBHÌSF ( x2 +Y G Y H Y $ 0 3 FöJUTJ[MJôJOJTBôMBZBOYUBNTBZMBSOOUPQMBN FöJUTJ[MJôJOJTBôMBZBOLBÀGBSLMQP[JUJGYUBNTB- LBÀUS ZTWBSES A) - # $ % & \" # $ % & 1. E 2. D 3. D 22 4. D 5. D 6. E
www.aydinyayinlari.com.tr %&/,-&.7&&õ÷54÷;-÷,4÷45&.-&3÷ 4. MODÜL 11. SINIF ÷,÷/$÷%&3&$&%&/#÷3#÷-÷/.&:&/-÷&õ÷54÷;-÷,-&3*** ÷MJöLJMJ,B[BONMBS 11.4.2.1 : öLJODJEFSFDFEFOCJSCJMJONFZFOMJFõJUTJ[MJLMFSJO¿Ë[ÐNLÐNFTJOJCVMVS Bölüm Şeklindeki Eşitsizliklerin ÖRNEK 4 Çözüm Kümesi _x -1i_4 - xi ÖRNEK 1 <0 x–4 #0 x+3 x+3 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ x – 1 = 0 &x = 1 x – 4 = 0 &x = 4 4 – x = 0 &x = 4 x + 3 = 0&x = – 3 x + 3 = 0 &x = – 3 x –3 4 x –3 1 4 x–4 + + (x – 1) (4 – x) + –+– x+3 – x+3 Ç. K = ( -3 , 1 ) , ( 4, 3) Ç. K = (–3, 4] ÖRNEK 2 ÖRNEK 5 x+2 $0 _-x +1 i_x – 5i 9 – x2 $0 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ _ 3 – x i_ x + 5 i FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ x + 2 = 0 &x = – 2 –x + 1 = 0 &x = 1 9 – x2 = 0 &x = 3, x = – 3 x – 5 = 0 &x = 5 3 – x = 0 &x = –3 x + 5 = 0 &x = –5 x –3 2 3 x –5 1 3 5 x+2 + –+– (–x + 1) (x + 5) +–+ –+ 9 – x2 mæY (x + 5) Ç. K = (-3 , –3) , [–2, 3) Ç. K = ( -3 , –5 ) , [ 1, 3 ) , [ 5, 3 ) ÖRNEK 3 ÖRNEK 6 x+1 <0 x2 - 16 < 0 4-x x2 - 2x - 15 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ x + 1 = 0 &x = – 1 x2 – 16 = 0 &x = 4, x = -4 4 – x = 0 &x = 4 x2 - 2x - 15 = 0 &( x – 5 )( x + 3) = 0 &x = 5, x = -3 x –1 4 x –4 –3 4 5 x+1 – +– 4–x x2mæ +–+ –+ x2mæYm Ç. K = ( -3 , –1 ) , ( 4, 3) Ç. K = ( –4 , –3 ) , ( 4, 5 ) 1. m > 2. (–3 , –3) , [–2, 3) 3. ( –3 , –1 ) , ( 4, 3) 23 4. (-3 , 1) , (4, 3) 5. (-3 , –5) , [1, 3) , [5, 3) 6. (–4 , –3) , (4, 5)
11. SINIF 4. MODÜL %&/,-&.7&&õ÷54÷;-÷,4÷45&.-&3÷ www.aydinyayinlari.com.tr ÖRNEK 7 ÖRNEK 10 _ x + 1 i_ x - 4 i x-1 <2 $0 x +1 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ x2 - 2x - 3 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ x – 4 = 0 &x = 4 x–1 <2& x–1 –2<0& –x – 3 <0 x + 1 = 0 &x = –1 x +1 x +1 x +1 x2 – 2x – 3 = 0 &( x – 3 ) · ( x + 1 ) = 0&x = +3, x = –1 – x – 3 = 0 &x = -3 x =mÀJGULBUMLÌL x + 1 = 0 &x = –1 x –1 3 4 x –3 –1 – +– (x + 1) (x – 4) + + – + –x – 3 x2mæYm x+1 Ç. K = ( –3 , 3 ) , [ 4, 3 ) - { -1 } Ç. K = ( -3 , -3 ) , ( -1, 3 ) ÖRNEK 8 ÖRNEK 11 x2 + x + 4 < 0 1#1 x3 - 4x2 - 5x x+3 x-2 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ 11 1 1 #0 #& – x+3 x–2 x+3 x–2 (x–2) (x + 3) x2 + x + 4 = 0 Ó<PMEVôVOEBOEBJNBQP[JUJG x3 – 4x2 – 5x = 0 x · ( x – 5 ) · ( x + 1 ) = 0 -5 #0 ^ x - 2 h^ x + 3 h x –1 0 5 x –3 2 – +–+ –5 – +– x2mæY (x – 2) (x + 3) x3mæY2 – 5x Ç. K = (-3 , -3 ) , ( 2, 3 ) Ç. K = ( -3 , -1 ) , ( 0, 5 ) ÖRNEK 9 ÖRNEK 12 _ x - 1 i4 _ x - 2 i7 x2 $1 #0 2-x FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ x +1 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ x2 x2 $ 1& – 1$ 0 2–x 2–x ( x – 1 )4 = 0 &Y ¦JGULBUM x2 + x – 2 ( x – 2 )7 = 0 &Y $0 2–x x2 +x – 2 = 0 & ( x + 2 )( x – 1 ) = 0 & x = -2, x =1 x + 1 = 0 &Y m 2 – x = 0& x = 2 x –1 1 2 x –2 1 2 (x – 1)4 (x – 2)7 x2 + x – 2 + –+– +– –+ x+1 2–x Ç. K = ( -1 , 2] Ç. K = ( –3 , –2 ] , [ 1, 2 ) 7. ( –3 , 3 ) , [ 4, 3 ) – { –1 } 8. ( –3 , –1 ) , ( 0, 5 ) 9. m > 24 10. (–3 , –3) , (–1, 3) 11. (–3 , –3) , (2, 3) 12. (–3 m>, [1, 2)
www.aydinyayinlari.com.tr %&/,-&.7&&õ÷54÷;-÷,4÷45&.-&3÷ 4. MODÜL 11. SINIF ÖRNEK 13 ÖRNEK 15 x + 5 · _ x2 - 4x - 5 i ôFLJMEF Z = G Y WF Z = H Y GPOLTJZPOMBSOO HSBGJLMF- $0 SJ¿J[JMNJõUJS 3x +1 ·_ x + 3 i y y = g(x) FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ | |x + 5 = 0 & x = - ¦JGULBU –5 x –1 O 3 x2 - 4x - 5 = 0 & ( x - 5 )( x + 1 ) = 0 y = f(x) 3x+1 = 0 & ,ÌLZPL x + 3 = 0 & x = -3 x –5 –3 –1 5 #VOBHÌSF |x + 5| (x2mæYm ––+ –+ f^ x h #0 3x + 1 · (x + 3) g^ x h Ç. K = { -5 } , ( -3 , -1 ] , [ 5, 3) FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ ÖRNEK 14 G Y = 0 & x = – 5, x =m ¦JGULBU Y= 3 H Y = 0 & x = – 1 .BUFNBUJLEFSTJJÀJOCJSQSPKFIB[SMBZBO;FZOFQ&Z- MÑMUBTBSMBEôCJSJöBSFUNBLJOFTJOJöVöFLJMEFQSPH- x –5 –1 3 SBNMBNöUS f(x) + – + – H Y Ç. K = [ –5 , –1 ) , [ 3, 3) \"Negatif” 4GSu \"Pozitif” –1 0 +1 ÖRNEK 15 r .BLJOFZFBUMBOTBZOFHBUJGJTF-TPOVDVOVCVMVS ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJ¿J[JMNJõUJS r .BLJOFZFBUMBOTBZTGSJTFTPOVDVOVCVMVS y r .BLJOFZFBUMBOTBZQP[JUJGJTF+TPOVDVOVCVMVS ;FZOFQ&ZMÑM NBLJOFZF 8 - x JGBEFTJOJBUUôOEB –4 O 5x –1 1 y = f(x) 2x + 3 +DFWBCOCVMEVôVOBHÌSF YJOBMBCJMFDFôJUBNTB- #VOBHÌSF f^ x + 2 h # 0 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNF- ZEFôFSMFSJOJOUPQMBNLBÀUS f^ x – 2 h 8-x TJOJCVMVOV[ > 0 PMNBMES 2x + 3 x –3/2 8 G Y+ OJOLÌLMFSJ{ –6, –3, –1, 3 } – G Ym OJOLÌLMFSJ{ –2, 1, 3, 7 } 8–x + + + 2x + 3 – + – –+ x –6 –3 –2 –1 1 3 7 f(x + 2) +–+ –+–– + f(x – 2) 3 Ç.K = d - , 8 n Ç. K = [ –6 , –3 ] , ( -2, -1 ] , ( 1, 7 ) – { 3 } 2 -1 + 0 + 1 + ... + 7 = 27 13. { –5 } , ( –3 , –1 ] , [ 5, 3) 14. 27 25 15. [ –5 , –1 ) , [ 3, 3) 16. [ –6 , –3 ] , ( -2, -1 ] , ( 1, 7 ) – { 3 }
TEST - 13 #ÌMÑNõFLMJOEFLJ&öJUTJ[MJLMFSJO¦Ì[ÑN,ÑNFTJ 1. x – 3 > 0 5. x2 – 1 # 0 x+4 x2 – 5x + 6 FöJUTJ[MJôJOJ TBôMBZBO FO CÑZÑL OFHBUJG Y UBN FöJUTJ[MJôJOJTBôMBZBOYUBNTBZMBSOOUPQMBN TBZTLBÀUS LBÀUS \" m # -5 C) -4 D) -3 E) -2 \" m # $ % & 2. 9 – x $ 0 _ x - 3 i_ 5 - x i x –1 6. > 0 FöJUTJ[MJôJOJTBôMBZBOLBÀGBSLMYUBNTBZTWBS- x2 - x - 2 ES FöJUTJ[MJôJOJTBôMBZBOLBÀGBSLMUBNTBZEFôFSJ WBSES \" # $ % & \" # $ % & 7. a <C< 0 <DPMNBLÐ[FSF cx + b $ 0 ax + b FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO IBOHJTJEJS 3. x - 3 # 0 A) f - 3, – b p # >- b , 3 i a c x2 – 25 FöJUTJ[MJôJOJTBôMBZBOJLJGBSLMUBNTBZOOUPQ- C) f - b , - b H D) >- a , - c n MBNFOÀPLLBÀUS ac bb E) R \" # $ % & 8. x2 + x –2 = x2 + x –2 x+3 x+3 4. x – 2 $ 0 FöJUTJ[MJôJOJTBôMBZBOYEFôFSMFSJOJOCVMVOEVôV FOHFOJöBSBMLBöBôEBLJMFSEFOIBOHJTJEJS –x2 – 1 A) [ -2, 1 ] FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO # ( -3, -3 ] , [ -2, 1 ] IBOHJTJEJS C) ( -3, 1 ] D) ( -3, -2 ] , [ 1, 3 ) A) ( - # -1, 2 ] C) ( 1, 2] E) ( - 3, 1 ] D) [ 2, 3 ) E) (-3, 2] 1. # 2. $ 3. # 4. E 26 5. # 6. $ 7. $ 8. D
#ÌMÑNõFLMJOEFLJ&öJUTJ[MJLMFSJO¦Ì[ÑN,ÑNFTJ TEST - 14 ^ 5 – x h1907 · ^ 3 – x h1988 5. 7 $ 1 1. 2 0 16 – x2 FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO x IBOHJTJEJS FöJUTJ[MJôJOJTBôMBZBOJLJGBSLMUBNTBZOOUPQ- MBNFOÀPLLBÀUS A) ( -3, 3 ) \" # $ % & # ( m3, -3 ) , [ 3, 3 ) C) ( -4, -3 ) D) ( -4, -3 ] , [ 3, 4 ) E) ( -4, 4 ) 2. x + a # 0 bx - 9 FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ [ - PMEVôVOB HÌSFB+CLBÀUS \" # $ % & 6. 2 - x $ 1 2x2 - 7x + 6 FöJUTJ[MJôJOJTBôMBZBOFOCÑZÑLUBNTBZEFôFSJ LBÀUS \" # $ % & 3. 1 < x 7. x - 2 # 0 x x2 - 4 FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO FöJUTJ[MJôJOJTBôMBZBOLBÀUBOFYUBNTBZEFôF- IBOHJTJEJS SJWBSES A) ( -1, 1 ) \" # $ % & # ( -1, 0 ) C) ( -3, -1 ) , ( 0, 1 ) D) ( -1, 0 ) , ( 1, 3 ) E) R 4. x – 5 $ x – 3 8. x2 + ax + 35 $ 0 x+3 x+8 x–b FöJUTJ[MJôJOJTBôMBZBOFOLÑÀÑLQP[JUJGUBNTBZ FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJ[ -7, -6 ) , [ -5, 3 ) LBÀUS PMEVôVOBHÌSFB+CLBÀUS \" # $ % & A) - # -6 C) 6 D) 12 E) 18 1. \" 2. D 3. D 4. # 27 5. D 6. # 7. \" 8. $
TEST - 15 #ÌMÑNõFLMJOEFLJ&öJUTJ[MJLMFSJO¦Ì[ÑN,ÑNFTJ 1. ^ x2 + 4 h · ^ x2 –13x –14 h # 0 5. ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS ^x –1h y FöJUTJ[MJôJOJ TBôMBZBO Y QP[JUJG UBN TBZMBS LBÀ 4 UBOFEJS \" # $ % & 3x O y = f(x) #VOBHÌSF f^ x h 20 2x + 5 x-3 +2 FöJUTJ[MJôJOJ TBôMBZBO Y QP[JUJG UBN TBZMBSOO UPQMBNLBÀUS 2. # 0 \" # $ % & _ x2 - 9 i a x + 2 - 3 k FöJUTJ[MJôJOJ TBôMBZBO Y UBN TBZ EFôFSMFSJOJO ÀBSQNLBÀUS \" m # m $ % & ^ x – 2 h2 –4 x – 2 + 3 3. < 0 x–4 FöJUTJ[MJôJOJTBôMBZBOLBÀGBSLMYUBNTBZEFôF- 6. ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS SJWBSES y \" # $ % & y = f(x) 1 x –2 O 4. 18 – 9x > 1 #VOBHÌSF x2 - 9 < 0 ^ x – 2 h2 f_ x i FöJUTJ[MJôJOJTBôMBZBOYUBNTBZMBSOOUPQMBN FöJUTJ[MJôJOJ TBôMBZBO Y QP[JUJG UBN TBZMBSOO LBÀUS UPQMBNLBÀUS A) - # -20 C) -15 D) -14 E) -6 \" m # m $ % & 1. D 2. # 3. # 4. # 28 5. $ 6. E
#ÌMÑNõFLMJOEFLJ&öJUTJ[MJLMFSJO¦Ì[ÑN,ÑNFTJ TEST - 16 1. ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS 3. ôFLJMEFZ=G Y WFZ=H Y GPOLTJZPOMBSOOHSB- y GJLMFSJWFSJMNJõUJS y = f(x) y y = g(x) O 3x –1 –2 O 4 y = f(x) #VOBHÌSF #VOBHÌSF f^ x h f^ x h·^ x + 4 h >0 $0 g^ x h 9 –x2 FöJUTJ[MJôJOJ TBôMBZBO Y UBN TBZMBS LBÀ UBOF- FöJUTJ[MJôJOJTBôMBZBOLBÀGBSLMYQP[JUJGUBNTB- EJS ZTWBSES \" # $ % & \" # $ % & 2. ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS 4. ôFLJMEFZ=G Y WFZ=H Y GPOLTJZPOMBSOOHSB- y GJLMFSJWFSJMNJõUJS –3 O x y –1 4 y = f(x) y = f(x) 4 #VOBHÌSF –1 23 x f^ x h·^ x2 –2x + 5 h O1 y = g(x) $0 x+2 #VOBHÌSF _ x2 - 25 i·f_ x i FöJUTJ[MJôJOJ TBôMBZBO Y HFSÀFL TBZMBSOO LÑ- >0 NFTJBöBôEBLJMFSEFOIBOHJTJEJS g_ x i A) [ -2, 1 ] FöJUTJ[MJôJOJTBôMBZBOLBÀGBSLMYUBNTBZTWBS- # [ -1, 3 ) ES C) [ -3, 4 ] D) ( -3, -3 ] , ( -2, -1 ] , { 4 } \" # $ % & E) R 1. \" 2. D 29 3. D 4. D
4*/*' 4. MODÜL %&/,-&.7&&õ÷54÷;-÷,4÷45&.-&3÷ www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/#÷3#÷-÷/.&:&/-÷&õ÷54÷;-÷,-&3*7 ÷MJöLJMJ,B[BONMBS 11.4.2.2 : öLJODJEFSFDFEFOCJSCJMJONFZFOMJFõJUTJ[MJLTJTUFNMFSJOJO¿Ë[ÐNLÐNFTJOJCVMVS ÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ&öJUTJ[MJL ÖRNEK 3 4JTUFNMFSJOJO¦Ì[ÑN,ÑNFTJ x2 - 4 > 0 TANIM x2 - 25 < 0 ö¿JOEF JLJODJ EFSFDFEFO CJS CJMJONFZFOMJ FO B[ FöJUTJ[MJLTJTUFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ CJS FõJUTJ[MJL CVMVOBO WF CJSEFO GB[MB FõJUTJ[MJL- UFOPMVõBOTJTUFNFJLJODJEFSFDFEFOCJSCJMJO- x2 - 4 = 0 j (x - 2) (x + 2) = 0 NFZFOMJFöJUTJ[MJLTJTUFNJEFOJS j x = 2, x = -2 x2 - 25 = 0 j (x - 5) (x + 5) = 0 %m/*m j x = 5, x = -5 &õJUTJ[MJLTJTUFNJOJO¿Ë[ÐNLÐNFTJUÐNFõJUTJ[- x –5 –2 2 5 MJLMFSJBZOBOEBTBóMBZBOOPLUBMBSLÐNFTJOEFO x2 – 4 + + – + + PMVõVS x2 – 25 + – – – + ÖRNEK 1 ÇK = (-5, -2) , (2, 5) x+4 $0 ÖRNEK 4 3x - 5 < 0 FöJUTJ[MJLTJTUFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ x2 + x - 2 > 0 3x2 - 4x - 7 > 0 x + 4 = 0 j x = -4 , 5 FöJUTJ[MJLTJTUFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ 3x - 5 = 0 j x = x –4 x2 + x - 2 = 0 j (x + 2) (x - 1) = 0 x+4 –+ 3 j x = -2, x = 1 5/3 3x2 - 4x - 7 = 0 j (x + 1) (3x - 7) = 0 + 7 j x = -1, x = 3x – 5 – – + 3 ÇK = [-4, 5 ) x –2 –1 1 7/3 x2 + x – 2 + – – + + 3 3x2 – 4x – 7 + + – – + ÖRNEK 2 7 ÇK = (-3, -2) , ( , 3) x- 3<0 x2 - 5x - 6 > 0 3 FöJUTJ[MJLTJTUFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ x - 3 = 0 j x = 3 x2 - 5x - 6 = 0 j (x - 6) (x + 1) =0 j x = 6, x = -1 x –1 3 x–3 – – + + x2mæYmæ + – – + ÇK = (-3, -1) 5 7 1. [–4, ) 30 3. (–5, –2) , (2, 5) 4. (–3, –2) , ( , 3) 2. (–3, –1) 3 3
www.aydinyayinlari.com.tr %&/,-&.7&&õ÷54÷;-÷,4÷45&.-&3÷ 4. MODÜL 11. SINIF ÖRNEK 5 ÖRNEK 7 x2 - 5x + 6 < 0 x2 - 6x + 15 > 0 -x2 + 2x + 8 > 0 3x2 + 5x - 2 < 0 FöJUTJ[MJLTJTUFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ FöJUTJ[MJL TJTUFNJOJO HFSÀFL TBZMBSEBLJ ÀÌ[ÑN LÑ- NFTJOJCVMVOV[ x2 - 5x + 6 = 0 j (x - 3) (x - 2) = 0 j x = 3, x = 2 x2 - 6x + 15 = 0 j Ó = 36 - 4 · 1 ·15 -x2 + 2x + 8 = 0 j (x - 4) (x + 2) = 0 = -24 < 0 j x = -2, x = 4 x –2 2 3 4 Ó <PMEVôVOEBOEBJNBQP[JUJGUJS x2mY + +– ++ 3x2 + 5x - 2 = 0 j x = 1 j x = - 2 3 x –2 1/3 –x2 + 2x + 8 – + + + – x2mY + ++ ÇK = ( 2, 3 ) 3x2 + 5x – 2 + – + ÇK = d - 2 , 1 n 3 ÖRNEK 6 ÖRNEK 8 2x2 - 5x - 3 T 0 -3x2 + 4x - 10 > 0 8x2 - 9x - 17 > 0 x2 - 7x + 10 < 0 FöJUTJ[MJLTJTUFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ FöJUTJ[MJL TJTUFNJOJO HFSÀFL TBZMBSEBLJ ÀÌ[ÑN LÑ- NFTJOJCVMVOV[ 2x2 - 5x - 3 = 0 j ( 2x + 1 ) ( x - 3 ) = 0 1 3x2 + 4x - 10 = 0 j Ó = 16 - 4 ( -3 )( -10 ) j x = – , x = 3 = -104 2 Ó <PMEVôVOEBOEBJNBOFHBUJGUJS 8x2 - 9x - 17 = 0 j ( 8x - 17 ) ( x + 1 ) = 0 x2 - 7x + 10 = 0 j (x - 5) (x - 2) = 0 17 x=5 x=2 j x = , x = -1 8 x 25 x –1 –1/2 17/8 3 –3x2 + 4x – 10 – – – 2x2 – 5x – 3 + + – – + x2 – 7x + 10 + – + 8x2 – 9x – 17 + – – + + 4JTUFNEFPSUBLÀÌ[ÑNPMNBEôOEBO¦,= QEJS ÇK = ( 17 , 3 ] 8 5. ( 2, 3 ) 17 31 1 8. q 6. ( > 7. ( –2, ) 3 8
11. SINIF 4. MODÜL %&/,-&.7&&õ÷54÷;-÷,4÷45&.-&3÷ www.aydinyayinlari.com.tr ÖRNEK 9 ÖRNEK 11 4x2 - 12x + 9 > 0 - 6 < x2 - 7x < 18 -x2 + 5x - 7 < 0 FöJUTJ[MJL TJTUFNJOJO HFSÀFL TBZMBSEBLJ ÀÌ[ÑN LÑ- NFTJOJCVMVOV[ FöJUTJ[MJL TJTUFNJOJO HFSÀFL TBZMBSEBLJ ÀÌ[ÑN LÑ- NFTJOJCVMVOV[ -6 < x2 - 7x x2 - 7x <18 0 0 4x2 - 12x + 9 = 0 j ( 2x - 3 )2 = 0 0 < x2 - 7x + 6 x2 - 7x - 18 < 0 3 x = ¦JGULBUM 0 < (x - 6)(x - 1) (x - 9)(x + 2) < 0 2 x –2 1 9 -x2 + 5x - 7 = 0 j Ó = 25-4 · ( -1 ) · ( -7 ) = -3 x2mY + +– ++ Ó <PMEVôVOEBOEBJNBOFHBUJGUJS x 3/2 x2 – 7x – 18 + – – – + 4x2 – 12x + 9 ++ ÇK = (-2, 1) b ( 6, 9 ) –x2 + 5x – 7 – – ÇK =3- { 3 } 2 ÖRNEK 10 ÖRNEK 12 ( x - 1 )2 > 0 m ! R olmak üzere 5x - x2 > 0 ( m - 3 ) x2 - ( 2 - m ) x + 6 - 3m = 0 x2 - x - 12 < 0 ikinci dereceden denkleminin kökleri x1 ve x2 dir. FöJUTJ[MJL TJTUFNJOJO HFSÀFL TBZMBSEBLJ ÀÌ[ÑN LÑ- x1 ve x2BSBTOEBY1 < 0 < x2CBôOUTPMEVôVOBHÌSF NFTJOJCVMVOV[ NEFôFSMFSJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ (x - 1)2 = 0 j x = ¦JGULBUM x1 < 0 < x2 PMEVôVOEBO EFOLMFNJO GBSLM SFFM LÌ- 5x - x2 = 0 j x ( 5 - x ) = 0 j x =WFZBY= 5 LÑWBSES x2 -x - 12 = 0 j ( x - 4 ) · ( x + 3 ) = 0 j x =WFZBY= -3 BJMFDUFSTJöBSFUMJPMEVôVOEBOD >PMVS x1 < 0 PMEVôVOEBOLÌLÀBSQNOFHBUJGUJS x –3 0 1 4 5 x2 > 0 (x – 1)2 + + + + + + 6 – 3m 5x – x2 – –+ ++ – <0 m–3 x2 – x – 12 + –– –+ + m 23 mN – + – m–3 ÇK = ( -3, 2 ) , ( 3, 3 ) ÇK = ( 0, 1 ) , ( 1, 4 ) 3 10. ( 0, 1 ) , ( 1, 4 ) 32 11. m b ( 6, 9 ) 12. m3, 2 ) , ( 3, 3 ) 9. 3m{ } 2
www.aydinyayinlari.com.tr %&/,-&.7&&õ÷54÷;-÷,4÷45&.-&3÷ 4. MODÜL 11. SINIF ÖRNEK 13 ÖRNEK 15 N`3PMNBLÐ[FSF N`3PMNBLÐ[FSF N+ 1 ) x2 - N+ 1 ) x -N- 3 = 0 N- 2 ) x2 + 4x +N< 5 JLJODJEFSFDFEFOEFOLMFNJOJOLËLMFSJY1WFY2EJS FöJUTJ[MJôJrx ` 3JÀJOTBôMBOEôOEBHÌSFNOJOÀÌ- | | | |,ÌLMFSBSBTOEBY1 < 0 < x2WF x1 < x2 CBôOUTPM- [ÑNLÑNFTJOJCVMVOV[ EVôVOB HÌSF N EFôFSMFSJOJO ÀÌ[ÑN LÑNFTJOJ CVMV- N- 2)x2 + 4x +N< 5 OV[ %FOLMFNJOJLJGBSLMLÌLÑWBSÓ > 0 Ó <PMNBMES- N- p N- 5 ) < 0 N2 -N+ 6 > 0 ,ÌLMFSÀBSQNY1 · x2 < 0 N- 2 <PMNBMES ,ÌLMFSUPQMBNY1 + x2 > 0 BJMFDUFSTJöBSFUMJPMEVôVOEBOD >PMVS m 12 m2mN -m- 3 3m + 1 + –– + <0 >0 m+1 m+1 m –3 –1 –1/3 m–2 – – + + –m – 3 – +– – ÇK = ( -3, 1 ) m+1 3m + 1 + + – + m+1 1 ÖRNEK 16 ÇK = ( -3, -3 ) , ( – , 3 ) 3 ôFLJMEF Z = G Y WF Z = H Y GPOLTJZPOMBSOO HSBGJLMF- SJWFSJMNJõUJS ÖRNEK 14 y y = f(x) y N!3PMNBLÐ[FSF 5 N- 1 ) x2 +NY+N- 3 = 0 y = g(x) JLJODJEFSFDFEFOEFOLMFNJOJOLËLMFSJY1WFY2EJS x1WFY2BSBTOEB< x1 < x2CBôOUTPMEVôVOBHÌSF –1 x 6x NEFôFSMFSJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ –5 bc Ó>0, – >0 , > 0 PMNBMES #VOBHÌSF aa G Y $ 0, H Y < 0 Ó =N2 - N- N- 1 ) > 0 FöJUTJ[MJLTJTUFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ =N- 12 > 0 (I) –2m m– 3 G Y GPOLTJZPOVJÀJOG -1 ) = 0 (II) > 0 (III) > 0 m–1 m– 1 m 0 3/4 1 3 H Y GPOLTJZPOVJÀJOH = 0 I– – + + + x –1 II – + + – – f(x) – + + III + + + – + g(x) – – + 3 ÇK = [ -1, 6 ) ÇK = ( , 1 ) 4 1 3 33 15. ( –3, 1 ) 16 [–1, 6 ) 13. (–3, –3 ) , ( – , 3 ) 14. ( , 1 ) 3 4
11. SINIF 4. MODÜL %&/,-&.7&&õ÷54÷;-÷,4÷45&.-&3÷ www.aydinyayinlari.com.tr ÖRNEK 17 ÖRNEK 19 ôFLJMEF Z = G Y WF Z = H Y GPOLTJZPOMBSOO HSBGJLMF- ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS y SJWFSJMNJõUJS y y = f(x) 4 5 –2 O y = f (x) –5 13 x –5 y = g(x) –3 x 47 #VOBHÌSF #VOBHÌSF G Y < 0 B G Y+ 3 ) > 0 G Y- 2 ) < 0 H Y > 0 FöJUTJ[MJLTJTUFNJOJOÀÌ[ÑN LÑNFTJOJCVMVOV[ FõJUTJ[MJL TJTUFNJOJ TBóMBZBO Y EFóFSMFSJOJO ¿Ë[ÐN LÐNFTJOFEJS G Y GPOLTJZPOVJÀJOG -5 ) =G = 0 C G Y- G Y+ 1) < 0 H Y GPOLTJZPOVJÀJOH -2 ) =H = 0 FõJUTJ[MJóJOJTBóMBZBOYEFóFSMFSJOJO¿Ë[ÐNLÐNFTJ OFEJS x –5 –2 1 3 f(x) + – – + + B G Y + ÑO LÌLMFSJ JÀJO G Y CS TPMB ÌUFMFOJS g(x) – – + + – ,ÌLMFS- HFMJS ÇK = ( -2, 1 ) G Y- OJOLÌLMFSJJÀJOG Y CSTBôBÌUFMFOJS,ÌLMFS - HFMJS ÖRNEK 18 x m –1 1 4 9 y f(x + 3) – + + – + + + y = g (x) f(x – 2) – – + + + – + 4 ÇK = ( -6, -1 ) , ( 6, 9 ) –3 –1 3 5x C G Y- JÀJOCSTBôBÌUFMFOJS –3 y = f (x) ,ÌLMFS- HFMJSG Y+ JÀJOCSTPMBÌUFMF- :VLBSEBZ=G Y WFZ=H Y GPOLTJZPOMBSOOHSBGJLMF- OJS,ÌLMFS- HFMJS SJWFSJMNJõUJS x –4 –2 3 5 8 #VOBHÌSF f(x + 1).f(x + 1) + – + – + – + G Y # 0 H Y > 0 Ç.K = (-4, -2) b (3, 5) b (6, 8) FöJUTJ[MJLTJTUFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ G Y GPOLTJZPOVJÀJOG -3 ) =G = 0 0 ¦JGULBUM H Y GPOLTJZPOVJÀJOH -1 ) =H = 0 x –3 –1 3 5 f(x) + + + + – g(x) + + – + + ÇK = { -3 } , [ 5, 3 ) 17. ( –2, 1 ) 18. { –3 } , [ 5, 3 ) 34 19. B ( –6, –1 ) , ( 6, 9 ) C (–4, –2) b (3, 5) b (6, 8)
www.aydinyayinlari.com.tr %&/,-&.7&&õ÷54÷;-÷,4÷45&.-&3÷ 4. MODÜL 11. SINIF ÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ ÖRNEK 21 &öJUTJ[MJLMFSJO(SBGJLMFSJ Z# -2x2 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJBOBMJUJLEÑ[- MFNEFHÌTUFSJOJ[ %m/*m y x O Z> ax2 +CY+DFõJUTJ[MJóJOJOHSBGJóJ Z= ax2 +CY+DQBSBCPMÐOÐOÐTUCËMHFTJOEFLJ OPLUBMBSES y = ax2+bx+c y y a>0 y = –2x2 Ox Ox y = ax2+bx+c ÖRNEK 22 a<0 Z$ -2 ( x - 1 )2 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJBOBMJ- Z< ax2 +CY+DFõJUTJ[MJóJOJOHSBGJóJ UJLEÑ[MFNEFHÌTUFSJOJ[ Z= ax2 +CY+DQBSBCPMÐOÐOBMUCËMHFTJOEFLJ OPLUBMBSES y = ax2 + bx + c y a>0 y y O1 x xx –2 OO y = –2 (x – 1)2 y = ax2 + bx + c ÖRNEK 23 a<0 Z# x2 - 4x + 3 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJBOBMJUJL Z$ ax2 +CY+DWFZ# ax2 +CY+DFõJUTJ[MJL- EÑ[MFNEFHÌTUFSJOJ[ MFSJOJO HSBGJLMFSJOEF QBSBCPMÐO Ð[FSJOEFLJ OPL- UBMBSEBHSBGJóFEBIJMFEJMJS y ÖRNEK 20 3 Z> x2 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJBOBMJUJLEÑ[MFN- EFHÌTUFSJOJ[ y y = x2 O1 3x x O 35
11. SINIF 4. MODÜL %&/,-&.7&&õ÷54÷;-÷,4÷45&.-&3÷ www.aydinyayinlari.com.tr ÖRNEK 24 ÖRNEK 27 Z# x + 1 Z# x2 Z$ x2 - 2 x +Z# 2 FöJUTJ[MJL TJTUFNJOJO ÀÌ[ÑN LÑNFTJOJ BOBMJUJL EÑ[- Z- x < 1 MFNEFHÌTUFSJOJ[ FöJUTJ[MJL TJTUFNJOJO ÀÌ[ÑN LÑNFTJOJ BOBMJUJL EÑ[- MFNEFHÌTUFSJOJ[ y 1 y = x2 – 2 y y = x2 – 2 –1 O ZmæY x 2 2 1 x –1 2 y=x+1 –2 x+y=2 ÖRNEK 25 Z< x2 - 1 Z$ -x2 + 2 FöJUTJ[MJL TJTUFNJOJO ÀÌ[ÑN LÑNFTJOJ BOBMJUJL EÑ[- MFNEFHÌTUFSJOJ[ y = x2–1 ÖRNEK 28 y A –2 2 y –1 O 1x E D x B y = –x2+2 C O F ÖRNEK 26 :VLBSEBHSBGJôJWFSJMFO x + 1 <Z< -x2 + 2x + 3 x2 - 2x <Z# x2 + 4x FöJUTJ[MJL TJTUFNJOJO ÀÌ[ÑN LÑNFTJOJ BOBMJUJL EÑ[- FöJUTJ[MJL TJTUFNJOJ \" # $ % & ' OPLUBMBSOEBO MFNEFHÌTUFSJOJ[ IBOHJMFSJTBôMBS y = x2+4x y x + 1 <Z< -x2 + 2x + 3 y = x2–2x %PôSVOVO1BSBCPMÑO ÑTUCÌMHFTJJÀCÌMHFTJ { \" # &} + { # $ '} = { #} –4 O 2 x 36 28. #
÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ&öJUTJ[MJL4JTUFNMFSJOJO¦Ì[ÑN,ÑNFTJ TEST - 17 1. x2 - 4x + 4 > 0 5. 3x + 1 $ 0 x2 - 5x - 14 < 0 3-x 1 #0 FöJUTJ[MJLTJTUFNJOJOÀÌ[ÑNLÑNFTJOEFLJYUBN TBZEFôFSMFSJOJOUPQMBNLBÀUS 4x2 - x - 3 \" # $ % & FöJUTJ[MJL TJTUFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJ- MFSEFOIBOHJTJEJS A) f - 1 , 2 p # >- 1 , 1 i C) f - 1 , 2 p 2 3 3 D) [ -4, -2 ) E) f 1 , 2 p 2 2. x2 + x + 2 > 0 x2 - 16 # 0 FöJUTJ[MJL TJTUFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJ- 6. x + 2 > 0 MFSEFOIBOHJTJEJS -x +1 A) [ -2, 4 ] # -4, 4 ) x2 - 3x - 4 # 0 C) [ -4, 4 ] D) R x2 - 4 E) ( - 3, -4 ] , [ 4, 3 ) FöJUTJ[MJL TJTUFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJ- MFSEFOIBOHJTJEJS \" # -2, 4 ) C) ( 2, 4 ] D) ( -2, -1 ] E) [ -1, 1 ) 3. x2 - 2 $ 0 7. x2 - 4 < 0 3-x x2 - x - 6 # 0 | |x - 2 $ 0 | |x3 · x + 1 2 $ 0 FöJUTJ[MJL TJTUFNJOJ TBôMBZBO LBÀ UBOF QP[JUJG Y UBNTBZTWBSES \" # $ % & FöJUTJ[MJL TJTUFNJOJ TBôMBZBO Y UBN TBZMBSOO UPQMBNLBÀUS \" # $ % & 4. x2 - x - 12 # 0 8. x2 + 2x + 3 > 0 _ x - 6 i3 x · ( x2 - 3x + 2 ) > 0 #0 BöBôEBLJMFSEFO IBOHJTJ FöJUTJ[MJL TJTUFNJOJO x -1 ÀÌ[ÑNLÑNFTJOJOBMULÑNFMFSJOEFOCJSJEJS FöJUTJ[MJLTJTUFNJOJTBôMBZBOLBÀUBOFUBNTBZ EFôFSJWBSES \" # $ % & \" # $ -1, 0 ) E) ( 0, 2 ) D) ( -3, 1 ) 1. \" 2. $ 3. \" 4. # 37 5. # 6. D 7. \" 8. \"
TEST - 18 ÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ&öJUTJ[MJL4JTUFNMFSJOJO¦Ì[ÑN,ÑNFTJ 1. x2 + 3x < 4 5. B`3PMNBLÑ[FSF 1 <1 ( a - 7 ) x2 - ( 2a - 8 ) x + a + 6 = 0 x EFOLMFNJOJO[UJöBSFUMJJLJLÌLÑPMEVôVOBHÌSF FöJUTJ[MJLTJTUFNJOJTBôMBZBOYEFôFSMFSJOJOÀÌ- BOOÀÌ[ÑNBSBMôBöBôEBLJMFSEFOIBOHJTJEJS [ÑNLÑNFTJBöBôEBLJMFSEFOIBOHJTJEJS A) ( - # 3 ) C) ( -4, 0 ) A) ( -3, - # 3 ) C) ( -6, 4 ) E) ( 0, 1 ) E) ( -8, -6 ) D) ( -3, -4 ) D) ( -6, 7 ) 2. x2 # -6x - 8 6. a `3PMNBLÐ[FSF x< 4 x2 - 3 ( a - 4 ) x - 5a = 0 x EFOLMFNJOJOLËLMFSJY1WFY2EJS FöJUTJ[MJL TJTUFNJOJ TBôMBZBO Y UBN TBZMBSOO UPQMBNLBÀUS | | | |x1WFY2BSBTOEBY1 < 0 < x2WF x1 > x2 CBôO- A) - # -9 C) -8 D) -7 E) -6 UMBS PMEVôVOB HÌSF B OO ÀÌ[ÑN BSBMô BöBô- EBLJMFSEFOIBOHJTJEJS A) ( -3 # 3 ) C) ( 0, 5 ) D) ( 0, 4 ) E) ( -4, 3 ) 3. 1 < 2 7. N`3PMNBLÐ[FSF x+2 x-2 NY2 +NY+ 4x + 1 =NY+ 5x 1 #2 EFOLMFNJOJOLËLMFSJY1WFY2EJS x +1 x1WFY2BSBTOEB< x1 < x2CBôOUTPMEVôVOB FöJUTJ[MJL TJTUFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJ- HÌSFNOJOFOLÑÀÑLUBNTBZEFôFSJLBÀUS MFSEFOIBOHJTJEJS A) ( -6, -2 ) , ( 2, 3 # [ 2, 3 ) C) [ 2, 4 ) D) ( 0, 2 ] A) - # $ % & E) ( -6, -1 ) , ( 2, 3 ) 8. a `3PMNBLÐ[FSF ax2 - 2 ( a - 4 )x + a - 6 = 0 4. -2 < x2 - 6 # 19 EFOLMFNJOJOLÌLMFSJOJOJLJTJOJOEFQP[JUJGPMNB- TJÀJOBIBOHJBSBMLUBPMNBMES FöJUTJ[MJôJOJTBôMBZBOLBÀUBOFYUBNTBZTWBS- A) (-3, 0 ) , # , ( 4, 6 ] ES C) ( 4, 6 ) , ( 6, 8 ] D) ( 6, 8 ) , ( 8, 3 ) \" # $ % & E) (-3, 0 ) , ( 6, 8 ] 1. $ 2. D 3. \" 4. \" 38 5. D 6. D 7. D 8. E
÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ&öJUTJ[MJL4JTUFNMFSJOJO¦Ì[ÑN,ÑNFTJ TEST - 19 1. y y y = g(x) 4. ôFLJMEFZ=G Y WFZ=H Y GPOLTJZPOMBSOOHSB- 4 3 GJLMFSJWFSJMNJõUJS O 5x y y = f(x) y = g (x) –3 O x 4 y = f (x) ôFLJMEFZ=G Y WFZ=H Y GPOLTJZPOMBSOOHSB- –5 –1 O 3 x GJLMFSJWFSJMNJõUJS #VOB HÌSF G Y # H Y # 0 FöJUTJ[MJL TJTUF- #VOBHÌSF G Y $ H Y > 0 FöJUTJ[MJLTJTUFNJ- NJOJTBôMBZBOFOCÑZÑLJLJUBNTBZOOUPQMBN OJTBôMBZBOLBÀUBOFYUBNTBZTWBSES LBÀUS \" # $ % & \" # -5 C) -9 D) -11 E) -13 2. ôFLJMEFZ=G Y WFZ=H Y GPOLTJZPOMBSOOHSB- 5. ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS GJLMFSJWFSJMNJõUJS y y = f(x) y y = f(x) 4 O 4 x –1 3 –1 O 4 –2 x #VOBHÌSF G - x ) < G Y > 0 FöJUTJ[MJLTJTUF- y =g(x) NJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO IBOHJTJ- EJS #VOBHÌSF G Y $ H Y $ 0 FöJUTJ[MJLTJTUFNJ- OJTBôMBZBOYUBNTBZMBSOOUPQMBNLBÀUS \" # $ % & A) ( -3, - # -3, -1 ) C) ( 2, 3 ) D) ( -3, -1 ) E) ( -1, 2 ) 3. ôFLJMEFZ=G Y WFZ=H Y GPOLTJZPOMBSOOHSB- 6. ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS GJLMFSJWFSJMNJõUJS y y –1 x –3 O x –2 O 2 3 5 –7 7 y = f (x) y = f (x) y = g (x) #VOBHÌSF G Y+ 3 ) > G - 2x ) < 0 FöJUTJ[- MJLTJTUFNJOJTBôMBZBOFOCÑZÑLJLJUBNTBZOO #VOBHÌSF G Y # H Y > 0 FöJUTJ[MJLTJTUFNJOJ UPQMBNLBÀUS TBôMBZBOYUBNTBZMBSLBÀUBOFEJS A) - # -3 C) -4 D) -5 E) -6 \" # $ D) 5 E) 6 1. $ 2. D 3. \" 39 4. D 5. D 6. D
TEST - 20 y ÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ&öJUTJ[MJLMFSJO(SBGJLMFSJ 12 1. 3. ôFLJMEFZ=G Y WFZ=H Y GPOLTJZPOMBSOOHSB- GJLMFSJWFSJMNJõUJS y y = f (x) 5 3 x –3 O 1 x –4 3 –4 O õFLJMEFLJ UBSBM CÌMHF BöBôEBLJ FöJUTJ[MJL TJT- –9 UFNMFSJOEFOIBOHJTJOJOÀÌ[ÑNLÑNFTJEJS A) Zã-x2 - x + 12 y = g (x) Zâ-x + 3 #VOBHÌSF öFLJMEFLJUBSBMCÌMHFBöBôEBLJFöJU- # Zâ-x2 - x + 12 TJ[MJL TJTUFNMFSJOEFO IBOHJTJOJO ÀÌ[ÑN LÑNF- TJEJS Zã-x + 3 C)Zâ-x2 + x + 12 \" Z$ x2 +Y # Z# x2 + 4x ZãY- 3 Z# -( x + 3 )2 Z$ -( x + 3 )2 D)Zâ-x2 - x - 12 $ Z> - ( x + 3 )2 % Z# - ( x + 3 )2 ZãY+ 3 E) Zâ-x2 - x + 12 Z< x2 +Y Z# x2 + 4x Z> -x + 3 & Z< x2 + 4x Z> -( x + 3 )2 2. 4. ôFLJMEFZ=G Y QBSBCPMÐJMFZ=YEPóSVTVOVOHSB- y GJóJWFSJMNJõUJS 9 y = x2 y y = f(x) 5T K y=x –3 O 3 x M L x y = g (x) O1 5 ôFLJMEFZ=G Y WFZ=H Y QBSBCPMMFSJOJOHSBGJóJ E WFSJMNJõUJS #VOBHÌSF, . - 5 &OPLUBMBSOEBOIBOHJTJ #VOBHÌSF öFLJMEFLJUBSBMCÌMHFBöBôEBLJFöJU- Z# x2 - 6x + 5 TJ[MJL TJTUFNMFSJOEFO IBOHJTJOJO ÀÌ[ÑN LÑNF- TJEJS Z- x $ 0 A) x2 #Z# -x2 + # -x2 + 9 #Z# x2 YpZ> 0 C) -x2 #Z# x2 D) x2 #Z# x2 - 9 LPöVMMBSOOIFQTJOJTBôMBS E) x2 #Z# x2 + 9 \" , # . $ 5 % - & & 1. # 2. \" 40 3. \" 4. \"
%FOLMFNWF&öJUTJ[MJL4JTUFNMFSJ KARMA TEST - 1 1. x2 +Z2 = 26 5. _ x4 - 4x2 i x $ 0 YpZ= 5 x2 - 2x EFOLMFNTJTUFNJOJTBôMBZBOGBSLMYHFSÀFLTB- FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO ZMBSOOUPQMBNLBÀUS IBOHJTJEJS A) - # - 3 C) 0 D) 5 E) 10 A) [ -2, 3 # -2, 3 ) C) [ 2, 3 ) D) [ -2, 2 ] E) [ -2, 3 ) - { 0, 2 } 2. #JS EJLEÌSUHFOJO LÌöFHFO V[VOMVôV 2 5 CS 6. 4x – 2 · _ x2 - 16 i # 0 EJLEÌSUHFOJOBMBOJTFCS2PMEVôVOBHÌSF CV _ x - 4 i3 · _ x + 3 i EJLEÌSUHFOJO ÀFWSFTJ BöBôEBLJMFSEFO IBOHJTJ- FöJUTJ[MJôJOJ TBôMBZBO Y UBN TBZ EFôFSMFSJ LBÀ UBOFEJS EJS \" # $ % & \" # $ % & 3. x2 - 8x + 7 < 0 7. 3 - x2 + 2x - 15 2 0 _ x - 2 i2 FöJUTJ[MJôJOJTBôMBZBOYUBNTBZEFôFSMFSJUPQMB- FöJUTJ[MJôJOJTBôMBZBOYUBNTBZMBSOOUPQMBN NLBÀUS LBÀUS A) - # -4 C) -2 D) 0 E) 3 \" # $ % & 4. x2 + ax - 18 $ 0 x+4· x-2 x+b 8. # 0 FöJUTJ[MJôJOÀÌ[ÑNLÑNFTJ[ -9, -3 ) , [2, 3 PM- x2 + 12x - 13 FöJUTJ[MJôJOJTBôMBZBOOFHBUJGUBNTBZEFôFSMFSJ- EVôVOBHÌSF B-CGBSLLBÀUS OJOUPQMBNLBÀUS A) - # - 2 C) -1 D) 1 E) 4 A) - # -13 C) -12 D) -11 E) -10 1. $ 2. $ 3. E 4. E 41 5. E 6. \" 7. $ 8. E
KARMA TEST - 2 %FOLMFNWF&öJUTJ[MJL4JTUFNMFSJ 1. #VSBLhOZÐSÐZÐõI[71NEL\"TMhOOZÐSÐZÐõI[ 5. B< 0 <CPMNBLÑ[FSF 72NELESEBLJLBCPZVODB#VSBLhOBMEóZP- ax2 + bx < 0 MVO \"TMhOO BMEó ZPMB PSBO JLJTJOJO UPQMBN BMEó ax2 - 4 FöJUTJ[MJôJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO ZPMVO#VSBLhOBMEóZPMBPSBOOBFõJUUJS IBOHJTJEJS #VOBHÌSF \"TMhOOI[OO#VSBLhOI[OBPSBO BöBôEBLJMFSEFOIBOHJTJEJS 1+ 5 5 –1 1– 5 A) f 0, - b p # f - 2 , 0 p C) f - 2 , - b p A) # C) a a aa 2 2 2 D) f 2 , - b p E) f - b , 2 p aa aa D) 1 E) 1 2 x2019 ^ x – 2 h2018 6. -1 < 1 # 4 #0 x+2 2. ^ x + 3 h2017 FöJUTJ[MJôJOJOFOHFOJöÀÌ[ÑNBSBMôBöBôEBLJ- MFSEFOIBOHJTJEJS FöJUTJ[MJôJOJOFO HFOJöÀÌ[ÑNLÑNFTJBöBôEB- A) f - 3, - 7 p # 3- 7- 3, - 7 p LJMFSEFOIBOHJTJEJS 4 4 A) [ - > # $ > C) 7- 3, - 7 p D) R - f - 3, - 7 H 4 4 D) [ -3, 0 ] E) ( -3, 0 ] , { 2 } E) R 3. ^ x2 + x + 1 h·^ x – 1 h2 7. ,PPSEJOBU EÐ[MFNJO *7 CËMHFTJOEF CVMVOBO CJS ) $0 3x2 - 12 OPLUBTOOLPPSEJOBUMBS) Y2- 5x -14, x2 + x -12) FöJUTJ[MJôJOJ TBôMBZBO Y UBN TBZ EFôFSMFSJOJO EJS UPQMBNLBÀUS #VOB HÌSF Y JO BMBCJMFDFôJ LBÀ GBSLM UBN TBZ EFôFSJWBSES \" # $ % & \" # $ % & 4. 3 $ 1 8. N`3PMNBLÐ[FSF x-2 x+1 NY2 - N+ 3 ) x + 1 = 0 FöJUTJ[MJôJOJTBôMBZBOFOLÑÀÑLQP[JUJGUBNTBZ EFOLMFNJOJOLËLMFSJY1WFY2PMTVO,ËLMFSBSBTOEB FOCÑZÑLOFHBUJGUBNTBZEBOLBÀGB[MBES | | | |x1 < 0 < x2WF x1 > x2 CBóOUTWBSES \" # $ % & #VOBHÌSFNOJOBMBCJMFDFôJUBNTBZEFôFSMFSJ UPQMBNLBÀUS A) - # - $ m % - 1 E) 2 1. # 2. E 3. # 4. $ 42 5. \" 6. # 7. # 8. $
%FOLMFNWF&öJUTJ[MJL4JTUFNMFSJ KARMA TEST - 3 1. y y 4. y = f (x) 3 5 x –1 x O –7 4 y = f(x) õFLJMEFLJZ=G Y GPOLTJZPOVOVOHSBGJôJOFHÌSF ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS ( x + pG Y ã #VOBHÌSF ( 5 -Y pG Y- ãFöJUTJ[MJôJOJTBô- MBZBO FO CÑZÑL QP[JUJG UBN TBZOO UPQMBN FöJUTJ[MJôJOJTBôMBZBOYUBNTBZMBSOOUPQMBN LBÀUS LBÀUS \" # $ % & \" # $ % & 5. x –3 –2 0 2 3 – – +–+ + + –– –– + 2. ôFLJMEFZ=G Y WFZ=H Y GPOLTJZPOMBSOOHSB- õFLJMEFLJ UBSBM CÌMHFMFSMF ÀÌ[ÑN LÑNFTJ WFSJ- MFO FöJUTJ[MJL TJTUFNJ BöBôEBLJMFSEFO IBOHJTJ GJLMFSJWFSJMNJõUJS PMBCJMJS y y = g (x) A) x2 > 9 # x2 < 9 C) x2 < 4 4 x< 4 x> 4 x< 3 y = f (x) x x x x –4 D) x2 < 4 E) x2 > 9 –3 O35 9 x> 4 #VOBHÌSF f^ x h < 0 FöJUTJ[MJôJOJTBôMBZBOLBÀ x< x x g^ x h GBSLMYUBNTBZTWBSES 6. y \" # $ % & 2 –1 O 2 x 3. y –2 3 –1 x :VLBSEB BOBMJUJL EÑ[MFNEF UBSBM CÌMHF JMF ÀÌ- O 7 [ÑN LÑNFTJ WFSJMFO FöJUTJ[MJL TJTUFNJ BöBôEB- LJMFSEFOIBOHJTJEJS y = f(x) A) Z# x2 - x - 2 # Z$ x2 - x - 2 ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS Z# 2x + 2 Z# 2x + 2 #VOBHÌSF C) Z# x2 + x - 2 D) Z$ x2 + x - 2 G Y+ 2 ) > G - x ) < 0 Z# 2x + 2 Z$ 2x + 2 FöJUTJ[MJL TJTUFNJOJO ÀÌ[ÑN LÑNFTJOEF LBÀ E) Z$ x2 - x - 2 GBSLMYUBNTBZTWBSES Z$ -2x + 2 \" # $ % & 1. $ 2. \" 3. \" 43 4. D 5. # 6. #
KARMA TEST - 4 %FOLMFNWF&öJUTJ[MJL4JTUFNMFSJ 1. G Y = -x2 + ax +CGPOLTJZPOVOVOLBUTBZMBSBSB- 4. U` R+PMNBLÐ[FSF TOEBC< 0 <BWF a < –b CBóOUTWBSES G Y =UY2 + U2 - 5 ) x - 25 2 GPOLTJZPOVJ¿JOG Y =FõJUMJóJOJTBóMBZBOEFóFSMFS #VOBHÌSF x1WFY2EJS x1WFY2 BSBTOEBY1 < 1 < x2CBôOUTPMEVôVOB * G Y =EFOLMFNJOJOHFS¿FLLËLMFSJUPQMBNB HÌSFUOJOBMBCJMFDFôJLBÀGBSLMUBNTBZEFôFSJ ES WBSES ** G - pG pG > 0 \" # $ % & *** | x2 - x -20 |pG Y $ 0FõJUTJ[MJóJOJTBóMBZBOJLJ GBSLMYUBNTBZEFóFSJWBSES JGBEFMFSJOEFOIBOHJMFSJEPôSVEVS \" :BMO[* # :BMO[** $ :BMO[*** % *WF*** & **WF*** 5. ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS y y = f(x) 7 2. G3Y3\" 3GPOLTJZPOV –5 O G Y Z = { YWFZEFOLпÐLPMBO} õFLMJOEFUBON- x MBOZPS 16 rx `3JÀJOG -x2 NY+ 4 ) ) = -x2PMEVôVOB HÌSFNOJOBMBCJMFDFôJGBSLMUBNTBZEFôFSMFSJ H Y = x2 + 6x UPQMBNLBÀUS PMEVôVOB HÌSF 0 < GPH Y < 7 FöJUTJ[MJôJOJ TBôMBZBOFOCÑZÑLUBNTBZJMFFOLÑÀÑLOFHB- A) - # -3 C) 0 D) 7 E) 9 UJGUBNTBZOOUPQMBNLBÀUS A) - # -6 C) -3 D) -2 E) 0 3. #JSGJSNB Y2 +NY- 2 ) 5-ZFBMEóCJSÐSÐOÐ 6. x2 + ax + 36 > 0 ( 2x - 5-ZFTBUNBLUBES –x2 + 2x + a FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJqPMEVôVOBHÌSFB #V GJSNB CV BMö WFSJöUFO EBJNB [BSBS FUUJôJOF HÌSF N OJO BMBCJMFDFôJ LBÀ GBSLM UBN TBZ EF- OOCVMVOEVôVFOHFOJöBSBMLBöBôEBLJMFSEFO ôFSJWBSES IBOHJTJEJS A) [ - > # <-6, -1 ] C) [ -12, -6 ] \" # $ % & D) [ -12, -1 ] E) [ -1, 12 ] 1. $ 2. $ 3. $ 44 4. \" 5. # 6. D
%FOLMFNWF&öJUTJ[MJL4JTUFNMFSJ YAZILI SORULARI 1. ( a - 2) x2 - 2ax - 1 < 0 4. y FöJUTJ[MJôJYJOIFSHFSÀFLTBZEFôFSJJÀJOTBôMB- –4 3 x OZPSTBBOOCVMVOEVôVFOHFOJöBSBMôCVMV- O OV[ y = f(x) rx !3JÀJOFöJUTJ[MJLTBôMBOZPSJTFÔ<WFB- 2 < 0 PMNBM ôFLJMEFZ=G Y QBSBCPMÐOÐOHSBGJóJWFSJMNJõUJS B2 -p B- 2 ) ( -1 ) <0 jB2 +B- 2 < B-2 < 0 #VOBHÌSF a –2 1 2 ^ x + 2 h f^ x h a2 + a – 2 + –+ + #0 a–2 – – – + x2 – 16 FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ B! ( -2, 1 ) x+2=0 x2 - 16 = 0 G -4 ) =G = 0 x = -2 x = 4, x = -4 2. x2 · ^ 3 – x h5 x –4 –2 3 4 #0 ^ 5 – x h3 (x + 2) f(x) + + – + – x2m FöJUTJ[MJôJOJTBôMBZBOYUBNTBZMBSOOLÑNFTJ- OJZB[O[ ÇK = [ -2, 3] , ( 4, 3 ) x 035 x2 (3 – x)5 + +– + (5 – x)3 5. x2 – 4x > 0 YUBNTBZMBSOOLÑNFTJ{ 0, 3, 4 } x+4 x2 – 16 > 0 x2 – 2x – 8 FöJUTJ[MJLTJTUFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ 3. ( a - 5 ) x2 - 4ax + a - 3 = 0 x2 - 4x = 0 x+4=0 x2 - 16 = 0 x = 4, x = -4 EFOLMFNJOJO LÌLMFSJOJO UFST JöBSFUMJ PMBCJMNFTJ x = 4, x = 0 x = -4 JÀJOBOOEFôFSBSBMôOCVMVOV[ x2 - 2x - 8= 0 a–3 x = 4, x = -2 5FSTJöBSFUMJJLJLÌLJÀJOY1 · x2 < 0 <0 –4 –2 0 4 a– 5 – + +–+ x + – +++ Ô>PMNBMB2 - B- B- 3 ) > 0 x2 – 4x OFHBUJGPMEVôVCJMJOJZPS x+4 a 35 x2m + –+ x2 – 2x – 8 a–3 a–5 B! ( 3, 5 ) ÇK = ( -2, 0 ) , ( 4, 3 ) 1. (–2, 1) 2. {0, 3, 4} 3. (3, 5) 45 4. <m >b ß 5. (–2, 0) b ß
YAZILI SORULARI %FOLMFNWF&öJUTJ[MJL4JTUFNMFSJ 6. 9 # ( x - 1 )2 # 49 9. FöJUTJ[MJLTJTUFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ y y = f(x) ã Y- 1)2 ( x - 1)2 ã –1 O x ã Y- 1)2 - 9 ( x - 1)2 -ã 3 ã Y- 4) · ( x + 2) ( x - 8) · ( x + ã y x m –2 4 8 –2 O 1 x (x – 4) (x + 2) + + – + + y = g(x) Y Ym + – – – + ÇK= [ -6, -2 ] , [ 4, 8 ] ôFLJMEFG Y WFH Y GPOLTJZPOMBSOOHSBGJLMFSJWF- SJMNJõUJS 7. x2 -YZ= 44 #VOBHÌSF YZ-Z2 = 19 ( x2 +Y pG Y pH Y $ 0 PMEVôVOBHÌSFY-ZJGBEFTJOJOQP[JUJGEFôFSJOJ FöJUTJ[MJôJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ CVMVOV[ x2 + x = 0 j x = 0, x = -1 x2 -YZ= 44 G =G -1 ) = 0 (x =ÀJGULBUM H -2 ) =H =H = 0 + -YZ-Z2 = 19 x –2 –1 0 1 3 x2 -YZ+Z2 = 25 (x2 + x)·f(x)·g(x) – + ++– – ( x -Z 2 = 25 x -Z= 5 ÇK = [ -2, 1 ] , { 3 } 8. x2–16 #0 10. Z $ x2 - 3x - 4 2x2 – 12x + 18 ZâY+ 1 YpZ$ 0 FöJUTJ[MJôJOJTBôMBZBOYUBNTBZEFôFSMFSJOJCV- FöJUTJ[MJLTJTUFNJOJOLPPSEJOBUEÑ[MFNJOEFCF- MVOV[ MJSUUJôJCÌMHFZJBOBMJUJLEÑ[MFNEFHÌTUFSJOJ[ x2 – 16 x=4 4 çift katl› =0& y = x2 – 3x – 4 y x=–4 1 2x2 - 12x + 18 = 0 j 2·(x – 3)2 –1 O 4 x y=x+1 x =ÀJGULBUM |x2m] –4 3 4 2x2 – 12x + 18 + ++ + ÇK = { -4, 4 } 6. <m m>b< > 7. 5 8. {–4, 4} 46 9. <m >b {3}
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