#VLJUBCOIFSIBLLTBLMESWF\":%*/:\":*/-\"3*OBBJUUJSTBZMZBTBOOIÐLÐNMFSJOF HËSFLJUBCOEÐ[FOJ NFUOJ TPSVWFõFLJMMFSJLTNFOEFPMTBIJ¿CJSõFLJMEFBMOQZBZNMBOB- NB[ GPUPLPQJZBEBCBõLBCJSUFLOJLMF¿PóBMUMBNB[ :BZO4PSVNMVTV $BO5&,÷/&- %J[HJ–(SBGJL5BTBSN *4#//P \"ZEO:BZOMBS%J[HJ#JSJNJ :BZOD4FSUJGJLB/P #BTN:FSJ ÷MFUJöJN &SUFN#BTN:BZO-UEõUJr \":%*/:\":*/-\"3* JOGP!BZEJOZBZJOMBSJDPNUS 5FMr 'BLT 0533 051 86 17 aydinyayinlari aydinyayinlari * www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 ·/÷7&34÷5&:&)\";*3-*, %¸O¾P.DSDáñ ÜNİVERSİTEYE HAZIRLIK 2. MODÜL MATEMATİK - 2 Alt bölümlerin Karma Testler ³İKİNCİ DERECEDEN DENKLEMLER EDĜOñNODUñQñL©HULU KARMA TEST - 1 ÷LJODJ%FSFDFEFO%FOLMFNMFS&öJUTJ[MJL1BSBCPM Modülün sonunda ³PARABOL tüm alt bölümleri ³EŞİTSİZLİKLER 1. x2 - 6x + 4 = 0 5. x2 + 2x + 12 = 7 L©HUHQNDUPDWHVWOHU x2 + 2x \\HUDOñU EFOLMFNJOJO LÌLMFSJOEFO CJSJ BöBôEBLJMFSE FO IBOHJTJEJS EFOLMFNJOJO LÌLMFSJOEFO CJSJ BöBôEBLJMFSEFO IBOHJTJEJS A) 2 - 5 B) 5 - 2 C) 3 - 5 ³ İkinci Dereceden Denklemler - I t 2 D) 5 - 3 E) 4 - 5 A) -3 B) -2 C) 2 D) 3 E) 4 ³ İkinci Dereceden Denklemler - II t 11 ³ İkinci Dereceden Denklemler - III t 21 ³ İkinci Dereceden Denklemler - IV t 24 ³ Parabol - I t 28 6ñQñIð©LðĜOH\\LĜ 2. ( a2 + 2a )2 - 18 ( a2 + 2a ) + 45 = 0 ³ Parabol - II t 36 %XE¸O¾PGHNL¸UQHN EFOLMFNJOJO FO CÑZÑL JMF FO LÑÀÑL LÌLÑOÑO 6. x + 34 - x = 4 VRUXODUñQ©¸]¾POHULQH UPQMBNLBÀUS EFOLMFNJOJO HFSÀFM TBZMBSEBLJ ÀÌ[ÑN LÑNFTJ ³ Parabol - III t 42·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 www.aydinyayinlari.com.tr DNñOOñWDKWDX\\JXODPDVñQGDQ BöBôEBLJMFSEFOIBOHJTJEJS XODĜDELOLUVLQL] ³ Parabol - IV t 48 ÷,÷/$÷%&3&$&%&/%&/,-&.-&3* A) -2 B) -1 C) 0 D) 1 E) 2 A) { -9 } B) { 2 } C) { –9, 2 } d) 5x2 - 3x = D) { 2, 9 } E) { 9 } x ( 5x - 3 ) = 0 ³ Eşitsizlikler - I t 56÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ%FOLMFN 33 TANIM x = 0 ve x = j Ç.K = { 0, } 55 ³ Eşitsizlikler - II t 61a, b, c `3WFBáPMNBLÐ[FSF e) 2x2 = -3x ³ Eşitsizlikler - III t 67ax2 + bx + c = 2x2 + 3x = 0 x ( 2x + 3) = 0 CJ¿JNJOEFLJB¿LËOFSNFMFSFJLJODJEFSFDFEFO 33 x = 0 ve x = - j Ç.K = { 0, - } ³ Eşitsizlikler - IV t 74CJSCJMJONFZFOMJEFOLMFNEFOJS 22 3. x + 4 x = 20 7. B`3PMNBLÑ[FSF <HQL1HVLO6RUXODU #VFõJUMJóJTBóMBZBOYHFS¿FLTBZMBSOBEBCV 0RG¾O¾QJHQHOLQGH\\RUXP ³ Karma TestlEeFrOLMFtNJOi7L8ÌLMFSJuEFOJS <(1m1(6m/6258/$5EFOLMFNJOJO HFSÀFM TBZMBSEBLJ ÀÌ[ÑN LÑN FTJ \\DSPDDQDOL]HWPHYE EHFHULOHUL¸O©HQNXUJXOX ³ Yeni Nesil Sorular t 87 BöBôEBLJMFSEFOI÷BLOJOHDJTJJ%EJFSSFDFEFO%FOLMFNMFS&öJUTJ[MJL1BSaB2CP+M3a + a2 + 3a + 5 = 7 VRUXODUD\\HUYHULOPLĜWLU ÖRNEK 1 $\\UñFDPRG¾OVRQXQGD A) { -5, 4 } 1B.) {\"4I}NFU#FZõFCL)JME{ 2F5W6F,S6JMF2O5 }BMBONP2MEPVMBôOVEOJBL-HÌSF B42.+(BÐO+Fõ )UPBQOMBNNhOO HOJUNEFôTJF SHJF SFLFO ZPMVO CJS LTN WDPDPñ\\HQLQHVLOVRUXODUGDQ NWFOHFS¿FLTBZMBSPMNBLÐ[FSF ROXĜDQWHVWOHUEXOXQXU N- 3 ) x3 + 2x3 -O + 4x - 3 = EËSUHFO CJ¿JNJOEFLJ CPõ BSTBZB CJS LFOBLSBÀYUSNFUSF BTGBMU CJSLTNUPQSBLUS\"TGBMUZPMLN UPQSBL D) { 256 } PMBO LBSFEC)J¿{J6N2J5OE}F CJS FW ZBQUSNBL JTUJZPS :BQ- ZPMLNV[VOMVóVOEBES\"TGBMUZPMEBLJI[ UPQ- MBOFWJOпUBSBGOEBõFSNFUSFCJSUBSBAG)O4EBNB) 8 CS)B9LZPMEBLDJ)I1[0OEBOE) 1L2NTEBIBGB[MBES CPõMVLCSBLMNõUS :PMDVMVLUPQMBNTBBUTÑSEÑôÑOFHÌSF UPQSBL EFOLMFNJ JLJODJ EFSFDFEFO CJS EFOLMFN PMEVôVOB f) x2 - 3x + 2 = 20 m ZPMEBLJI[TBBUUFLBÀLNEJS HÌSF N+OUPQMBNLBÀUS 1 20 m x m 20 m \" # $ % & N- 3 = 0 jN= 3 (x - 2) (x - 1) = 0 xm 3 -O= 2 jO= 1 x = 2 ve x = 1 j Ç.K = { 1, 2} N+O= 3 + 1 = 4 4. 3 x2 - 6 3 x + 8 = 0 x+2 2 x+2 8. f p - 4f p+3 = 0 x-1 x-1 EFOLMFNJOJOHFSÀFMLÌLMFSJOJOUPQMBNLBÀ3U5Sm EFOLMFNJOJTBôMBZBOYEFôFSJLBÀUS A) 6 B) 8 C#) V2O4BHÌSDF) 6F4WJOCJSEL)F7O2BSV[VOMVôVLAB)À3NFUSF- B) 2 5 7 C) E) ÖRNEK 2 g) x2 + 5x + 6 = D) 3 dir? 2 2 2 (x + 3) (x + 2) = 0 $OW%¸O¾P7HVWOHUL \"öBôEBLJ EFOLMFNMFSJO HFSÀFL TBZMBSEBLJ ÀÌ[ÑN x = -3 ve x = -2 j Ç.K = { -3, -2 } \" # $ % & Her alt bölümün LÑNFMFSJOJCVMVOV[ TEST - 1 ÷LJODJ%FSFDFEFO%FOLMFN,BWSBNWF¦Ì[ÑNÑ VRQXQGDRE¸O¾POHLOJLOL 1. C 2. A 3. D 4. E 78 5. A 6. B 7. E 8. C a) x2 - 4 = WHVWOHU\\HUDOñU 5. #JS TPLBóO EPóSVTBM PMBO ZPMVOVO SFGÐKÐ Ð[FSJOF ( x - 2 ) (x + 2) = 0 1. 3x2 - x + m - 1 = 0 5. Y`3PMNBLÑ[FSF FõJUBSBMLMBSMBBZEOMBUNBEJSFLMFSJEJLJMFDFLUJS x = 2 ve x = -2 x - 4 x = 2 Ç.K. = { -2, 2 } JEÀFJOOLNMFNOJFOJPOMNFöBJUMEJLJSHFSÀFLLÌhL)Ñ-OxÑ2O+CxVM+VO2N=BT EFOLMFNJOJOLBÀUBOFHFSÀFLLÌLÑWBSES \"SU BSEB EJLJMFO JLJ EJSFL BSBT NFTBGF UPQMBN EJSFL TBZTOB FöJU WF CBöUBLJ EJSFL JMF TPOEBLJ b) 4x2 - 36 = A) 11 B) 1 C) 13 D-) ( x72 - x - E2 )) 5 A) 0 B) 1 C) 2 D) 3 E) 4 EJSFLBSBTV[BLMLNFUSFPMEVôVOBHÌSF EJ- 12 12 =0 LJMFOEJSFLTBZTLBÀUS 63 2. #JS PLVMVO TBUSBO¿ UBLNOB TF¿JMFO L[ WF FSLFL -( x - 2 ) ( x + 1 ) = 0 ËóSFODJMFSEFO PMVõUVSVMBO LJõJMJL HSVQUBLJ L[ \" # $ % & ËóSFODJMFSJO TBZT JMF FSLFL ËóSFODJMFSJO TBZTOO 4 ( x - 3 ) ( x + 3) = 0 x = 2 ve x = -1 j Ç.K = { -1, 2 } ¿BSQNES x = 3 ve x = -3 j Ç.K. = { -3, 3 } 2x2 + 5x - 3 = (SVQUBLJ FSLFL ÌôSFODJMFSJO TBZT EBIB GB[MB PMEVôVOBHÌSF HSVQUBLBÀL[ÌôSFODJWBSES c) 3x2 + 6 = ( 2x -1 ) ( x + 3 ) = 0 \" # $ % & 3x2 = -6 11 x2 = -2 j Ç.K = Ø x = ve x = - 3 j Ç.K = { -3, } 22 6. Y`3PMNBLÑ[FSF 2. #JS\"#$пHFOJOEF\"LËõFTJOEFO#$LFOBSOB¿J[J- MFOZÐLTFLMJL#$LFOBSOEBODNEBIBLTBES 2x + 1 + x = 7 b)L\"{F–#3O$,B3}SÑOÀcBH) ØFBOJUJOPJMOBOBMZBÑOLTFLDMN2JL2LPBMÀEVdD)ôN{V0EO, JB53S HÌSF #$3 EFOLMFNJOJO ÀÌ[ÑN L1ÑNFTJ BöBôEBLJMFSEFO 6. ax2 + bx +D=JLJODJEFSFDFEFOEFOLMFNEF 1. 4 2. a) {–2,2} } e) { 0, - } f) { 1, 2 } g) { –3, –2 } h) { –1, 2} { –3, } * D JTFEFOLMFNJOTBOBMJLJLËLÐWBSES IBOHJTJEJS 2 ** C JTFEFOLMFNJOHFS¿FLJLJLËLÐWBSES 2 3. #JS\"#$пHFOJOEF\"LËõFTJOEFO#$LFOBSOB¿J[J- *** B WF D JTF EFOLMFNJO HFS¿FL JLJ LËLÐ WBSES A) 2 + 5 B) 1 + 5 C) 5 - 1 A) Ø B) { 4 } C) { 12 } | |MFOZÐLTFLMJL #$ LFOBSOEBOCSEBIBLTBES D) 5 - 2 E) 7 - 1 D) { 4, 12 } E) { 4, 7 } | | \"#$ÑÀHFOJOJOBMBOCS2PMEVôVOBHÌSF BC LFOBSOBBJUZÑLTFLMJLLBÀCSEJS \" # $ - 1 + 3 ZVLBSEBLJ JGBEFMFSEFO IBOHJMFSJ EBJNB EPôSV- dur? % 1 + 3 & - 1 + 2 \" :BMO[* # :BMO[*** $ *WF** % *WF*** & * **WF*** 3. ( x2 - x )2 + 8 ( x - x2 ) + 12 = 0 7. Y`3PMNBLÑ[FSF 1. B 2. C 3. C 87 4. A 5. C 6. D x - 1 + x + 4 = 5 EFOLMFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO EFOLMFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO IBOHJTJEJS IBOHJTJEJS A) { -2, 3 } B) { -1, 2 } C) { -2, -1, 2, 3 } A) { 2 } B) { 3 } C) { 4 } D) Ø E) R D) { 5 } E) { 5, 6 } 4. 9x + 27 =x + 1 8. Y`3PMNBLÑ[FSF EFOLMFNJOJOLÌLMFSUPQMBNLBÀUS A) 0 B) 1 C) 2 D) 3 | |x2 + x - 2 = 0 C C C % EFOLMFNJOJOLÌLMFSÀBSQNLBÀUS E) 4 A) -2 B) -1 C) 1 D) 2 E) 4 8 B B % B
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·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/%&/,-&.-&3* ÷LJODJ%FSFDFEFO#JS#JMJONFZFOMJ%FOLMFN d) 5x2 - 3x = TANIM a, b, c `3WFBáPMNBLÐ[FSF x ( 5x - 3 ) = 0 ax2 + bx + c = 33 CJ¿JNJOEFLJB¿LËOFSNFMFSFJLJODJEFSFDFEFO CJSCJMJONFZFOMJEFOLMFNEFOJS x = 0 ve x = j Ç.K = { 0, } #VFõJUMJóJTBóMBZBOYHFS¿FLTBZMBSOBEBCV 55 EFOLMFNJOiLÌLMFSJuEFOJS e) 2x2 = -3x ÖRNEK 1 2x2 + 3x = 0 NWFOHFS¿FLTBZMBSPMNBLÐ[FSF x ( 2x + 3) = 0 N- 3 ) x3 + 2x3 -O + 4x - 3 = EFOLMFNJ JLJODJ EFSFDFEFO CJS EFOLMFN PMEVôVOB 33 HÌSF N+OUPQMBNLBÀUS x = 0 ve x = - j Ç.K = { 0, - } 22 f) x2 - 3x + 2 = N- 3 = 0 jN= 3 (x - 2) (x - 1) = 0 3 -O= 2 jO= 1 x = 2 ve x = 1 j Ç.K = { 1, 2} N+O= 3 + 1 = 4 ÖRNEK 2 g) x2 + 5x + 6 = \"öBôEBLJ EFOLMFNMFSJO HFSÀFL TBZMBSEBLJ ÀÌ[ÑN (x + 3) (x + 2) = 0 LÑNFMFSJOJCVMVOV[ x = -3 ve x = -2 j Ç.K = { -3, -2 } a) x2 - 4 = h) -x2 + x + 2 = ( x - 2 ) (x + 2) = 0 x = 2 ve x = -2 -( x2 - x - 2 ) = 0 Ç.K. = { -2, 2 } -( x - 2 ) ( x + 1 ) = 0 b) 4x2 - 36 = x = 2 ve x = -1 j Ç.K = { -1, 2 } 4 ( x - 3 ) ( x + 3) = 0 x = 3 ve x = -3 j Ç.K. = { -3, 3 } c) 3x2 + 6 = 2x2 + 5x - 3 = 3x2 = -6 ( 2x -1 ) ( x + 3 ) = 0 x2 = -2 j Ç.K = Ø 11 x = ve x = - 3 j Ç.K = { -3, } 22 2 33 1 1. 4 2. a) {–2,2} b) {–3,3} c) Ø d) { 0, } e) { 0, - } f) { 1, 2 } g) { –3, –2 } h) { –1, 2} { –3, } 52 2
www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, J 6x2 - 7x - 3 = 7$1,0%m/*m ( 3x + 1 ) ( 2x - 3) = 0 ax2 + bx + c =EFOLMFNJOEFÔ= b2 - 4ac PMNBL Ð[FSF 13 13 x = - ve x = j Ç.K = { - , } 32 32 Ô> 0JTFEFOLMFNJOCJSCJSJOEFOGBSLMJLJLÌLÑ WBSES#VLËLMFS x1,2 = -b ! T CBóOUTJMFCVMVOVS 2a j) x2 - 3ax + 2a2 = Ô= 0JTFEFOLMFNJO (x - 2a) (x - a) = 0 x = 2a ve x = a j Ç.K = { a, 2a } a) &öJUJLJLÌLÑWBSES b) ÷GBEFCJSUBNLBSFEJS D %FOLMFNJOÀÌ[ÑNLÑNFTJCJSFMFNBOMES %FOLMFNJOLËLMFSJ x1 = x2 = - b ES 2a Ô JTFEFOLMFNJOHFSÀFLLÌLMFSJZPLUVS L ax2 + (a - 1) x - 1 = (ax - 1) (x + 1) = 0 ÖRNEK 3 11 \"öBôEBLJ EFOLMFNMFSJO HFSÀFL TBZMBSEBLJ ÀÌ[ÑN x = a ve x = -1 j Ç.K = { -1, a } LÑNFMFSJOJCVMVOV[ a) x2 - x - 3 = M ax2 + ( ab - b) x - b2 = b) 2x2 + 3x - 1 = c) x2 + 4x + 6 = ax -b d) 3x2 - 12x + 12 = xb B Ô= ( -1 ) 2 - 4.1. (-3) = 1 + 12 = 13 (ax - b) (x + b) = 0 bb x1.2 = 1 ! 13 x = a ve x = -b j Ç.K = { -b, a ) 2 b) Ô= 9 - 4 . 2 . ( -1 ) = 17 x1.2 = - 3 ! 17 4 N abx2 + (3a2 + b2) x - 6a2 - ab + 2b2 = c) Ô= 16 - 4 . 1. 6 = 16 - 24 = -8 <PMEVôVOEBO abx2 + ( 3a2 + b2 ) x-( 3a + 2b ) (2a - b) Ç.K = Ø ax - ( 2a - b ) bx ( 3a + 2b ) d) Ô= 144 - 144 = 0 12 ( ax - 2a + b ) ( bx + 3a + 2b ) = 0 x1.2 = 6 = 2 2a - b - 3a - 2b x = a ve x = b 2a - b - 3a - 2b Ç.K = { a , b } J ( - 1 3 2K \\B B^L) % - 1, 1 / M \\mC CB^N ( - 3a + 2b 2a - b 2 3 3. a) ( 1 ! 13 2 b) ( - 3 ! 17 2 c) Ø d) { 2 } , , 24 32 a ba
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 www.aydinyayinlari.com.tr ÖRNEK 4 ÷LJODJ%FSFDFEFO#JS%FOLMFNF %ÌOÑöUÑSÑMFCJMFO%FOLMFNMFS 2x2 - 3x +N- 2 = ÖRNEK 7 EFOLMFNJOJO HFSÀFM LÌLÑ PMNBEôOB HÌSF N OF PM NBMES \"öBôEBLJEFOLMFNMFSJOSFFM HFSÀFM TBZMBSEBLJÀÌ [ÑNLÑNFMFSJOJCVMVOV[ Ô<PMNBM a) x3 - x2 - 2x = 9 - N- 2 ) < 0 9 -N+ 16 < 0 x (x2 - x - 2 ) = 0 N> 25 x (x - 2 ) ( x + 1 ) = 0 x = 0, x = 2 ve x = -1 j Ç.K = { -1, 0, 2 } 25 N> 8 ÖRNEK 5 b) x2 - 3x + 2 = 0 x2 - 4 NY2 -NY+ 2 = EFOLMFNJOJO ÀÌ[ÑN LÑNFTJ CJS FMFNBOM PMEVôVOB ^ x - 2 h^ x - 1 h HÌSF NOJOBMBCJMFDFôJEFôFSMFSUPQMBNLBÀUS =0 Ô=PMNBM ^ x - 2 h^ x + 2 h N2 -N= 0 x = 2, x = 1 N N- 8 ) = 0 YâWFYâ-2 j Ç.K = { 1 } N=WFN= 8 N5PQ = 8 c) 2 - x + x - 1 = 1 x+1 x-2 2 ÖRNEK 6 1BZEBFöJUMFSTFL 3x2 - x -N+ 1 = EFOLMFNJOJOGBSLMJLJHFSÀFLLÌLÑOÑOPMNBTJÀJO N - 2 + 4x - 4 + 2 - 1 1 IBOHJBSBMLUBPMNBMES x x Ô>PMNBM 1 - 4 . 3 (-N+ 1 ) > 0 x2 - x - 2 = 2 1 +N- 12 > 0 N> 11 8x - 10 = x2 - x - 2 11 x2 - 9x + 8 = 0 N> (x - 8) (x - 1) = 0 36 x = 8 x = 1 j Ç.K = { 1, 8 } d) x4 -Y2 + 9 = x2 =UPMTVO U2 -U+ 9 = 0 U- U- 1 ) = 0 ( x2 - 9 ) ( x2 - 1 ) = 0 x=±3 x=±1 Ç.K = { -3, -1, 1, 3 } 25 11 4 7. a) { –1, 0, 2 } b) { 1 } c) { 1, 8 } d) { –3, –1, 1, 3 } 4. m > 5. 8 6. m > 8 36
www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, e) x6 - 16x3 + 64 = 4x -x + 1 + 8 = x3 =UPMTVO 2x =UPMTVO U2 -U+ 64 = 0 U2 -U+ 8 = 0 U- 8 )2 = 0 U- U- 2 ) = 0 ( x3 - 8 )2 = 0 2x = 4 2x = 2 x3 = 8 ve x = 2 x = 2 ve x = 1 Ç.K = { 2 } Ç.K = { 1, 2 } f) x3 - 2x2 - 9x + 18 = J 3x + 31 - x = 4 x2 ( x - 2 ) - 9 ( x - 2 ) = 0 3x =UPMTVO ( x2 - 9 ) (x - 2 ) = 0 3 x = ±3 ve x = 2 Ç.K = { -3, 2, 3 } t+ =4 t U2 -U+ 3 = 0 U- U- 1) = 0 3x = 3 3x = 1 x=1 x=0 Ç.K = { 0, 1 } g) ( x2 + x ) 2 - 8 ( x2 + x ) + 12 = x2 + x =UPMTVO j) x - 4 x - 72 = 0 U2 -U+ 12 = 0 4 x = t PMTVO U- U- 2 ) = 0 U2 -U- 72 = 0 ( x2 + x - 6 ) (x2 + x - 2 ) = 0 U- U+ 8) = 0 x = -3 x = -2 4 x = 9 j 4 x =-8 x=2 x=1 x = 38 q Ç.K = { -3, -2, 1, 2 } Ç.K = { 38 } h) c x 2 3x -4=0 3 x2 - 23 x-3=0 m- L x-2 2-x x 3 x = t PMTVO = t PMTVO U2 -U- 3 = 0 U- U+ 1) = 0 x-2 U2 +U- 4 = 0 3 x = 3 3 x =-1 x = 27 x = - 1 U+ U- 1 ) = 0 Ç.K = { -1, 27} d x + 4 nd x - 1 n = 0 x-2 x-2 8 q x= 5 e) { 2 } f) { –3, 2, 3 } g) { –3, –2, 1, 2 } h) ( 8 2 5 \\ ^J \\ ^K \\8^L \\m ^ 5
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 www.aydinyayinlari.com.tr M x2 - x + 2 + x2 - x - 4 = 0 | |Ì x2 + x - 6 = 0 2 Yãj x2 - x - 6 = 0 x - x + 2 = t PMTVO (x - 3) (x + 2) = 0 U2 +U- 6 = 0 x = -2 x2 + x - 6 = 0 U+ U- 2) = 0 Yäj 2 - x + 2 =-3 ve 2 - x + 2 = 2 (x + 3) (x - 2) = 0 x x Ø 2 x=2 x -x-2=0 (x - 2) (x + 1) = 0 Ç.K = { -2, 2 } x = 2 x =-1 Ç.K = { 2, -1} N 3x - 2 + x = 4 | |Q x2 = x - 2 3x - 2 = - x + 4 IFSJLJUBSBGOLBSFTJBMOSTB Yãj x2 + x - 2 = 0 3x - 2 = x2 - 8x + 16 x2 - 11x + 18 = 0 ( x + 2 ) ( x - 1) = 0 ( x - 9 ) (x - 2 ) = 0 YâY= 2 Yäj x = -2 ve x = 1 Ç.K = { 2 } x2 - x + 2 = 0 D < SFFMLÌLZPLUVS Ç.K = { -2, 1 } O x - x - 4 = 2 | | | |S x - 1 2 - x - 1 - 2 = x - x - 4 = 4 IFSJLJUBSBGOLBSFTJBMOSTB | x - 1| =UPMTVO x-4 = x-4 U2 -U- 2 = 0 x-4 = x2 - 8x + 16 x2 - 9x + 20 = 0 U- U+ 1 ) = 0 (x - 5) (x - 4) = 0 x = 5 ve x = 4 | x - 1| = 2 ve | x - 1| = -1 Ç.K = { 4, 5 } x - 1 = 2 ve x - 1 = -2 x=3 x = -1 Ç.K = { -1, 3 } P x - 2 + x + 3 = 5 | |T x2 - 3x = 2x - 4 x2 - 3x = 2x - 4 ve x2 - 3x = -2x + 4 x2 - 5x + 4 = 0 x2 - x - 4 = 0 )FSJLJUBSBGOLBSFTJBMOSTB x - 2 + 2 x2 + x - 6 + x + 3 = 25 (x - 4) (x - 1) = Ô= 1 + 16 = 17 2 + x - 6 = 12 - x x =Yâ x= 1 + 17 x≠ 1 - 17 x 22 12 22 x + x - 6 = 144 - 24x + x 1 + 17 25x = 150 Ç.K = { ,4} x=6 2 Ç.K = { 6 } M \\m ^N \\^O \\ ^P \\^ 6 Ì \\m ^Q \\m ^S \\m ^T ( 1 + 17 , 4 2 2
www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 8 ÖRNEK 10 x4 + 3x3 - 2x2 - 6x + 4 = x3 - x2 -4x + 4 = EFOLMFNJOJOFOLÑÀÑLLÌLÑLBÀUS EFOLMFNJOJOLÌLMFSJOJOFOCÑZÑLPMBOJMFFOLÑÀÑL PMBOBSBTOEBLJGBSLLBÀUS ,BUTBZMBSUPQMBNPMEVôVOEBO Y- CJSÀBSQBOES x2 (x - 1) - 4(x - 1) = 0 x4 + 3x3 – 2x2 – 6x + 4 x – 1 (x - 1) (x2 - 4) = 0 (x - 1) (x - 2) (x + 2) = 0 x4 – x3 x3 + 4x2 + 2x – 4 x = 1, x = 2, x = -2 0IBMEFFOCÑZÑLLÌL FOLÑÀÑLLÌL-PMEVôVOEBO 4x3 – 2x2 – 6x + 4 GBSLCVMVOVS 4x3 – 4x2 ÖRNEK 11 2x2 – 6x + 4 2x2 – 2x x ` Z+ PMNBL Ð[FSF LFOBS V[VOMVLBS CS EFO GBSL- M PMBO EJLEËSUHFOMFS QSJ[NBT CJ¿JNJOEFLJ CJS LVUVOVO –4x + 4 IBDNJOJWFSFOEFOLMFN –4x + 4 2x3 + 13x2 + 26x + 15 0 PMEVôVOB HÌSF CV LVUVOVO BZSUMBSOEBO FO V[VO PMBOIBOHJEFOLMFNMFJGBEFFEJMJS ÷GBEF Y- 1) (x3 + 4x2 + 2x - 4) = 0 x3 + 4x2 + 2x -JGBEFTJOJOÀBSQBOMBSOEBOCJSJ (x + PMEVôVOEBO x3 + 4x2 + 2x – 4 x+2 x3 + 2x2 x2 + 2x – 2 2x2 + 2x – 4 2x2 + 4x –2x – 4 –2x – 4 0 ÷GBEF Y- 1) (x + 2) (x2+ 2x - 2) =CJÀJNJOFEÌOÑ öÑS#VSBEBLÌLMFSY= 1, x = 2 ve x2+ 2x - 2 =EFOL MFNJOEF D = 4 – 4 (–2) = 12 x= - 2 ± 12 = - 1 ± 3 CVMVOVS 1,2 2 0IBMEFFOLÑÀÑLLÌL - 1 - 3 CVMVOVS ÖRNEK 9 x = -JÀJO-2 + 13 - 26 + 15 =PMEVôVOEBO Y+ 1) CJSÀBSQBOES x3 - 6x2 + 11x - 6 = EFOLMFNJOJOFOCÑZÑLLÌLÑLBÀUS 2x3 + 13x2 + 26x + 15 x+1 2x3 + 2x2 2x2 + 11x + 15 11x2 + 26x + 15 ,BUTBZMBSUPQMBNPMEVôVOEBO Y- CJSÀBSQBOES 11x2 + 11x x3 – 6x2 Ymæ x – 1 (x - 1) . (x2 - 5x + 6) = 0 15x + 15 15x + 15 x3 – x2 x2 – 5x + 6 (x - 1) (x - 2) (x - 3) = 0 0 –5x2 + 11x – 6 x = 1, x = 2, x = CV –5x2 + 5x MVOVS Ymæ &OCÑZÑLLÌLUÑS (x + 1) (2x3+ 11x + 15) = (x + 1) (x + 3) (2x + 5) 6x – 6 FOV[VOBZSUY+UJS 0 8. - 1 - 3 9. 3 7 10. 4 11. 2x + 5
TEST - 1 ÷LJODJ%FSFDFEFO%FOLMFN,BWSBNWF¦Ì[ÑNÑ 1. 3x2 - x +N- 1 = 5. x `3PMNBLÑ[FSF EFOLMFNJOJOFöJUJLJHFSÀFLLÌLÑOÑOCVMVONBT x-4 x=2 JÀJONOFPMNBMES EFOLMFNJOJOLBÀUBOFHFSÀFLLÌLÑWBSES A) 11 B) 1 C) 13 D) 7 E) 5 \" # $ % & 12 12 6 3 2. #JS\"#$пHFOJOEF\"LËõFTJOEFO#$LFOBSOB¿J[J- 6. x `3PMNBLÑ[FSF MFOZÐLTFLMJL#$LFOBSOEBODNEBIBLTBES 2x + 1 + x = 7 \"#$ÑÀHFOJOJOBMBODN2PMEVôVOBHÌSF #$ EFOLMFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO LFOBSOBBJUPMBOZÑLTFLMJLLBÀDNEJS IBOHJTJEJS A) 2 + 5 B) 1 + 5 C) 5 - 1 A) Ø B) { 4 } C) { 12 } D) 5 - 2 E) 7 - 1 D) { 4, 12 } E) { 4, 7 } 3. ( x2 - x )2 + 8 ( x - x2 ) + 12 = 7. x `3PMNBLÑ[FSF x-1+ x+4=5 EFOLMFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO EFOLMFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO IBOHJTJEJS IBOHJTJEJS A) { -2, 3 } B) { -1, 2 } C) { -2, -1, 2, 3 } A) { 2 } B) { 3 } C) { 4 } D) Ø E) R D) { 5 } E) { 5, 6 } 4. 9x + 27 =x + 1 8. x `3PMNBLÑ[FSF EFOLMFNJOJOLÌLMFSUPQMBNLBÀUS | |x2 + x - 2 = EFOLMFNJOJOLÌLMFSÀBSQNLBÀUS \" # $ % & A) -2 B) -1 C) 1 D) 2 E) 4 1. $ 2. $ 3. $ 4. D 8 5. # 6. # 7. D 8. #
÷LJODJ%FSFDFEFO%FOLMFN,BWSBNWF¦Ì[ÑNÑ TEST - 2 1. x2 - x - 3 x2 - x - 2 6 x2 - x = 0 5. x2 + 4x + 2 = EFOLMFNJOJOLBÀSFFMLÌLÑWBSES EFOLMFNJOJO LÌLMFSJOEFO LÑÀÑL PMBOl BöBôEB LJMFSEFOIBOHJTJEJS A) 1 B) 2 C) 3 D) 4 E) 5 A) - 2 - 2 B) - 2 + 2 C) 2 - 2 E) 4 + 2 D) 2 + 2 | |2. x = - x 6. x2 - 6x + 2 = EFOLMFNJOJO ÀÌ[ÑN BSBMô BöBôEBLJMFSEFO EFOLMFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO IBOHJTJEJS IBOHJTJEJS A) \" 3 - 2 7, 3 + 2 7 , B) \" 3 - 2 3, 3 + 2 3 , C) \" 6 - 2 7, 6 + 2 7 , D) \" 3 - 2 6, 3 + 2 6 , A) R– B) R+ C) R+ b\\^ E) \" 3 - 7, 3 + 7 , D) R– b\\^ & 3-\\^ 3. x4 - 4x2 - 2 = 7. NâPMNBLÑ[FSF EFOLMFNJOJOLBÀUBOFSFFMLÌLÑWBSES N- 1) x2 + N+ 2) x +N2 +N- 5 = \" # $ % & EFOLMFNJOJOLÌLMFSJOEFOCJSJPMEVôVOBHÌSF NLBÀUlS A) -4 B) -1 C) 2 D) 3 E) 4 4. x6 + 7x3 - 8 = 8. #JSEJLEËSUHFOJOV[VOLFOBSLTBLFOBSOEBODN EFOLMFNJOJOLBÀUBOFSFFMLÌLÑWBSES GB[MBES #V EJLEËSUHFOJO LTB LFOBS JLJ LBUOB ¿- LBSMS V[VO LFOBS DN LTBMUMSTB BMBO DN2 A) 2 B) 3 C) 4 D) 5 E) 6 B[BMS #VOBHÌSF JMLEJLEÌSUHFOJOLTBLFOBSLBÀDN EJS \" # $ % & 1. D 2. D 3. $ 4. A 9 5. A 6. & 7. A 8. #
TEST - 3 ÷LJODJ%FSFDFEFO%FOLMFN,BWSBNWF¦Ì[ÑNÑ 1. x2 - 6x - 5 = 5. BâPMNBLÑ[FSF EFOLMFNJOJO ÀÌ[ÑN LÑNFTJ BöBôEBLJMFSEFO ax 2 - x + a - 8 = IBOHJTJEJS EFOLMFNJOJOLÌLMFSJOEFOCJSJBPMEVôVOBHÌSF A) \" 3 + 14, 3 - 14 , B) \" 14, - 14 , EJÚFSLÌLÑLBÀUlS C) \" 6 + 2 14, 6 –2 14 , D) { } A) - 2 B) - 3 C) - 1 D) 2 E) 3 2 3 2 E) {5, 1} 2. 9x2 - 6x + 1 = ^ x - a h2 6. N2 -UN+ 2 = 9 EFOLMFNJOJOFöJUJLJLÌLÑOÑOPMNBTJÀJOQP[JUJG UEFôFSJLBÀPMNBMES FöJUMJôJOJTBôMBZBOBEFôFSJLBÀUS A) -1 B) - 1 C) - 1 D) 1 E) 1 A) 1 B) 3 C) 2 D) 2 6 E) 5 3 93 9 7. x2 - 2x + a - 1 = 3. x2 - 5x + 2 = EFOLMFNJOJO GBSLM JLJ HFSÀFL LÌLÑOÑO PMNBT JÀJO B OO EFôFS BSBMô BöBôEBLJMFSEFO IBOHJ EFOLMFNJOJOEJTLSJNJOBOULBÀUlS TJEJS A) a > 4 B) a > 2 C) a > A) 16 B) 17 C) 24 D) 25 E) 29 D) a < & B< 2 4. x2 - 6x +N= 8. a < -PMNBLÑ[FSF (a + 1) x2 + ( 2a + 1) x + a - 4 = EFOLMFNJOJOEJTLSJNJOBOUPMEVÚVOBHÌSF N EFOLMFNJOJOFúJUJLJHFSÀFLLÌLÑPMEVÚVOBHÌ LBÀUlS SF BLBÀUlS A) 1 B) 3 C) 2 5 E) 3 A) - 1 B) - 4 C) - 17 D) 2 5 20 22 D) - 17 E) - 9 16 8 1. A 2. D 3. # 4. $ 10 5. # 6. D 7. & 8. D
www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÷,÷/$÷%&3&$&%&/%&/,-&.-&3** Karmaşık Sayılar ÖRNEK 3 TANIM \"öBôEBLJEFOLMFNMFSJOÀÌ[ÑNLÑNFMFSJOJCVMVOV[ Y Z` R, i 2å=å-PMNBLÐ[FSF [å=Yå+åJZCJ¿J- a) x2 + 9 = b) x2 + x + 2 = NJOEFLJ TBZMBSB LBSNBöL LPNQMFLT TBZ MBSEFOJS c) x2 - 6x += d) x2 - 2x + 4 = Y [ LBSNBõL TBZTOO HFS¿FM SFFM LTNES a) x2 = -9 j x =J WF3F [ å=YJMFHËTUFSJMJS C Ô= 1 - 4 . 1 . 2 = -7 j x1, 2 = -1! -7 -1!i 7 Z [LBSNBõLTBZTOOTBOBM JNBKJOFS LTN- = ESWFIN [ å=ZJMFHËTUFSJMJS 2 2 $å=å{ x +JZY Z`å3 J2 = -1} 6 ± - 4 6 ± 2i LÐNFTJOF LBSNBöL LPNQMFLT TBZMBS LÑ D Ô= 36 -4.1.10 = -4 j x1, 2 = = =J 2 2 NFTJEFOJS E Ô= 4 - 4.1.4 = -12 a ` RJTF - a2 = i a ES j x1, 2 = 2 ± - 12 2 ± 2i 3 =J 3 = 2 2 ÖRNEK 1 ,BSNBöL4BZMBSO&öJUMJôJ \"öBôEBLJLBSNBöLTBZMBSOSFFMWFJNBKJOFSLTN 7$1,0%m/*m MBSOCVMVOV[ [1 = x +JZWF[2 = a +JCLBSNBõLTBZMBSJ¿JO a) [1 = 2 + 5i b) [2 = 3 + 4i [1 =[2 l ( x =BWFZ=C EJS c) [3 = -2i + 1 d) [4 = -4 e) z5 = 2 + 1 f) [6 = -4i ÖRNEK 4 B 3F [ = 2 , IN [ = 5 [1 = 3a -JWF[2 = 6 +CJJ¿JO C 3F [ = 3 , IN [ = 4 [1 =[2 PMEVôVOBHÌSF B C OFEJS D 3F [ = 1 , IN [ = -2 E 3F [ = -4 , IN [ = 0 3a = 6 -4 = 2b e) 3F [ = 2 + 1 , IN [ = 0 G 3F [ = 0 , IN [ = -4 a=2 b = -2 ( a, b ) = ( 2, -2 ) ÖRNEK 2 ÖRNEK 5 \"öBôEBLJTBZMBS JNBKJOFSTBZCJSJNJZMFZB[O[ a < b <PMNBLÑ[FSF - a2 + 2a - 1 + 2 a2 = 3 + - b2 + 4b - 4 PMEVôVOBHÌSF B C TSBMJLJMJTJOFEJS a) - 9 b) - 5 c) - 4 d) - 12 | a - 1 |J+ 2 | a | = 3 + | b - 2 |J | a - 1 | = | b - 2 | ve 2 | a | = 3 B J C 5 i -a + 1 = -b + 2 -2a = 3 c) J d) 2 3 i 3 b=a+1 1 a =- 2 b =- 2 d- 3 ,- 1 n 22 1. a) 2, 5 b) 3, 4 c) 1, -2 d) -4, 0 e) 2 + 1, 0 f) 0 , -4 11 3) a) {J -J} b) ( - 1 i 7 2 c) {J} d) {1±i 3 } 4. (2,-2) 5. (- - 2)B JC 5 iD JE 2 3 i ± 22
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 www.aydinyayinlari.com.tr J4BZTOO,VWWFUMFSJ ÖRNEK 8 7$1,0%m/*m P ( x ) = 2x + 3x99 - 4x + 2x84 PMEVôVOBHÌSF 1 J5 EFôFSJOFEJS O`/J¿JO i = i4 = i8 == i O = 1 1 J5 ) =1 J =J101 +J99 -J110 +J84 i1 = i5 = i9 == i O+ 1 = i =J+J3 -J2 + 2 i2 = i6 = i == i O+ 2 = -1 =J-J+ 4 + 2 i3 = i7 = i11 == i O+ 3 = -JPMVS = 6 -J ÖRNEK 6 ÖRNEK 9 \"öBôEBLJJGBEFMFSJOFöJUJOJCVMVOV[ - 3 . - 5 . - 15 a) i + i31 + i32 - 25 . - 8 . 2 b) i121 - i122 + i123 - i124 JGBEFTJOJOFöJUJOFEJS c) i + i2 + i3 + i4 + i5 ++ i d) i2 + i4 + i6 + i8 ++ i 3 i. 5 i. 15 i 15i 3i e) i + i3 + i5 + i7 ++ i121 == 5.4 4 B J2 -J3 +J4 = -1 -J+ 1 = -J 5i.2 2 i. 2 C J1 -J2 +J3 -J0 =J+ 1 -÷- 1= 0 c) 1i 4-414-2i +4 414+3 . . . . + i1 = i ÖRNEK 10 0 ( 1 - i3 ) (1 - i6 ) ( 1 - i9 ) … ( 1 - i300 ) JGBEFTJOJOFöJUJOFEJS d) -1 + 1 - 1 + 1 - ... -1 + 1 = 0 e) 9i - i + i - i + i - i + . . . . . + i = i ^ 1 + i h^ 1 + 1 h^ 1 - i h^>1 - 1 h . . . . ^ 1 - 1 h = 0 0 0 UYARI ÖRNEK 7 ( 1 + i )2 = 2i L`/PMNBLÑ[FSF (1 - i )2 = -2i JL+ 8 +JL+ 1 +JL+ 3 EFôFSJOFEJS ÖRNEK 11 J0 +J+J3 ( 1 + i )10 JGBEFTJOJOFöJUJOFEJS = 5 +J-J= 5 +J [ ( 1 +J 2 ]5 = J 5 =J5 =J 6. a) -JC D JE F J7. 5 +J 12 3i 10. 0 11.J 8. 6 -J9. 4
www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 12 ÷LJODJ%FSFDFEFO%FOLMFNMFSJO,ÌLMFSMF ,BUTBZMBS\"SBTOEBLJ#BôOUMBS ( 1 - i )9 LBSNBöLTBZTOOJNBKJOFSLTNLBÀUS [ ( 1 -J 2 ]4 . ( 1 -J = [ -J]4 . ( 1 -J TANIM = 16( 1 -J = 16 -J ax2 + bx + c =EFOLMFNJOJOLËLMFSJY1 ve x2 IN [ = -16 JTF ,BSNBöL4BZOO&öMFOJôJ r x1 + x2 = - b a r x1Y2 = c a 3 r x1 - x2 = a 7$1,0%m/*m ÖRNEK 14 [= x +JZLBSNBõLTBZTJ¿JO 2x2 - 4x + 1 =EFOLMFNJOJOLËLMFSJY1 ve x2EJS z = x - iy TBZTOB[OJOFöMFOJôJEFOJS #VOBHÌSF BöBôEBLJJGBEFMFSJOEFôFSMFSJOJCVMVOV[ ²[FMMJLMFS ^ z h = z, z1 + z2 = z1 + z2 a) x1 + x2 b) x1Y2 | |c) x1 - x2 f z1 p = z1 d) 1 + 1 e) x12 + x22 f) 1 + 1 z2 xx h) x1 + x2 2 2 z1.z2 = z1.z2, z2 ^ z2 ≠ 0h x 1 x 2 12 g) x31 + x33 ÖRNEK 13 b4 a) a = 2 b = -4 c = 1 j = x1 + x2 = - a = 2 = 2 \"öBôEBLJLBSNBöLTBZMBSOFöMFOJLMFSJOJCVMVOV[ c1 b) a = 2 b = -4 c = 1 j x1.x2 = a = 2 T 16 - 4.2.1 2 2 c) x1 - x2 = a = = =2 2 2 a) [= 5 + 3i b) [= -2 + i x +x 4 c) [= -5i + 1 d) [= -114 e) [= 6 + 2 f ) [= ^ - 7 + 5i h 1 1 12 2 =4 g) [= -3i h) [= 5 d) x +x = = x .x 1 12 12 2 e) 22 = ( x1 + x2 )2 -2x1.x2 j 22 - 2. 1 =4-1=3 2 x +x 12 22 x +x 11 3 f) + = 12 = = 12 a) z = 5 - 3i 2 2 ^ x .x 2 h2 d 1 2 b) z = - 2 - i c) z = 5i + 1 x x 1 n d) z = - 114 e) z = 6 + 2 1 2 2 f) z = - 7 + 5i H [=J g) x 3 + 3 =(x1+x2)3-3x1x2(x1+x2) = 3 - 3. 1 .2 = 8-3 = 5 I [= 5 1 2 x 2 2 h) a x+ x2 k 2 =x +x +2 x .x 1 12 12 a x+ x 2 = 2 + 2 1 2 1 2 k x + x = 2+ 2 12 12. -16 13. a) 5-JC -2-JD +JE -114 13 1 e) 6+ 2 f) -7+JH JI 14. a) 2 b) c) 2 d) 4 e) 3 f) 12 g) 5 h) 2 + 2 2
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 www.aydinyayinlari.com.tr ÖRNEK 15 ÖRNEK 18 3x2 - 2x - 4 =EFOLMFNJOJOLÌLMFSJY1 ve x2PMEV x2 -NY+N+ 1 = ôVOBHÌSF EFOLMFNJOJOLÌLMFSJOEFOCJSJ-PMEVôVOBHÌSF EJ ( 2x - 3 ) ( 2x - 3 ) ôFSJLBÀUS 1 2 %JôFSLÌLY1PMTVO x - 2 = 2m JGBEFTJOJOFöJUJLBÀUS 1 (2x1 - 3) (2x2 - 3) = 4x1.x2 - 6 ( x1+ x2 ) + 9 - 2/ x .^ - 2 h = m + 1 = 4.d - 4 n - 6d 2 n + 9 33 1 16 12 27 1 5x - 2 = - 2 =- - + =- 333 3 1 x1 = 0 ÖRNEK 16 ÖRNEK 19 2x2 - x - 3 =EFOLMFNJOJOLÌLMFSJY1 ve x2PMEV x2 - N- 1 ) x +N= ôVOBHÌSF EFOLMFNJOJO LÌLMFSJOJO HFPNFUSJL PSUBMBNBT PM EVôVOBHÌSF BSJUNFUJLPSUBMBNBTLBÀUS x 2 . x32 + x 3 . x22 G.O = x1.x2 = & 2m = 2 1 1 JGBEFTJOJOFöJUJLBÀUS 2m = 4 & m = 2 x +x 2m - 1 3 12 A.O = = = 3 2 1 9 2 22 ^ x .x h2 ^ x + x h = d - 2 2 = 8 12 1 2 n. ÖRNEK 17 ÖRNEK 20 x2 - N+ 1) x +N=EFOLMFNJOJOLÌLMFSJBSBTOda, NY2 - N- 1 ) x - 2 = EFOLMFNJOJOTJNFUSJLJLJLÌLÑOÑOCVMVONBTJÀJON x21 + x 2 = 5 OFPMNBMES 2 1 CBôOUTOOCVMVONBTJÀJONOJOBMBCJMFDFôJEFôFS 3m - 1 = 0 & m = MFSLÑNFTJOFEJS 3 ^ x + x h2 - 2x .x = 5 & ^ m + 1 h2 - 2m = 5 UYARI 1 2 12 ax2 + bx + c = EFOLMFNJOJO TJNFUSJL JLJ LËLÐOÐO CVMVONBT J¿JO 2 + 1 = 5 x1+ x2 =WFY1Y2 <PMNBMES :BOJYMJUFSJNJOLBUTBZTPMBOCOJOTGSPMNBTHF- m SFLJS m =\"2 {-2, 2} 1 9 14 18. 0 3 1 15. - 16. 17. { –2, 2 } 19. 20. 3 8 2 3
www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 21 ÖRNEK 24 2x2 - 4x +N- 3 =EFOLMFNJOJOLÌLMFSJY1 ve x2 2x2 -NY+O=EFOLMFNJOJOLÌLMFSJBSBTOEB PMNBLÑ[FSF x 2 + x 2 1 2 =2 x1 . x2 2x - x = 4 1 2 PMEVôVOBHÌSF NOFPMNBMES CBôOUTWBSTBOJONDJOTJOEFOFöJUJOFEJS 2x - x = 4 ^ x + x 2 h2 - 2x .x 2 12 1 1 =2 + x +x =2 x .x 1 2 ,ÌLEFOLMFNJTBôMBS 12 3x = 6 &d m 2 n = 2. n 1 2.4 - 4.2 +N- 3 = 0 jN= 3 n - 2. x =2 2 22 1 22 mm = 2n & n = 48 ÖRNEK 22 ÖRNEK 25 x2 - 4x + m - 2 = 0 4 x2 -OY+N=EFOLMFNJOJOCJSLËLÐ-2, 3x2 - 2x + 3m - 1 = 0 x2 +QY+ q =EFOLMFNJOJOCJSLËLÐUÐS #VJLJEFOLMFNJOEJôFSLÌLMFSJPSUBLPMEVôVOBHÌSF EFOLMFNMFSJOJOCJSFSLÌLMFSJPSUBLPMEVôVOBHÌSF N 2m + q LBÀUS LBÀUS - 3/ 2 - 4x + m - 2 = 0 n+p x + 2 - 2x + 3m - 1 = 0 3x 10x + 5 = 0 0SUBLLÌLY1PMTVO 2/ x .^ - 2 h = m 1 x -2=n 1 x =- 1 2 x .4 = q -/ x +4 =-p ,ÌLEFOLMFNJTBôMBS +1 1 11 -6=n+p 0 = 2m + q + 2 +m- 2 = 0 & m =- 44 2m + q 0 = =0 n+p -6 ÖRNEK 23 ÖRNEK 26 x2 -NY+N- 4 = x2 - 3x + 1 =EFOLMFNJOJOLËLMFSJY1 ve x2PMNBLÐ[F- SF x21 - 4x1 - x2 LBÀUS EFOLMFNJOJOLËLMFSJY1 ve x2PMTVO x1 ve x2BSBTOEBLJCBôOUOFEJS x + x = 2m ,ÌLMFSEFOLMFNJTBôMBS 12 2 - 2/ x .x = m - 4 x - 3x = - 1 ve x + x = 3 11 12 12 x + x - 2x .x = 8 2 - 4x - x = 2 - 3x - ^ x + x h = - 1 - 3 = - 4 12 12 x 1 2 x 1 1 2 1 1 2x .x + 8 = x + x 12 12 21. 3 1 23. 2x1.x2 + 8 = x1 + x2 15 2 25. 0 26. –4 22. - m 4 24. 8
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 www.aydinyayinlari.com.tr ÖRNEK 27 ÖRNEK 30 x2 - N+ 1 ) x + 3 =EFOLMFNJOJOLËLMFSJY1 ve x2EJS x2 - 3x +N=EFOLMFNJOJOLËLMFSJY1 ve x2EJS x21 - x22 = 6 8 – x1 = 1 x2 + 1 PMEVôVOBHÌSF NLBÀUS PMEVôVOBHÌSF NLBÀUS 8 - x .x - x |x1 - x2| . |x1 + x2| = 6 8 12 1 D ·d - b n=6 -x =1 & a a =1 x +1 1 x +1 2 2 &8-3-x =x +1 9 - 4m · 3 = 6 5 12 9 -N= 4 jN= 5 jN= 5-1=x +x 4 12 4=m+1 m=3 ÖRNEK 31 ÖRNEK 28 x2 - N- 4 ) x - 15 =EFOLMFNJOJOLËLMFSJY1 ve x2EJS x2 + ax = 2x +EFOLMFNJOJOLËLMFSJY1 ve x2EJS x1 = -PMEVôVOBHÌSF N- x2GBSLLBÀUS x1 = x22 x1 . x2 = -15 ve x1= -3 j x2= 5 PMEVôVOBHÌSF LÌLMFSUPQMBNLBÀUS x2 + ( a - 2 ) x - 8 = 0 x + x2=N- 4 = 2 1 x1.x2 = - 8 N= 6 jN= 3 2 x2 = -JTFY1= 4 N- x2 = 3 - 5 = -2 x2.x2 = - 8 x1 + x2 = 2 3 x2 =-8 x2 =-2 ÖRNEK 29 ÖRNEK 32 x2 + N- 1 ) x +O=EFOLMFNJOJOLÌLMFSJ x2 - 3x - 1 =EFOLMFNJOJOLÌLMFSJOEFOCJSJY1PM x2 + N+ 1 ) x -N- 3 = EVôVOBHÌSF EFOLMFNJOJOLÌLMFSJOJOöFSLBUJTFOLBÀUS x13 - x12 - 7x1 - 4 EFôFSJLBÀUS 2x + 2x = - 3m + 1 4x1.x2 = n 3 - 3x 2 + 2x 2 - 6x -x -4 12 x - 2/ x + x = - m - 1 1 1 1 11 + 12 - 4/ x1.x2 = - 6 + 0=-m+3 22 m=3 = x (x - 3x ) + 2 (x - 3x ) - x - 4 0 = n + 24 1 1 441 2 4413 1 441 2 4413 1 n = - 24 11 = x1 + 2 - x1 - 4 = -2 27. 3 28. 2 29. –24 16 5 31. –2 32. –2 30. 4
÷LJODJ%FSFDFEFO%FOLMFNJO,BSNBöL4BZ,ÌLMFSJ TEST - 4 1. i18 - 4i16 + 3i21 5. [= ( -3 +J + i ) JöMFNJOJOTPOVDVBöBôEBLJMFSEFOIBOHJTJEJS LBSNBöLTBZTOOTBOBMLTNLBÀUS A) i B) - å+åJ $ -5 A) -8 B) -4 C) -1 D) 1 E) 8 D) -5 -åJ & 6. x2 - 6x + 13 = 2. 5 - 32 . 3 - 27 . - 25 EFOLMFNJOJO LPNQMFLT TBZMBS LÑNFTJOEFLJ LÌLMFSJOEFOCJSJBöBôEBLJMFSEFOIBOHJTJEJS JöMFNJOJOTPOVDVOFEJS & J A) 3 + i B) 3 - 2i C) 1 + 3i A) - # - 5i C) - % J D) 1 - 3i E) -3 - 3i 3. 1 Y å= x3å- 3x2å+åYPMNBLÐ[FSF 7. x2 - 2x + 4 =EFOLMFNJJMFJMHJMJ P ( 1 -J JGBEFTJOJOEFôFSJBöBôEBLJMFSEFOIBO * öLJGBSLMTBOBMLËLÐWBSES HJTJEJS ** 4BOBMLËLMFSJOEFOCJSJ1 - 3 iEJS *** ,ËLMFSJCJSCJSJOJOFõMFOJóJEJS A) 1 + i B) 1 - i C) -i JGBEFMFSJOEFO IBOHJTJ ZB EB IBOHJMFSJ EPôSV EVS D) i E) 1 \" :BMO[* # :BMO[** $ *WF** D) I ve III E) I, II ve III 4. ^ - 12 - - 3 h2 + ^ - 48 - - 27 h2 8. 5PQMBNMBS WF ÀBSQNMBS PMBO JLJ TBZEBO JöMFNJOJOTPOVDVOFEJS CJSJBöBôEBLJMFSEFOIBOHJTJEJS A) - 3i B) -6i C) - % & \" å-J # å+åJ$ -å-åJ % å-åJ & å+åJ 1. # 2. & 3. A 4. $ 17 5. D 6. # 7. & 8. #
TEST - 5 ÷LJODJ%FSFDFEFO%FOLMFNJO,ÌLMFSJWF,BUTBZMBS\"SBTOEBLJ÷MJöLJ 1. 2x2 - x - 3 =EFOLMFNJOJOLÌLMFSJY1 ve x2 5. 2x2 -NY+N+ 1 = PMNBLÑ[FSF EFOLMFNJOJOLÌLMFSJOEFOCJSJ-PMEVôVOBHÌ SF EJôFSJLBÀUS ( 3x1 - 1 ) ( 3x2 - 1 ) ÀBSQNOOFöJUJLBÀUS A) 1 B) 1 C) 3 D) 2 E) 5 22 2 A) - # -12 C) -14 D) -16 E) -18 2. 3x2 - 2x - 1 =EFOLMFNJOJOLÌLMFSJY1 ve x 2 PMEVôVOBHÌSF 6. x2 -NY+N- 4 = x13. x2 + 2x12 . x22 + x23 . x1 EFOLMFNJOJO LÌLMFSJOJO BSJUNFUJL PSUBMBNBT PMEVôVOBHÌSF HFPNFUSJLPSUBMBNBTLBÀUS JGBEFTJOJOFöJUJLBÀUS A) 1 B) 2 C) 3 D) 2 E) 3 A) - 2 B) - 1 C) - 2 9 27 27 D) - 4 27 E) - 1 9 3. x2 -NY+N- 1 =EFOLMFNJOJOLÌLMFSJBSBTO 7. N+ 1 ) x2 -NY+N- 3 = da, EFOLMFNJOJOTJNFUSJLJLJLÌLÑWBSTBNLBÀUS 1 + 1 =5 2 2 4 x 1 x 2 A) -2 B) - $ % & CBôOUTWBSTBNOJOBMBCJMFDFôJEFôFSMFSUPQMB NLBÀUS \" # $ % & 8. 3x2 - 6x +N- 1 = 4. 2 - 1 - 1 = 0 EFOLMFNJOJOY1 ve x2LÌLMFSJBSBTOEB x2 x EFOLMFNJOJO LÌLMFS UPQMBN BöBôEBLJMFSEFO 3x1 - x2 =CBôOUTWBSTBNLBÀUS IBOHJTJEJS \" # $ % & A) -3 B) -1 C) 1 D) 2 E) 3 1. $ 2. D 3. A 4. # 18 5. A 6. # 7. $ 8. #
÷LJODJ%FSFDFEFO%FOLMFNJO,ÌLMFSJWF,BUTBZMBS\"SBTOEBLJ÷MJöLJ TEST - 6 1. x2 -NY+N- 2 = 4. x2 - N- 1 ) x + 8 +N= EFOLMFNJOJOLÌLMFSJBSBTOEB NZFCBôMPMNB EFOLMFNJOJO LÌLMFSJ GBSL PMEVôVOB HÌSF N ZBOCJSCBôOUBöBôEBLJMFSEFOIBOHJTJEJS OJOBMBCJMFDFôJEFôFSMFSUPQMBNLBÀUS A) x1 + x2 + 3x1 x2 = 6 A) 7 B) 6 C) 5 D) -5 E) -6 B) x1 + x2 - x1 x2 = 3 C) x1 + x2å- 2x1 x2 = 3 D) x1 + x2 - 3x1 x2 = 6 E) x1 + x2 + x1 x2 = 6 2. x2 - 2x + m - 1 = 0 4 5. x 2 - 12 x + 8 = x2 - x + 2m + 1 = 0 EFOLMFN JO JOLËLMFSJBWFCEJS EFOLMFNMFSJOJOCJSFSLÌLMFSJPSUBLPMEVôVOBHÌ SF NOJOBMBCJMFDFôJEFôFSMFSUPQMBNLBÀUS #VOBHÌSF 1 + 12 A) -8 B) -7 C) -6 D) -4 E) -3 a+ 9 a+ 1 b JGBEFTJOJOEFôFSJLBÀUS A) - # $ % & 3. Ná Y2 + 4x +N=WFY2 +NY+ 4 =EFOL- 6. x2 +QY+ q =EFOLMFNJOJOCJSLËLÐ MFNMFSJOJOCJSFSLËLMFSJPSUBLUS x2 +NY+O=EFOLMFNJOJOCJSLËLÐ-EJS ÷MLEFOLMFNJOPSUBLPMNBZBOLÌLÑLBÀUS #V JLJ EFOLMFNJO EJôFS LÌLMFSJ PSUBL PMEVôVOB A) -5 B) -4 C) -3 D) -2 E) -1 HÌSF Q-NLBÀUS A) -4 B) -3 C) -2 D) 2 E) 4 1. D 2. # 3. A 19 4. # 5. D 6. A
TEST - 7 ÷LJODJ%FSFDFEFO%FOLMFNJO,ÌLMFSJWF,BUTBZMBS\"SBTOEBLJ÷MJöLJ 1. x 2 - 4 x +N= 5. x2 - (a + b) x + 2a + b = EFOLMFNJOJOLËLMFSJY1 ve x2PMEVóVOBHËSF EFOLMFNJO JOLËLMFSJY1 ve x 2EJS NOJOIBOHJEFôFSJJÀJOY1 . x 2 + x 1 + x 2 = 7 ,ÌLMFSBSBTlOEB PMVS A) -3 B) -2 C) 2 D) 3 E) 4 x 1 + x 2 = 9 ve 1+1= 9 x1 x2 10 CBôOUMBSPMEVôVOBHÌSF CLBÀUS A) -8 B) - $ % & 2. x 2 + U- 1 ) x + 8 = 6. ,FOBS V[VOMVLMBS DN WF DN PMBO EJLEËSU- EFOLMFN JO JOLËLMFSJY1 ve x2EJS HFOCJ¿JNJOEFLJLBSUPOVOLËõFMFSJOEFOCJSLFOBSY ,ÌLMFSBSBTlOEBY1 = x22 CBôOUTPMEVôVOBHÌ DNPMBOLBSFMFSLFTJMJQ¿LBSMZPS SF ULBÀUS A) -5 B) -4 C) -3 D) 5 E) 6 x x x x 80 cm xx 100 cm xx (FSJZFLBMBOLBSUPOVOBMBODN2PMEVôVOB 3. x 2 - N+ 4) x +N- 2 = HÌSF YLBÀUS EFOLMFN JO JOLËLMFSJY1 ve x 2EJS \" # 10 2 C) 16 D) 16 2 & 6x1 = x1 - 6 x2 PMEVôVOBHÌSF NLBÀUS 7. Z2 -Z+L= A) 26 B) 12 C) -18 D) -24 E) – 26 EFOLMFNJOJOLËLMFSJZ1WFZ2EJS y21 + y22 = 5PMEVôVOBHÌSF LLBÀUS A) 1 B) 3 C) 2 D) 5 E) 3 22 4. ax2 + bx + 6 =EFOLMFNJOJOLËLMFSJY1 ve x2EJS 8. a2 + 5a +N- 2 = * a, b ` R EFOLMFNJOJOLËLMFSJB1 ve a2EJS ** x1 + x2 = 3 a 2 + a1.a2 = 5 PMEVôVOBHÌSF NLBÀUS *** x1Y2 = –3 2 PMEVôVOBHÌSF B+CUPQMBNLBÀUS A) - # $ % & A) 6 B) 5 C) 4 D) 3 E) 2 1. D 2. A 3. & 4. $ 20 5. D 6. # 7. $ 8. A
www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÷,÷/$÷%&3&$&%&/%&/,-&.-&3*7 ,ÌLMFSJ7FSJMFO÷LJODJ%FSFDFEFO%FOLMFNJO ÖRNEK 3 :B[MNBT ,ÌLMFSJOEFOCJSJ 1 PMBOEFSFDFEFOSBTZPOFM 7$1,0%m/*m 3-2 ¥Ë[ÐNLÐNFTJ{ x1, x2 }PMBOEFSFDFEFOEFOLMFN LBUTBZMEFOLMFNJOJCVMVOV[ x2 - ( x1 + x2 ) x + x1Y2 =ES 1 -2- 3 x = = =-2- 3 1 -2+ 3 4-3 (-2- 3) ÖRNEK 1 ise x = - 2 + 3 2 \"öBôEBÀÌ[ÑNLÑNFMFSJWFSJMFOEFSFDFEFOEFOL MFNMFSJZB[O[ x +x =-4 12 x .x = 1 12 2 x + 4x + 1 = 0 a) { -2, 3 } b) *- 1 , 2 c) { 3 } 4 23 d) \" 0, 2 , ÖRNEK 4 a) x + x = 1 ve x . x = -6 j x2 - x - 6 = 0 ,ÌLMFSJ Y2 - 4x + 1 =EFOLMFNJOJOLÌLMFSJOEFO ÑÀFSFLTJLPMBOEFSFDFEFOEFOLMFNJCVMVOV[ 1 2 1 2 11 b) x1 + x2 = 6 ve x .x = - 12 3 x1 + x2 = 4 x1.x2 = 1 x2 - 1 x- 1 =0 & 2 - x - 2 = 0 63 6x c) x1 + x2 = 6 ve x1 . x2 = 9 j x2 - 6x + 9 = 0 ,ÌLMFSJY1 - 3 ve x2 -PMBOEFOLMFNJÀJO T = x1+ x2 - 6 = 4 - 6 = - 2 d) x1 + x2 = 2 ve x1.x2 = 0 j x2 - 2 x = 0 Ç = x . x - 3( x + x ) + 9 = 1 - 12 + 9 = -2 1 2 1 2 x2 + 2x - 2 = 0 UYARI ÖRNEK 5 3BTZPOFMLBUTBZMEFSFDFEFOCJSEFOLMFNJOLËL- x2 + 3x - 2 =EFOLMFNJOJOLÌLMFSJY1 ve x2JTFLÌL MFSJOEFOCJSJp - q JTFEJóFSJp + q EVS MFSJ Y1 + 2 ve 3x +PMBOEFSFDFEFOEFOL 2 MFNJCVMVOV[ x1 + x2 = -3 ÖRNEK 2 x . x = -2 ,ÌLMFSJOEFO CJSJ 3 - 2 PMBO SBTZPOFM LBUTBZM 1 2 EFSFDFEFOEFOLMFNJCVMVOV[ ,ÌLMFSJY1 + 2 ve 3x2 +PMBOEFOLMFNJÀJO T = 3( x1 + x2 ) + 4 = 3 . ( -3 ) + 4 = -5 Ç = ( 3x + 2) (3x + 2 ) 1 2 x = 3 - 2 ve x = 3 + 2 = 9x1 . x2 + 6 ( x1 + x2 ) + 4 12 x +x =6 = 9 . ( -2 ) + 6 ( -3 ) + 4 12 x .x = ^ 3 - 2 h ^ 3 + 2 h = 9 - 2 = 7 = -18 -18 + 4 = -32 x2 + 5x - 32 = 0 12 2 - 6x + 7 = 0 x 1. a) x2 – x – 6 = 0 b) 6x2– x – 2 = 0 c) x2 – 6x + 9 = 0 21 3. x2 + 4x + 1 = 0 4. x2 + 2x – 2 = 0 5. x2 + 5x – 32 = 0 2 2x = 0 2. x2 – 6x + 7 = 0 d) x -
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 www.aydinyayinlari.com.tr ÖRNEK 6 ÖRNEK 9 3x2 - 4x - 1 =EFOLMFNJOJOLËLMFSJY1 ve x2EJS x2 - 4x - 7 = ,ÌLMFSJ 3 ve 3 PMBO EFSFDFEFO EFOLMFNJ EFOLMFNJOJOLÌLMFSJOJOFSGB[MBTOLÌLLBCVMFEFO x1 x2 JLJODJEFSFDFEFOEFOLMFNJCVMVOV[ CVMVOV[ 4 x1 + x2 = 4 x +x = x1 . x2= -7 1 23 x .x = - 1 ,ÌLMFSJY1+ 3 ve x +PMBOEFOLMFNJÀJO 12 3 2 4 T = x1+ 3 + x2+ 3 = 4 + 6 = 10 3^ x1 + x2 h 3. Ç = (x1 + 3) (x2 + 3) = -7 + 12 + 9 = 14 3 x2 - 10x + 14 =CVMVOVS T= = = - 12 x .x 1 12 - 3 99 = - 27 j x2 + 12x - 27 = 0 Ç= = x .x 1 - 12 3 ÖRNEK 7 ÖRNEK 10 ,ÌLMFSJBSBTOEB x2 - 2x - 4 =EFOLMFNJOJOLËLMFSJY1 ve x2EJS x1+ x2 + x1Y2= 11 3x1 + 3x2 + 3 = x1Y2 x21 2 1+1 2 x1 x2 CBôOUT CVMVOBO JLJODJ EFSFDFEFO EFOLMFNJ CVMV OV[ ,ÌLMFSJ + x ve PMBO EFOLMFNJ CVMV ^ x 1 + x 2 h + x .x 2 = 11 1 OV[ 3^ x + x h - x .x = - 3 1 2 12 + x + x = 2 1 2 4 ( x + x) = 8 x1 . x2 = -4 1 2 22 x1 + x2 = 2 ,ÌLMFSJ x + x ve 11 x1 . x2 = 9 j x2 - 2x + 9 = 0 + PMBOEFOLMFNJÀJO 12 xx 12 2 + x 2 = ^ x + x 2 h2 - 2x .x 2 2 x 1 1 1 = 4 + 8 = 12 ÖRNEK 8 1 1 x +x 2 1 ++ = =- #JSLÌLÑY1 = 4 + 3 PMBOJLJODJEFSFDFEFOSBTZP 12 OFMLBUTBZMEFOLMFNJCVMVOV[ xx x .x -4 2 12 12 1 23 T = 12 - = 22 x =4+ _ Ç = 12 . d - 1 n = - 6 3 bb 2 1 ` j T = 8, Ç = 13 2 23 3 bb %FOLMFN - x-6=0 x =4- a x 2 2 %FOLMFNY2- 8x + 13 = 0 2x2 - 23x - 12 =CVMVOVS 6. x2 + 12x – 27 = 0 7. x2 – 2x + 9 = 0 8. x2– 8x + 13 = 0 22 9. x2 – 10x + 14 = 0 10. 2x2 – 23x – 12 = 0
,ÌLMFSJ7FSJMFO÷LJODJ%FSFDFEFO%FOLMFNJO:B[MNBT TEST - 8 1. KÌLMFSJOEFOCJSJ2 2 - 1PMBO SBTZPOFMLBUTBZ 4. ÷LJODJ EFSFDFEFO CJS EFOLMFNJO LÌLMFSJ BSBTO MEFSFDFEFOEFOLMFNBöBôEBLJMFSEFOIBOHJ EB TJPMBCJMJS 3x ( 2 - x) + x ( 6 - x ) = 2 A) x2 + 2x - 7 = # Y2 - 2x - 7 = 1 2 2 1 C) x2 + 2x + 7 = % Y2 + 2x - 5 = x1 ( 2 - x2 ) + x2 ( 2 + 3x1 ) = 14 E) x2 + 2x - 9 = CBôOUMBS CVMVOEVôVOB HÌSF CV EFOLMFN BöBôE BLJMFSEFOIBOHJTJEJS A) x2 - 4x + 3 = B) x2 + 4x + 3 = C) x2 - 3x + 4 = % Y2 - 3x - 4 = E) x2 + 3x + 4 = 2. x2 - 2x - 4 = 5. ,ÌLMFSJOEFO CJSJ 3 – 2 PMBO JLJODJ EFSFDFEFO EFOLMFNJOJOLÌLMFSJY1 ve x2JTFLÌLMFSJ SBTZPOFM LBUTBZM EFOLMFN BöBôEBLJMFSEFO 2x1 - 1 ve 2x2 -PMBOJLJODJEFSFDFEFOEFOL IBOHJTJEJS MFNBöBôEBLJMFSEFOIBOHJTJEJS A) x2 - 19x - 2 = B) x2 + 2x + 19 = A) x2 + 4x + 1 = # Y2 + 4x - 1 = C) x2 - 2x + 19 = % Y2 + 2x - 19 = C) x2 - 4x + 1 = % Y2 - 2x + 1 = E) x2 - 2x - 19 = E) x2 - 2x - 1 = 3. x2 - 2x - 1 =EFOLMFNJOJOLËLMFSJY1 ve x2EJS 6. \"SJUNFUJLPSUBMBNBMBS HFPNFUSJLPSUBMBNBMB ,ÌLMFSJ 2 ve 2 PMBOEFSFDFEFOEFOLMFN S 15 PMBOJLJTBZZLÌLLBCVMFEFOJLJODJEF x1 x2 SFDFEFOEFOLMFN BöBôEBLJMFSEFOIBOHJTJEJS BöBôEBLJMFSEFOIBOHJTJEJS 2 2 A) x - 6x + 15 = 0 B) x - 6x + 15 = 0 A) x2 - 4 = # Y2 + 4x + 4 = 2 2 C) x2 - 4x + 4 = % Y2 + 4x - 4 = C) x - 12x + 15 = 0 D) x - 12x + 15 = 0 E) x2 - 4x - 4 = 2 E) x + 12x + 15 = 0 23 1. A 2. & 3. D 4. $ 5. A 6. D
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/%&/,-&.-&3*7 ÷LJODJ%FSFDFEFO÷LJ#JMJONFZFOMJ%FOLMFN ÖRNEK 3 4JTUFNMFSJOJO¦Ì[ÑNÑ x2 + y2 = 7 4 TANIM x2 - 2y2 = - 5 B C D E F G`3PMNBLÐ[FSF EFOLMFNTJTUFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ ax2 +CYZ+DZ2 +EY+FZ+G= x2 = 7 -Z2ZB[MSTB CJ¿JNJOEFLJEFOLMFNMFSF JLJODJEFSFDFEFOJLJ 7 -Z2 = -5 jZ2 = 12 jZ2 = 4 CJMJONFZFOMJEFOLMFNEFOJS Z= ± 2 x2 = 3 j x = ! 3 #VEFOLMFNJTBóMBZBO Y Z JLJMJMFSJEFOLMFNJO Ç .K = % a - 3, - 2 k,a - 3, 2 k,a 3, - 2 k,a 3, 2 k / ¿Ë[ÐNLÐNFTJOJPMVõUVSVS ÖRNEK 1 ÖRNEK 4 x2 +Z2 - 4x +Z+ 13 = x+y = 3 EFOLMFNJOJTBôMBZBO Y Z JLJMJTJOJCVMVOV[ x2 + 2xy = 5 4 22 x - 4x + 4 + y + 6y + 9 = 0 EFOLMFNTJTUFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ (x - 2) 2 + ^ y + 3 h2 = 0 Z= 3 -YZB[MSTB x-2=0 y+3=0 x2 + 2x ( 3 - x ) = 5 j x2 + 6x - 2x2 = 5 jx2 - 6x + 5 = 0 j ( x - 5)(x -1) = 0 x=2 y = - 3 & Ç .K = \" ^ 2, - 3 h , x=5 x=1 TANIM x = 5 jZ= -2 x = 1 jZ= 2 öLJCJMJONFZFOJ¿FSFOCJSJODJEFSFDFEFOFOB[JLJ Ç.K = { ( 5, -2 ) , ( 1, 2 )} EFOLMFNJOPMVõUVSEVóVTJTUFNF CJSJODJEFSFDF- EFOJLJCJMJONFZFOMJEFOLMFNTJTUFNJEFOJS ÖRNEK 5 %FOLMFNMFSEFO FO B[ CJS UBOFTJ JLJODJ EFSFDF- 2x - y = 3 EFO JTF TJTUFNF JLJODJ EFSFDFEFO JLJ CJMJONF ZFOMJEFOLMFNEFOJS x2 + y2 = 5 4 %FOLMFNMFSJO PSUBL ¿Ë[ÐN LÐNFTJ EFOLMFN EFOLMFNTJTUFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ TJTUFNJOJO¿Ë[ÐNLÐNFTJEJS Z= 2x -ZB[MSTB ÖRNEK 2 x2 + 4x2 -12x + 9 = 5 x+y = 5 5x2 - 12x + 4 = 0 x2 - y2 = 15 4 ^ 5x - 2 h^ x - 2 h = 0 & x = 2 EFOLMFNTJTUFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ ,x=2 5 ^>x - y h^>x + y h = 15 35 2 11 x+y=5 x = & y =- + x-y=3 55 2x = 8 j x = 4 jZ= 1 j Ç = {( 4, 1 )} x=2 & y=1 Ç .K = ( d 2 11 n,^ 2, 1 h 2 ,- 55 1. ( 2, –3 ) 2. { ( 4, 1) } 24 3. Ç .K = \" ^ - 3, - 2 h,^ - 3, 2 h,^ 3, - 2 h,^ 3, 2 h , 4.{ ( 5, –2 ), ( 1, 2 ) } 5. ( d 2 11 n,^ 2, 1 h 2 ,- 55
www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 6 ÖRNEK 9 x2 + 2x - y = 4 4 5PQMBNMBS LBSFMFSJGBSLPMBOJLJTBZEBOCÑZÑL - 4x + y = - 5 PMBOLBÀUS EFOLMFNTJTUFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ x+y=8 2 2 - 2 = 48 & ^>x - y h^>x + y h = 48 x + 2x - y = 4 x y + - 4x + y = - 5 68 2 - 2x + 1 = 0 x -Z= 6 + x +Z= 8 x ^ x - 1 h2 = 0 & x = 1 & y = - 1 2x = 14 j x = Z= 1 Ç .K = \" ^ 1, - 1 h , ÖRNEK 7 ÖRNEK 10 x+y = 4 x+y=5 4 x.y = 6 x2 + y2 = 3 4 EFOLMFNTJTUFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ EFOLMFNTJTUFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ y=5-x Z= 4 -YZB[MSTB x2 + ( 4 - x )2 = 3 j2x2 - 8x + 13 = 0 x^ 5 - x h = 6 & 2 - 5x + 6 = 0 Ô= 64 - 4 . 2 . 13 Ô= -40 <PMEVôVOEBOSFFMLÌLZPLUVS x Ç.K = Ø ^ x - 2 h^ x - 3 h = 0 x=2 x=3 x = 2 & y = 3 4Ç .K = \" ^ 2, 3 h, ^ 3, 2 h , x=3&y=2 ÖRNEK 8 ÖRNEK 11 x2 - y2 = 48 4 x.y = 0 x +y=6 EFOLMFNTJTUFNJOJOÀÌ[ÑNLÑNFTJOJCVMVOV[ x2 - 4x - 5 = y 4 Z= 6 - | x |ZB[MSTB TJTUFNJOJOÀÌ[ÑNLÑNFTJLBÀFMFNBOMES 2 - 36 + 12 x - 2 = 48 x =JÀJOZ= -5 j (0, -5) x x Z=JÀJOY2- 4x - 5 = 0 (x - 5) (x + 1) = 0 12 x = 84 x = 5 ve x = - 1 (5, 0) , (-1, 0) x = 7 & x =!7 & y =-1 ¦Ì[ÑNLÑNFTJFMFNBOMES Ç .K = \" ^ - 7, - 1 h^ 7, - 1 h , 25 9. 7 10. \" ^ 2, 3 h, ^ 3, 2 h , 11. 3 6. \" ^ 1, - 1 h , 7. Ø 8. \" ^ - 7, - 1 h,^ 7, - 1 h ,
TEST - 9 ÷LJODJ%FSFDFEFO÷LJ#JMJONFZFOMJ%FOLMFN4JTUFNMFSJOJO¦Ì[ÑNÑ 1. ÷LJTBZOOGBSL LBSFMFSJGBSLPMEVôVOBHÌ 4. 2x2 + 3y2 = 29 4 SF CVTBZMBSOÀBSQNLBÀUS x2 + y2 = 14 A) -4 B) -3 C) -2 D) 3 E) 4 EFOLMFN TJTUFNJOJO ÀÌ[ÑN LÑNFTJOEFLJ FMF NBOMBSEBOCJSJBöBôEBLJMFSEFOIBOHJTJEJS A) ( 1, 3 ) B) ( -1, 3 ) C) ^ 1, 13 h D) ^ 1, - 13 h E) ^ - 13, - 1 h 2. x2 - y2 - 2xy = - 14 4 5. x2 - y2 = 9 4 y - 3x = 0 3x2 + y2 = 55 EFOLMFN TJTUFNJOJ TBôMBZBO Y EFôFSMFSJOEFO EFOLMFNTJTUFNJOJOÀÌ[ÑNLÑNFTJOEFLJZFMF CJSJBöBôEBLJMFSEFOIBOHJTJEJS NBOMBSOEBOCJSJBöBôEBLJMFSEFOIBOHJTJEJS A) -2 B) - $ % & A) - 7 B) - 5 C) - 3 D) 1 E) 3 3. x2 - y2 + 2x - 3 = 0 4 6. x2 - x + y2 - y = 0 4 y2 + 3x - 3 = 0 x+y = 1 EFOLMFNTJTUFNJOJTBôMBZBOFOCÑZÑLZEFôFSJ EFOLMFNTJTUFNJOJTBôMBZBOFOLÑÀÑLYEFôFSJ LBÀUS LBÀUS A) 13 B) 15 C) 17 D) 19 E) 21 A) -2 B) - $ % & 1. # 2. # 3. & 26 4. & 5. A 6. $
÷LJODJ%FSFDFEFO÷LJ#JMJONFZFOMJ%FOLMFN4JTUFNMFSJOJO¦Ì[ÑNÑ TEST - 10 1. x.y = 3 4 4. a2 + b2 = 80 4 x+y = 4 3a - b = 4 EFOLMFNTJTUFNJOJOÀÌ[ÑNLÑNFTJBöBôEBLJ EFOLMFNTJTUFNJOEFCLBÀPMBCJMJS MFSEFOIBOHJTJEJS A) -4 B) -2 C) 2 D) 4 E) 8 A) { ( 1, 3 ) , ( -1, -3 ) } B) { (-1, -3 ), ( 3, 1 ) } C) { (-3, -1 ), (3, 1 ) } D) { (1, 3 ), ( 3, 1 ) } E) { (-1, 3 ), ( 1, -3 ) } 5. a2 + b2 = 13 _ b 1 1 13 b a2 b2 36 ` 2. a2 + 3b2 = 7 4 + = bb a 3a2 - 2b2 = 10 EFOLMFNTJTUFNJOJTBôMBZBOBEFôFSMFSJOEFOCJ EFOLMFNTJTUFNJOJTBôMBZBOBEFôFSMFSJOEFOCJ SJBöBôEBLJMFSEFOIBOHJTJEJS SJBöBôEBLJMFSEFOIBOHJTJEJS A) -3 B) -2 C) - % & A) -4 B) -1 C) 2 D) 4 E) 6 3. a2 + b2 + 4ab = 52 4 6. ^ x - y h2 - 2^ x - y h - 3 = 0 4 a2 + b2 - ab = 12 2x + y = 4 EFOLMFNTJTUFNJOJTBôMBZBOBEFôFSJLBÀPMBCJ EFOLMFNTJTUFNJOJTBôMBZBOYEFôFSMFSJOEFOCJ MJS SJBöBôEBLJMFSEFOIBOHJTJEJS A) -5 B) -4 C) -3 D) -1 E) 1 A) 1 B) 2 C) 3 D) 4 E) 5 1. D 2. # 3. # 27 4. & 5. $ 6. A
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 www.aydinyayinlari.com.tr 1\"3\"#0-* ÷LJODJ%FSFDFEFO'POLTJZPOMBS ÖRNEK 3 TANIM a, b, c `3WFBáPMNBLÐ[FSF G Y = ( 7 -N Y2 + N- 1 ) x + 2 QBSBCPMÑOÑOLPMMBSZVLBSEPôSVPMEVôVOBHÌSF N Z= ax2 + bx + c OJOFOHFOJöEFôFSBSBMôOFEJS CJ¿JNJOEFLJ GPOLTJZPOMBSB JLJODJ EFSFDFEFO a >PMNBM GPOLTJZPOMBS EFOJS #V GPOLTJZPOMBSO HSBGJóJ- 7 -N> 0 jN< 7 OFQBSBCPMBEWFSJMJS jN` ( -ß ÖRNEK 1 ÖRNEK 4 G Y = N- 2 ) x3 + xO+ 3 + 2x - 1 NWFOCJSFSEPôBMTBZPMNBLÑ[FSF G Y = N-O Y2 + 5x - 13 GPOLTJZPOVOVO HSBGJôJ CJS QBSBCPM CFMJSUUJôJOF HÌSF QBSBCPMÑOÑOLPMMBSOOZVLBSEPôSVPMNBTOTBôMB NOÀBSQNLBÀUS ZBO O EFôFSMFSJOJO UPQMBN PMEVôVOB HÌSF N FO B[LBÀUS N- 2 = WF O+ 3 = 2 N-O> 0 jN>O N= O= -1 O n^ n + 1 h NO= -2 = 28 & n^ n + 1 h = 56 jO= 7 ÖRNEK 2 2 N>PMEVôVOEBONNJO =EJS N2 -N- 12 ) x3 + xN2 +N-1+ 3x + 2 GPOLTJZPOVOVO CFMJSUUJôJ FôSJ CJS QBSBCPM PMEVôVOB 7$1,0%m/*m HÌSF NLBÀUS G Y = ax2 + bx +DGPOLTJZPOVOVOHSBGJóJOJO N2 -N- 12 = 0 N- N+ 3 ) = 0 x =J¿JOZFLTFOJOJLFTUJóJOPLUB Z= c ) N=N= -3 G Y =J¿JOYFLTFOJOJLFTUJóJOPLUBMBS WBSTB N2 +N- 1 = 2 N2 +N- 3 = 0 CVMVOVS N+ N- 1 ) = 0 N= -N=PMEVôVOEBO ÖRNEK 5 N= -CVMVOVS G Y = x2 +NY-N- 1 7$1,0%m/*m GPOLTJZPOVOVO HSBGJôJOJO Z FLTFOJOJ LFTUJôJ OPLUB - PMEVôVOB HÌSF WBSTB Y FLTFOJOJ LFTUJôJ OPLUB G Y = ax2 + bx +DGPOLTJZPOVOVOHSBGJóJOEF MBSLBÀUS a > 0JTFQBSBCPMÐOLPMMBSZVLBS -N- 1 = -4 jN= 3 f ( x ) = x2 + 3x - 4 a < 0JTFQBSBCPMÐOLPMMBSBõBóEPóSVEVS f ( x ) = 0 j x2 + 3x - 4 = 0 (x + 4) (x - 1) = 0 x = -4 x =CVMVOVS 1. –2 2. –3 28 3. ( –Þ, 7 ) 4. 8 5. { –4, 1 }
www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 6 7$1,0%m/*m G Y = x2 - N2 - Y+N G Y = ax2 + bx +DGPOLTJZPOVOVOHSBGJóJOJO5FQF QBSBCPMÑOÑO Y FLTFOJOJ LFTUJôJ OPLUBMBSO BQTJTMFSJ /PLUBT5 S L PMNBLÐ[FSF UPQMBN PMEVôVOB HÌSF Z FLTFOJOJ LFTUJôJ OPLUB OOPSEJOBULBÀUS r = - b ve k = f (r)EJS 2a N2 - 10 = 6 jN2 - 16 = 0 N- N+ 4 ) = 0 #VSBEB k = f (r) = 4ac - b2 GPSNÐMÐJMFEFCVMVOB- N=N= -4 4a N= 4 jZ=PMEVôVOEBQBSBCPMYFLTFOJOJLFTNF[ N= -4 jZ= -CVMVOVS CJMJS 7$1,0%m/*m ÖRNEK 9 G Y = ax2 + bx +DGPOLTJZPOVOVOHSBGJóJ\" Y1 Z1) G Y = x2 - 4x + 3 OPLUBTOEBOHF¿JZPSTB QBSBCPMÑOÑOUFQFOPLUBTLPPSEJOBUMBSOFEJS G Y1) =Z1EJS :BOJOPLUBEFOLMFNJTBóMBS 5 S L PMNBLÑ[FSF b4 r = - = = 2 ve k = f (r) j L= f ( 2 ) jL= -1 2a 2 T ( 2, - CVMVOVS ÖRNEK 7 ÖRNEK 10 G Y = N- 2 ) x2 + 4x +N2 -N- 2 G Y =NY2 -NY+N- 1 QBSBCPMÑ CBöMBOHÀ OPLUBTOEBO HFÀUJôJOF HÌSF N QBSBCPMÑOÑO UFQF OPLUBT LPPSEJOBUMBS 5 PM LBÀUS EVôVOBHÌSF NLBÀUS 0 EFOLMFNJTBôMBS f ( 1 ) =PMNBM N2 -N- 2 = 0 N-N+N- 1 = 3 N- N+ 1 ) = 0 N= 4 N=N= - 1 N=CVMVOVS N=BMOSTBG Y JLJODJEFSFDFEFOGPOLTJZPOPMBNB ZBDBôOEBON= -EJS ÖRNEK 8 ÖRNEK 11 G Y = x2 -NY+ 3 -N G Y = 3 ( x - 2 )2 + 4 QBSBCPMÑ\" OPLUBTOEBOHFÀUJôJOFHÌSF QBSB QBSBCPMÑOÑOUFQFOPLUBTLPPSEJOBUMBSOFEJS CPMZFLTFOJOJIBOHJOPLUBEBLFTFS f ( x ) = 3x2 - 12x + 12 + 4 = 3x2 - 12x + 16 f ( 1 ) = 10 j 1 -N+ 3 -N= 10 b 12 N= - 6 N= -3 5 S L PMNBLÑ[FSF r = - = = 2 f ( x ) = x2 + 3x + 6 2a 6 f ( 0 ) =QBSBCPMZFLTFOJOJZ=OPLUBTOEBLFTFS L=G S = f ( 2 ) = 4 j5 CVMVOVS 6. –16 7. –1 8. 6 29 9. T ( 2, –1 ) 10. 2 11. T ( 2, 4 )
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 www.aydinyayinlari.com.tr 7$1,0%m/*m d) Z= 2x2 + 4 Z= ax2 + bx +DGPOLTJZPOVOVO QBSBCPMÐOÐO HSB- y GJóJ¿J[JMJSLFO y = 2x2 + 4 B JTFLPMMBSZVLBS 4 B JTFLPMMBSBõBóEPóSVEVS Ox x =J¿JOZFLTFOJOJLFTUJóJOPLUBCVMVOVS e) Z= 2 (x - 1 )2 + 4 Z=J¿JOYFLTFOJOJLFTUJóJOPLUBMBSCVMVOVS y y = 2(x–1)2 + 4 ax2 + bx + c =EFOLMFNJOEF 6 x 1. Ó JTFLËLZPLUVS:BOJQBSBCPMYFLTFOJOJ 4 LFTNF[ O1 2. Ó=JTF¿BLõLJLJLËLWBSES:BOJQBSBCPMY f) Z= -3 (x + 2 )2 + 3 FLTFOJOFUFQFOPLUBTOEBUFóFUUJS 3. Ó>JTFGBSLMJLJLËLWBSES y 1BSBCPMYFLTFOJOJCVJLJOPLUBEBLFTFS 3 5 S L CVMVOVS(SBGJL¿J[JMJS O x ÖRNEK 11 –3 –2 –1 \"öBôEBLJGPOLTJZPOMBSOHSBGJLMFSJOJÀJ[JOJ[ y = –3(x+2)2 + 3 a) Z= x2 g) Z= 2 (x - 2 )2 y y = x2 y y = 2(x–2)2 8 x Ox O2 b) Z= -2x2 x h) Z= x2 - 3x + 2 y = –2x2 y y O y = x2 – 3x + 2 2 O1 2 x c) Z= x2 - 1 Z= -2x2 + 8x - 6 y y O1 x y = x2 –1 –6 y = –2x2 + 8x – 6 3 O x –1 1 –1 30
www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, J Z= 4x - x2 y ÖRNEK 14 O 4x \"õBóEBLJ õFLJMEF Z = 4x2 GPOLTJZPOVOVO HSBGJóJ WFSJM- NJõUJS y y = 4x2 y = 4x – x2 CB OA x ÖRNEK 12 \"OBMJUJL EÐ[MFNEF (SBGJLUF0\"#$LBSFTJOJO#LÌöFTJQBSBCPMÑ[FSJOEF WFCJSLÌöFTJPSJKJOEFPMEVôVOBHÌSF LBSFOJOÀFWSF y FLTFOMFSJ \" # WF $ TJLBÀCJSJNEJS A OPLUBMBSOEB LFTFO L2 =LjL L- 1 ) = 0 B Z = -x2 + x + 6 1 O GPOLTJZPOVOVO HSB- L= 0 k = 4 x GJóJWFSJMNJõUJS 1 C L=PMBNBZBDBôOEBOk = UÑS #VOBHÌSF \" \"#$ LBÀCJSJNLBSFEJS 4 1 ¦ 0\"#$ =L= 4. = 1br CVMVOVS 4 y A^ ABC h = 5.6 = 15 2 br 6 2 –2 3x ÖRNEK 15 O ôFLJMEFLJ QBSBCPMÐO 5FQF OPLUBT BOBMJUJL EÐ[MFNJO * ÖRNEK 13 CËMHFTJOEFEJS ôFLJMEFLJ#LËõFTJZ= x2 - 6x -QBSBCPMÐÐ[FSJOEF y PMBO0\"#$LBSFTJWFSJMNJõUJS T y x O O Cx y = ax2 + bx + c AB #VOBHÌSF BöBôEBLJMFSEFOLBÀUBOFTJEPôSVEVS * a < ** b2 > 4ac *** c > *7BCD< 7 b < #VOBHÌSF \" 0\"#$ LBÀCJSJNLBSFEJS * 1BSBCPMÑO HSBGJôJOJO LPMMBS BöBô EPôSV PMEV ôVOEBOB<ES #OPLUBTQBSBCPMÑTBôMBS x2 - 6x - 14 = -x ** 1BSBCPMYFLTFOJOJJLJOPLUBEBLFTUJôJJÀJO x2 - 5x - 14 = 0 D > 0 j b2 - 4ac >ES (x - 7) (x + 2) = 0 x = 7 *** 1BSBCPMZFLTFOJOJQP[JUJGUBSBGUBLFTUJôJJÀJO \" 0\"#$ =CS2 c > 0 ES *7 B< 0, b > 0 ve c >PMEVôVOEBOBCD< 0 b 7 Y1 + x2 > 0 j - a > 0 & b > 0 EJS 12. 15 13. 49 31 14. 1 15. 4
TEST - 11 y 1BSBCPM,BWSBN T 1. 5. G ( x ) = L- 3 ) x2 + 4x + 5 x1 x2 x QBSBCPMÑOÑOLPMMBSZVLBSEPôSV O H Y = L- 8 ) x2 + 2x - 3 y = f(x) QBSBCPMÑOÑOLPMMBSBöBôEPôSVPMEVôVOBHÌ (SBGJôJWFSJMFOG Y = ax2 + bx +DQBSBCPMÑJÀJO SF LOJOBMBCJMFDFôJUBNTBZEFôFSMFSJOJOUPQMB BöBôEBLJMFSEFOIBOHJTJLFTJOMJLMFZBOMöUS NLBÀUS \" # $ % & A) a < # C< C) c > % - b < 0 2a E) ff - b p < 0 2a 2. G Y = 3xN- 4 + 2x - 1 6. G Y = x2 + N- 3 ) x - 8 GPOLTJZPOVOVO CFMJSUUJôJ FôSJ CJS QBSBCPM PMEV QBSBCPMÑOÑO Y FLTFOJOJ LFTUJôJ OPLUBMBSO BQ ôVOBHÌSF NLBÀUS TJTMFSJUPQMBNPMEVôVOBHÌSF NLBÀUS A) 3 B) 4 C) 5 D) 6 E) 7 A) 1 B) 2 C) 3 D) 4 E) 5 3. G Y = N- 2 )x3 + xO+ 4 + 2x - 3 7. G Y = x2 - ( 3 -N Y+N-N2 GPOLTJZPOVOVO CFMJSUUJôJ FôSJ CJS QBSBCPM PMEV QBSBCPMÑOÑO Z FLTFOJOJ LFTUJôJ OPLUBOO PSEJ ôVOBHÌSF N+OUPQMBNLBÀUS OBU-PMEVôVOBHÌSF YFLTFOJOJLFTUJôJOPL UBMBSOBQTJTMFSJUPQMBNFOB[LBÀUS A) -4 B) -3 C) -2 D) -1 E) 1 A) -2 B) - $ % & 4. G Y = x2 + N+ 1 ) x +N3 - 7 8. G Y = x2 - 2x + 2b - 3 GPOLTJZPOVOVO Z FLTFOJOJ LFTUJôJ OPLUBOO PS QBSBCPMÑOÑOZFLTFOJOJLFTUJôJOPLUB EJOBUPMEVôVOBHÌSF NLBÀUS ( 2a - 6, b - 1 ) OPLUBTPMEVôVOBHÌSF B+CUPQ MBNLBÀUS A) 5 B) 3 C) 2 D) -1 E) -3 A) - # $ % & 1. & 2. D 3. $ 4. D 32 5. $ 6. A 7. $ 8. A
1BSBCPM,BWSBN TEST - 12 1. G Y = N+ 2 ) x2 - 3x +N2 -N- 6 5. G Y = x2 - N- 2 ) x +O- 7 QBSBCPMÑPSJKJOEFOHFÀUJôJOFHÌSF NLBÀUS QBSBCPMÑOÑO UFQF OPLUBT 5 - PMEVôVOB A) 4 B) 3 C) 2 D) 1 E) -2 HÌSF N+OUPQMBNLBÀUS A) -6 B) -4 C) -3 D) 1 E) 3 2. G Y = ax2 + 3x - 5 6. G Y = x2 - 6x +N+ 4 QBSBCPMÑ B OPLUBTOEBO HFÀUJôJOF HÌSF B QBSBCPMÑOÑOUFQFOPLUBTYFLTFOJÑ[FSJOEFPM LBÀUS EVôVOBHÌSF NLBÀUS A) -3 B) -2 C) -1 D) 1 E) 2 A) 2 B) 3 C) 4 D) 5 E) 6 3. G Y = x2 + 2x - 4 7. G Y = 2 ( x + 3 )2 + 4 QBSBCPMÑ Ñ[FSJOEF CVMVOBO WF BQTJTJ PSEJOBU QBSBCPMÑOÑOUFQFOPLUBT5 S L PMEVôVOBHÌSF OO ZBST PMBO OPLUBMBSO PSEJOBUMBS UPQMBN S+LUPQMBNLBÀUS LBÀUS A) -2 B) - $ % & A) - # $ % & 8. G Y =NY2 + N+ 1 ) x - 4 4. G Y = x2 + N+ 1 ) x -N QBSBCPMÑ OPLUBTOEBOHFÀUJôJOFHÌSF UF QFOPLUBTOOPSEJOBULBÀUS QBSBCPMÑ OPLUBTOEBOHFÀUJôJOFHÌSF N LBÀUS A) -7 B) - 25 C) - 23 D) -5 4 4 A) 7 B) 6 C) 5 D) 4 E) 3 E) - 17 4 1. # 2. & 3. # 4. & 33 5. $ 6. D 7. D 8. #
TEST - 13 1BSBCPM¦J[JNJ 1. Z= ax2 + bx +DQBSBCPMÐJ¿JO 3. y a > C>WFD< T PMEVôVOBHÌSF QBSBCPMÑOHSBGJôJBöBôEBLJMFS EFOIBOHJTJPMBCJMJS x O A) y B) y T :VLBSEBLJHSBGJLG Y = ax2 + bx +DGPOLTJZPOV- O OBBJUUJS C) y xO x T T #VOBHÌSF BöBôEBLJMFSEFOIBOHJTJ D) y Z = cx2 - bx - B GPOLTJZPOVOB BJU PMBO HSBGJL PMBCJMJS A) y B) y Ox Ox Ox O x T E) y C) y D) y x OT Ox O x x 2. Z= ax2 + bx +DQBSBCPMÐJ¿JO E) y 4ac - b2 > 0 ve a <PMEVôVOBHÌSF QBSBCP O MÑOHSBGJôJBöBôEBLJMFSEFOIBOHJTJPMBCJMJS A) y B) y T O x O x y D) T DC C) y y 4. T AO B x O T x x O E) y x y = 8 – x2 OT õFLJMEFZ= 8 - x2QBSBCPMÑWFJLJLÌöFTJQBSB CPMÑ[FSJOEF JLJLÌöFTJEFYFLTFOJÑ[FSJOEFÀJ [JMFO\"#$%LBSFTJOJOBMBOLBÀCS2EJS A) 16 B) 8 C) 4 D) 2 E) 1 1. # 2. & 34 3. $ 4. A
www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, 1\"3\"#0-** %JTLSJNJOBOUO÷öBSFUJJMF1BSBCPM\"SBTOEBLJ ÖRNEK 3 ÷MJöLJ G Y = 3x2 - 6x +N+ 6 7$1,0%m/*m QBSBCPMÑYFLTFOJOJLFTNFEJôJOFHÌSF NOJOBMBCJ MFDFôJFOLÑÀÑLUBNTBZEFôFSJLBÀUS Z= ax2 + bx +DGPOLTJZPOVOVOYFLTFOJOJLFT- UJóJOPLUBMBSCVMVOVSLFOBY2 + bx + c =EFOLMF- Ô<PMNBM NJOJOLËLMFSJOFCBLMS1BSBCPMÐOUFQFOPLUBTOO 36 - N+ 6 ) < 0 BQTJTJSPMNBLÐ[FSF 1. Ó>JTFQBSBCPMYFLTFOJOJGBSLMJLJOPLUBEB 36 -N- 72 < 0 -36 <N LFTFS -3 <NjNNJO = -2 2. a) Ó=WFS>JTFQBSBCPMYFLTFOJOFQP[JUJG ÖRNEK 4 UBSBGUBUFóFUUJS b) Ó=WFS<JTFQBSBCPMYFLTFOJOFOFHBUJG UBSBGUBUFóFUUJS 3. Ó<JTFQBSBCPMYFLTFOJOJLFTNF[ ÖRNEK 1 N>PMNBLÑ[FSF G Y = x2 +NY+ 4 G Y = x2 - 2x +N- 1 QBSBCPMÑ JÀJO BöBôEB WFSJMFO JGBEFMFSEFO LBÀ UBOF QBSBCPMÑYFLTFOJOJGBSLMJLJOPLUBEBLFTUJôJOFHÌSF TJEPôSVEVS NOJOBMBCJMFDFôJFOCÑZÑLUBNTBZEFôFSJLBÀUS * YFLTFOJOJLFTNF[ ** ZFLTFOJOJOPLUBTOEBLFTFS Ô>PMNBM *** 5FQFOPLUBTBOBMJUJLEÐ[MFNJO*CËMHFTJOEFEJS ( -2 )2 - N- 1 ) > 0 *7 YFLTFOJOJJLJGBSLMOPLUBEBLFTFS 7 ,PMMBSBõBóEPóSVEVS 4 -N+ 4 > 0 8 >N **WF*7NBEEFMFSEPôSV EJôFSMFSJZBOMöUS 2 >NjNNBY = 1 ÖRNEK 2 ÖRNEK 5 G Y = x2 + 12x -N+ 3 G Y = x2 - N- 1 ) x +N+ 1 QBSBCPMÑYFLTFOJOFUFôFUPMEVôVOBHÌSF NLBÀUS GPOLTJZPOVOVO HSBGJôJ Y FLTFOJOF UFôFU PMEVôVOB Ô=PMNBM HÌSF NOJOBMBCJMFDFôJEFôFSMFSUPQMBNLBÀUS 144 - 4 ( -N+ 3 ) = 0 Ô=PMNBM 144 +N- 12 = 0 N- 1 )2 - N+ 1 ) = 0 N= -132 N2 -N+ 1 -N- 4 = 0 N= -CVMVOVS N2 -N- 3 = 0 N1 +N2 = 10 1. 1 2. –11 35 3. –2 4. 2 5. 10
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 www.aydinyayinlari.com.tr ÖRNEK 6 ÖRNEK 9 G Y = x2 + N+ 3 ) x +N2 - 3 Z= 2x2 -NY+O- 1 GPOLTJZPOVOVOHSBGJôJYFLTFOJOJLFTNFEJôJOFHÌSF NOJOBMBCJMFDFôJFOCÑZÑLEFôFSLBÀUS GPOLTJZPOVOVOHSBGJôJYFLTFOJOF - OPLUBTO EBUFôFUPMEVôVOBHÌSF N+OEFôFSJLBÀUS Ô<PMNBM N+ 3 )2 - N2 - 3 ) < 0 5 S L = T ( -1, 0 ) 22 bm =-1 & m=-4 16m + 24m + 9 - 16m + 12 1 0 r=- & N< -21 2a 2.2 7 L= f ( -1 ) =PMNBMES m 1- 2 ( -1 )2 + 4 ( -1 ) +O- 1 = 0 8 2 - 4 +O- 1 = 0 m = -1 O=CVMVOVS max #VSBEBON+O= -EJS ÖRNEK 7 7$1,0%m/*m G Y = x2 -NY+N2 -3 Z BY2 CY D QBSBCPMÐOÐO UFQF OPLUBTOO GPOLTJZPOVOVO HSBGJôJ Y FLTFOJOJ GBSLM JLJ OPLUBEB LFTUJôJOFHÌSF NOJOEFôFSBSBMôOFEJS BQTJTJ JMF WBSTB LËLMFS UPQMBNOO ZBST CJSCJSJOF Ô>PMNBM FöJUUJS N 2 - N2 - 3 ) > 0 y N2 -N2 + 12 > 0 r =- b = x1 + x2 12 >PMEVôVOEBO 2a 2 N`3 O x1 // r // x 7$1,0%m/*m x2 Z BY2 CY D GPOLTJZPOVOVO HSBGJóJOJO TJNFUSJ T(r, k) FLTFOJ x = - b EPóSVTVEVS ÖRNEK 10 2a Z= x2 - 4x + 3 GPOLTJZPOVOVOHSBGJóJYFLTFOJOJLFTUJóJOPLUBMBS\"WF# EJS &ôSJOJOUFQFOPLUBT5PMEVôVOBHÌSF \"#5ÑÀHFOJ OJOBMBOLBÀCSJNLBSFEJS ÖRNEK 8 Z= x2 - 4x +GPOLTJZPOVOEB Z= 0 j x2 - 4x + 3 = 0 (x - 3) (x - 1) = 0 Z= x2 - N- 2 ) x - 7 x = 3, x =EJS GPOLTJZPOVOVO HSBGJôJOJO TJNFUSJ FLTFOJ Y = EPô #VEVSVNEB\" # ES SVTVPMEVôVOBHÌSF NLBÀUS Td - b b n n & T^ 2, - 1 h bulunur. , fd - 2a 2a b 3m - 2 y & AB . HT ABT x=- & =2 A^ h = 2a 2 2 3m - 2 = 4 2 2.1 2 = = 1br 3m = 6 OA H B x 13 2 m=2 –1 T 6. –1 7. 38. 2 36 9. –1 10. 1
www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 11 ÖRNEK 14 ôFLJMEF Y FLTFOJOJ - WF Z FLTFOJOJ - OPLUBTOEB Z=LY2 + ( 4 -L2 ) x + 18 LFTFO5FQFOPLUBT5PMBOQBSBCPMÐOHSBGJóJWFSJMNJõUJS QBSBCPMÑYFLTFOJOJPSJKJOFHÌSFTJNFUSJLJLJOPLUBEB y LFTUJôJOFHÌSF LLBÀUS –2 O A B6 x 4 -L2 = 0 ( 2 -L +L = 0 –4 D C L=WFZBL= -EJS L=JÀJOY2 + 18 =SFFMLÌLÑZPLUVS T L= -JÀJO-2x2 + 18 = 0 x = ±PMEVôVOEBO L= -CVMVOVS #VOBHÌSF \"#$%EJLEÌSUHFOJOBMBOLBÀCJSJNLBSF ÷LJODJ%FSFDFEFO#JS'POLTJZPOVO(ÌSÑOUÑ EJS ,ÑNFTJOJO&O#ÑZÑLWFZB&O,ÑÀÑL&MFNBO \"OPLUBTTJNFUSJFLTFOJÑ[FSJOEFPMEVôVOEBO TANIM | OA | = |\"#| =CJSJNEJS \" \"#$% = 4 . 2 =CJSJNLBSFEJS a >PMNBLÑ[FSF Z= ax2 + bx +DGPOLTJZP- OVOVOHËSÐOUÐLÐNFTJOJOFOLпÐLEFóFSJ UF- ÖRNEK 12 QFOPLUBTOOPSEJOBUES:BOJ k = 4ac - b2 4a Z= x2 - N- 2 ) x + 4 ES&OCÐZÐLEFóFSJZPLUVS GPOLTJZPOVOVOHSBGJôJOJO5FQFOPLUBT5 PMEV a <PMNBLÑ[FSF Z= ax2 + bx +DGPOLTJZP- ôVOBHÌSF EFOLMFNJOLÌLMFSUPQMBNLBÀUS OVOVOHËSÐOUÐLÐNFTJOJOFOCÐZÐLEFóFSJ UF- b x +x & 2= x +x QF OPLUBTOO PSEJOBUES :BOJ k = 4ac - b2 r =- = 12 12 2a 2 2 ES&OLпÐLEFóFSJZPLUVS 4a &x +x =4 12 7$1,0%m/*m ÖRNEK 15 Z= ax2 + bx +DGPOLTJZPOVOVOHSBGJóJZFLTF- Aşağıdaki fonksiyonların görüntü kümelerinin varsa, OJOFHËSFTJNFUSJLJTFC= en büyük veya en küçük elemanlarını bulunuz. ( yani ax2 + bx + c = 0 denkleminin kökler top- a) Z= 2x2 MBNY1+ x2 = ES a >PMEVôVOEBO 4.2.0 - 02 k = = 0 & FOLÑÀÑL 4.2 ÖRNEK 13 Z= L- 2 ) x2 + L+ 2 ) x +L+ 8 QBSBCPMÑZFLTFOJOFHÌSFTJNFUSJLPMEVôVOBHÌSF Y b) Z= -3x2 FLTFOJOJIBOHJOPLUBMBSEBLFTFS a <PMEVôVOEBO 2 4. (- 3) .0 - 2 k + 2 = 0 & k = - 2 & - 4x + 6 = 0 0 k = = 0 & FOCÑZÑL 4. (- 3) 26 3 36 x = = & x =\" =\" 42 22 11. 8 6 37 14. –2 15. B FOLÑÀÑLC FOCÑZÑL 12. 4 13. \" 2
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 www.aydinyayinlari.com.tr c) Z= 3x2 + 2 ÖRNEK 16 a >PMEVôVOEBO FOLÑÀÑL Z= N- 1 ) x2 - N+ 1 ) x + 5 2 GPOLTJZPOVOVOHÌSÑOUÑLÑNFTJOJOFOLÑÀÑLEFôFSJ PMEVôVOBHÌSF NLBÀUS 4.3.2 - 0 k= =2 & 4.3 d) Z= -x2 - 3 4^ m - 1 h.5 - ^ m + 1 h2 =3 a <PMEVôVOEBO 4^ m - 1 h 2 N- 20 -N2 -N- 1 =N- 12 N2 -N+ 9 = 0 4. (- 1) . (- 3) - 0 N- 3 )2 = 0 k = = - 3 & FOCÑZÑL-3 N= 3 4. (- 1) e) Z= x2 - 5x + 2 ÖRNEK 17 a >PMEVôVOEBO - 17 N`3+PMNBLÑ[FSF 4.1.2 - (- 5) 2 - 17 & en küçük G Y =NY2 +NY-N+ 3 GPOLTJZPOVOVOFOLÑÀÑLEFôFSJPMEVôVOBHÌSF N k= = 4 LBÀUS 4.1 4 - 2m f) Z= -2x2 + x - 3 r= = -1 a <PMEVôVOEBO 2m f ( -1 ) =PMNBM 4. (- 2) . (- 3) - (1) 2 - 23 - 23 N -1 )2 +N -1 ) -N+ 3 = 1 k= = & en büyük N-N-N+ 3 = 1 4. (- 2) 8 8 -N= -2 N= 1 g) Z= 2 ( x - 3 ) 2 - 4 ÖRNEK 18 5 S L = T ( 3, - PMEVôVOEBO FOLÑÀÑL- 4 ôFLJMEFWFSJMFOEJLEËSUHFOJOLFOBSV[VOMVLMBSY+ 6 ve 8 -YCJSJNEJS h) Z= -3 ( x - 1 ) 2 + 2 8–x 5 S L =5 PMEVôVOEBO FOCÑZÑL x+6 #VEJLEÌSUHFOJOBMBOOOFOCÑZÑLEFôFSJLBÀCJSJN LBSFEJS \"MBO= ( x + 6) ( 8 - x ) = -x2 + 2x + 48 b r =- = 1 2a f ( 1 ) = -1 + 2 + 48 = 49CS2 17 38 16. 3 17. 1 18. 49 15. D FOLÑÀÑLE FOCÑZÑLmF FOLÑÀÑL- 23 4 G FOCÑZÑL- H FOLÑÀÑLmI FOCÑZÑL 8
%JTLSJNJOBOUWF5FQF/PLUBTOO²[FMMJLMFSJ TEST - 14 1. G Y =NY2 - N+ 1 ) x + 11 5. ôFLJMEFZ=G Y = ax2 + bx +DQBSBCPMÐOÐOHSB- GPOLTJZPOVOVO HÌSÑOUÑ LÑNFTJOJO FO LÑÀÑL GJóJWFSJMNJõUJS EFôFSJPMEVôVOBHÌSF NOJOEFôFSJBöBôEB y LJMFSEFOIBOHJTJPMBCJMJS 6 A) -3 B) -2 C) 1 D) 2 E) 3 –4 12 x –5 #VOBHÌSF BDLBÀUS A) - 11 B) -7 C) - 11 5 8 2. G Y = ( 1 -N Y2 -NY+ 1 D) - & -18 QBSBCPMÑOÑOTJNFUSJFLTFOJY+ 3 =EPôSVTV PMEVôVOBHÌSF NLBÀUS 10 8 6 6 5 A) B) C) D) E) 11 9 7 5 3 6. ,FOBSV[VOMVLMBSY+CJSJNWF-YCJSJN PMBOCJSEJLEÌSUHFOJOBMBOFOÀPLLBÀCS2EJS A) 141 B) 145 C) 147 D) 149 E) 151 3. G Y = x2 + N+ 1 ) x +N+ 3 QBSBCPMÑZ=EPôSVTVOBOFHBUJGUBSBGUBUFôFU PMEVôVOB HÌSF N OJO EFôFSJ BöBôEBLJMFSEFO IBOHJTJEJS \" # $ % -1 E) -2 y 7. x C –2 T 4. Z= x2 + 2x +N- 3 ôFLJMEFLJZ=G Y = 2ax2 - 4x + 2a2 - 4 QBSBCPMÐOÐOUFQFOPLUBT5PMEVóVOBHËSF GPOLTJZPOVOVO HSBGJôJOJO Y FLTFOJOJ JLJ GBSLM OPLUBEBLFTNFTJJÀJO NOJOFOCÑZÑLUBNTBZ | |$5LBÀCS2EJS EFôFSJLBÀUS A) 2 B) 5 C) 7 D) 3 E) 2 2 A) 1 B) 2 C) 3 D) 4 E) 5 1. & 2. $ 3. # 4. $ 39 5. $ 6. $ 7. #
TEST - 15 %JTLSJNJOBOUWF5FQF/PLUBTOO²[FMMJLMFSJ 1. Z=G Y = x2 -NY+N2 - 1 5. G3Z R QBSBCPMMFSJOJO UFQF OPLUBT BOBMJUJL EÑ[MFNEF G Y = 2x2 - 12x -O+ 1 ***CÌMHFEFPMEVôVOBHÌSF NIBOHJBSBMôOFMF GPOLTJZPOVOVOBMBCJMFDFôJFOLÑÀÑLEFôFS- 10 NBOES PMEVôVOBHÌSF OLBÀUS A) ( - # $ - A) -7 B) -6 C) -5 D) -4 E) -3 D) ( -2, -1 ) E) ( 1, 2 ) 2. Z= N- 2 ) x2 - 2x +N- 2 6. G[ -2, 3 ] Z R QBSBCPMMFSJOJO UFQF OPLUBT Z = Y EPôSVTV G Y = x2 - 4x + 5 Ñ[FSJOEFPMEVôVOBHÌSF NOJOBMBCJMFDFôJEF ôFSMFSUPQMBNLBÀUS GPOLTJZPOVOVOBMBCJMFDFôJ FOCÑZÑL ve FOLÑ ÀÑLEFôFSMFSJOUPQMBNLBÀUS A) 2 B) 3 C) 4 D) 6 E) 8 A) 19 B) 18 C) 17 D) 16 E) 15 3. x `3PMNBLÑ[FSF 7. G Y = x2 - 5x + 7 ( 2x - 3 )2 + ( x + 1 )2 QBSBCPMÑ Ñ[FSJOEFLJ CJS OPLUBOO LPPSEJOBUMBS UPQMBNOOBMBCJMFDFôJFOLÑÀÑLEFôFSLBÀUS UPQMBNOO FOLÑÀÑLEFôFSJLBÀUS A) 1 B) 2 C) 3 D) 4 E) 5 A) 1 B) 2 C) 3 D) 4 E) 5 4. G Y = x2 + N- 3 ) x +O- 1 8. YHFSÀFLTBZPMNBLÑ[FSF QBSBCPMÑOÑO UFQF OPLUBT 5 - PMEVôVOB A = 2x2 - 5x + 4 HÌSF N+OUPQMBNLBÀUS B = -x2 + 3x + 6 A) -1 B) 1 C) 2 D) 3 E) 4 PMEVôVOBHÌSF \"+#UPQMBNFOB[LBÀUS \" # $ % & 1. $ 2. $ 3. & 4. # 40 5. A 6. # 7. $ 8. D
%JTLSJNJOBOUWF5FQF/PLUBTOO²[FMMJLMFSJ TEST - 16 1. G Y = N- 2 ) x2 + 4x +N+ 3 5. Z= x2 - 6x + 5 GPOLTJZPOVOVO HSBGJôJOJO Y FLTFOJOJ LFTUJôJ GPOLTJZPOVOVOHSBGJóJOJOYFLTFOJOJLFTUJóJOPLUBMBS OPLUBMBSOBQTJTMFSJÀBSQN 7 PMEVôVOBHÌSF \"WF#PMTVO 2 &ôSJOJOUFQFOPLUBT5PMEVôVOBHÌSF \"#5ÑÀ NLBÀUS HFOJOJOBMBOLBÀCJSJNLBSFEJS A) -1 B) 1 C) 2 D) 3 E) 4 A) 15 B) 8 C) 17 D) 9 E) 19 2 2 2 2. G Y = 2x2 + 6x +N- 3 GPOLTJZPOVOVOHSBGJôJYFLTFOJOJLFTUJôJOFHÌ 6. Z= N- 2 ) x2 + N2 - 4) x + 8 SF NOJOFOHFOJöEFôFSBSBMôOFEJS QBSBCPMÑOÑOUFQFOPLUBTZFLTFOJÑ[FSJOEFPM A) f - 3 , 15 p B) f - 3 , 9 H EVôVOBHÌSF NLBÀUS 2 2 C) > 15 , 3 p D) f - 3 , 15 H A) -2 B) -1 C) 1 D) 2 E) 8 2 2 E) f 15 , 3 p 2 3. Z= 2x2 -LY+N- 1 7. G Y = x2 + N+ 3 ) x +N+ 2 GPOLTJZPOVOVO HSBGJôJ Y FLTFOJOF - OPL QBSBCPMÑOÑOUFQFOPLUBTYFLTFOJÑ[FSJOEFPM UBTOEBUFôFUPMEVôVOBHÌSF L+NUPQMBNLBÀ EVôVOBHÌSF ZFLTFOJOJLFTUJôJOPLUBBöBôEB US LJMFSEFOIBOHJTJEJS A) -4 B) -2 C) -1 D) 1 E) 2 A) -3 B) -2 C) -1 D) 1 E) 2 4. Z= L- 2 ) x2 +LY+L- 4 GPOLTJZPOVOVOHSBGJôJEBJNBYFLTFOJOJOBMUO 8. #JSTBUDYMJSBZBBMEóCJSNBMZMJSBZBTBUNBLUBES EBPMEVôVOBHÌSF NOJOÀÌ[ÑNBSBMôOFEJS YJMFZBSBTOEB A) ( -R, 2 ) B) f 4 , 2 p Z= -x2 + 9x + C) f - 3 , 4 p 3 CBôOUT PMEVôVOB HÌSF TBUDOO L»S FO ÀPL 3 D) f - 4 , 3 p LBÀMJSBES 3 A) 4 B) 16 C) 32 D) 56 E) 72 E) f - 4 , - 2 p 3 1. & 2. D 3. $ 4. $ 41 5. # 6. A 7. D 8. D
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 www.aydinyayinlari.com.tr 1\"3\"#0-*** (SBüôJ7FSJMFO1BSBCPMÑO%FOLMFNJOJ:B[NB b) y 7$1,0%m/*m YFLTFOJOJLFTFOOPLUBMBSCJMJOJZPSTB y y = f(x) 1BSBCPMÐOEFOLMFNJ –1 O x x2 x –2 3 O G Y =B Y- x1 ) ( x - x2 ) x1 EJS1BSBCPMÐ[FSJOEFLËL- Z= a( x + 1 ) ( x - 3), MFS EõOEB WFSJMFO IFS- IBOHJ CJS OPLUB QBSBCPM ( 0, - OPLUBTEFOLMFNJTBôMBS EFOLMFNJOEF ZB[MBSBL B EFóFSJCVMVOVS -2 = a.1. (-3 ) 1BSBCPMÐO UFQF OPLUBTOO LPPSEJOBUMBS CJMJOJ- a = 2 UÑS ZPSTB 3 y 1BSBCPMÐOEFOLMFNJ 2 G Y =B Y-S 2 +LEJS y = (x + 1) (1 - 3) 3 1BSBCPM Ð[FSJOEF UFQF c) y O OPLUBT EõOEB WFSJMFO 2 A(2, 2) T(r,k) x IFSIBOHJ CJS OPLUB QBSB- –1 CPMEFOLMFNJOEFZB[MBSBL O BEFóFSJCVMVOVS 24 x ÖRNEK 1 Z= a( x + 1 ) ( x - 4) \"öBôEBWFSJMFOQBSBCPMMFSJOEFOLMFNMFSJOJCVMVOV[ \" OPLUBTEFOLMFNJTBôMBS a) y 2 = a.3.( -2 ) j a = - 1 3 y = - 1 ^ x + 1 h^ x - 4 h 3 2 x d) y O1 2 O 45 x Z= a ( x - 1 ) ( x - 2), –5 A(5, –5) OPLUBTEFOLMFNJTBôMBS 2 = a (-1 ) ( -2 ) Z= a( x - 0 ) ( x - 4) a =EJS A ( 5, - OPLUBTEFOLMFNJTBôMBS Z= ( x - 1 ) ( x - 2 ) -5 = a.5.1 j a = -1 Z= -x ( x - 4) 1. B Z Ym Ym 42 b) y = 2 ^ x + 1 h^ x - 3 h c) y = - 1 ^ x + 1 h^ x - 4 hE Z mY Ym 33
www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, e) h) 5FQFOPLUBT5 - WFZFLTFOJOJLFTUJôJOPL y UB - PMBOQBSBCPMÑOEFOLMFNJOFEJS –1 O x Z=a ( x + 2 )2 ( 0, - OPLUBTEFOLMFNJTBôMBS T(–1, –2) –1 -4 = a.4 j a = -1 –2 Z= -( x + 2)2 Z=a ( x + 1 )2 - 2 y ( 0, - OPLUBTEFOLMFNJTBôMBS -1 = a - 2 j a = 1 O1 x Z= ( x + 1)2 - 2 –2 f) y –4 A(1, –4) T(3, 4) Z=a.x2 - 2 4 A ( 1, - OPLUBTEFOLMFNJTBôMBS -4 = a - 2 j a = -2 O x Z= -2x2 - 2 3 ÖRNEK 2 y –2 2k –4 Z=a ( x - 3 )2 + 4 O kx ( 0, - OPLUBTEFOLMFNJTBôMBS -2 = a(-3)2 + 4 j a = - 2 3 y = - 2 ^ x - 3 h2 + 4 3 g) y :VLBSEBWFSJMFOZ=G Y GPOLTJZPOVJÀJO f^ k + 1 h 2 =-1 f^ -1 h O1 PMEVôVOBHÌSF LLBÀUS 5 PMEVôVOEBOZ= a(x - 1)2+ 0 OPLUBTEFOLMFNJTBôMBS x 2 = a(-1)2 j a = 2 Z= 2(x - 1)2 f (x ) = a (x -L Y+ 4 ) f (k + 1) a (k + 1 - k) (k + 5) = =-1 f (- 1) a (- 1 - k) .3 L+ 5 =L+ 3 L= 2 jL= 1 F Z Y 2 – 2 f) y = - 2 ^ x - 3 h2 + 4 H Z 2(x – 1)2 43 I Z m Y 2 Z mY2 – 2 2. 1 3
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 www.aydinyayinlari.com.tr ÖRNEK 3 ÖRNEK 5 y y = f(x) y x y = x2 – mx – m – 5 O B A AO Bx C ôFLJMEFHSBGJóJWFSJMFOGPOLTJZPOVOEFOLMFNJ | |:VLBSEBLJHSBGJLUF \"# =PMEVôVOBHÌSF NOJO | | | | | |f(x) = ax2 - bx - 1 , 12 0\" = 4 0# = 3 0$ EJS BMBCJMFDFôJEFôFSMFSJCVMVOV[ 4 :VLBSEBLJWFSJMFSFHÌSF B+CLBÀUS x2 -NY-N- 5 =EFOLMFNJOEF f (x ) = a (x +L Y-L -L EFOLMFNJTBôMBS | x - x | = T =5 1 2 a 4 2 -L=B L -L j a.k = m + 4m + 20 = 5 m2 + 4m + 20 = 25 3 2 1 1 64 m + 4m - 5 = 0 x = 0 j-L= - j k = , a = 4 16 3 f (x) = 64 2 - 8 x- 1 (m + 5) (m - 1) = 0 x 3 34 56 m = - 5 ve m = 1 bulunur. a+b= 3 CVMVOVS ÖRNEK 4 ÖRNEK 6 %FOJ[TFWJZFTJOEFONZÐLTFLMJLUFLJCJSLBZBMLUBOEF- y y = 2x2 – 8x – 4m – 2 OJ[F BUMBO CJS UBõ QBSBCPM HSBGJóJ PMVõUVSBSBL EÐõNFL- UFEJS5BõOEFOJ[TFWJZFTJOEFOFOZÐLTFLUFCVMVOEVóV OPLUB5OPLUBTES T AO B x C 12 m # | | | |:VLBSEBLJHSBGJLUF 0# = 5 AO PMEVôVOBHÌSF $ \"H OPLUBTOOLPPSEJOBUMBSOCVMVOV[ \"OPLUBTOOBQTJTJ-L #OPLUBTOOBQTJTJL | | | | | |TH =NWF AH =NPMEVôVOBHÌSF \"# LBÀ x + x =L= 4 jL= 1 NFUSFEJS 1 2 y Z= a(x - 2)2 + 16 PIBMEFLÌLMFSY1 = -1 ve 16 12 = a(-2)2+ 16 j a = -1 x2 =UJS ,ÌLMFSEFOLMFNJTBôMBS 12 Z= -x2 + 4x - 4 + 16 x1 = -JÀJO x Z= -x2 + 4x + 12 2 + 8 -N- 2 = 0 -(x - 6) (x + 2) = 0 O2 N= 8 j N=CVMVOVS x =JÀJO$OPLUBTOOPSEJOBU-CVMVOVS x = 6 j |\"#| =N 56 4. 6 44 5. –5 ve 1 6. (0, –10) 3. 3
www.aydinyayinlari.com.tr ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 2. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 7 ÖRNEK 9 Bx \" `G f(x) = ax2 + bx + c # `G ôFLJMEFLJ QBSBCPM FLTFOMFSJ LFTUJóJ OPLUBMBS JMF CFMJSMFO- y $ N `G NJõUJS#VQBSBCPMÐOÐTUZBSTÐ[FSJOEFEFóJõLFOCJS$ OPLUBTBMOZPS A [ CA ] m [AB] y C O 2 C BO A :VLBSEBLJ WFSJMFSF HÌSF B + b + D UPQMBNOO EF –1 x ôFSJLBÀUS 4 0MVöBDBL\"0$ÑÀHFOJOJOBMBOOOBMBCJMFDFôJNBLTJ \"#$EJLÑÀHFOJOEFÌLMJECBôOUTOEBO-N= 32 NVNEFôFSLBÀCS2PMVS N= -CVMVOVS 1BSBCPMÑOEFOLMFNJZB[MSTB #VSBEBOQBSBCPMÑOEFOLMFNJG Y = a(x + 1) (x - 9), Z= a ( x + 1 ) ( x - 4 ) OPLUBTEFOLMFNJTBôMBS 1 1 2 = a . 1. ( -4 ) j a = - 3 = a.1. (-9) j a = - 2 3 Z= - 1 ^ x + 1 h^ x - 4 h f^ x h =- 1 ^ x + 1 h ^ x - 9 h =- 1 x2 + 8 x + 3 2 3 33 \"MBOOO NBLTJNVN PMNBT JÀJO $ OPLUBT 5FQF OPL 18 16 a+b+c =- + +3 = UBTPMNBMES 33 3 3 25 Td , n 28 25 25 bulunur. 4. = A^ AOC h = 8 4 2 ÖRNEK 8 ÖRNEK 10 #JSZÐ[ÐDÐ\"OPLUBTOEBOEFOJ[FHJSJQQBSBCPMJLCJSSPUB ôFLJMEFUFQFOPLUBT5PMBO Z= x2 - 4x +DGPOLTJZPOVO JMFZÐ[FSFL#OPLUBTOBVMBõZPS HSBGJóJWFSJMNJõUJS y y y = f(x) Sahil K A B x –2 8 LT –5 x O Deniz :Ñ[ÑDÑTBIJMEFOFOÀPLLBÀCJSJNV[BLMBöNöUS | KL | = | -0| PMEVôVOBHÌSF DLBÀUS Z= a ( x + 2 ) ( x - 8 ) | KL | = | LO |PMEVôVOEBO -5 = a. 2. ( -8 ) j 5 c a= = c - 4 & c = 2c - 8 j c = 8 125 16 2 Td 3, 16 n 16 ZÑ[ÑDÑFOÀPL 125 CJSJNV[BLMBöS 45 9. 10. 8 16 3 25 125 7. 8. 4 16
TEST - 17 (SBGJôJ7FSJMFO1BSBCPMÑO%FOLMFNJOJ:B[NB 1. y f(x) 4. y = –x y õFLJMEFLJ 1\"0 ÑÀHF 2x OJOJOBMBOLBÀCJSJN –1 O O LBSFEJS x P A y = –3x2 –4 :VLBSEBHSBGJôJWFSJMNJöPMBOQBSBCPMÑOEFOL 1 1 1 1 1 MFNJBöBôEBLJMFSEFOIBOHJTJEJS A) B) C) D) E) \" G Y = x2 + x - # G Y = x2 - x - 2 C G Y = x2 - 2x - % G Y = 2x2 - 2x - 4 2 6 9 18 24 & G Y = 2x2 + 2x - 4 y 5. y 2. CB x f(x) OA 1 x –2 O :VLBSEBLJHSBGJLUF QBSBCPMJLGPOLTJZPOVOUF QFOPLUBTOOBQTJTJWF0\"#$EJLEÌSUHFOJOJO :VLBSEBHSBGJôJWFSJMNJöPMBOQBSBCPMÑOEFOL BMBOCS2PMEVôVOBHÌSF EJLEÌSUHFOJOÀFWSF MFNJBöBôEBLJMFSEFOIBOHJTJEJS TJLBÀCJSJNEJS \" G Y = x2 + x + # G Y = x2 - 4x + 4 A # $ % & $ G Y = 1 x2 + x + 1 % G Y = 1 x2 - x + 1 24 & G Y = 1 x2 + x + 1 4 6. y y = f(x) = x2 + px + c 55FQFOPLUBT 3. y 16 T A A OD x OB x –1 5 –2 BC ôFLJMEFG Y = x2 +QY+ c QBSBCPMÐOÐOHSBGJóJWFSJM- NJõUJS õFLJMEFLJ QBSBCPMÑO UFQF OPLUBT $ PMEVôVOB HÌSF \"#$%EJLEÌSUHFOJOJOBMBOLBÀCS2EJS \"0#5 EJLEÌSUHFOJOJO ÀFWSFTJ CS PMEVôVOB HÌSF QDLBÀUS \" # $ % & A) -96 B) - $ - % -& -48 1. D 2. & 3. D 46 4. D 5. # 6. A
(SBGJôJ7FSJMFO1BSBCPMÑO%FOLMFNJOJ:B[NB TEST - 18 1. y 4. ,PPSEJOBUEÐ[MFNJOEFZ=G Y QBSBCPMGPOLTJZPOV- y = f(x) OVOHSBGJóJWFSJMNJõUJS O x y T –3 1 3 2 x –2 1 O ,PPSEJOBUEÑ[MFNJOEFHSBGJôJWFSJMFOQBSBCPMJL y = f(x) GPOLTJZPOJÀJOG G - EFôFSJLBÀUS A) -3 B) -2 C) -1 D) 1 E) 2 #VOB HÌSF GPOLTJZPOVO Z ZFLTFOJOJ LFTUJôJ OPLUBOOLPPSEJOBUMBSBöBôEBLJMFSEFOIBOHJTJ EJS \" - # - $ -4 ) % - & -6 ) 2. \" # WF$ - OPLUBMBSOEBO HFÀFO QBSBCPMÑO EFOLMFNJ BöB 5. ôFLJMEFLJTVLFNFSJOJOUFQFOPLUBTOOZFSEFOZÐL- ôEBLJMFSEFOIBOHJTJEJS TFLMJóJNFUSF BZBLMBSOOJ¿LTNMBSBSBTOEBLJ \" Z= -x2 + # Z= -x2 - 4 NFTBGFNFUSFEJS $ Z= x2 - % Z= -x2 - x + 4 12 m & Z= x2 + x + 4 6m 3. 5FQFOPLUBTLPPSEJOBUMBS5 - PMBO #VOBHÌSF ZVLBSEBLJHJCJNPEFMMFOFOQBSBCP MÑOEFOLMFNJ TJNFUSJFLTFOJZFLTFOJPMBDBLöF Z=G Y QBSBCPMGPOLTJZPOV\" -4, - OPLUB LJMEF BöBôEBLJMFSEFOIBOHJTJEJS TOEBOHFÀUJôJOFHÌSF CVGPOLTJZPOVOZFLTF OJOJLFTUJôJOPLUBOOPSEJOBULBÀUS A) y = - 4 ^ x2 - 9 h B) y = 4 ^ x2 - 9 h 3 3 A) -9 B) -7 C) -5 D) -3 E) -1 C) y = - 3 ^ x2 - 9 h D) y = - 4 ^ x2 - 3x h 4 3 E) y = - 4 ^ x2 + 3x h 3 1. # 2. A 3. # 47 4. & 5. A
·/÷7&34÷5&:&)\";*3-*, 2. MODÜL ÷,÷/$÷%&3&$&%&/%&/,-&.-&31\"3\"#0-&õ÷54÷;-÷,-&3 www.aydinyayinlari.com.tr 1\"3\"#0-*7 #JS1BSBCPMJMF#JS%PôSVOVO#JSCJSJOF(ÌSF ÖRNEK 2 %VSVNMBS Z= 2x2 - 4x + 2 7$1,0%m/*m QBSBCPMÑJMFZ= 2x +EPôSVTVOVOLFTJNOPLUBMB SOOBQTJTMFSJUPQMBNLBÀUS Z= ax2 + bx +DQBSBCPMÐJMFLY+QZ+S= EPóSVTVOVO LFTJN OPLUBMBS CVMVOVSLFO CV JLJ 2x2 - 4x + 2 = 2x + 1 EFOLMFNJOPSUBL¿Ë[ÐNÐOEFOFMEFFEJMFDFLPMBO 2x2 - 6x + 1 = 0 EFSFDFEFOEFOLMFN¿Ë[ÐMÐS#VEFOLMFNEF Ó JTFEPóSV QBSBCPMÐGBSLMJLJOPLUBEBLF- 6 TFS x + x = = 3 CVMVOVS Ó=JTFEPóSV QBSBCPMFUFóFUUJS 1 22 Ó JTFEPóSV QBSBCPMÐLFTNF[ Q=JTFEPóSV QBSBCPMÐCJSOPLUBEBLFTFS ÖRNEK 1 ÖRNEK 3 \"öBôEBLJ QBSBCPM WF EPôSVMBSO CJSCJSJOF HÌSF EV Z= x2 - N+ 1 ) x - 4 SVNMBSO JODFMFZJOJ[ WBSTB LFTJN OPLUBMBSO CVMV QBSBCPMÑJMFZ=NY+EPôSVTVOVOLFTJNOPLUBMB OV[ SOOBQTJTMFSJUPQMBNPMEVôVOBHÌSF NLBÀUS a) Z= x2 - 3x +JMFZ= 3 - 2x x2 - N+ 1 ) x - 4 =NY+ 2 x2 - 3x + 1 = 3 - 2x x2 - N+ 1) x - 6 = 0 x2 - x - 2 = 0 x1 + x2 =N+ 1 =PMNBM (x - 2) (x + 1) = 0 #VSBEBON=CVMVOVS x = 2 x = -1 ÖRNEK 4 x = 2 jZ= -1 jCwF-1) x = -1 jZ= 5 j( -1, 5 ) Z= x2 - 4x + 2 1BSBCPMEPôSVJMF -1) ve (- OPLUBMBSOEBLFTJöJS QBSBCPMÑJMFZ= 2x +EPôSVTVOVOLFTJNOPLUBMBS b) Z= x2 - x +JMFZ= x + 1 \"WF#PMEVôVOBHÌSF [\"#]EPôSVQBSÀBTOOPSUB x2 - x + 2 = x + 1 OPLUBTOCVMVOV[ x2 - 2x + 1 = 0 x2 -4x + 2 = 2x + 3 ( x - 1 )2 = 0 x=1 x2 - 6x - 1 = 0 x = 1 jZ=QBSBCPMEPôSVJMF OPLUB x +x 6 TOEBUFôFUUJS 12 = =3 c) Z= 3x2 + x +JMFZ= 4 - x 22 3x2 + x + 5 = 4 - x x =OPLUBTEPôSVEFOLMFNJOEFZB[MSTBZ=CVMV 3x2 + 2x + 1 = 0 Ô= 4 - 4.3.1 = -8 <PMEVôVOEBOSFFMLÌLZPLUVS OVS0IBMEFPSUBOPLUB EVS 1BSBCPMJMFEPôSVLFTJöNF[ 1. B m m C OPLUBTOEBUFôFUUJSD ,FTJöNF[MFS 48 2. 3 3. 1 4. ( 3, 9 )
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