Tiêu đề higher-math 2019.pdf ✅

m„Rbkxj D”PZi MwYZ : beg-`kg †kÖwY  19 19 XvKv †evW© 2019 welq †KvW : 1 2 6 mgq25 wgwbU D”PZi MwYZ eûwbe©vPwb Afxÿv c~Y©gvb25 [we‡kl `aóe ̈ : mieivnK...Z eûwbe©vPwb Afxÿvi DËic‡Î cÖ‡kœi μwgK b¤^‡ii wecix‡Z cÖ`Ë eY©msewjZ e„Ëmg~n n‡Z mwVK/ m‡e©vrK...ó Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki| cÖwZwU cÖ‡kœi gvb 1| mKj cÖ‡kœi DËi w`‡Z n‡e| cÖkœc‡Î †Kv‡bv cÖKvi `vM/wPý †`Iqv hv‡e bv|] 1. wb‡Pi †Kvb we›`ywU 2x + 3y – 3 > 0 AmgZvi AšÍM©Z? K (– 3, 3) L (2, 5) M (0, 1) N (2, – 1) 2. F(x) = x x – 3 dvskbwUi †Wv‡gb KZ? K (x : xÑ Ges x > 3} L {x : xÑ Ges x < 3} M {x : xÑ Ges x = 3} N {x : xÑ Ges x  3} 3. f(x) = ( ) 1 2 x GKwU m~PKxq dvskb n‡j i. GwU (0, 1) we›`yMvgx ii. Gi †Wv‡gb (– , ) iii. Gi †iÄ (0, ) wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii  DÏxcKwU c‡o 4 I 5 bs cÖ‡kœi DËi `vI : A(1, – 1) B(2, 2) Ges C(– 2, 2) wZbwU we›`y| 4. AB †iLvi Xvj KZ? K 3 L 1 3 M – 1 3 N – 3 5. A, B, C we›`y wZbwU Øviv MwVZ wÎfzR‡ÿ‡Îi †ÿÎdj KZ? K 2 eM© GKK L 4 eM© GKK M 6 eM© GKK N 12 eM© GKK 6. 3.27y = 9y + 4 n‡j, y Gi gvb KZ? K 7 5 L 9 5 M 4 N 7 7. – x2 + 4x – 3 = 0 mgxKi‡Yi wbðvqK KZ? K 4 L 12 M 20 N 28 8. 10C4 = KZ? K 210 L 1260 M 3150 N 30240 9. C A B ABC Gi †ÿ‡Î i.  AB +  BC =  AC ii.  AB +  AC =  BC iii.  AC –  AB =  BC wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii  DÏxcKwU c‡o 10 I 11 bs cÖ‡kœi DËi `vI : GKwU K ̈vcmy‡ji m¤ú~Y© ˆ`N© ̈ 21 †m. wg. Ges Gi wmwjÛvi AvK...wZ As‡ki e ̈vmva© 3 †m.wg.| 10. K ̈vcmy‡ji wmwjÛvi AvK...wZ As‡ki ˆ`N© ̈ KZ? K 3 †m.wg. L 9 †m.wg. M 15 †m.wg. N 18 †m.wg. 11. K ̈vcmyjwUi AvqZb KZ? K 207  Nb †m.wg. L 171  Nb †m.wg. M 135 Nb †m.wg. N 36 Nb †m.wg. 12. cuvP UvKvi PviwU gy`av GKmv‡_ wb‡ÿc Kiv n‡j, bgybv we›`y KqwU n‡e? K 4 L 8 M 16 N 32 13. wb‡Pi †KvbwUi Rb ̈ A I B †mUØq mgvb n‡e? K A\B Ges B\A L AB Ges BA M AB Ges B  A N A  B Ges B  A 14. x, 7 I 11 †m.wg. e ̈vmva©wewkó wZbwU e„Ë ci ̄úi‡K ewnt ̄úk© Ki‡j †K›`aÎq Øviv Drcbœ wÎfz‡Ri cwimxgv 52 †m.wg. nq| x Gi gvb KZ? K 5 †m.wg. L 8 †m.wg. M 16 †m.wg. N 34 †m.wg. 15. wP‡Î  AC = KZ? A 2a B b C K 2a + b L 2a – b M b – 2a N – b – 2a 16. sin A = 3 2 Ges 0 < A < 2 n‡j i. A =  3 ii. A = 2 3 iii. A = 4 3 wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii  DÏxcKwU c‡o 17 I 18 bs cÖ‡kœi DËi `vI : ABC Gi ga ̈gvÎq AD = 6 †m.wg., BE = 5 †m.wg. CF = 4.5 †m.wg. ci ̄úi O we›`y‡Z †Q` K‡i‡Q| A B C F O E D 17. OA = KZ? K 2.25 †m.wg. L 2.5 †m.wg. M 3 †m.wg. N 4 †m.wg. 18. AB, BC Ges AC evûi e‡M©i mgwó KZ? K 27.08 eM© †m.wg. L 60.94 eM© †m.wg. M 81.25 eM© †m.wg. N 108.33 eM© †m.wg. 19. f(a) = a2 + 5a – 4 eûc`xi a Gi †Kvb gv‡bi Rb ̈ f(a) = 2 n‡e? K – 4 L 1 M 6 N 10 20. wb‡Pi †KvbwU mggvwÎK eûc`x? K ax2 + 2xy + cy L ax2 + 2bxy + c2 M ax2 + 2bxy + cy2 N a2 x + 2abxy + c2 y2 21. ( ) y + 1 y 4 Gi we ̄Í...wZ‡Z y ewR©Z c` †KvbwU? K 10 L 6 M 4 N 1  DÏxcKwU c‡o 22 I 23 bs cÖ‡kœi DËi `vI : 2 + 0.2 + 0.02 + 0.002 + 0.0002 + .... 22. avivwUi `kg c` KZ? K 10–9 L 109 M 2  109 N 2  10–9 23. avivwUi AmxgZK mgwó KZ? K 9 5 L 10 9 M 20 9 N 20 11 24. X X Y Y B(4, 0)  O A(4, 3) cosec( – ) + sec ( – ) = KZ? K – 5 12 L – 35 12 M 1 5 N 7 5 25. †Kv‡bv NUbv NUvi m¤¢vebv P n‡j, wb‡Pi †KvbwU mwVK? K 0 < P < 1 L 0  P < 1 M 0 < P  1 N 0  P  1 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN 14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN Self test 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1 L 2 N 3 N 4 K 5 M 6 N 7 K 8 K 9 L 10 M 11 L 12 M 13 M 14 L 15 K 16 K 17 N 18 N 19 L 20 M 21 L 22 N 23 M 24 K 25 N
†mUK 20 ivRkvnx †evW© 2019 welq †KvW : 1 2 6 mgq25 wgwbU D”PZi MwYZ eûwbe©vPwb Afxÿv c~Y©gvb25 [we‡kl `aóe ̈ : mieivnK...Z eûwbe©vPwb Afxÿvi DËic‡Î cÖ‡kœi μwgK b¤^‡ii wecix‡Z cÖ`Ë eY©msewjZ e„Ëmg~n n‡Z mwVK/ m‡e©vrK...ó Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki| cÖwZwU cÖ‡kœi gvb 1| mKj cÖ‡kœi DËi w`‡Z n‡e| cÖkœc‡Î †Kv‡bv cÖKvi `vM/wPý †`Iqv hv‡e bv|] 1. 15x2 + 24x3 – 3x4 + 2x + 6, eûc`xi gyL ̈ mnM KZ? K – 3 L 4 M 6 N 15 2. A = {x  Z : 9  x2  36} n‡j, A Gi Dc‡mU KqwU? K 4 L 16 M 32 N 256 3. x3 + px2 + 3x – 15 Gi GKwU Drcv`K (x – 5) n‡j, p Gi gvb KZ? K – 5 L – 31 5 M 5 N 31 5 4. N D O M MD evû‡Z MO Gi j¤^ Awf‡ÿc, K DO L MD M MN N ND 5. PQR Gi PRQ i. ̄’~j‡KvY n‡e, hLb PQ2 > PR2 + QR2 ii. m~2‡KvY n‡e, hLb PQ2 > PR2 + QR2 iii. mg‡KvY n‡e, hLb PQ2 = PR2 + QR2 wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 6. MNO Gi MN = MO = NO, MD  NO Ges MD = 6 cm n‡j, MNO Gi cwie„‡Ëi e ̈vmva© KZ? K 2 †m.wg. L 2 3 †m.wg. M 4 †m.wg. N 4 3 †m.wg. 7. 4x2 + 8x – 18 = 0 mgxKi‡Yi g~jØq i. ev ̄Íe-mgvb ii. ev ̄Íe-Amgvb iii. ev ̄Íe-Ag~j` wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 8. 3x . 2y = 72, 32x. 2y = 648 mgxKiY †Rv‡Ui g~jØq K (2, 3) L (2, 2) M ( 2,  3) N (3, 2) 9. – 4x + 6 > – 12 AmgZvwUi i. GKwU iƒc 2x – 3 < 6 ii. mgvavb †mU, S = {xR : x > 9 2 } iii. mgvavb †mU, S = {xR : x < 9 2 } wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii  5 + 5 4 + 5 16 + 5 64 + ..... D3 Z‡_ ̈i Av‡jv‡K 10 I 11 bs cÖ‡kœi DËi `vI : 10. avivwUi AmxgZK mgwó KZ? K 4 L 5 M 25 4 N 20 3 11. avivwUi 7g c` KZ? K 5 47 L 5 46 M 20 3 ( ) 1 – 1 47 N 20 3 ( ) 1 – 1 46 12. sin2  = 1 4 hLb     3 2 ,  Gi gvb †KvbwU? K  6 L 5 6 M 7 6 N 4 3 13. sin( ) 25 2 –  †Kvb PZzf©v‡M? K 1g L 2q M 3q N 4_© 14. log 8 4 Gi gvb KZ? K 8 L 2 M 4 3 N 3 4 15. 3 a5 = 2 6 a7 n‡j a Gi gvb KZ? K 1 L 2 M 2 N 4 16. ( ) x – a 4 6 Gi we ̄Í...wZ‡Z x3 Gi mnM – 540 n‡j a Gi gvb KZ? K – 12 L – 6 M 6 N 12 17. ( ) x + a x 8 Gi we ̄Í...wZ‡Z c` KZwU? K 2 L 4 M 8 N 9 18. A(3, – 6) we›`y †_‡K x A‡ÿi `~iZ¡ Ges B(a, – 4) we›`y †_‡K g~j we›`yi `~iZ¡ mgvb n‡j, a Gi gvb KZ? K – 6 L 20 M 6 N 52 19. 3x + 2y = 6 mgxKi‡Yi Xvj KZ? K – 3 2 L 3 2 M 3 N 6  DÏxcKwU c‡o 20 I 21 bs cÖ‡kœi DËi `vI : B C P Q A D a d b c P, Q h_vμ‡g  AB I  DC Gi ga ̈we›`y| 20. P we›`yi Ae ̄’vb †f±i †KvbwU? K a + b – c 2 L b – a 2 M a – b 2 N a + b 2 21.  PQ Gi †ÿ‡Î i. PQ 7 BC 7 AD ii.  PQ = 1 2 ( )  BC –  AD iii.  PQ = 1 2 ( )  AD +  BC wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 22. GKwU wbi‡cÿ Q°v wb‡ÿ‡c †gŠwjK I †Rvo DVvi m¤¢vebv KZ? K 1 6 L 1 2 M 2 3 N 5 6 23. GKwU mylg PZz ̄Íj‡Ki †h †Kv‡bv av‡ii ˆ`N© ̈ 3 cm n‡j Gi mgMÖZ‡ji †ÿÎdj KZ? K 9 3 4 cm2 L 27 4 cm2 M 9 3 cm2 N 27 cm2 24. GKRb †jv‡Ki åg‡Yi m¤¢vebvi gva ̈‡g probability tree wb‡¤œ †`Iqv n‡jv XvKv 3 7 4 7 KzwgÍÏv evGm| KzwgÍÏv evGm bq| wmGjU ˆUÇGb bq| wmGjU ˆUÇGb| wmGjU ˆUÇGb bq| wmGjU ˆUÇGb| 4 9 5 9 4 9 5 9 †jvKwUi Kzwgjøv ev‡m bq Ges wm‡jU †Ua‡b hvIqvi m¤¢vebv K 12 63 L 15 63 M 16 63 N 20 63 25. 15 †m.wg. evûwewkó eM©vKvi f‚wgi Dci Aew ̄’Z GKwU wcivwg‡Wi D”PZv 20 †m.wg. n‡j Gi AvqZb KZ? K 4500 Nb †m.wg. L 2250 Nb †m.wg. M 1500 Nb †m.wg. N 1200 Nb †m.wg. 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN 14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN Self test 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1 K 2 N 3 K 4 L 5 L 6 M 7 M 8 K 9 L 10 N 11 L 12 M 13 K 14 M 15 N 16 N 17 N 18 L 19 K 20 N 21 L 22 K 23 M 24 N 25 M
m„Rbkxj D”PZi MwYZ : beg-`kg †kÖwY  21 †mUK 21 h‡kvi †evW© 2019 welq †KvW : 1 2 6 mgq25 wgwbU D”PZi MwYZ eûwbe©vPwb Afxÿv c~Y©gvb25 [we‡kl `aóe ̈ : mieivnK...Z eûwbe©vPwb Afxÿvi DËic‡Î cÖ‡kœi μwgK b¤^‡ii wecix‡Z cÖ`Ë eY©msewjZ e„Ëmg~n n‡Z mwVK/ m‡e©vrK...ó Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki| cÖwZwU cÖ‡kœi gvb 1| mKj cÖ‡kœi DËi w`‡Z n‡e| cÖkœc‡Î †Kv‡bv cÖKvi `vM/wPý †`Iqv hv‡e bv|] 1. x  A\B Gi cwie‡Z© wb‡Pi †KvbwU †jLv hvq? K x  A Ges x  B L x  A Ges x  B M x  A Ges x  B N x  A Ges x  B 2. 4x3 – 3x2 + 2a + 6 eûc`xi GKwU Drcv`K (x + 2) n‡j a Gi gvb KZ? K – 19 L 7 M 13 N 19 3. P(x, y, z) = (x + y) (y + z) (z + x) + zyz n‡j i. P(x, y, z) PμμwgK ivwk ii. P(x, y, z) cÖwZmg ivwk iii. P(– 2, 1, 2) = – 4 wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 4. PQR Gi gva ̈gvÎq PM, QN Ges RL n‡j i. PR2 > PQ2 + QR2 hLb Q ̄’~j‡KvY ii. PR2 < PQ2 + QR2 hLb Q m~2‡KvY iii. 4(PQ2 + QR2 + RP2 ) = 3(PM2 + QN2 + RL2 ) wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 5. DEF-Gi cwi‡K›`a O Ges DM  EF| DE = 5 †m.wg. DF = 3.5 †m.wg., DM = 2 †m.wg. n‡j, wÎfzRwUi be we›`y e„‡Ëi e ̈vmva© KZ? K 8.75 †m.wg. L 4.38 †m.wg. M 2.19 †m.wg. N 1.50 †m.wg.  wb‡Pi Z‡_ ̈i Av‡jv‡K 6 I 7 bs cÖ‡kœi DËi `vI : p = x2  3x  36 6. p = 0 n‡j, mgxKiYwUi wbðvqK KZ? K 135 L 153 M 135 N 153 7. p – 2 = 0 n‡j, mgxKiYwUi mgvavb wb‡Pi †KvbwU? K 8, – 5 L – 8, 5 M 8 N – 5 8. x 4 + x 5 + x 12  16 15 AmgZvwUi mgvavb †m‡Ui msL ̈v‡iLv wb‡Pi †KvbwU? K –2 –1 0 1 2 3 4 L –2 –1 0 1 2 3 4 M –2 –1 0 1 2 3 4 N –2 –1 0 1 2 3 4 9. †Kv‡bv ̧‡YvËi avivi mvaviY AbycvZ 1 2x + 3 Ges AmxgZK mgwó 1 2(x + 1) n‡j avivwUi 1g c` wb‡Pi †KvbwU? K 1 2x – 3 L 1 2(x – 1) M 1 2x + 2 N 1 2x + 3 10. †Kv‡bv Abyμ‡gi n-Zg c` 3n – 5, n  N n‡j AbyμgwUi beg c` wb‡Pi †KvbwU? K – 2 L 22 M 27 N 32 11. 3 †m.wg., 4 †m.wg. Ges 5 †m.wg. e ̈vmva© wewkó wZbwU e„Ë ci ̄úi‡K ewnt ̄úk© Ki‡j †K›`aÎq Øviv Drcbœ wÎfz‡Ri cwimxgv KZ? K 60 †m.wg. L 24 †m.wg. M 12 †m.wg. N 6 †m.wg. 12. wb‡Pi †Kvb dvskbwU GK-GK? K F(x) = x2 + 3 L F(x) = x2 – 3 M F(x) = 1 x – 3; x  3 N F(x) = 3 x ; x  0 13. – 785 Gi Ae ̄’vb †Kvb PZzf©v‡M Aew ̄’Z? K 1g L 2q M 3q N 4_© 14. sin  = – 1 2 Ges sin  I cos  GKB wPýhy3 n‡j  †KvYwU †Kvb PZzf©v‡M Aew ̄’Z n‡e? K 1g L 2q M 3q N 4_© 15. GKwU mge„Ëf‚wgK †KvY‡Ki D”PZv 3 †m.wg. Ges e ̈vmva© 5 †m.wg. n‡j †KvY‡Ki AvqZb KZ? K 78.54 Nb †m.wg. (cÖvq) L 62.83 Nb †m.wg. (cÖvq) M 47.12 Nb †m.wg. (cÖvq) N 37.70 Nb †m.wg. (cÖvq)  DÏxcKwU c‡o 16 I 17 bs cÖ‡kœi DËi `vI : ( ) x4 + 1 x4 – 2 3 GKwU exRMvwYwZK ivwk| 16. ivwkwUi we ̄Í...wZ‡Z c` msL ̈v KZwU? K 3 L 4 M 6 N 7 17. ivwkwUi we ̄Í...wZ‡Z x ewR©Z c‡`i gvb KZ? K – 20 L – 1 M 15 N 20  wb‡Pi DÏxcKwU c‡o 18 I 19 bs cÖ‡kœi DËi `vI : P(5, 6), Q(– 3, 8) Ges R(– 3, 2) we›`y wZbwU Nwoi KuvUvi wecixZ w`‡K AvewZ©Z nq| 18. PQR Gi †ÿÎdj KZ? K 6 eM© GKK L 24 eM© GKK M 48 eM© GKK N 96 eM© GKK 19. PQ Gi i. ˆ`N© ̈ 2 17 GKK ii. Xvj – 1 4 iii. mgxKiY 4x + y = 26 wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 20. y + 3x + 5 = 0 †iLvwU x-Aÿ‡K P we›`y‡Z †Q` Ki‡j P Gi ̄’vbv1⁄4 KZ? K (– 5, 0) L (0, – 5) M ( ) – 5 3  0 N ( ) 0  5 3 21. G D E O F DEFG mvgvšÍwi‡Ki `yBwU KY© DF Ges EG n‡j i.  EO =  OG = 1 2  EG ii.  DG = 1 2  DF + 1 2  EG iii.  OF –  OE =  EF wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 22. 6 †m.wg. e ̈vmwewkó GKwU †MvjKvK...wZ ej GKwU NbKvK...wZi ev‡· wVKfv‡e Gu‡U hvq| ev·wUi AbwaK...Z As‡ki AvqZb KZ? K 102.90 Nb †m.wg. (cÖvq) L 688.78 Nb †m.wg. (cÖvq) M 823.22 Nb †m.wg. (cÖvq) N 1614.90 Nb †m.wg. (cÖvq) 23. 1 + logp(qr) = 0 n‡j wb‡Pi †KvbwU mwVK? K pqr + 1 = 0 L pqr – 1 = 0 M qr – 1 = 0 N pqr = 0  wb‡Pi DÏxcKwU c‡o 24 I 25 bs cÖ‡kœi DËi `vI : `yBwU wbi‡cÿ gy`av GKmv‡_ GKevi wb‡ÿc Kiv n‡jv| 24. `yBwU †Uj cvIqvi m¤¢vebv KZ? K 1 4 L 3 8 M 1 2 N 3 4 25. eo‡Rvi GKwU †nW cvIqvi m¤¢vebv KZ? K 1 4 L 1 2 M 3 4 N 7 8 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN 14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN Self test 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1 L 2 N 3 N 4 K 5 M 6 N 7 K 8 K 9 N 10 L 11 L 12 M 13 N 14 M 15 K 16 N 17 K 18 L 19 K 20 M 21 N 22 K 23 L 24 K 25 M
†mUL 22 Kzwgjøv †evW© 2019 welq †KvW : 1 2 6 mgq25 wgwbU D”PZi MwYZ eûwbe©vPwb Afxÿv c~Y©gvb25 [we‡kl `aóe ̈ : mieivnK...Z eûwbe©vPwb Afxÿvi DËic‡Î cÖ‡kœi μwgK b¤^‡ii wecix‡Z cÖ`Ë eY©msewjZ e„Ëmg~n n‡Z mwVK/ m‡e©vrK...ó Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki| cÖwZwU cÖ‡kœi gvb 1| mKj cÖ‡kœi DËi w`‡Z n‡e| cÖkœc‡Î †Kv‡bv cÖKvi `vM/wPý †`Iqv hv‡e bv|] 1. Y = 70 n‡j Y Gi m¤ú~iK †Kv‡Yi A‡a©‡Ki gvb KZ? K 35 L 55 M 70 N 110 2. x2 – x – 13 = 0 n‡j, mgxKiYwUi GKwU g~j wb‡Pi †KvbwU? K 1 + 53 2 L 1 + – 51 2 M – 1 + 51 2 N – 1 – 53 2 3. p, q Ges r GKwU ev ̄Íe msL ̈v| p > q Ges r  0 n‡j i. pr > qr, hLb r > 0 ii. pr < qr, hLb r < 0 iii. p r < q r , hLb r > 0 wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 4. †Kv‡bv Abyμ‡gi n-Zg c` = 1 – (– 1) n 2 , Gi 20 Zg c` †KvbwU? K 2 L 1 M 0 N – 1  DÏxcKwU c‡o 5 I 6 bs cÖ‡kœi DËi `vI : M b N P a 5. sin P + cos M = KZ? K 2b a L 2a b M b + a2 – b2 a N a + a2 – b2 a 6. tan M Gi gvb †KvbwU? K b a2 – b2 L a a2 – b2 M a2 – b2 b N a2 – b2 a 7. GKwU AvqZvKvi Nbe ̄‘i ˆ`N© ̈, cÖ ̄’ I D”PZv h_vμ‡g 5 †m.wg. 4 †m.wg. I 3 †m.wg. n‡j, Gi KY© KZ? K 5 2 †m.wg. L 25 †m.wg. M 25 2 †m.wg. N 50 †m.wg. 8. M O N wP‡Î ON = 3 GKK, MN = 5 GKK n‡j, †ÿÎwUi AvqZb KZ? K 48 Nb GKK L 36 Nb GKK M 16 Nb GKK N 12 Nb GKK 9. –3 0 3 Dc‡ii msL ̈v‡iLvi e ̈ewa n‡jv K [– 3, 3] L [– 3, 3[ M ] – 3, 3[ N ] – 3, 3] 10. GKwU Q°v wb‡ÿ‡c 2 Avmvi m¤¢vebv KZ? K 1 6 L 1 2 M 2 3 N 1 11. GKwU _‡j‡Z PviwU mv`v ej I cuvPwU jvj ej Av‡Q| ˆ`efv‡e GKwU ej Zz‡j Avbv n‡j, ejwU mv`v nIqvi m¤¢vebv KZ? K 1 9 L 4 9 M 5 9 N 1 12. sec  = 2, 3 2 <  < 2 n‡j i. tan  = – 3 ii. sin  = – 3 2 iii. cos  = 1 2 wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 13. hw` A  B nq Z‡e wb‡Pi †KvbwU mwVK? K B  A = A L B  A = B M A  B = A N A  B 14. f(x) = 3x – 5 dvskbwUi †Wv‡gb wb‡Pi †KvbwU? K {xR : x > 3 5 } L {xR : x  3 5 } M {xR : x > 5 3 } N {xR : x  5 3 } 15. a3 – a2 – 10a – 8 eûc`xi GKwU Drcv`K wb‡Pi †KvbwU? K a + 4 L a + 2 M a – 1 N a – 2 16. x x2 – 9 Gi AvswkK fMœvsk †KvbwU? K 1 x + 3 + 1 x – 3 L 1 x + 3 – 1 x – 3 M 1 2(x + 3) + 1 2(x – 3) N 1 2(x + 3) – 1 2(x – 3) 17. D C F G B A mgevû wÎfzR ABC Gi fi‡K›`a G n‡j i. AG = 2 3 AD ii. BG : GF = 3 : 2 iii. AC2 – CD2 = AD2 wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii  wb‡Pi wP‡Îi Av‡jv‡K 18 I 19 bs cÖ‡kœi DËi `vI : R Q S P 18. PS Gi j¤^ Awf‡ÿc †KvbwU? K PQ L QR M QS N RS 19. R m~2‡KvY n‡j PS2 Gi gvb †KvbwU? K PR2 + RS2 – 2RS.SQ L PR2 + RS2 – 2RS.RQ M PR2 + RS2 + 2PR.SQ N PR2 + RS2 + 2PR.PQ 20. hw` m, n, p > o Ges m  1, n  1 nq, Z‡e i. logmp = lognp  logmn ii. logm m  logn n  logp p = 1 8 iii. xlogmy = ylogmx Dc‡ii Z‡_ ̈i Av‡jv‡K wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 21. (1 – x) ( ) 1 + x 2 8 Gi we ̄Í...wZ‡Z x Gi mnM KZ? K 3 L 1 2 M – 1 N – 1 2  DÏxcKwU c‡o 22 I 23 bs cÖ‡kœi DËi `vI : x + y = 1 GKwU mij‡iLvi mgxKiY| 22. †iLvwU x-Aÿ‡K †h we›`y‡Z †Q` K‡i Zvi ̄’vbvsK †KvbwU? K (1, 1) L (1, 0) M (0, 1) N (0, 0) 23. †iLvwU Aÿ؇qi mv‡_ †h †ÿÎwU ˆZwi K‡i Zvi †ÿÎdj KZ? K 2 eM© GKK L 1 eM© GKK M 1 2 eM© GKK N 1 4 eM© GKK 24. D E Q R P D I E h_vμ‡g PQ I PR Gi ga ̈we›`y n‡j, wb‡Pi †KvbwU mwVK? K  QR = 2 ( PQ +  PR) L  QR = 2 ( PE +  PD) M  QR = 2 ( PD   PE) N  QR = 2 ( PE   PD) 25. F D E  DF †f±‡ii gvb KZ? K  EF +  ED L  DE +  EF M  DE +  FE N  DE   EF 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN 10 KLMN 11 KLMN 12 KLMN 13 KLMN 14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN Self test 19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1 L 2 K 3 K 4 M 5 K 6 M 7 K 8 N 9 N 10 K 11 L 12 N 13 M 14 N 15 L 16 M 17 L 18 M 19 L 20 N 21 K 22 L 23 M 24 N 25 L