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†mUK 10 XvKv †evW© 2020 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. hw` A B = , n(A) = 2 Ges n(A B) = 10 n‡j n(B) = ? K 2 L 6 M 8 N 10 2. †mU An = {n, 2n, 3n, ...} Gi Rb ̈ i. A1 A1 ii. A1 A2 iii. A1 A3 wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii wb‡Pi DÏxc‡Ki Av‡jv‡K 3 I 4 bs cÖ‡kœi DËi `vI : Pn = {2n , 22n, 23n, ......} mKj n N. 3. P2 P4 Gi gvb wb‡Pi †KvbwU? K P1 L P2 M P3 N P4 4. P2 Gi Dc‡mU †KvbwU? K P1 L P3 M P4 N P5 5. P X P A B Y A C B XY Gi Dci AA Gi j¤^ Awf‡ÿc n‡jv K A L AA M AB N BC 6. cx2 + bx + a = 0 mgxKi‡Yi mv‡_ 2x2 + 3x + 5 = 7x + 1 mgxKiY Zzjbv Ki‡j 'a' Gi gvb KZ? K 1 L 2 M 3 N 4 wb‡Pi DÏxc‡Ki Av‡jv‡K 7 I 8bs cÖ‡kœi DËi `vI : a x + 3 = ay .a4 Ges x + y = 1 GKwU mgxKiY †RvU| 7. x Gi gvb KZ? K 3 L 4 M 3 N 5 8. y Gi gvb KZ? K 2 L 3 M 4 N 6 9. p, q Ges r ev ̄Íe msL ̈v †hLv‡b p 0, q > r Gi Rb ̈ i. p + q > p + r hLb p > 0 ii. pq < pr hLb p < 0 iii. q p > r p hLb p > 0 wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 10. 2 + 5 + 8 + .... avivi n-Zg c` KZ? K n + 1 L 3n 1 M 2n N 4n 2 wb‡Pi DÏxc‡Ki Av‡jv‡K 11 I 12bs cÖ‡kœi DËi `vI : 1 + 2 3 + 4 3 + 8 3 3 + 16 9 + ... GKwU aviv| 11. cÖ`Ë avivi 7g c` †KvbwU? K 32 9 3 L 32 27 M 64 27 N 64 27 3 12. cÖ`Ë avivi †hvMdj KZ? K 3 2 3 L 3 3 2 M 2 3 2 N 2 2 3 [we:`a: cÖ`Ë avivwUi AmxgZK mgwó bvB| 13. cos (300) Gi gvb KZ? K 1 2 L 3 2 M 1 2 N 3 2 14. C A B 2 1 Dc‡ii wP‡Îi Rb ̈ i. AB2 = 5 GKK ii. 2 cos + 3 = 2 3 iii. sin + 2 cos2 = 2 wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 15. hw` log 27 x = 11 3 nq Zvn‡j x = ? K 3 L 9 M 1 3 N 1 9 16. 4 x+ 1 = 4.6y = 4.9z †hLv‡b x 0, y 0, z 0. DÏxc‡Ki Av‡jv‡K †KvbwU mwVK? K 4 = 9 z x + 1 L 4 = 9 z x M 4 = 6 y x + 1 N 4 = 6 x + 1 y 17. px = qy = rz Ges q2 = pr n‡j, wb‡Pi †KvbwU mwVK? K 1 x + 1 z = y 2 L 1 x + 1 y = 2 z M 1 x + 1 z = 2 y N 1 x + 1 z = 1 y 18. ivwk ̧‡jvi Rb ̈ i. 6! = 720 ii. 6 C1 = 6 C5 = 6 iii. 6 C0 = 0! wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 19. (1 2x + x2 ) 4 Gi we ̄Í...wZ‡Z x2 Gi mnM KZ? K 28 L 16 M 28 N 56 wb‡Pi DÏxc‡Ki Av‡jv‡K 20 I 21 bs cÖ‡kœi DËi `vI : 6 †m.wg. e ̈vmwewkó GKwU avZe †MvjK Mwj‡q 2 †m.wg. e ̈vmva©wewkó GKwU mge„Ëf‚wgK wb‡iU wmwjÛvi ˆZwi Kiv n‡jv| 20. DÏxc‡Ki wmwjÛv‡ii D”PZv KZ? K 72 cm L 36 cm M 27 cm N 9 cm 21. wmwjÛv‡ii m¤ú~Y© c„ôZ‡ji †ÿÎdj KZ? K 44 L 40 M 17 N 12 22. 6x 3y 12 = 0 mij‡iLvi Rb ̈ i. Xvj = 2 ii. mij‡iLvwU x-Aÿ‡K (2, 0) we›`y‡Z †Q` K‡i iii. mij‡iLvwU y-Aÿ‡K (0, 4) we›`y‡Z †Q` K‡i wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 23. ABC Gi AB I AC evûi ga ̈we›`y h_vμ‡g D I E n‡j AD + DE = ? K 1 2 BC L 1 2 AC M 1 2 AB N 1 2 BE wb‡Pi Z‡_ ̈i Av‡jv‡K 24 I 25 bs cÖ‡kœi DËi `vI : 50 Rb †jv‡Ki i‡3i MÖæc wb¤œiƒc : 10 R‡bi Av‡Q A cÖKv‡ii i3, 12 R‡bi Av‡Q B cÖKv‡ii i3, 13 R‡bi Av‡Q O cÖKv‡ii i3 Ges 15 R‡bi Av‡Q AB cÖKv‡ii i3| 24. GKRb †jv‡Ki B cÖKv‡ii i3 _vKvi m¤¢vebv KZ? K 1 5 L 6 25 M 13 50 N 3 20 25. GKRb †jv‡Ki AB cÖKv‡ii i3 bv _vKvi m¤¢vebv KZ? K 2 3 L 4 5 M 3 10 N 7 10 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 M 2 N 3 L 4 M 5 K 6 N 7 M 8 M 9 N 10 L 11 M 12 * 13 K 14 M 15 L 16 L 17 M 18 N 19 K 20 N 21 K 22 K 23 L 24 L 25 N
†mUK 12 h‡kvi †evW© 2020 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. hw` (x) = 4x 1 Ges 0 x 3 nq Zvn‡j dvsk‡bi †iÄ KZ? K {y Ñ : 0 y 3} L {y Ñ : 1 y 11} M {y Ñ : 1 y 13} N {y Ñ : 1 y 11} 2. 40 x 3x 30 +2x U P Q U = P Q Ges n(U) = 90 n‡j Dc‡ii †fbwPÎ Abymv‡i P\Q Gi gvb KZ? K 15 L 20 M 35 N 50 wb‡Pi Z‡_ ̈i Av‡jv‡K 3 I 4 bs cÖ‡kœi DËi `vI : P(x) = x3 mx2 + 3x 1 GKwU eûc`x| 3. x 1, P(x) Gi GKwU Drcv`K n‡j, m Gi gvb KZ? K 5 L 3 M 3 N 5 4. eûc`xwU‡Z i. gyL ̈ mnM I aaæec‡`i mgwó k~b ̈ ii. eûc`xi gvÎv 3 iii. k~b ̈ gvÎvhy3 c`‡K aaæec` e‡j wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 5. hw` 5x 7 (x 1)(x 2) A x 1 + B x 2 nq, †hLv‡b A I B g~j` msL ̈v, Z‡e B Gi gvb wb‡Pi †KvbwU? K 3 L 2 M 2 N 3 wb‡Pi Z‡_ ̈i Av‡jv‡K 6 I 7 bs cÖ‡kœi DËi `vI : PQRS e„Ë ̄’ PZzf©y‡Ri PR †K›`aMvgx Ges PT QS, PQ = 8, PS = 5, PT = 4. P O S Q R T 6. e„ËwUi e ̈vmva© KZ? K 3 L 5 M 10 N 20 7. OR = 6 n‡j PQR wÎfz‡Ri bewe›`ye„‡Ëi †ÿÎdj KZ? K 3 L 6 M 9 N 36 8. L M O N LMN G LOM = 130, LO = 10, MO = 7, NO = 5 n‡j LM Gi ˆ`N© ̈ KZ? K 8.88 L 14.79 M 79 N 219 9. ax2 + bx + c = 0 mgxKi‡Y b2 4ac > 0 Ges c~Y©eM© bv n‡j, g~j `yBwU i. ev ̄Íe ii. mgvb iii. Ag~j` wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 10. xy = y2 Ges y2y = x4 mgxKiY `yBwU mgvavb wb‡Pi †KvbwU? †hLv‡b x 1. K (2, 2) L (2, 1) M (2, 2) N (2, 2) 11. Avwjd 7 UvKv `‡i ywU †cwÝj Ges 9 UvKv `‡i (y + 3)-wU LvZv wK‡b‡Q| †gvU g~j ̈ Ab~aŸ© 171 UvKv n‡j Avwjd me©vwaK KqwU †cwÝj wK‡b‡Q? K 9 L 10.50 M 10.80 N 12.40 12. 2 1 + 1 2 1 4 + ..... My‡YvËi avivwUi AmxgZK mgwó KZ? K 4 3 L 3 4 M 4 3 N 3 4 13. 7 : 35 am G NÈvi KuvUv Ges wgwb‡Ui KuvUvi ga ̈eZ©x †KvY KZ? K 17.5 L 15.5 M 15.5 N 17.5 14. = 3 2 n‡j i. cot( ) + 3 = 3 ii. sin ( ) 6 = 3 2 iii. cosec ( ) 4 = 2 wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 15. hw` cot ( ) n 2 + = 1 Ges = 4 nq Z‡e n Gi gvb KZ? K 1 L 2 M 4 N 0 16. y = 1 5x Gi wecixZ dvskb wb‡Pi †KvbwU? K log5 (1 x) L 1 5x M log5 ( ) 1 1 x N log5(x 1) 17. (x) = x |x| dvskbwUi †iÄ wb‡Pi †KvbwU? (hLb x 0) K Rf = {1} L Rf = {1} M Rf = {x : x Ñ} N Rf = {1, 1} wb‡Pi Z‡_ ̈i Av‡jv‡K 18 I 19 bs cÖ‡kœi DËi `vI : ( ) p3 + 1 p3 2n ; †hLv‡b n GKwU abvZ¥K c~Y©msL ̈v| 18. cÖ`Ë we ̄Í...wZi c`msL ̈v wb‡Pi †KvbwU? K 2n 1 L 2n M 2n + 1 N 2n + 6 19. hw` n = 2 nq Zvn‡j cÖ`Ë we ̄Í...wZi p gy3 c‡`i gvb †KvbwU? K 6 L 12 M 24 N 2 wb‡Pi Z‡_ ̈i Av‡jv‡K 20 I 21 bs cÖ‡kœi DËi `vI : 3x + 4y = 12 Ges x + 4y = 5 `yBwU mij‡iLvi mgxKiY| 20. ca`Ë †iLv؇qi Xvj؇qi ̧Ydj KZ? K 3 16 L 1 3 M 3 N 16 3 21. 4x + 3y = 12 †iLvwU I Aÿ؇qi mv‡_ MwVZ wÎfz‡Ri †ÿÎdj wbY©q Ki| K 6 L 8 M 12 N 4 22. A B P Q D C wP‡Î P I Q h_vμ‡g BD I AC Gi ga ̈we›`y, †hLv‡b AB || CD Ges AB = 5 cm, CD = 7 cm Zvn‡j PQ Gi gvb KZ? K 6 cm L 4 cm M 2 cm N 1 cm 23. A B D C wP‡Î AB = CD = 6 cm Ges Bnvi wmwjÛvi AvK...wZi As‡ki e ̈vmva© 2 cm n‡j K ̈vcmy‡ji AvqZb KZ? K 92.15 cc L 108.90 cc M 152.66 cc N 180.90 cc 24. 15wU jvj ej I 4wU Kv‡jv ej n‡Z ˆ`efv‡e GKwU ej wbe©vPb Kiv n‡j i. ejwU jvj nIqvi m¤¢vebv 15 19 ii. ejwU Kv‡jv nIqvi m¤¢vebv 11 19 iii. ejwU Kv‡jv bv nIqvi m¤¢vebv 15 19 wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 25. AvenvIqvi `ßi n‡Z cvIqv wi‡cvU© Abyhvqx †m‡Þ¤^i gv‡m 19 w`b e„wó n‡q‡Q| Zvn‡j 8B †m‡Þ¤^i e„wó nIqvi m¤¢vebv KZ? K 8 30 L 19 31 M 19 30 N 8 31 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 N 2 M 3 M 4 N 5 N 6 L 7 M 8 L 9 L 10 M 11 K 12 M 13 N 14 K 15 K 16 M 17 N 18 M 19 K 20 K 21 K 22 N 23 L 24 L 25 M
m„Rbkxj D”PZi MwYZ : beg-`kg †kÖwY 13 †mUK 13 Kzwgjøv †evW© 2020 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 †KvbwU eûc`x? K x + 1 x L x2 + x M x2 x x3 x N x3 + x2 2. (1 2y)4 Gi we ̄Í...wZ‡Z y3 Gi mnM KZ? K 32 L 8 M 16 N 32 3. 1 4 + 1 42 + 1 43 + .......AbšÍ ̧‡YvËi avivwUi AmxgZK mgwó KZ? K 1 5 L 1 3 M 4 5 N 4 3 4. tan = 3 4 Ges < < 3 2 n‡j cos Gi gvb KZ? K 5 4 L 4 5 M 4 5 N 5 4 wb‡Pi Z‡_ ̈i Av‡jv‡K 5 I 6 bs cÖ‡kœi DËi `vI : X X P(3, 0) O Y Y Q(0, 4) 5. PQ †iLvi Xvj KZ? K 4 3 L 3 4 M 3 4 N 4 3 6. PQ †iLvi mgxKiY wb‡Pi †KvbwU? K 3x 4y = 9 L 4x 3y = 12 M 4x + 3y = 12 N 3x 4y = 9 7. F(x) = 1 4x Gi †Wv‡gb †KvbwU? K {x R : x } 1 4 L {x Ñ : x } 1 4 M {x Ñ : x > } 1 4 N {x Ñ : x < } 1 4 8. PQR Gi S, QR Gi ga ̈we›`y Ges PM QR n‡j A ̈v‡cv‡jvwbqv‡mi Dccv` ̈ Abyhvqx wb‡Pi †KvbwU mwVK? K PQ2 + PR2 = 2QM2 L PQ2 + PR2 = 2RM2 + 2PS2 M PQ2 + PR2 = 2QS2 + 2MS2 N PQ2 + PR2 = 2 QS2 + 2 PS2 9. 5x 1 x2 = 0 Gi wbðvqK †KvbwU? K 21 L 24 M 26 N 29 10. 2295 †KvYwU †Kvb PZzf©v‡M Aew ̄’Z? K ca_g L wØZxq M Z„Zxq N PZz_© 11. log 27 y = 1 1 3 n‡j y Gi gvb KZ? K 3 1 3 L 3 3 2 M 9 N 36 12. wP‡Î P Q R O 10 ˆm.wg. 8 ˆm.wg. i. QR = 6 †m.wg. ii. eμZ‡ji †ÿÎdj = 60 eM© †m.wg. iii. AvqZb = 96 Nb †m.wg. wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 13. A = {x N : 4x < 20}, n‡j A Gi Dc‡mU msL ̈v wb‡Pi †KvbwU? K 32 L 16 M 8 N 4 14. bewe›`y e„‡Ëi e ̈vmva© 6 †m.wg. n‡j cwie„‡Ëi †ÿÎdj KZ? K 144 eM© †m.wg. L 36 eM© †m.wg. M 12 eM© †m.wg. N 9 eM© †m.wg. 15. O †K›`awewkó ABP e„‡Ëi OB = KZ? O P A B 8 ˆm.wg. 2c K 2 †m.wg. L 4 †m.wg. M 8 †m.wg. N 16 †m.wg. wb‡Pi Z‡_ ̈i Av‡jv‡K 16 I 17 bs cÖ‡kœi DËi `vI : P A B O D wP‡Î, O e„‡Ëi †K›`a, BD = 8 †m.wg. Ges OP = 12 †m.wg.| 16. AB Gi j¤^ Awf‡ÿ‡ci gvb KZ? K 3 †m.wg. L 4 †m.wg. M 6 †m.wg. N 8 †m.wg. 17. PA Gi ˆ`N© ̈ KZ? K 8 †m.wg. L 11.31 †m.wg. M 12 †m.wg. N 12.65 †m.wg. 18. ( ) y2 + 1 y2 4 Gi we ̄Í...wZ‡Z y ewR©Z c‡`i gvb KZ? K 2 L 4 M 6 N 8 19. F(p, q, r) = p3 + q3 + r3 3pqr n‡j i. Gi GKwU Drcv`K (p + q + r) ii. F(2, 2, 1) = 27 iii. F(p, q, r) cÖwZmg wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 20. S = {(2, 3), (4, 1), (5, 0), (6, 3)} n‡j i. S Gi †iÄ {3, 1, 0} ii. S GKwU GK-GK dvskb iii. S1 = {(3, 2), (1, 4), (0, 5), (3, 6)} wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii 21. cv‡ki wP‡Î G, ABC Gi fi‡K›`a n‡j A B D C F E G i. AG : GD = 1 : 2 ii. AB + AC = 2 AD iii. AD + BE + CF = 0 wb‡Pi †KvbwU mwVK? K i I ii L i I iii M ii I iii N i, ii I iii wb‡Pi Z‡_ ̈i Av‡jv‡K 22 I 23 bs cÖ‡kœi DËi `vI : GKwU gy`av wZbevi wb‡ÿc Kiv n‡jv| 22. `yBwU H Ges GKwU T cvIqvi m¤¢vebv KZ? K 1 8 L 3 8 M 1 2 N 5 8 23. Kgc‡ÿ GKwU H cvIqvi m¤¢vebv KZ? K 7 8 L 5 8 M 3 8 N 1 8 24. 4x + y 4 = 0 mij‡iLvwU x Aÿ‡K †h we›`y‡Z †Q` K‡i Zvi ̄’vbv1⁄4 KZ? K (0, 4) L (0, 4) M (1, 0) N (1, 0) 25. P(y) = y3 + 2y2 5y 6 n‡j, wb‡Pi †KvbwU P(y) Gi GKwU Drcv`K? K y 3 L y 1 M y + 2 N y + 3 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 L 4 L 5 K 6 L 7 L 8 N 9 K 10 M 11 M 12 M 13 L 14 K 15 L 16 L 17 L 18 M 19 N 20 L 21 M 22 L 23 K 24 N 25 N
Tiếng Việt