Title Real Numbers and Inequalities MCQ Practice Sheet Solution.pdf ✅

ev ̄Íe msL ̈v I AmgZv  MCQ Practice Sheet Solution 1 ev ̄Íe msL ̈v I AmgZv Real Numbers and Inequalities cÖ_g Aa ̈vq 1. wb‡Pi †KvbwU mwVK? [B.B 19] Z  N N  R QІ  QІ  = R QІ  QІ    DËi: N  R e ̈vL ̈v: ev ̄Íe msL ̈vi Dc‡mU ̧wji †ÿ‡Î: N  Z  R ; QІ  QІ  = R ; QІ  QІ  =  2. wb‡Pi †Kvb m¤úK©wU mwUK? [S.B 17] Z  N QІ  N QІ  Z QІ  R DËi: QІ  R e ̈vL ̈v: Avgiv Rvwb, R n‡”Q ev ̄Íe msL ̈vi †mU, QІ n‡”Q g~j` msL ̈vi †mU, Z n‡”Q c~Y©msL ̈vi †mU, N n‡”Q ̄^vfvweK msL ̈vi †mU| G‡`i g‡a ̈ m¤úK© wb¤œiƒc: N  Z  QІ  R  QІ  R 3. wb‡Pi †KvbwU mwVK? [Ctg. B 17] N  Z  QІ  CІ  R N  QІ  Z  R  CІ N  Z  QІ  R  CІ N  QІ  Z  CІ  R DËi: N  Z  QІ  R  CІ 4. – 3  2x < 8 Gi mgvav‡b c~Y©msL ̈v KqwU? [J.B 17] 3 4 5 6 DËi: 5 e ̈vL ̈v: †`Iqv Av‡Q, – 3  2x < 8  – 3 2  x < 8 2 – 1.5  x < 4 hvi †jLwPÎ wb¤œiƒcÑ –2 –1.5 –1 0 1 2 3 4  cÖ`Ë e ̈ewa‡Z c~Y© msL ̈v: –1, 0, 1, 2, 3 AZGe c~Y©msL ̈v = 5wU 5. wb‡Pi †KvbwU g~j` msL ̈v? [Ctg.B 17]  e 1 5 5 125 DËi: 5 125 e ̈vL ̈v: 5 125 = 5 25  5 = 5 5 5 = 1 5 hv g~j`| 6. [1, 3) e ̈ewai AmgZv iƒc wb‡Pi †KvbwU? [C.B 17] 1 < x < 3 1  x < 3 1 < x  3 1  x  3 DËi: 1  x < 3 e ̈vL ̈v: [1, 3) e ̈ewai AmgZv iƒc: 1  x < 3. 7. a = 3, b = –7 Ges c = – 9 n‡j ||a – b| – c| Gi gvb †KvbwU? [C. B 17] 1 5 13 19 DËi: 19 e ̈vL ̈v: a = 3, b = –7 Ges c = – 9 n‡j ||a – b| – c| = ||3 – (– 7)| – (– 9)| = ||10| + 9| = |10 + 9| = 19 8. x < – 1 n‡j wb‡Pi †KvbwU mwVK? [Ctg.B 19] 3x > – 3 3x > 3 3x < 3 – 3x > 3 DËi: – 3x > 3 e ̈vL ̈v: x < – 1  3x < – 3  – 3x > 3 ; [– 1 Øviv ̧Y K‡i] 9. |x – 1|  1 AmgZvi mgvavb †mU †KvbwU? [All B 18] [– 1, 1] [0, 2] (0, 2] (– 1, 1) DËi: [0, 2] e ̈vL ̈v: |x – 1|  1  – 1  x – 1  1  – 1 + 1  x  1 + 1  0  x  2 hvi e ̈ewaiƒc = [0, 2] 10. |x – 3|  1 AmgZvi mgvavb †KvbwU? [B. B 17] – 4  x  4 – 4 < x < 4 2 < x < 4 2  x  4 DËi: 2  x  4 e ̈vL ̈v: |x – 3|  1  – 1  x – 3  1  – 1 + 3  x  1 + 3  2  x  4
2  Higher Math 2nd Paper Chapter-1 11. |2x + 1|  3 AmgZvi mgvavb †KvbwU? [C. B 19] – 2  x  1 – 2 < x < 1 – 1  x  2 – 1 < x < 2 DËi: – 2 < x < 1 e ̈vL ̈v: |2x + 1|  3  – 3  2x + 1  3  – 4  2x  2  – 2  x  1 12. |2x – 7|  3 n‡j wb‡Pi †KvbwU mwVK? [R.B 17] – 7  x  – 3 – 5  x  – 2 2  x  5 3  x  7 DËi: 2  x  5 e ̈vL ̈v: |2x – 7|  3  – 3  2x – 7  3  – 3 + 7  2x  3 + 7  4  2x  10 2  x  5 13. |x|  3 AmgZvi mgvavb †KvbwU? [J. B 19] (–  3]  [– 3, ) (– , – 3]  [3, ) [– 3, 3] [3, ] DËi: (– , – 3]  [3, ) e ̈vL ̈v: |x|  3  x  – 3 A_ev x  3  mgvavb †mU = {x  R : x  – 3 A_ev x  3} = (– , – 3]  [3, ) 14. |2x – 7| > 5 AmgZvwUi ev ̄Íe msL ̈vq mgvavb wK? [S.B 17] x < 1 x > 6 x > 6 A_ev x < 1 x > 6 Ges x < 1 DËi: x > 6 A_ev x < 1 e ̈vL ̈v: |2x – 7| > 5  2x – 7 > 5 A_ev 2x – 7 < – 5  2x > 5 + 7 A_ev 2x < – 5 + 7  x > 6 A_ev x < 1  wb‡Y©q mgvavb: x > 6 A_ev x < 1 15. |2x – 9| > 7 AmgZvwUi mgvavbÑ [R.B 19] (– , 1) (8, ) (– , 1)  (8, ) (– , 1)  (8, ) DËi: (– , 1)  (8, ) e ̈vL ̈v: |2x – 9| > 7  2x – 9 < – 7 A_ev 2x – 9 > 7  x < 1 A_ev x > 8  (– , 1) A_ev (8, )  (– , 1)  (8, ) 16. |2x – 1| Gi †ÿ‡Î †KvbwU mwVK? [Ctg. B 17] 1 – 2x hLb x < 1 2 2x – 1 hLb x < 1 2 1 + 2x hLb x < 1 2 2x + 1 hLb x < 1 2 DËi: 1 – 2x hLb x < 1 2 e ̈vL ̈v: |2x – 1| Gi gvb me©`v abvZ¥K A_©vr 2x – 1 n‡e hLb 2x – 1 > 0  x > 1 2 nq| Avevi, – (2x – 1) n‡e hLb 2x – 1 < 0  x < 1 2 nq| 17. – 7 < x + 3 < 5 †K cig gv‡bi mvnv‡h ̈ cÖKvk: [D.B 19] |x + 2| < 4 |x + 4| < 6 |x + 3| < 6 |x + 1| < 3 DËi: |x + 4| < 6 e ̈vL ̈v: – 7 < x + 3 < 5  – 7 + 1 < x + 3 + 1 < 5 + 1    ⸪  – 7 + 5 2 = – 1  – 6 < x + 4 < 6  |x + 4| < 6 18. –1 0 1 2 3 4 5 Dc‡iv3 msL ̈v‡iLvi cig gv‡b cÖKvk †KvbwU? [B. B 19] |x – 1| < 4 |x + 4| < – 1     x – 3 2 < 5 2 =     x – 3 2 > 5 2 DËi:     x – 3 2 < 5 2 e ̈vL ̈v: msL ̈v‡iLv n‡Z: – 1 < x < 4  – 1 – 3 2 < x – 3 2 < 4 – 3 2    ⸪  – 1 + 4 2 = 3 2  – 5 2 < x – 3 2 < 5 2      x – 3 2 < 5 2 19. a I b abvZ¥K ev ̄ÍemsL ̈v n‡j wb‡Pi †KvbwU mwVK? [C. B 17] |a – b| > |a| + |b| |a + b| < |a| + |b| |a + b| = |a| + |b| |a – b| = |a| + |b| DËi: |a + b| = |a| + |b| e ̈vL ̈v: a I b abvZ¥K nIqv a‡i †bB a = 3, b = 4  |a + b| = |3 + 4| = |7| = 7 Ges |a| + |b| = |3| + |4| = 3 + 4 = 7  |a + b| = |a| + |b|
ev ̄Íe msL ̈v I AmgZv  MCQ Practice Sheet Solution 3 20.       1  1 2  1 3  .... 1 n  ..... †mUwUi GKwU wb¤œmxgvÑ [Ctg.B 19] 1 2 0 2 1 DËi: 0 e ̈vL ̈v: n   n‡j 1 n  0        1  1 2  1 3  .... 1 n  ..... =       1  1 2  1 3  ....0  †mUwUi wb¤œmxgv = 0 21.      1  2  1 4  1 6  1 8 ..... Gi Bbwdgvg KZ? [B.B 17] 0 1 8 1 2  DËi: 0 e ̈vL ̈v: g‡b Kwi, S =      1  2  1 4  1 6  1 8 ..... †mUwUi Bbwdgvg Inf(S) = 1  = 0 22. S =      1  3  1 9  1 27  1 81 ..... Gi mywcÖgvg KZ? [D.B 19] 0 1 81 1 3  DËi: 1 3 e ̈vL ̈v: S =      1  3  1 9  1 27  1 81 ..... =      1  3  1 9  1 27  1 81 ..... 0  sup S = 1 3 23. S =      1  2  2 3  3 4  4 5  .... n n + 1  ..... †mUwUi sup S = ? [D. B 19] 1 2 4 5 1  DËi: 1 e ̈vL ̈v: n   n‡j, n n + 1 = n n     1 + 1 n = 1 1 + 1 n = 1 1 + 1  = 1 1 + 0 = 1  Sup S = 1 24. S = {x  N : 5  x 2 + 1  82} Gi mywcÖgvg KZ? [J. B 19] 2 4 9 81 DËi: 9 e ̈vL ̈v: S = {x  N : 5  x 2 + 1  82} = {x  N : 4  x 2  81} = {x  N : 2  x  9}  Sup S = 9 25. S = {x  R : x – x 2 + 6 > 0} n‡j, sup S = KZ? [C.B 19] – 2 – 3 2 3 DËi: 3 e ̈vL ̈v: S = {x  R : x – x 2 + 6 > 0} = {x  R : x2 – x – 6 < 0} = {x  R : – 2 < x < 3}  Sup S = 3 26. S = {x : x  Z Ges 8  x 2  27} Gi Mwiô wb¤œmxgv wb‡Pi †KvbwU? [R.B 19] – 5 – 3 3 5 DËi: – 5 e ̈vL ̈v: †`Iqv Av‡Q, S = {x : x  Z Ges 8  x 2  27}  S = {x : x  Z Ges – 3 3  x  – 2 2 A_ev 2 2  x  3 3}  S = {x : x  Z Ges – 5.2  x  – 2.8 A_ev 2.8  x  5.2} = {– 5, – 4, –3, 3, 4, 5} Inf S = – 5 27. ev ̄Íe msL ̈vi ̄^xKv‡h©i †ÿ‡Îi ms‡hvM wewai D`vniY †KvbwU? [All B 18] 2 + 3 = 3 + 2 (2 + 3) + 4 = 2 + (3 + 4) 2 + 0 = 2 2(3 + 4) = 2.3 + 2.4 DËi: (2 + 3) + 4 = 2 + (3 + 4) e ̈vL ̈v: a, b, c  R Gi Rb ̈ ms‡hvM wewa n‡”Q: (a + b) + c = a + (b + c) a = 2, b = 3, c = 4 n‡j, (2 + 3) + 4 = 2 + (3 + 4) 28. (x – 4) (x – 5) > 0 Gi mgvavb †KvbwU? [J.B 17] x > 4 Ges x < 5 x < 4 A_ev x > 5 x < 4 Ges x < 5 x > 4 A_ev x < 5 DËi: x < 4 A_ev x > 5 e ̈vL ̈v: †`Iqv Av‡Q, (x – 4) (x – 5) > 0 AmgZvwU mZ ̈ n‡e hw` I †Kej hw`, x – 4 > 0, x – 5 > 0 A_ev x – 4 < 0, x – 5 < 0 nq  x > 4, x > 5 A_ev x < 4, x < 5 A_©vr, x > 5 A_ev x < 4
4  Higher Math 2nd Paper Chapter-1 29. x 2 + 2x – 3 < 0 AmgZvwUi mgvavb wb‡Pi †KvbwU? [S.B 19] 1 < x < 3 x < – 3 ev x > 1 – 1 < x < 3 – 3 < x < 1 DËi: – 3 < x < 1 e ̈vL ̈v: x 2 + 2x – 3 < 0  x 2 + 3x – x – 3 < 0  (x + 3) (x – 1) < 0  – 3 < x < 1 30. 1 x(x – 1) < 0 Gi mgvavb wb‡Pi †KvbwU? [D.B 19] x < 0 A_ev x > 1 x > 0 Ges x < 1 x > 0 A_ev x > 1 x < 0 Ges x < 1 DËi: x > 0 Ges x < 1 e ̈vL ̈v: a < b n‡j 1 (x – a)(x – b) < 0 AmgZvwUi mgvavb: a < x < b  1 x(x – 1) < 0 ev 1 (x – 0)(x – 1) < 0 AmgZvwUi mgvavb: 0 < x < 1 31. (2x – 5) 2  0 Gi mgvavb †KvbwU? [D.B 19] x = 2.5 x  2.5 x  2.5 0  x  2.5 DËi: x = 2.5 e ̈vL ̈v: bs Ack‡bi Rb ̈ x = 2.5 n‡j, (2  2.5 – 5)2 = 0  0 hv mZ ̈ bs Ack‡bi Rb ̈ x = 1 n‡j x  2.5 (2  1 – 2.5)2 = 0.25  0 hv mZ ̈ bq bs Ack‡bi Rb ̈ x = 3 n‡j x  2.5  (2  3 – 2.5)2 = 1  0 hv mZ ̈ bq bs Ack‡bi Rb ̈ x = 1 n‡j 0  x  2.5  (2  1 – 2.5)2 = 0.25  0 hv mZ ̈ bq 32. (i) N = {1, 2, 3, 4......} [S.B 17] (ii) Z = {..., – 3, – 2, – 1, 0, 1, 2, 3, ....} (iii) QІ = {, 1, e, ......} wb‡Pi †KvbwU mwVK? i I ii i I iii ii I iii i, ii I iii DËi: i I ii e ̈vL ̈v: ̄^vfvweK msL ̈vi †mU, N = {1, 2, 3, 4......}  (i) mwVK c~Y© msL ̈vi †mU, Z = {..., – 3, – 2, – 1, 0, 1, 2, 3, ....}  (ii) mwVK Ag~j` msL ̈vi †mU, QІ  = { e 2 3 5 ...}}  (iii) mwVK bq 33. – 2  x  3 GiÑ [D.B 17] (i) g‡a ̈ 6wU c~Y©msL ̈v i‡q‡Q (ii) EaŸ©mxgv 15 (iii) cig AvKvi |2x – 1|  5 wb‡Pi †KvbwU mwVK? i I ii i I iii ii I iii i, ii I iii DËi: i, ii I iii e ̈vL ̈v: – 2  x  3 Gi g‡a ̈ c~Y© msL ̈v: {– 2, – 1, 0, 1, 2,3} A_©vr 6wU|  (i) mwVK Avevi, – 2  x  3 Gi EaŸ©mxgv 3, 4, 5, 6,.... BZ ̈vw`  (ii) mwVK Avevi, – 2  x  3  – 4  2x  6  – 4 – 1  2x – 1  6 – 1  – 5  2x – 1  5  |2x – 1|  5  (iii) mwVK 34. ̄^vfvwK msL ̈vi †mU N Ave×Ñ [S.B 17] (i) †hv‡Mi †ÿ‡Î (ii) we‡qv‡Mi †ÿ‡Î (iii) ̧‡Yi †ÿ‡Î wb‡Pi †KvbwU mwVK? i I ii i I iii ii I iii i, ii I iii DËi: i I iii e ̈vL ̈v: (i) `yBwU ̄^vfvweK msL ̈vi †hvMdj me©`vB ̄^vfvweK msL ̈v| A_©vr ̄^vfvweK msL ̈vi †mU N †hv‡Mi †ÿ‡Î Ave×|  (i) mwVK (ii) `yBwU ̄^vfvweK msL ̈vi we‡qvMdj ̄^vfvweK msL ̈vi bvI n‡Z cv‡i| A_©vr N we‡qv‡Mi †ÿ‡Î Ave× bq|  (ii) mwVK bq (iii) `yBwU ̄^vfvweK msL ̈vi ̧bdj me©`vB ̄^vfvweK msL ̈v A_©vr ̄^vfvweK msL ̈vi †mU N ̧‡Yi †ÿ‡Î Ave×|  (iii) mwVK 35. c~Y©msL ̈vi †mU Z Ave×Ñ [D.B 17] (i) †hv‡Mi †ÿ‡Î (ii) we‡qv‡Mi †ÿ‡Î (iii) ̧‡Yi †ÿ‡Î wb‡Pi †KvbwU mwVK? i i I ii ii I iii i, ii I iii DËi: i, ii I iii e ̈vL ̈v: c~Y©msL ̈vi †mU Z †hvM, we‡qvM Ges ̧Y cÖwμqvq Ave×| A_©vr `yBwU c~Y© msL ̈vi †hvMdj, we‡qvMdj Ges ̧bdj I GKwU c~Y© msL ̈v|  (i), (ii) Ges (iii) mwVK|