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

ev ̄Íe msL ̈v I AmgZv  Engineering Practice Sheet 1 ev ̄Íe msL ̈v I AmgZv Real Numbers and Inequalities cÖ_g Aa ̈vq WRITTEN weMZ mv‡j BUET-G Avmv cÖkœvejx 1| 1 |3x – 5| > 2 AmgZvwU KLb AmsÁvwqZ? AmgZvwU mgvavb Ki Ges mgvavb †mU msL ̈v †iLv‡Z †`LvI| [BUET 19-20] mgvavb: 1 |3x – 5| > 2  |3x – 5| < 1 2 [wecixZKiY]  – 1 2 < 3x – 5 < 1 2  – 1 2 + 5 < 3x – 5 + 5 < 1 2 + 5 [mKj c‡ÿ 5 †hvM K‡i]  9 2 < 3x < 11 2  3 2 < x < 11 6 [mKj c‡ÿ 3 Øviv fvM K‡i] cÖ`Ë AmgZvwU msÁvwqZ n‡e hw` 3x – 5  0 nq,  x  5 3 nq|  wb‡Y©q mgvavb, 3 2 < x < 11 6 †hLv‡b, x  5 3  mgvavb †mU, S =       x  R : 3 2 < x < 11 6 †hLv‡b x  5 3 msL ̈v‡iLvq cÖKvk: 0 1 3 2 2 5 3 11 6 2| mgvavb Ki: 1 |5x + 2|  5 [BUET 16-17] mgvavb: 1 |5x + 2|  5  |5x + 2|  1 5 [wecixZKiY]  – 1 5  5x + 2  1 5  – 1 5 – 2  5x  1 5 – 2 [mKj cÿ †_‡K 2 we‡qvM K‡i]  – 11 5  5x  – 9 5  – 11 25  x  – 9 25     x  – 2 5 (Ans.) weMZ mv‡j KUET-G Avmv cÖkœvejx 3| ev ̄Íe msL ̈vi Dc‡mU s = {x : 5x2 – 16x + 3 < 0} Gi e„nËg wb¤œmxgv I ÿz`aZg EaŸ©mxgv wbY©q Ki| [KUET 04-05] mgvavb: 5x2 – 16x + 3 < 0  5x2 – x – 15x + 3 < 0  x (5x – 1) – 3 (5x – 1) < 0  (5x – 1) (x – 3) < 0  1 5 < x < 3  ÿz`aZg EaŸ©mxgv = 3 Ges e„nËg wb¤œmxgv = 1 5 (Ans.) weMZ mv‡j RUET-G Avmv cÖkœvejx 4| cÖgvY Ki †h, |a – b|  |a| + |b| [RUET 12-13] mgvavb: (|a| + |b|) 2 = (|a|) 2 + 2 |a| |b| + (|b|) 2  (|a| + |b|) 2 = a 2 + 2 |ab| + b2 [ |a| |b| = |ab|]  (|a| + |b|) 2  a 2 + 2ab + b2 [ |ab|  ab]  (|a| + |b|) 2  (a + b)2  (|a| + |b|) 2  |a + b|2  |a| + |b|  |a + b| awi, b = – b  |a| + |– b|  | |a + (– b)|  |a – b|  |a| + |b| (Proved) 5| gvb wbY©q Ki: |3x – 4| < 2 [RUET 11-12] mgvavb: |3x – 4| < 2  – 2 < 3x – 4 < 2  – 2 + 4 < 3x < 2 + 4  2 < 3x < 6  2 3 < x < 2 6| mgvavb Ki: 1 |3x + 1|  5 [RUET 10-11] mgvavb: 1 |3x + 1|  5  |3x + 1|  1 5  – 1 5  3x + 1  1 5  – 6 5  3x  – 4 5  – 2 5  x  – 4 15 Avevi, 3x + 1  0  x  – 1 3  wb‡Y©q mgvavb, – 2 5  x  – 4 15 Ges x  – 1 3 (Ans.)
2  Higher Math 2nd Paper Chapter-1 7| mgvavb Ki: 1 |3x – 5| > 2 [RUET 09-10] mgvavb: 1 |3x – 5| > 2  |3x – 5| < 1 2 ; x  5 3  – 1 2 < 3x – 5 < 1 2 ; x  5 3  – 1 2 + 5 < 3x < 1 2 + 5 ; x  5 3  9 2 < 3x < 11 2 ; x  5 3  3 2 < x < 11 6 ; x  5 3 (Ans.) 8| mgvavb Ki: 2  1 |x – 1| [RUET 05-06] mgvavb: 2  1 |x – 1| hLb x  1  |x – 1|  1 2  – 1 2  x – 1  1 2  1 2  x  3 2 (Ans.) 9| mgvavb Ki: 2  1 |x – 1| hLb x  1 [RUET 04-05] mgvavb: 2  1 |x – 1|  |x – 1|  1 2 hLb, x  1  – 1 2  x – 1  1 2 ; 1 2  x  3 2 hLb, x  1 (Ans.) weMZ mv‡j MIST-G Avmv cÖkœvejx 10| 1 |3x – 5| > 2 AmgZvwU KLb AmsÁvwqZ? AmgZvwU mgvavb Ki Ges mgvavb †mU msL ̈v †iLv‡Z †`LvI| [MIST 19-20] mgvavb: 1 |3x – 5| > 2  |3x – 5| < 1 2 [wecixZKiY]  – 1 2 < 3x – 5 < 1 2  – 1 2 + 5 < 3x – 5 + 5 < 1 2 + 5 [mKj c‡ÿ 5 †hvM K‡i]  9 2 < 3x < 11 2  3 2 < x < 11 6 [mKj c‡ÿ 3 Øviv fvM K‡i] cÖ`Ë AmgZvwU msÁvwqZ n‡e hw` 3x – 5  0 nq,  x  5 3 nq|  wb‡Y©q mgvavb, 3 2 < x < 11 6 †hLv‡b, x  5 3  mgvavb †mU, S =       x  R : 3 2 < x < 11 6 †hLv‡b x  5 3 msL ̈v‡iLvq cÖKvk: 0 1 3 2 2 5 3 11 6 weMZ mv‡j BUTex-G Avmv cÖkœvejx 11| A = {0, 1, 2, 3, 4, 5} n‡j Gi ÿz`aZg EaŸ©mxgv KZ? [BUTex 09-10] mgvavb: 5 12| AmgZv x 2  x Gi mgvavb Kx n‡e? [BUTex 09-10] mgvavb: x 2  x  x 2 – x  0  x (x – 1)  0  0  x  1 13| 5x – x 2 – 6 > 0 n‡j, x Gi gvb wbY©q Ki| [BUTex 07-08] mgvavb: x 2 – 5x + 6 < 0  x 2 – 3x – 2x + 6 < 0  x(x – 3) – 2(x – 3) < 0  (x – 3) (x – 2) < 0 hLb (x – 3) (x – 2) x < 2 2 < x < 3 x > 3 – – + – + + wb‡Y©q mgvavb: 2 < x < 3 mxgvt 0 1 2 3 MCQ weMZ mv‡j KUET-G Avmv cÖkœvejx 1. x + 4 x + 3 > x – 6 x – 7 AmgZvwUi mgvavb n‡jvÑ [KUET 17-18] – 4 < x < 3 – 4  x  6 x < – 3 and x > 7 x < – 4 and x > 6 – 3 < x < 7 DËi: – 3 < x < 7 e ̈vL ̈v: x + 4 x + 3 > x – 6 x – 7  x + 4 x + 3 – x – 6 x – 7 > 0  (x + 4) (x – 7) – (x + 3) (x – 6) (x + 3) (x + 7) > 0  (x 2 + 4x – 7x – 32) – (x 2 + 3x – 6x – 18) (x + 3) (x – 7) > 0  – 14 (x + 3) (x – 7) > 0  (x + 3) (x – 7) < 0  – 3 < x < 7
ev ̄Íe msL ̈v I AmgZv  Engineering Practice Sheet 3 weMZ mv‡j RUET-G Avmv cÖkœvejx 2. 0 < |x – a| < p n‡j x Gi mKj gvb wbY©q Ki| GLv‡b a †h‡Kvb ev ̄Íe msL ̈v Ges p GKwU abvZ¥K msL ̈v| [RUET 14-15] (a – p, a) (a, a + p) a – p  x  a a  x  a + p a – p  x  a + p (a, p – a)  (a + p, a) DËi: (a – p, a) (a, a + p) e ̈vL ̈v: 0 < |x – a| < p  nq, 0 < (x – a) < p A_©vr, a < x < p + a A_ev, 0 < – (x – a) < p A_ev, – p < (x – a) < 0 A_ev, a – p < x < a  a < x < a + p A_ev, a – p < x < a  x  (a – p, a)  (a, a + p) 3. |x – 1| < 1 10 n‡j, |x2 – 1| Gi gvb †KvbwU? [RUET 12-13] 1 100 7 100 14 100 21 10 21 10 DËi: 21 10 e ̈vL ̈v: |x – 1| < 1 10 |x + 1| = |x – 1 + 2|  |x – 1| + 2  |x + 1| < 1 10 + 2  |x + 1| < 21 10 GLb, |x2 – 1| = |x – 1| |x + 1| < 1 10 . 21 10  |x2 – 1| < 21 100 4. (x + 2) (x + 3)  0 Gi me©vwaK mwVK DËi †KvbwU? [RUET 09-10] x  – 3 x  – 3 or x  – 2 x  – 2 All real numbers – 3  x  2 DËi: x  – 3 or x  – 2 e ̈vL ̈v: (x + 2) (x + 3)  0  nq x + 2  0, x + 3  0 ev, x  – 2, x  – 3 A_©vr, x  – 2 A_ev, x + 2  0, x + 3  0 ev, x  – 2, x  – 3 A_©vr, x  – 3 myZivs, x  – 2 A_ev, x  – 3 weMZ mv‡j CUET-G Avmv cÖkœvejx 5. hw` a, b  R n‡j |a – b| Gi gvb KZ? [CUET 10-11] > ||a| – |b|| = ||a| – |b||  ||a| – |b|| None of these DËi:  ||a| – |b|| e ̈vL ̈v: a, b  R n‡j |a – b|  ||a| – |b|| -----