Nội dung text Matrices Varsity Practice Sheet Solution.pdf
g ̈vwUa· I wbY©vqK Varsity Practice Sheet Solution 1 01 g ̈vwUa· I wbY©vqK Matrices and Determinants weMZ mv‡j DU-G Avmv cÖkœvejx 1. 1 2 1 3 0 – 1 2 3 p g ̈vwUa·wU e ̈wZμgx n‡j, p Gi gvb KZ? [DU 23-24] 4 3 3 4 5 3 3 5 DËi: 4 3 e ̈vL ̈v: e ̈wZμgx n‡j, 1 2 1 3 0 – 1 2 3 p = 0 1(0 + 3) – 3(2p – 3) + 2(– 2 – 0) = 0 3 – 6p + 9 – 4 = 0 8 – 6p = 0 p = 8 6 = 4 3 2. A = 1 2 2 5 n‡j, det(AA–1 ) Gi gvb KZ? [DU 21-22] 1 – 1 0 – 1 2 DËi: 1 e ̈vL ̈v: AA–1 = I [I n‡jv A‡f`K g ̈vwUa·] det(AA–1 ) = det(I) = 1 Note: A‡f`K g ̈vwUa‡·i wbY©vq‡Ki gvb 1 3. hw` A, B, C g ̈vwUa· wZbwUi AvKvi h_vμ‡g 4 5, 5 4 Ges 4 2 nq, Z‡e (AT + B)C g ̈vwUa·wUi AvKvi wK? [DU 20-21] 4 2 5 4 2 5 5 2 DËi: 5 2 e ̈vL ̈v: B g ̈vwUa‡·i μg 5 4 A T + B g ̈vwUa‡·i μg 5 4 C g ̈vwUa‡·i μg 4 2 (AT + B)C Gi †ÿ‡Î, (5 4) (4 2) (AT + B)C Gi μg: 5 2 4. A = 3 2 – 4 – 3 n‡j, det(2A–1 ) Gi gvb n‡jvÑ [DU 19-20] 1 4 – 4 4 – 1 4 DËi: – 4 e ̈vL ̈v: |A| = 3 2 – 4 – 3 = – 9 + 8 = – 1 det(2A–1 ) = 2 n |A| [†hLv‡b, n gvÎv] = 2 2 – 1 = – 4 5. A = a – 2 – 5 2 b 3 5 – 3 c GKwU eμ cÖwZmg g ̈vwUa· n‡j, a, b, c Gi gvb ̧‡jvÑ [DU 17-18] – 2, – 5, 3 0, 0, 0 1, 1, 1 2, 5, 3 DËi: 0, 0, 0 e ̈vL ̈v: AcÖwZmg/wecÖwZmg/eμ cÖwZmg g ̈vwUa‡·i †ÿ‡Î gyL ̈ K‡Y©i mKj fzw3 0 nq| 6. k Gi †Kvb gv‡bi Rb ̈ 1 1 1 1 k k 2 1 k 2 k 4 wbY©vqKwUi gvb k~b ̈ n‡e bv? [DU 17-18] k = 1 k = – 1 k = 3 k = 0 DËi: k = 3 e ̈vL ̈v: wbY©vq‡Ki ag© ̧‡jv g‡b ivL‡Z n‡e| k = 1 n‡j wbY©vq‡Ki wZbwU Kjvg B mgvb nq| gvb 0 k = – 1 n‡j wbY©vq‡Ki 1g I 3q Kjvg mgvb nq| gvb 0 k = 0 n‡j wbY©vq‡Ki 2q I 3q Kjvg mgvb nq gvb 0 wKš‘, k = 3 n‡j wbY©vq‡Ki †Kv‡bv Kjvg ev mvwi ci ̄úi mgvb n‡e bv| gvb k~b ̈ n‡e bv|
2 Higher Math 1st Paper Chapter-1 7. x x = 0 n‡j, x = ? [DU 14-15] , , , , , DËi: , e ̈vL ̈v: x = n‡j 1g I 3q Kjvg mgvb nq| d‡j wbY©vq‡Ki gvb k~b ̈| x = n‡j 1g I 2q Kjvg mgvb nq| d‡j wbY©vq‡Ki gvb k~b ̈| wKš‘ x = n‡j †Kv‡bv mvwi ev Kjvg ci ̄úi mgvb nqbv| ZvB x = , n‡j wbY©vq‡Ki gvb k~b ̈ n‡e| 8. cos – sin sin cos Gi wecixZ g ̈vwUa·Ñ [DU 13-14] cos – sin – sin cos cos sin – sin – cos cos sin – sin cos cos sin sin cos DËi: cos sin – sin cos e ̈vL ̈v: cos – sin sin cos –1 = 1 cos2 + sin2 cos sin – sin cos = cos sin – sin cos [⸪ cos2 + sin2 = 1] 9. 0 2 0 3 7x 0 2x + 7 9 + 5x 2x + 5 = 0 n‡j, x Gi gvb KZ? [DU 13-14] – 9 5 – 7 2 – 5 2 0 DËi: – 5 2 e ̈vL ̈v: (2x + 5)(0 – 3 2) = 0 2x + 5 = 0 x = – 5 2 10. BA Gi gvb wbY©q Ki, hw` A = 1 – i i 1 Ges B = i – 1 – 1 – i I i = – 1 nq| [DU 12-13; CU 18-19; RU 17-18] – 1 – i i – 1 1 7 1 3 1 0 0 1 2i – 2 – 2 – 2i DËi: 2i – 2 – 2 – 2i e ̈vL ̈v: BA = i – 1 – 1 – i 1 – i i 1 = i + i – 1 + i2 i 2 – 1 – i – i = 2i – 2 – 2 – 2i [⸪ i 2 = – 1] 11. hw` A = – 2 3 2 1 – 1 2 nq, Z‡e A –1 mgvbÑ [DU 10-11; JU 14-15] 1 2 3 4 1 0 0 1 3 1 4 2 1 3 2 4 DËi: 1 3 2 4 e ̈vL ̈v: A –1 = 1 (– 2) – 1 2 – 3 2 – 1 2 – 3 2 – 1 – 2 = 1 1 – 3 2 – 1 2 – 3 2 – 1 – 2 = 1 3 2 4 12. hw` A = 1 0 0 5 , B = 5 0 0 1 nq, Z‡e AB n‡jvÑ [DU 05-06, 03-04; JU 18-19, 16-17, 14-15; RU 08-09, 06-07] 5 0 0 5 5 10 0 5 10 0 0 5 0 5 5 10 DËi: 5 0 0 5 e ̈vL ̈v: AB = 1 0 0 5 5 0 0 1 = 5 + 0 0 + 0 0 + 0 0 + 5 = 5 0 0 5 13. hw` A = 2 3 – 3 2 nq, Z‡e A 2 gvbÑ [DU 04-05; JU 14-15; RU 08-09] – 5 – 12 12 5 5 – 12 – 12 5 – 5 12 12 – 5 – 5 12 – 12 – 5 DËi: – 5 12 – 12 – 5 e ̈vL ̈v: A 2 = 2 3 – 3 2 2 3 – 3 2 = 4 – 9 6 + 6 – 6 – 6 – 9 + 4 = – 5 12 – 12 – 5
4 Higher Math 1st Paper Chapter-1 weMZ mv‡j Agri-G Avmv cÖkœvejx 1. (x + 5, 2y + 1) = (2y + 4, 3y) n‡j, x Gi gvb KZ? [Agri. Guccho 20 -21] – 1 0 1 2 DËi: 1 e ̈vL ̈v: x + 5 = 2y + 4 x = 2y – 1 .... (i) 2y + 1 = 3y y = 1 (i) G y Gi gvb ewm‡q, x = 2 1 – 1 x = 1 weMZ mv‡j JU-G Avmv cÖkœvejx 1. wb‡Pi †KvbwU cÖwZmg g ̈vwUa·? [JU 22-23; RU 17-18] 0 – b b 0 b 0 0 – b b – b 0 0 0 0 – b b DËi: b 0 0 – b e ̈vL ̈v: cÖwZmg g ̈vwUa‡·i †ÿ‡Î A T = A nq| GLv‡b, bs Ack‡b A T = A kZ©wU cÖ‡hvR ̈ n‡q‡Q| Note: cÖwZmg g ̈vwUa· n‡j, aij = aji eμ cÖwZmg g ̈vwUa· n‡j, aij = – aji 2. A = 3 4 1 5 0 6 1 2 4 n‡j, A – 2I Gi gvb †KvbwU? [JU 22-23] 5 4 1 5 2 6 1 2 6 1 4 1 5 – 2 6 1 2 2 1 4 1 3 0 6 – 1 2 4 1 2 – 1 3 – 2 4 1 2 6 DËi: 1 4 1 5 – 2 6 1 2 2 e ̈vL ̈v: A – 2I = 3 4 1 5 0 6 1 2 4 – 2 1 0 0 0 1 0 0 0 1 = 3 4 1 5 0 6 1 2 4 – 2 0 0 0 2 0 0 0 2 = 1 4 1 5 – 2 6 1 2 2 3. A g ̈vwUa‡·i μg 2 3 Ges B g ̈vwUa‡·i μg 3 2 n‡j, AB Gi μg †KvbwU? [JU 22-23] 2 2 2 3 3 2 3 3 DËi: 2 2 e ̈vL ̈v: AB g ̈vwUa‡·i †ÿ‡Î, μg: 2 2 (2 3) (3 2) 4. A = 1 4 7 2 5 8 3 6 9 Ges B = 0 2 4 1 3 5 n‡j, A + B = ? [JU 21-22] 1 6 11 3 8 13 3 6 9 1 4 7 2 7 12 3 9 14 1 6 7 3 8 8 Am¤¢e DËi: Am¤¢e e ̈vL ̈v: A + B wbY©q Kiv hv‡e hw`, A I B g ̈vwUa‡·i mvwi msL ̈v I Kjvg msL ̈v ci ̄úi mgvb nq| GLv‡b, A I B g ̈vwUa·Ø‡qi mvwi msL ̈v I Kjvg msL ̈v mgvb bq| A + B wbY©q Am¤¢e| 5. A = 1 2 3 Ges B = (4 5 6) n‡j, AB = ? [JU 21-22] (4 10 18) 4 10 18 4 8 12 5 10 15 6 12 18 Am¤¢e DËi: 4 8 12 5 10 15 6 12 18 e ̈vL ̈v: A Gi μg 3 1 B Gi μg 1 3 AB Gi μg 3 3 GLv‡b ïaygvÎ bs Ack‡b 3 3 μ‡gi g ̈vwUa· we` ̈gvb| 6. k Gi †Kvb gv‡bi Rb ̈ k – 2 3 4 9 g ̈vwUa·wU Ae ̈wZμgx bq? [JU 21-22] 10 3 30 3 9 4 DËi: 10 3 e ̈vL ̈v: kZ©vbyhvqx, k – 2 3 4 9 = 0 9k – 18 – 12 = 0 9k = 30 k = 10 3