Title Complex Number MCQ Suggestion HSC 23.pdf ✅

RwUj msL ̈v  Higher Math MCQ Suggestion HSC 2023 1 RwUj msL ̈v Complex Number Gravitation and Gravity Z...Zxq Aa ̈vq Board Questions Analysis eûwbe©vPwb cÖkœ †evW© mvj XvKv gqgbwmsn ivRkvnx Kzwgjøv h‡kvi PÆMÖvg ewikvj wm‡jU w`bvRcyi 2022 4 4 5 4 4 4 3 4 4 2021 mswÿß wm‡jev‡m GB Aa ̈vq AšÍfz©3 wQj bv| †evW© cixÿvi Rb ̈ ̧iæZ¡c~Y© eûwbe©vPwb cÖkœ 1. n  N n‡j i 8n + 5 Gi gvb KZ? [wm. †ev. Õ22] 1 –1 i – i DËi: i e ̈vL ̈v: i 4n = 1 GLb, i 8n+5 = i8n . i5 = (i4n) 2 . i5 = i A_ev, n = 1 a‡i cvB i 8n + 5 = i13 = i Note: EX/CW Calculator w`‡q Direct i5 / i 13 Gi gvb †ei K‡i †djv hvq| Calculator Aek ̈B Complex Mode G ivL‡Z n‡e| 2. i m + im+1 + im+2 + im+3 = KZ? (m  Z) [iv. †ev. Õ17] – 1 –i 0 i DËi: 0 e ̈vL ̈v: i Gi PviwU μwgK cvIqvi m¤^wjZ c‡`i †hvMdj = 0  i m + im+1 + im+2 + im+3 = 0 Note: i = – 1 i 2 = – 1 i 3 = – i i 4 = 1  i + i2 + i3 + i4 = 0 i 4 = 1 i 8 = 1 i 12 = 1  i 4n = 1 3. i 7 + i9 + i11 + i13 Gi gvb KZ? [w`. †ev. Õ22] –1 1 –i 0 DËi: 0 e ̈vL ̈v: i 7 + i9 = 0 Ges i 11 + i13 = 0  i 7 + i9 + i11 + i13 = 0 i Gi cvIqvi `ywU μwgK †Rvo msL ̈v/μwgK we‡Rvo msL ̈v n‡j G‡`i †hvMdj k~b ̈ nq| †hgb i 2 + i4 = 0 ; i + i3 = 0 Note: EX/CW Calculator w`‡q Direct i7 + i9 + i11 + i13 Gi gvb †ei K‡i †djv hvq| Calculator Aek ̈B Complex Mode G ivL‡Z n‡e| 4. i –70 + 1 Gi gvb †KvbwU? [h. †ev. Õ17] 0 2 1 – i 1 + i DËi: 0 e ̈vL ̈v: 1 i 70 + 1 = 1 i 2 + 1 = 1 –1 + 1 = 0 Note: EX/CW Calculator w`‡q Direct gvb †ei K‡i †djv hvq| Aek ̈B Complex Mode e ̈envi Ki‡Z n‡e| 5. KvíwbK msL ̈v i Ges n  I NGi Rb ̈ i 4n – i + i4n + 1 – 1 Gi gvb KZ? [mw¤§wjZ †ev. Õ18] – i i 0 1 DËi: 0 e ̈vL ̈v: i 4n – i + i4n+1 – 1 = 1 – i + i4n  i – 1 [i4n = 1] = 0 – i + i = 0 6. –3 × –1 Gi gvb †KvbwU? [iv. †ev. Õ22] 3i ± 3 – 3 3 DËi: – 3 e ̈vL ̈v: Using Calculator 7. hw` 2 + 3i 2 – i = A + iB Ges A I B ev ̄Íe msL ̈v nq, Z‡e B = KZ? [e. †ev. Õ19] –8 5 1 5 – 1 5 8 5 DËi: 8 5 e ̈vL ̈v: 2 + 3i 2 – i = 1 5 + 8 5 i [Using Calculator] = A + Bi  A = 1 5 ; B = 8 5 8. i 2 = – 1 n‡j – i – i –5 2i–5 + i Gi gvbÑ [P. †ev. Õ22] –2 0 1 2 2 DËi: 0 e ̈vL ̈v: – i – 1 i 5 2 i 5 + i = – i – 1 i 2 i + i = – i 2 – 1 2 + i2 = 1 – 1 2 –1 = 0 Note: EX/CW Calculator w`‡q Direct gvb †ei K‡i †djv hvq| Aek ̈B Complex Mode e ̈envi Ki‡Z n‡e|
2  Higher Math 2nd Paper Chapter-3 9. i Gi Av ̧©‡g›U KZ? [Xv. †ev. Õ17] 0  2 1  4 DËi:  2 e ̈vL ̈v: Using Calculator 10. – i Gi gWzjvm I Av ̧©‡g›UÑ [P. †ev. Õ22] 1 I 0 1 I –  2 1I  1 I  2 DËi: 1 I –  2 e ̈vL ̈v: Using Calculator 11. z = 1 1 + i Gi Av ̧©‡g›U †KvbwU? [Kz. †ev. Õ17] – 3 4 – 4  4 3 4 DËi: – 4 e ̈vL ̈v: Using Calculator 12. z1 = 1 + i Ges z2 = 2 + i n‡j, z1 – z2 Gi gWzjvmÑ [w`. †ev. Õ22] tan–1 2 2 5 5 2 10 DËi: 10 e ̈vL ̈v: Using Calculator 13. z = – 1 + i n‡j, – z Gi Av ̧©‡g›U KZ? [h. †ev. Õ19] – 3 4 – 5 4  4 –  4 DËi: – 3 4 e ̈vL ̈v: Using Calculator 14. 1 – 3i Gi mvaviY Av ̧©‡g›U KZ? [mw¤§wjZ †ev. Õ18] 2n –  3 ; n  Z 2n +  3 ; n  Z 2n – 5 3 ; n  Z 2n + 5 3 ; n  Z DËi: 2n –  3 ; n  Z e ̈vL ̈v: 1 – 3i Gi gyL ̈ Av ̧©‡g›U = –  3  mvaviY Av ̧©‡g›U = 2n +    –   3 = 2n –  3 Note: mvaviY Av ̧©‡g›U = 2ngyL ̈ Av ̧©‡g›U  DÏxcKwUi Av‡jv‡K 15 bs cÖ‡kœi DËi `vI: z = – 2i GKwU RwUj msL ̈v| 15. – z Gi cÖwZiƒcx we›`y †KvbwU? [iv. †ev. Õ19] (–2, 0) (0, – 2) (2, 0) (0, 2) DËi: (0, 2) e ̈vL ̈v: – z = 2i = 0 + 2i cÖwZiƒcx we›`y(x, y)  (0, 2) 16. a = – 1 + 3i 2 Ges Gi AbyeÜx – a n‡j †KvbwU mZ ̈? [mw¤§wjZ †ev. Õ18] a – a = a2 a + – a = 2a a + – a = –1 – a + a2 = –1 DËi: a + – a = –1 e ̈vL ̈v:  I  2 Gi gvb memgq gyL ̄’ ivLev| GLv‡b, a =   – a =  2  a + – a =  +  2 = –1 Note:  = – 1 + 3i 2 ;  2 = – 1 – 3i 2  DÏxcKwUi Av‡jv‡K 17 bs cÖ‡kœi DËi `vI: z = 3i 17. – z Øviv MwVZ we›`y †KvbwU? [iv. †ev. Õ22] (0, –3) (0, 3) (–3, 0) (3, 0) DËi: (0, –3) e ̈vL ̈v: – z = –3i = 0 – 3i MwVZ we›`y = (x, y) = (0, –3) 18. –7 + 24i Gi eM©g~j n‡e:  (–3 + 4i)  (3 + 4i)  (3 – 4i)  (–3 – 4i) DËi:  (3 + 4i) e ̈vL ̈v: †h Option †K eM© Ki‡j –7 + 24i n‡e †mUvB mwVK DËi| weKí mgvavb: awi, –7 + 24i =  (x + iy) GLv‡b, r = (–7) 2 + (24) 2 = 25  x = r + a 2 = 25 –7 2 = 3  y = r – a 2 = 25 + 7 2 = 4  wb‡Y©q eM©g~j =  (3 + 4i) Note: a + ib =  (x + iy) ai‡e Ges a – ib =  (x – iy) ai‡e 19. 11 – 60i Gi eM©g~j KZ? [w`. †ev. Õ19] ± (5 – 6i) ± (6 + 5i) ± (6 – 5i) ± (6i – 5) DËi: ± (6 – 5i) e ̈vL ̈v: †h Option †K eM© Ki‡j 11 – 60i n‡e †mUvB mwVK DËi|

4  Higher Math 2nd Paper Chapter-3 28. 4 – 16 = ? 2  ( 3  i)  3 2 (1  i)  2 (1  i) 1 3 DËi:  2 (1  i) e ̈vL ̈v: 4 – n 2 =  n 2 (1  i)  4 – 16 = 4 – 4 2 =  4 2 (1  i) = 2 (1  i) A_ev, Option ̧‡jvi g‡a ̈ †hUvi cvIqvi 4 w`‡j – 16 Av‡m IUvB DËi| Use EX/CW Calculator. 29. z = x + iy n‡j z – z = 1 mgxKi‡Yi R ̈vwgwZK iƒc †KvbwU? [Kz. †ev. Õ19] Awae„Ë e„Ë cive„Ë Dce„Ë DËi: e„Ë e ̈vL ̈v: z – z = 1  (x + iy) (x – iy) = 1  x 2 – i 2 y 2 = 1  x 2 + y2 = 1, hv e„Ë wb‡`©k K‡i| 30. z = x + iy n‡j |z – 1| = 5 mgxKiYwU Kx wb‡`©k K‡i? [Kz. †ev. Õ22] mij‡iLv e„Ë cive„Ë Dce„Ë DËi: e„Ë e ̈vL ̈v: |x + iy – 1| = 5  (x – 1)2 + y2 = 25  (x – 1)2 + (y – 0)2 = 25  e„‡Ëi mgxKiY| A_ev, |z + a| = k AvK...wZi mgxKiY e„Ë wb‡`©k K‡i Gfv‡eI g‡b ivL‡Z cv‡iv| 31. z = x – 2iy n‡j z – z = 7 Gi mÂvic_ GKwUÑ [P. †ev. Õ19] cive„Ë Dce„Ë e„Ë Awae„Ë DËi: Dce„Ë e ̈vL ̈v: z = x – 2iy; – z = x + i2y z – z = 7  (x – 2iy) (x + i2y) = 7  x 2 – (2iy)2 = 7  x 2 + 4y2 = 7  x 2 7 + y 2 7 4 = 1 ; hv GKwU Dce„‡Ëi mgxKiY| Note: hviv Conics c‡ov bvB Zv‡`i GKUz mgm ̈v n‡Z cv‡i †KvbUv †Kvb mgxKiY †mUv| Z‡e simple wKQz K_v e‡j †`B ax2 + by2 = c GB UvB‡ci mgxKiYÑ (i) e„Ë n‡Z n‡j x 2 I y 2 Gi mnM mgvb n‡Z n‡e| (ii) Dce„‡Ëi mgxKi‡Yi x 2 I y 2 Gi mnM Amgvb Ges Df‡qi mn‡Mi wPý GKB n‡e| (iii) Awae„‡Ëi mgxKi‡Y x 2 I y 2 Gi †h‡Kvb GKwUi mnM Positive Ges Ab ̈wUi mnM Negative n‡e| 32. z = 2x + i.3y ; x I y ev ̄Íe msL ̈v n‡j |z| = 1 Øviv wK wb‡`©wkZ nq? [Xv. †ev. Õ19] e„Ë Dce„Ë cive„Ë Awae„Ë DËi: Dce„Ë e ̈vL ̈v: |z| = 1  |2x + i.3y| = 1  (2x) 2 + (3y) 2 = 1  4x2 + 9y2 = 1  x 2 1 4 + y 2 1 9 = 1, hv Dce„Ë wb‡`©k K‡i| 33. hw` a = 1 + i 2 nq, Z‡e a 6 Gi gvb n‡eÑ – 1 i 1 – i DËi: – i e ̈vL ̈v: EX/CW Calculator w`‡q Direct gvb †ei K‡i †djv hvq| A_ev, a = 1 + i 2  a 2 = (1 + i) 2 2  a 2 = 1 + 2i + i2 2  a 2 = 1 + 2i – 1 2  a 2 = i  a 6 = (a2 ) 3 = i3 = – i 34. n Gi abvZ¥K me©wb¤œ ALÐ gvb †ei Ki hvi Rb ̈     1 + i 1 – i n = 1 2 3 6 4 DËi: 4 e ̈vL ̈v:     1 + i 1 – i n = 1  i n = 1    Using Calculator  1 + i 1 – i = i  n Gi me©wb¤œ gvb 4 Note: i 4 = 1 35. x = 2 – i n‡j x 3 – 3x2 + x + 10 Gi gvb wbY©q Ki| 4 2 5 –2 DËi: 5 e ̈vL ̈v: x 3 – 3x2 + x + 10 = 5 [x = 2 – i ewm‡q Using Calculator] 36. x + iy = i–2021 + 2() –2019 n‡j y x = ? [iv. †ev. Õ22] 1 2 – 1 2 2 –2 DËi: – 1 2