Title Complex Number Practice Sheet HSC FRB 24.pdf ✅

RwUj msL ̈v  Final Revision Batch '24 1 03 RwUj msL ̈v Complex Number Board Questions Analysis m„Rbkxj cÖkœ †evW© mvj XvKv gqgbwmsn ivRkvnx Kzwgjøv h‡kvi PÆMÖvg ewikvj wm‡jU w`bvRcyi 2023 1 1 1 1 1 1 1 1 1 2022 1 1 1 1 1 1 1 1 1 eûwbe©vPwb cÖkœ †evW© mvj XvKv gqgbwmsn ivRkvnx Kzwgjøv h‡kvi PÆMÖvg ewikvj wm‡jU w`bvRcyi 2023 4 4 4 4 5 4 4 4 5 2022 4 4 5 4 4 4 3 4 4 weMZ mv‡j †ev‡W© Avmv m„Rbkxj cÖkœ 1| DÏxcK-1: x = a + b + c 2 , y = a + b 2 + c DÏxcK-2: 7 + i8 = (p + iq)3 [XvKv †evW©- Õ23] (K) GK‡Ki GKwU KvíwbK Nbg~j  n‡j †`LvI †h,     1 +  + 3  6 = 64 (L) DÏxcK-1 Gi mvnv‡h ̈, hw` x 3 + y3 = 0 nq, Z‡e †`LvI †h, b = 1 2 (c + a) (M) DÏxcK-2 Gi mvnv‡h ̈ cÖgvY Ki †h, p 2 – q 2 = 7 4p + 2 q 2| `„k ̈Kí-1: z1 = – 1 + 3i Ges z2 = 1 – 3i `„k ̈Kí-2: g(x) = l + mx + nx2 [ivRkvnx †evW©- Õ23] (K) i Gi eM©g~j wbY©q Ki| (L) cÖgvY Ki †h, arg (z1z2) = arg(z1) + arg(z2) (M) DÏxcK-2 G, l + m + n = 0 n‡j, cÖgvY Ki †h, {g()}3 + {g( 2 )}3 = 27 lmn DËi: (K) i =  1 2 (1 + i) 3| i. x + y + z = R ii. p = x + iy [h‡kvi †evW©- Õ23] (K) (– 1 – 3i) msL ̈vwUi Av ̧©‡g›U wbY©q Ki| (L) p RwUj msL ̈vwUi AbyeÜx RwUj msL ̈v q n‡j |p + 3i| = |q + 4| Øviv wb‡`©wkZ mÂvic_ wbY©q Ki| (M) hw` R = 0 Ges  GK‡Ki GKwU KvíwbK Nbg~j nq Z‡e, cÖgvY Ki †h, (x + y + z 2 ) 3 + (x + y 2 + z) 3 = 27xyz DËi: (K) – 2 3 ; (L) 8x – 6y + 7 = 0 4| `„k ̈Kí-1: z = rcos + irsin [Kzwgjøv †evW©- Õ23] (K) (1 – i)–2 – (1 + i)–2 Gi gvb wbY©q Ki| (L) `„k ̈K‡í  = 45 I r = 1 n‡j, z 8 + z6 + z4 + z2 + 1 Gi gvb wbY©q Ki| (M) `„k ̈Kí n‡Z cÖgvY Ki †h, Arg(z2 ) = 2Arg(z) DËi: (K) i ; (L) 1 5| Z1 = 1 – ix Ges Z2 = a + ib †hLv‡b a, b  [PÆMÖvg †evW©- Õ23] (K) x = 3 n‡j, Z1 †K †cvjvi AvKv‡i cÖKvk Ki| (L) cÖgvY Ki †h, x Gi GKwU ev ̄Íe gvb Z1 — Z1 – = Z2 – mgxKiY‡K wm× K‡i †hLv‡b a 2 + b2 = 1 (M) 3 Z2 = p + iq n‡j, cÖgvY Ki †h, – 2 (p2 + q2 ) = a p – b q DËi: (K) 2     cos  3 – isin  3 6| z = x + iy RwUj msL ̈vwUi AbyewÜ RwUj msL ̈v z –– | [wm‡jU †evW©- Õ23] (K) 4 – 49 Gi gvb wbY©q Ki| (L) x = 2 Ges y = 2 n‡j, z Gi eM©g~j wbY©q Ki| (M) |z + 4| – |z –– – 4| = 10 Øviv wb‡`©wkZ mÂvic‡_i mgxKiY wbY©q Ki| DËi: (K) x =  7 2 (1  i) ; (L)  ( 2 + 1 + i 2 – 1) ; (M) x 2 5 2 + y 2 3 2 = 1
2  Higher Math 2nd Paper Chapter-3 7| z1 = – 1 – i 3, z2 = 3 – i. [ewikvj †evW©- Õ23] (K) z1 Gi eM©g~j wbY©q Ki| (L) †`LvI †h, Arg    z1 z2 = Arg z1 – Arg z2 (M) cÖgvY Ki †h,    1 2 z1 –– n +     1 2 z1 n = 2, hLb n Gi gvb 3 Øviv wefvR ̈ A_ev, – 1, hLb n Gi gvb Ab ̈ †Kv‡bv c~Y©msL ̈v| DËi: (K)  1 2 (1 – 3i) 8| P = 1 + 5i 1 + i , Q = 3 – 2i, 2x = – 1 + – 3 , 2y = – 1 – – 3 [w`bvRcyi †evW©- Õ23] (K) – 3 + 4 – 1 Gi eM©g~j wbY©q Ki| (L) Q –– – 2P Gi gWzjvm I Av ̧©‡g›U wbY©q Ki| (M) cÖgvY Ki †h, 3x4 + x3 y + xy2 + y4 = – 3 DËi: (K)  (1 + 2i) ; (L) 13 ; –  + tan–1     2 3 9| z = x + iy Ges p 2 + p + 1 = 0 mgxKi‡Yi g~jØq  I | [gqbgwmsn †evW©- Õ23] (K) 4 – 2401 Gi gvb wbY©q Ki| (L) |z + 4| + |z – 4| = 10 Øviv wb‡`©wkZ mÂvic‡_i bvg D‡jøLmn mgxKiYwU wbY©q Ki| (M) cÖgvY Ki †h,  s +  s = – 1, hLb s Gi gvb 3 Øviv wefvR ̈ bq Giƒc c~Y©msL ̈v| DËi: (K)  7 2 (1  i) ; (L) x 2 5 2 + y 2 3 2 = 1 10| DÏxc‡K: z = x + iy [XvKv †evW©- Õ22] (K) – 1 + 3i Gi gWzjvm I Av ̧©‡g›U wbY©q Ki| (L) 3 p + iq = z n‡j, †`LvI †h, 3 p – iq = z – (M) 3|z –1| = 2|z – 2| Øviv wb‡`©wkZ mÂvic‡_i mgxKiY wbY©q Ki| DËi: (K) 2 ; 2 3 ; (M) 5x2 + 5y2 – 2x – 7 = 0 11| a = 4, b = – 4, z = 1 n (1 + im) GKwU RwUj msL ̈v [ivRkvnx †evW©- Õ22] (K) 2 – 3i 4 – 4i †K A + iB AvKv‡i cÖKvk Ki| (L) a + b wbY©q Ki| (M) l = m = 3, n = 18 n‡j |z| Gi Nbg~j ̧‡jvi †hvMdj wbY©q Ki| DËi: (K) 5 8 +     – 1 8 i ; (L)  ( 2 + 5 + i 5 – 2) (M) 0 12| DÏxcK-1: z = – 1 + i GKwU RwUj msL ̈v| DÏxcK-2: z = x + iy [Kzwgjøv †evW©- Õ22] (K) z = i n‡j – z Gi eM©g~j wbY©q Ki| (L) DÏxcK-1 G DwjøwLZ RwUj msL ̈vi gWzjvm I Av ̧©‡g›U AvM©Û wP‡Î †`LvI| (M) DÏxcK-2 Gi mvnv‡h ̈ |z + 2| = 5 e„‡Ëi †K›`a I e ̈vmva© wbY©q Ki| DËi: (K)  1 2 (1 – i) ; (L) 2 ; 3π 4 (M) †K›`a (– 2, 0); e ̈vmva© 5 13| M = – 5 + 12 –1, p = 3 a + ib Ges q = x + iy [h‡kvi †evW©- Õ22] (K) 1 + 2i †K AvM©Û wP‡Îi mvnv‡h ̈ cÖKvk Ki| (L) M Gi eM©g~j wbY©q Ki| (M) p = q n‡j, cÖgvY Ki †h, 4(x2 – y 2 ) = a x + b y DËi: (L)  (2 + 3i) 14| `„k ̈Kí-1: |z + 6| + |z – 6| = 20 †hLv‡b z = x + iy `„k ̈Kí-2: (1 + y)n = b0 + b1y + b2y 2 + b3y 3 + ..... + bny n [PÆMÖvg †evW©- Õ22] (K) 6 – 2 3i RwUj msL ̈vi gWzjvm I Av ̧©‡g›U wbY©q Ki| (L) `„k ̈Kí- 1 Øviv wb‡`©wkZ mgxKiYwUi mÂvi c_ Ges Dnvi bvg D‡jøL K‡i wPÎ AsKb Ki| (M) `„k ̈Kí-2 Gi mgxKiY n‡Z †`LvI †h, (b0 – b2 + b4 ......)2 = (b0 + b1 + b2 + b3 + ....) – (b1 – b3 + b5 – ......)2 DËi: (K) gWzjvm 4 3 ; Av ̧©‡g›U –  6 ; (L) x 2 102 + y 2 8 2 = 1 15| z1 = 1 + ia, z2 = a + i Ges |z + 2| + |z – 2| = 6 GKwU KwYK, †hLv‡b z = x + iy. [ewikvj †evW©- Õ22] (K) –1 Gi eM©g~j wbY©q Ki| (L) a = 3 n‡j †`LvI †h, arg     z1 z2 = arg(z1) – arg(z2) (M) KwYKwUi Aÿ؇qi •`N© ̈ wbY©q Ki| DËi: (K) i =  1 2 (1 + i) (M) e„n`v‡ÿi •`N© ̈ = 6 GKK ; ÿz`av‡ÿi •`N© ̈ 2 5 GKK