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
RwUj msL ̈v  Final Revision Batch '24 3 16| f(x) = 2x 1 + x2 Ges g(x) = p + qx + rx2 `yBwU dvskb| [wm‡jU †evW©- Õ22] (K) Z = 1 + 2i 1 – 3i Gi gWzjvm †ei Ki| (L) f(1) Gi Nbg~j wbY©q Ki| (M) p + q + r = 0 n‡j cÖgvY Ki †h, {g()}2 + {g( 2 )}2 = 3(p2 + 2qr), †hLv‡b  GK‡Ki Nbg~j ̧‡jvi GKwU RwUj g~j| DËi: (K) 1 2 ; (L) f(1) Gi Nbg~j ̧‡jv 1, –1+ 3i 2 , –1– 3i 2 17| `„k ̈Kí-1: z1 = 1 – 3i, z2 = 1 – i `„k ̈Kí-2: |z – 3| – |z + 3| = 4 [w`bvRcyi †evW©- Õ22] (K) (2 + i) (x + iy) = 1 + 3i n‡j x, y wbY©q Ki| (L) `„k ̈Kí-1 n‡Z z1z2 wbY©q Ki| (M) `„k ̈Kí-2 n‡Z mÂvic‡_i mgxKiY wbY©q Ki hLb z = x + iy DËi: (K) wb‡Y©q gvb: x = 1, y = 1 ; (L)  ( 5 – 1 – i 5 + 1) ; (M) x 2 4 – y 2 5 = 1 18| z1 = 1 + ix, z2 = a + ib Ges z3 = x + iy wZbwU RwUj msL ̈v| [gqgbwmsn †evW©- Õ22] (K) i – 3 Gi Av ̧©‡g›U wbY©q Ki| (L) |z2| 2 = 1 n‡j, †`LvI †h, x Gi GKwU ev ̄Íe gvb – z1 z1 = – z2 mgxKiY‡K wm× K‡i| (M) 3 z2 = z3 n‡j cÖgvY Ki †h, |z3| = b 2y – a 2x DËi: (K) 5 6 19| `„k ̈Kí-1: f(x) = |bx – c|. `„k ̈Kí-2: 2x = –1 + –3 Ges 2y = –1 – –3. [ivRkvnx †evW©- Õ19] (K) – 5 + 12 –1 Gi eM©g~j wbY©q Ki| (L) `„k ̈Kí-1 G, b = 1, c = 2 Ges f(x) < 1 4 n‡j †`LvI †h, f(x2 – 2) < 17 16 [kU© wm‡jev‡mi AšÍf~©3 bq] (M) `„k ̈Kí-2 Gi Av‡jv‡K cÖgvY Ki, x 4 + x3 y + x2 y 2 + xy 3 + y4 = – 1 DËi: (K) ± (2 + 3i) 20| DÏxcK: g‡b Ki g(x) = 2x – 1, x  R GKwU ivwk Ges A = {a : a  c~Y©msL ̈v Ges |g(a)| < 4} I {t : t  ̄^vfvweK msL ̈v Ges 2 < t < 4} `ywU †mU| [Kzwgjøv †evW©- Õ19] (K) –3 < g(x) < 7 †K ciggvb wP‡ýi mvnv‡h ̈ cÖKvk Ki| [kU© wm‡jev‡mi AšÍf~©3 bq] (L) |g(x) + 2iy| = t Øviv wb‡`©wkZ mÂvic‡_i †K›`a I e ̈vmva© wbY©q Ki| (M) A †mUwUi mywcÖgvg Ges Bbwdgvg †ei Ki| [kU© wm‡jev‡mi AšÍf~©3 bq] DËi: (K) |2x – 3| < 5 ; (L) †K›`a     1 2  0 ; e ̈vmva© = 3 2 (M) Inf(A) = – 1 Ges Sup(A) = 2 21| Z GKwU RwUj msL ̈v Ges f(x) = 5x + 1. [h‡kvi †evW©- Õ19] (K) S = {x : x  , – 9 < f(x) < 16} – Gi mywcÖgvg wbY©q Ki| [kU© wm‡jev‡mi AšÍf~©3 bq] (L) 1 |f(x)| > 1 9 , x  – 1 5 mgvavb K‡i mgvavb †mU msL ̈v‡iLvq Dc ̄’vcb Ki| [kU© wm‡jev‡mi AšÍf~©3 bq] (M) |2z + 3| = |3z + 1| Øviv wb‡`©wkZ mÂvic_ wbY©q Ki| DËi: (K) mywcÖgvg, sup(S) = 3 ; (M) 5x2 + 5y2 – 6x – 8 = 0 22| `„k ̈Kí-1: p(x) = a + bx + cx2 `„k ̈Kí-2: GK‡Ki GKwU KvíwbK Nbg~j | [ewikvj †evW©- Õ19] (K) – 3 – 4i Gi eM©g~j wbY©q Ki| (L) `„k ̈Kí-1 Gi mvnv‡h ̈ hw` {p()}3 +       P     1  3 = 0 nq, Z‡e †`LvI †h, a = 1 2 (b + c) A_ev c = 1 2 (a + b). (M) `„k ̈Kí-2 n‡Z cÖgvY Ki †h, 1 +  +  2 = 0. DËi: (K)  (1 – 2i) 23| f(x) = x – 2 [wm‡jU †evW©- Õ19] (K) – 1 ≤ f(x) ≤ 11 AmgZvwU ciggvb wP‡ýi mvnv‡h ̈ cÖKvk Ki| [kU© wm‡jev‡mi AšÍf~©3 bq] (L) f(x) f(x + 2) > f(x + 3) f(x + 4) AmgZvi mgvavb †mU msL ̈v‡iLvq †`LvI| [kU© wm‡jev‡mi AšÍf~©3 bq] (M) z = p + iq n‡j, |f(z + 6)| + |f(z – 2)| = 10 Øviv wb‡`©wkZ mÂvi c‡_i mgxKiY wbY©q Ki| DËi: (K) |x – 7| ≤ + 6 ; (M) p 2 5 2 + q 2 3 2 = 1
4  Higher Math 2nd Paper Chapter-3 24| `„k ̈Kí-1: z = 3x + 4y kZ©mg~n: x + y ≤ 450, 2x + y ≤ 600, y ≤ 400, x, y ≥ 0 `„k ̈Kí-2: y 2 + y + 1 = 0 [wm‡jU †evW©- Õ19] (K) 5i -Gi eM©g~j wbY©q Ki| (L) `„k ̈Kí-1 n‡Z †jLwP‡Îi mvnv‡h ̈ z -Gi m‡e©v”P gvb wbY©q Ki| [kU© wm‡jev‡mi AšÍf~©3 bq] (M) `„k ̈Kí-2 Gi mgxKiYwUi g~jØq p, q n‡j, †`LvI †h, p m + qm =   2 hLb m Gi gvb 3 Øviv wefvR ̈ –1 hLb m Aci †Kv‡bv c~Y©msL ̈v DËi: (K)  5 2 (1 + i) ; (L) Zmax = 1750 25| `„k ̈Kí-1: f(x) = 3x + 1 `„k ̈Kí-2: |z – 5| = 3 [XvKv, h‡kvi, wm‡jU I w`bvRcyi †evW©- Õ18] (K) I C| Øviv Kx †evSvq? G‡`i g‡a ̈ m¤úK© Kx? (L) 2|f(x – 2)| ≤ 1 Gi mgvavb †mU msL ̈v‡iLvq †`LvI| [kU© wm‡jev‡mi AšÍf~©3 bq] (M) z = x + iy n‡j `„k ̈Kí-2 Gi mÂvic_ R ̈vwgwZKfv‡e Kx wb‡`©k K‡i? wPÎ AuvK| DËi: (K) m¤úK© n‡”Q  CI (M) (x – 5)2 + y2 = 32 ; e„‡Ëi mgxKiY wb‡`©k K‡i 26| `„k ̈Kí-1: |z + 1| + | z – 1| = 4; †hLv‡b z = x + iy. `„k ̈Kí-2 a = p + q, b = p + q Ges c = p +  2 q. [ivRkvnx, Kzwgjøv, PÆMÖvg I ewikvj †evW©- Õ18] (K)     1 + i 1 – i 3 †K A + iB AvKv‡i cÖKvk Ki| (L) `„k ̈Kí-1 n‡Z cÖgvY Ki †h, 3x2 + 4y2 = 12 (M) `„k ̈Kí-2 n‡Z †`LvI †h, a 3 + b3 + c3 = 3(p3 + q3 ) DËi: (K) – i = 0 + (– 1).i 27| z1 = 2 + 3i, z2 = 1 + 2i, a = p 2 + q + r Ges b = p + q + r 2 , †hLv‡b  GK‡Ki Nbg~j ̧wji GKwU RwUj Nbg~j| [Kzwgjøv †evW©- Õ17] (K) 1 2 – i Gi Av ̧©‡g›U wbY©q Ki| (L) DÏxc‡Ki Av‡jv‡K –––– z1 – z2 Gi eM©g~j wbY©q Ki| (M) DÏxc‡Ki mvnv‡h ̈ a 2 + b3 = 0 n‡j, cÖgvY Ki †h, 2p = q + r, 2q = r + p Ges 2r = p + q DËi: (K) tan–1 1 2 ; (L) ± 1 2 {( 2 + 1) } 1 2 – i( 2 – 1) 1 2 28| `„k ̈Kí-1: x + iy = 2e–i `„k ̈Kí-2: F = y – 2x. kZ© ̧wj: x + 2y ≤ 6, x + y ≥ 4, x, y ≥ 0. [h‡kvi †evW©- Õ17] (K) z = x + iy n‡j, |z + i| = | | – z + 2 Øviv wb‡`©wkZ mÂvic_ wbY©q Ki| (L) `„k ̈Kí-1 n‡Z cÖgvY Ki †h, x 2 + y2 = 4. (M) `„k ̈Kí-2 G ewY©Z †hvMvkÖqx †cÖvMÖvgwU n‡Z •jwLK c×wZ‡Z F Gi m‡e©v”P gvb wbY©q Ki| [kU© wm‡jev‡mi AšÍf~©3 bq] DËi: (K) 4x – 2y + 3 = 0 29| z = x + iy ; |z + 5| + |z – 5| = 15 ........ (i) 2x + 3 x – 3 < x + 3 x – 1 .............. (ii) [PÆMÖvg †evW©- Õ17] (K) GK‡Ki Nbg~jmg~n wbY©q Ki| (L) DÏxcK-1 n‡Z, mÂvic‡_i mgxKiY wbY©q Ki| (M) DÏxcK-2 G ewY©Z AmgZvwUi mgvavb Ki Ges msL ̈v‡iLvq †`LvI| [kU© wm‡jev‡mi AšÍf~©3 bq] DËi: (K) 1, –1 + 3i 2 , –1– 3i 2 (L) 20x 2 + 36y2 = 1125; hv wb‡Y©q mÂvic‡_i mgxKiY| 30| f(x) = |x – 3| g(x) = p + qx + rx2 [ewikvj †evW©- Õ17] (K) 15 + 8i Gi eM©g~j wbY©q Ki| (L) f(x) < 1 7 n‡j cÖgvY Ki †h, |x2 – 9| < 43 49 [kU© wm‡jev‡mi AšÍf~©3 bq] (M) p + q + r = 0 n‡j cÖgvY Ki †h, g{()}3 + {g( 2 )}3 a x pqr †hLv‡b  GK‡Ki KvíwbK Nbg~j Ges a = x = 3. DËi: (K)  (4 + i)