MvwYwZK m~Îvewj

RULES OF MATHEMATICSRULES OF MATHEMATICSRULES OF MATHEMATICSRULES OF MATHEMATICS

exRMwYZ (ALGEBRA) eM©, Nb, ¸b, Drcv`K, Abywm×vš— I gvb wbY©‡qi m~Ε (a + b) 2 = a2 + 2ab + b2 • (a + b) 2 = (a – b) 2 + 4ab• (a – b) 2 = a2 – 2ab + b2 • (a – b) 2 = (a + b) 2 – 4ab• a2 + b2 = (a + b) 2 – 2ab • a2 + b2 = (a – b) 2 + 2ab• a2 – b2 = (a + b) – (a – b) • 2 (a2+b2) = (a + b) 2 + (a – b) 2• (a + b + c) 2 = (a2 + b2 + c2) + 2 (ab + bc + ca)• (a2 + b2 + c2) = (a + b + c) 2 – 2(ab + bc + ca)• 2 (ab + bc + ca) = (a + b + c) 2 – (a2 + b2 + c2)• (a + b) 3 = a3 + 3a2b + 3ab2 + b3

• (a + b) 3 = a3 + b3 + 3ab (a + b)• (a – b) 3 = a3 – 3a2b + 3ab2 – b3

• (a – b) 3 = a3 – b3 – 3ab (a – b)• a3 + b3 = (a + b) (a2 – ab + b2)• a3 + b3 = (a + b) 3 – 3ab (a + b)• a3 – b3 = (a – b) (a2 + ab + b2)• a3 – b3 = (a – b) 3 + 3ab (a – b)• (a + b + c) 3 = a3 + b3 + c3 + 3 (a + b) (b + c) (c + a)• a3 + b3 + c3 – 3abc = (a + b + c) (a2 + b2 + c2 – ab – bc – ca)• a3 + b3 + c3 – 3abc = �� (a + b + c) { (a – b) 2 + (b – c) 2 + (c – a) 2 }• 4ab = (a + b) 2 – (a – b) 2 • ab = (���

� )� � (� � �� )�

• (x + a) (x + b) = x2 + (a + b) x + ab• (x + a) (x – b) = x2 + (a – b) x – ab• (x – a) (x + b) = x2 + (b – a) x – ab• (x – a) (x – b) = x2 – (a + b) x + ab• (x + p) (x + q) (x + r) = x3 + (p + q + r) x2 + (pq + qr + rp) x +pqr• bc (b – c) + ca (c – a) + ab (a – b) = – (b – c) (c – a) (a – b)• a2 (b – c) + b2 (c – a) + c2 (a – b) = – (b – c) (c – a) (a – b)• a (b2 – c2) + b (c2 – a2) + c (a2 – b2) = (b – c) (c – a) (a – b)• a3 (b – c) + b3 (c – a) + c3 (a – b) = – (b – c) (c – a) (a – b) (a + b + c)• b2c2(b2–c2) + c2a2(c2–a2) + a2b2(a2–b2) = – (b–c) (c–a) (a–b) (b+c) (c+a) (a+b)• (ab + bc + ca) (a + b + c) – abc = (a + b) (b + c) (c + a)• (b + c) (c + a) (a + b) + abc = (a + b +c) (ab + bc + ca)

ev¯—e mgm¨v mgvav‡b exRMvwYwZK m~Î • Rb cÖwZ †`q ev cÖvc¨ q UvKv n‡j, n R‡bi †`q ev cÖvc¨, A = qn UvKv• ˆ`wbK m¤cvw`Z Kv‡Ri cwigvY q n‡j, d w`‡b m¤cvw`Z Kv‡Ri cwigvY, W = qd• MwZ‡eM NÈvq q wgUvi n‡j, t NÈvq AwZµvš— `~iZ¡ , D = qt wgUvi• q % e„wׇZ ev nªv‡m a Gi ewa©Z ev nªvmK…Z gvb, A = a (1 ± &

�'')[ e„w×i †¶‡Î + wPý I nªv‡mi †¶‡Î – wPý cÖ‡hvR¨ ]

• GKK mg‡q GKK g~ja‡bi gybvdv r UvKv n‡j, P UvKv wewb‡qv‡M n mgqv‡š— gybvdv I Ime„w× g~jab A n‡e †hLv‡b,

mij gybvdvi †¶‡Î, I = Pnr UvKv Ges A = P (1 + nr) UvKvPµe„w× gybvdvi †¶‡Î, A = P (1 + r)n UvKv

m~PK [ a ≠0, b ≠ 0 Ges m, n mKj c~Y© msL¨vi †m‡Ui GKwU Dcv`vb ]

• am . an = am+n•

�/

�0 = am–n • (am)n = amn

• (ab)n = anbn• 1�

�23

= �0

�0• a0 = 1

• a–n = ��0 • a

40 = 5a0

• a/0 = 5a60

jMvwi`g [ a > 0 Ges a ≠ 1 ]

• 89:� ;< = r 89:� ; • 89:�(;=) = 89:� ; + 89:� =• 89:�(>

?) = 89:� ; � 89:� = • 89:� ; = 89:� ; @ 89:� b• 89:� ; = ABCD >

ABCD �• 89:� 1 = 0 • 89:� b = �

ABCD �• 89:� a = 1 • 89:� 5;0 = �3 89:� ;• a > 0 Ges ax = ay n‡j , x = y • x > 0 Ges ax = bx n‡j , a = b

aviv GLv‡b, a = cÖ_g c`, p = †kl c`, d = mvavib Aš—i, r = mvaviY AbycvZ mgvš—i avivi †¶‡Î, • n Zg c` = a + (n – 1) d• n msL¨K c‡`i mgwó = 3� {2a + (n – 1) d}• c` msL¨v = (FG�)

� + 1 • a I b Gi mgvš—i ga¨K =

(���)�

• 1 + 2 + 3 + . . . . . + n = 3(3��)�

• 1 + 3 + 5 + . . . . . + n = n2

• 2 + 4 + 6 + . . . . . + n = n (n + 1)• 12 + 22 + 32 + . . . . . + n2 = 3(3��)(�3��)

J

• 13 + 23 + 33 + . . . . . + n3 = K3(3��)� L

�

¸‡YvËi avivi †¶‡Î, • n Zg c` = arn–1

• n msL¨K c‡`i mgwó = �(<0G�)<G� ; r > 1

• n msL¨K c‡`i mgwó = �(�G<0)�G< ; r < 1

• a + ar + ar2 + arn = ��–<

w·KvYwgwZ (Tri:9n9metry)(Tri:9n9metry)(Tri:9n9metry)(Tri:9n9metry) • sin θ = • c9s θ = . • tan θ =

• c9t θ = • sec θ = . • c9sec θ =

• sin θ = �

SBTUS θ

• c9sec θ = �TV3 θ

• c9s θ = �TUS θ

• sec θ = �SBT θ

• tan θ = �SBW θ

• c9t θ = �W�3 θ

• sin2θ + c9s2θ = 1 • sin2θ = 1 – c9s2θ • c9s2θ = 1 – sin2θ• sec2θ − tan2θ = 1 • sec2θ = 1 + tan2θ • tan2θ = sec2θ – 1• c9sec2θ − c9t2θ = 1 • c9sec2θ = c9t2θ + 1 • c9t2θ = c9sec2θ – 1

†KvY 0000°°°° 30303030°°°° 45454545°°°° 60606060°°°° 90909090°°°°sin 0 1

21

52 532

1

c9s 1 532

152 1

20

tan 0 153

1 53 AmsÁvwqZ

c9t AmsÁvwqZ 53 1 153 0

sec 1 253

52 2 AmsÁvwqZ

c9sec AmsÁvwqZ 2 52 253 1

• 60” †m‡KÛ = 1’ wgwbU • 60’ wgwbU = 1° wWwMÖ • 90° wWwMÖ = 1 mg‡KvY

j¤ AwZfzR

f‚wg AwZfzR

j¤^ f‚wg

f‚wg j¤^

AwZfzR f‚wg

AwZfzR j¤^

• 1° = 1 [�\'2

S• 1c = (�\'

[ )]• DbœwZ †KvY = tan θ • AebwZ †KvY = sin θ• e„‡Ëi e¨vmva© r, †K‡› ªª Pv‡ci †iwWqvb †KvY θ n‡j Pv‡ci ˆ`N¨©, s = rθ GKK

• †KvY = (n @ 90° ± θ )• n we‡Rvo n‡j, sin θ ^ c9s θ , tan θ ^ c9t θ , sec θ ^ c9sec θ• 1g PZzf©v‡M cÖ‡Z¨K †KvY abvZœK (+)• 2q PZzf©v‡M sin θ I c9sec θ abvZœK (+) Ges evwK¸‡jv FYvZœK (–)• 3q PZzf©v‡M tan θ I c9t θ abvZœK (+) Ges evwK¸‡jv FYvZœK (–)• 4_© PZzf©v‡M c9s θ I sec θ abvZœK (+) Ges evwK¸‡jv FYvZœK (–)• sin (–θ) = –sin θ • c9s (–θ) = c9s θ • tan (–θ) = –tan θ• sec (–θ) = sec θ • c9t (–θ) = –c9t θ • c9sec (–θ) = –c9sec θ

cwiwgwZ (;easurement)(;easurement)(;easurement)(;easurement) • AvqZ‡¶‡Îi ˆ`N¨© a GKK I cÖ ’ b GKK n‡j,

†¶Îdj, A = ab eM©GKKcwimxgv, s = 2(a +b) GKK

KY©, d = 5a� + b� GKK• eM©‡¶‡Îi GK evûi ˆ`N © a GKK n‡j,

†¶Îdj, A = a2 eM©GKKcwimxgv, s = 4a GKK

KY©, d = a52 GKK• i¤‡mi GK evûi ˆ`N © a GKK I KY©Øq d1, d2 n‡j,

†¶Îdj, A = �� (d� @ d�) eM©GKK

cwimxgv, s = 4a GKK• mvgvš—wi‡Ki f‚wg a GKK I D”PZv h GKK n‡j,

†¶Îdj, A = ah eM©GKK• mvgvš—wi‡Ki `yBwU mwbœwnZ evû a, b GKK I Zv‡`i Aš—f©z³ †KvY θ n‡j,

†¶Îdj, A = ab.sinθ eM©GKK• mvgvš—wi‡Ki GKwU KY© d I wecixZ kxl©we›`y n‡Z K‡Y©i Dci j¤^ h n‡j,

†¶Îdj, A = dh eM©GKK• UªvwcwRqv‡gi mgvš—ivj evûØq a, b GKK I D”PZv ev j¤^ `~iZ¡ h GKK n‡j,

1g PZzf©vM 2q PZzf©vM

3q PZzf©vM 4_© PZzf©vM

†¶Îdj, A = �� h(a+b) eM©GKK

• wÎfz‡Ri f‚wg a GKK I D”PZv h GKK n‡j,

†¶Îdj, A = �� ah eM©GKK

• wÎfz‡Ri wZb evû a, b, c GKK I a, b Gi Aš—f©zw³ †KvY θ n‡j,cwimxgv = a + b + c GKK

Aa©cwimxgv, s = ����S

� GKK

†¶Îdj, A = as (s � a) (s � b) (s � c) eM©GKK

†¶Îdj, A = �� ab sinθ eM©GKK

• mgevû wÎfz‡Ri GKwU evû a GKK n‡j,cwimxgv = 3a GKK

†¶Îdj, A = 5bc a� eM©GKK

• mgwØevû wÎfz‡Ri mgvb evûØq a GKK I Aci evû b GKK n‡j,cwimxgv = 2a + b GKK

†¶Îdj, A = �c 54a� � b� eM©GKK

• e„‡Ëi e¨vmva© r GKK, †K‡›`ª Pv‡ci †KvY θ n‡j,cwiwa, C = 2er GKK

†¶Îdj, A = er2 eM©GKK

e„ËKjvi †¶Îdj = θ

bJ' @ er� eM©GKK

Pv‡ci ˆ`N ©, s = θ

bJ' @ 2er GKK [ θ = †Kv‡Yi wWwMÖ cwigvc ]

Pv‡ci ˆ`N ©, s = rθ GKK [ θ = †Kv‡Yi †iwWqvb cwigvc ] • AvqZvKvi Nbe¯‘i ˆ`N © a GKK, cÖ ’ b GKK I D”PZv c GKK n‡j,

KY©, d = 5a� + b� + c� GKKmgMÖZ‡ji †¶Îdj = 2(ab + bc + ca) eM©GKK

AvqZb, V = abc NbGKK• Nb‡Ki GK avi a GKK n‡j,

KY©, d = a53 GKKc„ôZ‡ji K‡Y©i ˆ`N © = a52 GKKmgMÖZ‡ji †¶Îdj = 6a2 eM©GKK

AvqZb, V = a3 NbGKK• mge„Ëf‚wgK †KvY‡Ki f‚wgi e¨vmva© r GKK, D”PZv h GKK I †njvb DbœwZ llll n‡j,

†njvb DbœwZ, llll = 5h� + r� GKKeµZ‡ji †¶Îdj = er llll eM©GKK

mgMÖZ‡ji †¶Îdj = er(llll+r) eM©GKK

AvqZb, V = �b er�h NbGKK

• mge„Ëf‚wgK †ej‡bi f‚wgi e¨vmva© r GKK I D”PZv h GKK n‡j,eµZ‡ji †¶Îdj = 2erh eM©GKK

mgMÖZ‡ji †¶Îdj = 2er(h+r) eM©GKKAvqZb, V = er2h NbGKK

• †Mvj‡Ki e¨vmva© r GKK n‡j,Z‡ji †¶Îdj = 4er2 eM©GKK

AvqZb, V = cb erb NbGKK

†f±i (Vect9r)(Vect9r)(Vect9r)(Vect9r) • Avw`we›`y A I Aš—we›`y B n‡j, H w`Kwb‡`©kK †iLvsk ABgggggh Øviv m~wPZ Kiv nq, Gi ˆ`N ©

iABggggghi Ges ABgggggh = �BAgggggh • †f±i †hv‡Mi wÎfzR wewa t ACgggggh = ABgggggh + BCgggggh• †f±i we‡qv‡Mi wÎfzR wewa t ACgggggh � ABgggggh = BCgggggh• wÎfz‡Ri evû·qi GKB µg Øviv m~wPZ †f±i·qi †hvMdj k~b¨

GLv‡b, ABgggggh + BCgggggh = ACgggggh = �(CAgggggh)A_©vr, ABgggggh + BCgggggh + CAgggggh = CAgggggh � CAgggggh = 0

• †f±i †hv‡Mi mvgvš—wiK wewa t ACgggggh = ABgggggh + ADgggggh• †f±i †hv‡Mi wewbgq wewa t †h‡Kv‡bv u, v †f±‡ii Rb¨ u+v = v+u• †f±i †hv‡Mi ms‡hvM wewa t †h‡Kv‡bv u, v, w Gi Rb¨ (u+v)+w = u+(v+w)• †f±i †hv‡Mi eR©b wewa t †h‡Kv‡bv u, v, w Gi Rb¨ u+v = v+w n‡j, v=w• †f±‡i mvsL¨¸wYZK msµvš— eÈb myÎ t m, n `yBwU †¯‹jvi I u, v `yBwU †f±i n‡j

(m+n)u=mu + nu Ges m(u+v) = mu + mv • Aš—we©fw³KiY myÎ t A, B we›`yi Ae¯’vb †f±i h_vµ‡g a, b n‡j

Ges AB †iLvsk C we›`y‡Z m:n Abycv‡Z Aš—we©f³ n‡j,

C we›`yi Ae¯’vb †f±i, c = 3��6�

6�3

>>>> A B A

B C

A B

CD

O Aa

bc

B

Cm

n

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