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0% found this document useful (0 votes)
185 views67 pagesAll Maths Formulae
Uploaded by
Vrushabh DhoteCopyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
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/ 67
_“Tijgonomentey _
form ulee =
sin? @ + Cos*@=]
sin*?@ = [~ Cos*@
cs? @ = l= sin?@
$ec"@ ~fan*@ St
= [+t fanr®
cot *O =
_ sin Ch + p) = sing. corp F Cosd. sing
— © sin CxK-p) = sind *tosp = C03K- sing
= cos CXHFP) = Cosx-cos B = §fhk=sinB
7 cos (o-B) = CosX.cosp + Sind sing
9 tan (&k#p) = tang + tanB 4 |
-1— fang. tanp
= fan c— +anB
Seem ait aL ot) pe
MOS
eo
ft ttanr®
COS*#O — sf
fo eaineom
tia tain tz i) sin @= 2519@ - (Oso
a LL -
_ ©) cine = a tan Ar \y
tof Fan® co ’
om
(os @ = cos*@f = sin? Cf
Loa 3tan*@
c#e = 1 (1 Fc0s2@)
z
G
|
eo Co = qo ° ZINN Oy,
oy cos O = 2Cos*ay, — i a
73) Cos@ Ss La dant @fe
| Lt FAR > Of |
Gi yeno pss u tan 6/63 ya i
: Loa tant@s,
| GY FOS70 = 2cost@— 15 | F C052 O= 2 cosH
GA vt = cos 2O = sinr@~
ey) [i Cos 2O VlEsEO
nip} f.Cos @ = 9 cos?*O/y,
Qe ewes @ Ee sin of :
% [oF sin2@= (cos@ + sin@)e
es) EL — sinz@ = € COS @~ s(n@)ye
Oy 1 4+ StIN@ = Coos Of, +5in @A) =
COL f — sin@ = ( 8s Of — stnofryr
G3). gin 3@ =. 3 sin@ — YSjn&®
(a) Os 3 OVS 4 ¢os°O — 36050
Ug) _jyans@ = _37an©@ — fan3@
@
C
Sin ~o= ube C4 = ¢os 2@)
L Cee ton) + Cos(¢t>)]
7 Tm Ean CHd) . cos[c=d\ |
~ © me pads aah Tas) LED)
) gine = stab = 2008/7 C FPN o sin [C-9\
il = ar C x z z)
%
cos © oF COSP= 2eos/CHp\ (os /l-D \
es [CH BELEB
_@& sc esac (.c+d) + sinf c-D)
a : a0 crew
G3) gin (—@ )= — SIN@
a) co(-O JE (ose
ea) tan (-@) = — jane
® qonf or ~o@)\ = [- Fano
: C4 ) | + Fane claccnute
eae
sin (1 -@) = : nila
os (7 = @) = -cos@ °
qan (CW -@ ) = —tane Fj -
Cinema) fe )=- SINO- +0 ay - ©
cas Ct +e )= = Cose
ton Cr +@)= fane
@__ Sin (37, -e)s - oso
cep (30%, -&)= — sine
tan (30/2 ~oF = Cote =
Ga). in ( xy fo) =-603@
@ cee (374, +o) = Sino
Ga_4an (39/4, to) = — Cot© “BR. NILESH M. PARDE
BARDE MATHS ACADEMY |
, 24) 1 cos
2 35)
9 i
4) | 26)
5) sec’@=1 + tan"o a)
|
6) casee"0=1 + cot’
28) i
7) 1 J
8) Bb) 29) sinA+h cos Beas A sin B
» Lal ,
| Aj 30) sin(A-Bj=sin A.cos B
10) ial
| | 31) cos(A +Bj=eos’A cos B~ sine sin
ry a
Fe | 32). cos(A—1B)= cos A cos B+ sin sin 8
12) {Fy
Lol sca ey -aznatims
13) | OD ORAS SB) ean atan 8
7 ee Te
15) cos (-0) = cos 6 Ze i 4
16) gin 0.c0s 0 a Ss | 5} tama +0) =
in 0 cos 0 = ante, ye | 335) wine +0) = a
rr) S-sin'0 jase satan
Jc) 36) tana =a
cos") Heal
TAT
a-tanta | cap
5 ial Ein Cea = Jain ee ces
aE } >| 37) sin C+ sin D =2 sin === cos
; araif c+E CD
Is} tan29 = ae | 38) sinC~sin D#2 cos S22 yin S=P
19) 1 = ca820 2sin*0 Cre
" 39) cos C + cus b=2 cas S22
2o0sA sin’ = sing\ + B}~ sin( A
A cos = cos(A + B} + cas, 0
= By caster t by 30)
Cosine rile
cos A=
roje:
axbeosC
u
e=acesB
pete?
ction rete
coos?
os Abacos C
boos A,
56)
3)
58)
DIRECTOR : ER. NILESH N. anave SIR MS. NG. : 8007525257
ii)
iii)
(atbP = (a? + 2ab +b?)
(a= by 2ab +t?)
(ee atb)(a-b)
(at bP =a) + 30% + Jab? +b?
— by =a) Sab + 3b? — bP
—b = (aba? ab +b?)
(a +o')= (at) (aah +b ) PE i . ‘ =
a — a =
5 | V.S.A. = Vertical Surface Area
59)
i J | C.S.A, = Curved Surface Area
i | TSA. = Total Surface Area
— ir —
VS.AICS.A, TSA, | Volume {
|
|
ee a |
2(1+b)b 2CTb+ bh “bly Lbh |
ar | oP B |
2arh | 2a(r+hy ; xh
| ‘ Lady |
ail | mrt) a '
i |
: | y |
dae | = '
| |
i J
|
|
|
8
A
68)
Area gtjanile 7
Area,of equilateral & i: . al + Dx = . j
Are(of circle = mr* || a I
Age Of Parallelogram = bx h | i
!
‘y/ A [ee cea
‘Area of trapazsie > (sum of sides) x h | > by | :
Perimeter of rectangle = 2 (1 +b) E ao |
Perimeter of squar uM bos 3) When a detertaires. fins two Sdention!
Perinneter of triangle =a + b+e ie, sum \¥ oY or colsain tha P= 0
of sides | ii) When ww sows oF ¢
cireumferce of eiveie = 2nr | of tie waterrainan
. | & vinant fs
; iF two lines ¢
ivovo lines ers pornes
yee tne nes ten
ii) external x= 2
#i) Midpoire formula
= MUN | eve
7 YES .
re 7
_ Mt Ve Ys Ne
3 SRR Sta Dis trates
n n roots are
nol
nat 7
yeas
iim (Lex
SI) Ate 97. BG
roots =
inpe porst form y= yy, }
wo Slope interes: from. y
465 Two points f
x
82) Deuble tatercept fom = +E = |
: a’ b
86) Cenesel equation of line as + by +e = 0
Ty at by
Siepe ="), Stanx
&
eV,
logfax+b)
arn
09) SL cotx=
109) oot =
110) jy seex = see tan x
13a) fekdx
iby fee ade
135)
136; f sinzdx
137) feosxd
138)
139)
|
|
|
|
|
|
_ 3)
Eijr
a
, a | 145)
Bt
c | i46)
MT 147)
~ 2-2 e ; vl]
dx du dx LU
138)
|
149)
ae 150)
123, Approximate Value
7 rae J=T() +h Fa) st)
126)., Approximate :vor in y
Yay 182)
ay
dy= Ex bx
153)
127) Relative error in y
|
I r
in Ps Gnde =~ | fede
ny
162) cus¢—8" = con
$64; tan (—@) ~~ tant! se
136) see (=8 = ser | | |
i
| | i
i i Me | i
| |
pies iG | me
Hai ' i
| jo | Wg te MA
| |
, om,
wt (foods = [ fat b—xax
vf fOodx +f (ax
sax 2 ears | pta~ayde
iseven, a Gato
|\—__4 JES (ue ele
i tans ~ ———
Ave. Che » 4
Bocg 2 4
2,
me :
Bind. GP+ Bd. SinB
ys
1 Sin_(l-B)_= Si ol. GB = Chel. Siu P
Les) (4B). Gol. @B- Sind. SKE
Lt. hol -B) = God. @B + Sin of. Sima B
65-1 2S MGB = Bu + Ain (A-B)
2GA.SinB_~ Bin CAtB) - Sh (A-B)
1204-8 = cy(AtB) + @(A-BY
2 Sard. SenB > Co (A-B) ~ G& (AB) eC ED es
SLE.
¢a.| Ain Ct Sin ® = 25in (CB CPN |
aan CL
to | Bin C- Sin D: aS gn_(P\
AU
1 tos (Cee
@
i +8 GLB)
Ae.
ltac- (hf = - 2 Si (ci®\ St. {t8\
EAB) Bin Cs}
43. |Bm 26 = 25m - SG _
% | oye - Gieo Ge
2 2
4: [Anze = 38in9 - 4Sin%@
+c. | cb 26 = 8"o- Sin’ _
. = 20e-1
sz 4+ 28mg
qi} oe = Ge ~ Sho 7
2 a
> ate oi
28 -
=. 1. 28m.
—z
45. | (39 . 4@30- 3%
49. | Air 26. 2fime _ oe
____ + amie, a eas?
% | @26. 2 \-donig a
[4+ Jtanre oe :
f62\ | 7
a Pe
7 py S| dom 39__ = 34ome=tomio
(ca) | | atoms
, $21 tom A+B) = time + tam B
GP) - (net. dan
——| 83.1 ton (o-p) = tim, of - Zam B
SON | I tdonl tom
& = tai! “8
4| 4s. ta { AB, dow A tn
14AB.
86] 1-CosKo = 2Sin* Ka e
L S| bare: “Sa
| 4] 4+ @ 26 = 25h 0
| a es ashi s
| hs ko 2 2G? ke
4 2
eee G | _ 1+@26 = 2a2q
ee | 44 GRe = sare
—_____] 2
aT 421 4+ Sings = (GO +Sno)*
Pe 4z| 1-Sin 20 > ( Go - Sine)*
ma 9 | 4+ 2434 Ms MCMtL)
a 2
2 | qs] 7+ o2e tee. t= p(mtt) @ntt)
—-1 G
aes ==
Pe |
eae yee, yn? = [nent | *
R A
z
_ 3 on (nay aie
‘4
ag | SEO _
G41 tong = Sin® —s a
Ge
ag.| mre = Be. aa
Sino T
ga (ayby?_> o2+ 2ab+2*
Yoo! Ca-byt = at —2ab+b*
joi, G2 b% = (utd) (a-b)__,
or} (a+b)? = P+ 3q7b+ Ba + bE
sl (a-b)? = 0? -aeh+ Sab? —b —t
wlio? b? = (asd) (abt B*)
os |g? ~ b? = (a-b) (a°*+ab+*) __
Jb. eg 40) = b mn
a
dot log {2 E Log sm = bog
Jo n= sn Goa 2m i
109, nym 5 loge
lo zk . _ ee
Uy | Pon t = ©. _
Ut Who =D if >
Asin (sine) = ©
2) cs" (ese) = ©
\aan*(yare) =O —S
“)_sin*(coo) = pO
_§) cos "* (sing) =
Ft lete)s ome
3) fe“ leone) =n
1
sin Cone) =
ed) tan (tin“*@) = ©
see ( e¥0) = © a1) cos ( sino} yn Na-o™ a an
4) fa = con't) i -
b a
») tan “(4) = col t/t) es
14) cos =f Z
tan*® + cof to =
ay
2)
See ~*@ + cosee™4 =
sin “2 (-*) = ~ sir tx “AL tanh Gp) Eo teh
8) cost ex) = Me cor tx _Devivative of Trijonomedvic finefon
2)
Ux cx tene
4$x0 es
Le Ux
) x= fano A ppl ca tion s of Derivatives.
Approxi methions =>
_fca)_+ hf ica)
Taga =
em wocetion ays ent to curve y= Fr) at
_P Fahy
Y- “tae fae) cx —x1) ~
and The equettion «) Pe given by
Y~nH= st (x-*)
(2
Ox
; ,
note 3- @D_ TF tute fine cue perpenclreeele fhen
mime. = -1 7
@ 16 fue ines ae then
m= my “Rofles fheowm i>
rey = FX) TF veal vetlec of funclron
_— of rect veenietb le. such fret - a
—F) FON ts confinuous_on Carb] _
BR) Fx) fs cbhenn Heth om (6) | an
ii?) fea) = FCb) then thee exists ec rea] num ber ~
- Fuse €Cetyb) such thet ¥'Ce) =8
LITTV C Levrgy tn ges Pein veelue fheovon)
mn funcfien fx) ps sarel Have cesfitfied iain
tes sett hes Bleue: _conditen ee -
® fix) ps Confinctows” on carne
Sor 7s ORreatiecble on (a,b)
~ Forke ext = € (76) i Incvcecsing ad ctr fanctian o oo ON
re funclisn y= tix) on a,b sy, ce febeg
o Innceaig funebien fax YO 6s
; faanetisn rt ft ax LO
clecreZe( =
|“ axima and’ Minima — Sees
ACG e Shel Sei 9
_f' fe th) <0. [awe
fl (c~h) £0” SS a
fi Ccth)yo0. = es wee
aahbo ame
at _-edtto said tobe firs} qferivativve Pop et) Rute: consiele viven ft y
<5 hen fica cron 1 rulths rreegaimeeen ov
nung
i saicl fo bea point ef toflechon
in fhis case fend leiv cétioo fh yet
fixie! thu cppty Fost ebeivective Het
——~Suenel aleaverice Fest >
n) find Fe) Fr)
3) put i(k) =e and obfaincd fhe oneor me
- expert vetltct oF bouncloy Yeeletey = 7
a) IE x= & be fhe one of oder value then
Orta) a SIN X dx = — osx tc
1) ( Cos X cle = sinx te
ca See xd tenn { + co
J) x [ tanth | +
Dn feet x dy:
¥)
i)
1)
j
j
es i Side et pe
ae
tein x te
corey dx =
seek Ferny efx = = ste + c
soe tote a= = cores: eG
-3)~
| coeeK — saa ‘bo — pul aA
= bo a C
SW) ers mabe -aabese
a sinB gia.
q a @ oe _
— Cosine “pale 2
4) av =br $c*_rbe cos A
ft) pe: bes qrder_ ae cosB
ion Rule =—
sina UL usiry projection mule cesinrale. 2
a T Tt a oo oe
By.
tI
Cog G= at tbor_cr
_ rab |
ems Cos A= beter Qe *
a zbe 2
Cos B=
Hol Angle formula 7
3) sin Ah = [ tsb) Gs)
== Besos
5
Hy] sin By eS | (S-a
VY
cle ape te as
sem
Tpeindey - aa & re =
be ac ac sin
a ai es
ii) A C4AB¢) = 1 ab sinc
u
Heroes formule: 3 oe
A (DABC) = \f's (6-1) C5 B)GHC)
ey Anal loyie S 3 ES
fon( B-c\ = (b-c \ cota
[oN es b te os
oa (4) cot 8
L
etd u 4. pair of Straight Jine
joint eqtecetron =
(A+ bry)=O F (a + b>y)=0
joint €an fs
Caretery) (artt bey) =o
Foy slope of (ing S-
© eartel
@® pespunclccecles 2- stope= &
ee a
A eg 1
af point ancl tinedare gryen euhichns IS
dren
[Cy-9n) =m Geer]
Separcite equation of line :- |
vit imys oO
°
eo cand! tt 3g=O
389
cay eo =
ee eceeecltm of IE any
To tine jor equecttrern rine Fhrough oxtgINd Je bing mo
i mo > combine Ce
by ) gee ey 439%
Caine
ma-yso | thim
ho owygin fomel |
tq? of line
Combinect eq” is axttohegt by Zo
| rr" gr sere re slope sum =
| G& Thay procluct4m,.m,—q |
5
ancl ciererey v= (mbtn JE Ctr ©,
, “1 — 4mim
pen ener = 4 m,
Faery .
of che, crpta beter lin oa
Fan @= =
M caserre
atib |
sfope tine only treet eccn aniby to°
Grenenec| eqyeea tty ef paw 4F line meen nos
an» Fahey tpyt pdyrt +afy 4 «2d
th ©, <@ bex
hg
a
eek
g fo
ich ©, be THE cEete
given by euch be ubey Fy rp gato fute= eo
y a
fans > cS?
atb
oe pew percliceclers dain
=O
te unyle beturin pero & lft?
Te tato IMS
atb=o
af tuto yinet tet ane! ewinerelens fhe
ancl fhe point of intersecfron is gmen bY
bh-by gh-af
ab= he ? Gb he
Bee me 9 eg peri of fine cacth Anel cuuh
ido, pa@ h? cry
=P EB) arin tt wis
.
sO
tef ego I NT NI NIN
‘pv cetrO orn,
7, Vides
) Sectton Form ule for _intunaly |
ys mbt na SSS
Mh
oun
Y
i atly Fo ratio 5
i) Secflon (Foe: mule Foy my be
yo me na L
ne A
$$
=p ome vector oot coplang +nen
+ = —_— oy
ira Bal =o]
Scalat Porple product *—~
(go x procuct
> (exe) oO” {a bt) = GJ, A As
a-( ) br by b3
cy Cz C3 pe petits of s.teP 37 -
) y \
G2— b clock ue
pabel= (ecal= peak] olmoys
if) positive
uw
_ Anticlockurse . -
epavcjJ=-[b ae] = -(é 5 aj=-[4* 6]
climays Ny cet Ht
an sectlar triple prectucd
=P any tucd vector vepreedeel Hen
Cau bj =o
=> cabéel= a. (bx2) cCaxb)°s
_
veors are col{fnecy éhen fined c= 4,
a= Pre ei 4
bo at tbujt Gk
af
oe
ome eee
olunes ot fetrechedeon => TAs Ac AD)
z |
ot poreutstopipic ot ca & 6]
yolcrt
colfiness ev coplaney inn
ap tutto Vetos Or
fab é] =o ov Re Points 4B,c |
PAB HEAT] =0 | ieee _
5D - Geomeriby
Cdcs) direction cosines = cosy , cosB cos TH Lim,
n
Cdys) qirution reuttos = 4, 62¢
© Maton ratio fe gin ane Ane
direfion cosins ” Use
irepion cost”
2 4, mons _ a
: 6 ie
Vatipye Vaypie Vatpre~
© for Co-orclinate. frnelrey
A Ceig ie) = tes40 -
@ Suately ctr of piven tine Z
oe
consis A btranen
cy tmv) > (z£n¥)
7a?
anima |
@ pus _betesen dove [ine
coi © = [LZ,Lrt TUM
© ef yutco Lind are flee thee
a- Ba
Qi Be ee
@ Th dee find ane perpenclvculay then
at Qrtbtthrtata= Shortes! test befeereer
fines
1) Vedor eeperafion
Fo G4 Obi
Y5 Get Ube
d = | hy xe). (a.-%')
( & xbo|
a) Cartesceen eq cceehro?
AUK a ye 2 Ett
Ao a
“m1
Xr 2 y-th 22,
ne
4) finel Vector eqn from
Cartsean ect?
Mer le ,Octty , mee
dinchon yatiors Asm
Ven ear
yr atnKb
PAny lo bute [ines
be) find «Hf fine avel™
Fo ecech othe
Ss
c) Ante peturen plane
ay Angle (se
)
x vector ec)
ne ante betucer
plane
ns giver
oo)
Ar oe
d= Kitt
At me
Ac me
Cami — ey nye t
Chim — hans)
3) For infesting Cyd
Cartaven eM 1s ve
finel Kit Hines evn tntesetny
Mets Ya-Yy Vane
ai mm,
ae me
By. 7 Uy 2%
coo = bebe
ler{B.l
& ContOo1con ee aie)
tener fee to tach ore.
find k
ad. fhroxt artes O|
Ryze,
6)
1 et
yD, Pw
tan of pla
nw
RyRy
Tri ttral
=0 d) My yeu
. Cavdess1cn egtUe> geen .
. ; D Pistene bel? HL Hines
plane eqn Se EE
y
seep} Comp oeiny anel put valu rm sae Buna
fomula
piace, 0. A= [ctar-cu) vB
. -| ag,+ boi tor |
SIND = I ; —TH El
Via pyre Nantes tc)
2) Ab YR) BOn Ie 2)
ee ee
"ial Aivechron ret o+
oY fo [Ines ied 24
of iat Boon ery Pe
S : 3) Velo eqyucctiono Plan,
as custhy Inve h
b P ? 4
Act tort) B Ct ot ot),
clr
oO ee
J. CAB XAC)S & (ABA)
ABYAC
B
c
) collineos pein’
ACt yi zr)
Blais ye zs)
6 Cmts Y325)
Aine A 34 4 B4em
Collmmcor
eee C ——
VG = 0P4 >
\ siu)evnreq_\
oe y
5 E Cx) Se
Vix) ay
ie 7)
+) Keto, 6002 5
Pa?
3) pro 6 E€u=é
VO) = 2
2 = 20
= ; NTC
Vix) = >
Sd) ZY)
Steud ang
oo
ni P=?
cl new, P= 6-4
EOQ S72 Ue ear7 fe eQuine ot Boole
Gi Ha Fa =
AB = [aubn sank,
at Bit Fy, bs F4,yb31 at Bt daw Bagh Gag fete. aan
G53 eat
| Fens gnvas- of cofetdev matix ge m
Colma, .. nek Th thats ay
: Se eae E
= (an A )
yw ay
ies
An Ar 5
lagu ay Gay Fi
gr | 43 | aoa
S41 (las %93)\- (a
a 3) — (r3* a o bogerins
= igen’ - = an ae =
“ge ee loy B= lojffaey
2) WG A — 1h p= te CA/B)_
Tt) ley 0 = Not dehine —
4) Wey (ye) = vot define
7 9) é eA ee ee
a __te) ‘os e=i- -
4 WW ys (A) = = =! (%)
7 19 Ba fa Ba
ne = = aan o
kg:
2-303 lege (omphye Number
Q -=-Real zi the comple ee
bio= image nevy
2) wr fort] Zo wie 2 - a
: W4eu =~ 1
uUt+ y -S-utr>
aie fy = HU
) af oc and am the compl Llihe
we of undy flacn : At=P, prs oC
3 Bs
3 & x
7
DoD
yl
] [_imits Pe
ape fance
Ao et Cexknds toa)
f [put value athe |
limit _funelion £= Seamer Conca cricrem
(ct hand fiof CLH-t) | -
I _
= xm a
oo Ry ht hand jimit (ReHet) -
+
fee pel as 1g) theiseune@
xq Subs multi cien
[tim Foo Hy hin q Oe \ as _
Vere —a Cx a J —
Formulae claserate,
C)
(O
pose
&
F =
4) eg Ween ce and Series.
formulas te
Dine a ¢(n-i)d in =n [rattn-~i)d]
DQ tn = Sn-sn- |
dy = aynol Sala) y>1
4) dn = yn : sn =a(t-¥) +Z |
na Lifer
(| Fe gp
Cz
Shi = a (ley) +k]
= Niza
at sequne ft dt% 5 ta... tn Set fobe
aqip if “dnt = leenrfanr)
g° +n .
— @& sum of _/ntinite ¢ 3)
SoO = q 1
a a dor classiate
Date J
pone
ipygeesion (Hep) ~
A sequence! “GF fy »trity ... FS Collen >
a 4 7
- AP = wot
ifoi4 oo fee aa for
ti Rta) Mai tn
- an Hep \
—
A Avthmatre Mam CA) :
| 9 ALY Or thie numb IN A®
A= x+y |
| ae
a
® Gomiuotic meen Gie ) “vadjoncll numbw Cin
oO age ba gs)
_to- oor, +o. 000033 ie
s Greomenbr< Puprssion JP) g} Aetna Gteomen hic Open
[dy = Cag = “Dd y= i}
Se
_ ; sno g 4+ dy G=v"-T) Zn }
— Jur Cl=xye ]
: Coeein-De fx? |
I~ i
xx t
VF tee one i EP “Thad oom oa
a Qs Cw
=
a 4 » ay ar?
x3 Y c
q q a *) AY gy >
as ui
bre ore three hymi ig F the
bra ac, ¢ ~ Campus Education
3.Co-ordinate Geometry
(i) If Bis the inclination of a line where @ + 5, then slope of the line is tan 8.
Itis denoted by ‘m’.Thus, slope = m = tan@.
(ii) HfAGx, y2) and B(x2, y2) are any two points on a line,
then slope of line AB = 24—¥2 (x1 # x2)
Xi Xz
1) Slopes of parallel lines are equal.
If slopes of two lines are equal then the lines are parallel
2) Slope of X-axis or a line parallel to X— axis is zero.
If the slope of a line is zero then the line is either X- axis or Parallel to X — axis
3) The slope of y — axis or a line parallel to y-axis does not exist.
If the slope of a line does not exist then the line is either Y- axis or parallel to y — axis
) Equation of a line :
(i) Slope —Point form :
The equation of a line having slope m and passing through the point (x: , y2)
isy—yi = m(x=x:)
(ii) Slope — intercepts form :
The equation of the line with slope and y-intercepts cis y = mx +c
(iii) Two point form :
The equation of a line passing through the points (x1, y2) and (x2, y2) is
YoY LYrY2 gy YM LM
X-X. 0 Xa Xz Yi-Y2 Xi Xz
(iv) Double intercepts form :
The equation of a line making intercepts a on X — axis and b on y — axis
!) Slope of a line :
Xe¥e
sot 1
If P(x1, Y2) and Q(x2 , y2) are two points, then co-ordinates of mid-point of segment
(@ +%2 Wit 2)
of PQare 2
-19- By Mungle Sir Then,
Volume
6. Mensuration
(1) Length of the arc = 575 X 2nr.
(2) Area of the sector = = Xx mr.
(3) Area of the minor segment. = Area of the sector — Area of triangle.
(4) Euler's formula : F+V = E + 2, where V: no. of vertices, F : no. of faces,
E: no. of edges.
Surface areas and volume of solid figure :
1) Cuboid :
1) Total surface area of cubiod = 2(Ib + bh + Ih) fq
2) Vertical surface area of the cubiod = 2(I+b) h
3) Diagonal of of cubiod avi? + b? + h?
4) Volume of cubiod =IxXbxh.
2) Cube
1) Plane surface area =6l*
2) Total surface area of cube = 6l iC
3) Diagonal of of cube = v3l ———
4) Volume of cube =P
3) Cylinder
1) Plane surface area = 2nr’
2) Curved surface area of cylinder = 2nrh
3) Total surface area of cylinder = 2nr(h+r)
4) Volume of cylinder =arh
Note :-All the right primes have top and base congruent and
parallel to each other.
Then lateral faces are perpendicular to the base.
-20- = By Mur 1
2)
3)
4)
5)
1
2
3
4)
1)
2
Then,
Volume of right prism =
Plane surface area
Slant Height (I)
Curved surface area of cone
Total surface area of cone
Volume of cone
Slant Height (I)
Curved surface area
Total surface area
Volume
Total surface area of sphere
Volume of sphere
Curved surface area
Total surface area
Volume of hemisphere
Area of base x height
= tr (l+r)
"
dy
Sm?h
3™
5) Frustum of Cone
WPF GF
= 1(r +r)
= (ry + ra) + wr2+ mr?
= Snr + P+ ran)h
6) Sphere
= 4nr2
= tne
7) Hemisphere
= 2nr*
)
A
A
Ss
7
- Campus Education
= By Mungie Sit | - Campus Ec
Problem set -1
Find ts from the following A.P. 4,9,14, ..
Sol. Herea = 4, =5
= at(n- 1d
= 44 (0-15
=445n-5
t= Sn—1
t= 5(1) —
t= 55 —
t= 54
2. Find first negative term from following
Sol. Here a = 122,
n= at (n= 1d
ta= 122 + (n-1)(-6)
122-6n+6
t= 128-6n
Now t,< 0,
128-6n<0
122, 116, 110, .... (Note: Find the smallest n such that t,<0)
‘Smallest n = 22
The first negative term of the A.P. is 22.
3. Find the sum of first 11 positive numbers which are multiples of 6.
Sol. The positive multiple of 6 are, 6 , 12, 18, 24, 30, 36.
Which is an A.P. with a = 6,
2 $= (2a + (n-1)d)
-: Sum of 1* 11 Positive multiple of 6 ere
Su= Z[266)+ 1-0)
2 [2 +60] Le
sBxn
= 11 (98)
Su = 306
The sum of the first 11 positive numbers which are multiple of 6 fe 396.
4. Inthe AP. 7, 14, 21 How many terms are we have to consider for
(eting sum 5740, ~~
Sol. Hore a=7,d=7 Sy = 5740
We have to find n, which gives. Sn = 5740
Now, Sn= =[2a+ (n—1)d]
8740 = 3[2(7) | (n- 197)
11480 = n (14 + 7n-7)
\
-ByMu ppebraic OR Expansion Formulae
(at) easzabi and (a-b}'+ab-
2(abj'atdab+b? and (atb/-4ab
3.a°4b°= (atb)'-2ab; 4. a' +a" = (a-b)'+2ab;
5.a°b" = (atb) (aby 6. teria eal
Tabs (et ao (eth =4ob
9. heb) 10 b+ (eb= 48)
MH.(a+b/=a'+ adh Sb'+b'= b+ Sab a+)
{2.(a-bf = a-3db+3at/-b'=2-b Jb (2-b)
‘RULES FOR FINDING MISSING TERMS
FOR PERFECT QUADRATIC EXPRESSION
4, (a) First term {S928 (Third term leet
(0) Second term =2 xaos x (am
2 LeasConmen Mail (LCM) and ght Conon Factor WCF)
putt nan iso po er La
Fintona scolar =LCX et erunbes HF of cnn,
Lethe
CF #Deeneaes
SHOE ad L.CM of polpomis.
WCE eter oon sy Caos is et
AEA eet cman tt
on rent
HEE
LCM RCE Pdeet hemi
Lem rn:
ues acre
4. (aby’ = 2°:
ae eagle = tay"
DIVISION ALGORITHM.
1. Dividend =
Divisor x Quotient + Remainder
Dividend - Remainder
Quotient
Dividend - Remainder
Divisor
2. Divisor =
3. Quotient =
4, Remainder = Dividend -
(Divisor x Quotient)
QUADRATIC EQUATION
1. Qundatc equations define a th equation ate =o}
‘Roots ofthe equation a'Yoxt = ae gan by
b. Vbrdae
2a
roots ofthe equation ax'toxte = 0
_ Govaticlnt ok
Coticant of
Constant wr
— Costlent of
“4ittheootsce_andB._ are known then the equation ie
given by se. (a+p) x+ (EB) =0
5. Nature ofthe roots :fora given quaatic equation
ax'sbxre = 0,D= bifac is known a discriminate
ofthe equation
Nature
(yifD>o
(@ifD=0
@iro => then a, bic, and d are said to
be in proportion and writter'as a:
d where 'a and d' are called the
extremes while 'b and c’ are called
the mean of the proportion.
Conditions :-
1. (i) fad = be the
(ii) If ad > be then
(invertendo)
(alternendo)
7 ss (componendo)
od (dividendo) —
(e) ab = ord. (compenendo & dividendo)
oe + k a=bk, c =dk (k method)
(g) for continued proportion $=
(h) Theorem of equal ration
Wh= $=
then each ratio
_ la +me+n
atso- each ratio = {2 +MGEne*
“(i) Consecutive numbers are x, x+1, x+2,.
() Consecutive odd number are x, x+2, X+4,,
(k) Consecutive even number are x+1, x+3, x+5,-. Pythagoras Theorem —
| Perimeter [Area | Nay reelictary
are congruent.
v) Opposite sides of a
rectangle are parallel
i) Allsides of a square
are congruent
ii) Each angle of a square
is a right angle
ill) The diagonals of a
square bisect each
other_at right angle
iv) ‘The diagonal of a
rectangle are congruent.
v)_ The opposite sides of
a square are parallel 1. Speed
time taken
Distance covered = speed x tine taken
18 km per hour = § meters por second
ARITHMETIC PROGRESSION
Sequence =
a,(a+d),(a+2d),
Where ais the frst term and dis the common
Aiference ofthe sequence ofan AP. then
1. the pl tenn of an AP.
=at(n-l)d
where m = last term = 1, 2, 3.
2. The sum of the first n terms of an A.P let the
sequence = 0+ ad a+2d-boudr4(-I) dy
“Sy 5 Ra+(m-Na] is the sum of|
the sequence,
[STATISTICAL FORMULAE
1. Combined M
=x tks tly fix
: f +f +f, fi
2. Median -
(0) For grouped data M = 1+ + h
were {= lower limit of the class in which
‘median lies
f= frequency of the class
(C= Cumulative frequency
n= Width ofthe class - interval
N
al frequency
(b) Mode : For continuous series
where = lower limit of the modal-class
f= feequeney ofthe modal class
F, = freqweney ofthe premodal class
1, frequency of the post «mod class
1h sizeof the elas interval
TRICK OF TRIGONOMETRIC RATIO]
wy Xx
(cab, sho)
A. Trigonometrieal ratios and identities
Perpendicular
Hypotenuse
Hypotenuse
1, Sind = asco =
[sino x cosec 6 = 1]
Hypotenuse
See 9 ~ Hypotenuse
Base
cos Oxsee 0 =1
3. Tang = Perpendl
Base
Cora = —Base__
Perpendicular
Tan 0x Co
(To team above =
and Bade Prasad
Har” Har” Bole
tano = 28 andcora Se?
en ox Oe in
5. sia 8608 8-1 sin #=1-v08
sin = Vi-cos'd
6. see! @tan' B= se
an
7. com’ -co¥ O= cof 8
B. Relation between trigonomentrical ratios
and complementary angles.
1. sin@ = cos (90-8)
=sin (90-8)
3. tan @ =cot (90-8)
+. cot@ = tan (90-0)
3. sec = cosec (90-8)
6. cosec6 = sec (90-8)
>. tan xtan (90-6)
s.cot@xcot(90-0)=1
2. cos|
C. Signs of Trigonometric ratios in
different quadrants.
90" <4 <180°
ul
Only Sin, Cosec (+)ve
180° <6< 270°
WU
Only tan, corre
0