12 Maths Good Notes

8/18/2019 12 Maths Good Notes

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REVIEW TEAM : 2014-15

Sl. No. Name Designation

Dr. Vandita Kalra SKV, Moti Nagar

Vice Principal / HOS

1. Sh. Jogindr Arora SBV Subhash Nagar, Delhi

(PGT Maths)

2. M!. Ra"ni Arora SKV Ramesh Nagar, Delhi

(PGT Maths)

3. Sh. A!ho# K$%ar &$'ta GBSSS, SU Block, itam ura, Delhi

(PGT Maths)


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()ASS *II +2014 - 2015,

MATEMATI(S

nit! /o. o Wightag

riod! +Mar#!,

!i" Relatio#s a#$ %u#ctio#s 3& 1&

!ii" 'lgebra (& 13

!iii" )alculus *& ++

!i" Vector a#$ -hree Dime#sio#al Geometr 3& 1/

!" 0i#ear rogrammi#g 2& 1

!i" robabilit 3&

Total : 240 100

nit I : RE)ATI/S A/D 3/(TI/S

1. Rlation! and 3$ntion! +15 riod!,

T'! o Rlation! : Releie, smmetric, tra#sitie a#$ e4uiale#ce

relatio#s. %u#ctio#s. 5#e to o#e a#$ o#to u#ctio#s, com6osite u#ctio#s,

i#erse o a u#ctio#. Bi#ar o6eratio#s.

2. In6r! Trigono%tri 3$ntion! +15riod!,

Dei#itio#, ra#ge, $omai#, 6ri#ci6al alue bra#ches. Gra6hs o i#erse

trigo#ometric u#ctio#s. 7leme#tar 6ro6erties o i#erse trigo#ometric

u#ctio#s.

nit II : A)&E7RA

1. Matri! +25 riod!,

)o#ce6t, #otatio#, or$er, e4ualit, t6es o matrices, 8ero a#$ i$e#tit

8(la!! *II : Math!9 829


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matri, tra#s6ose o a matri, smmetric a#$ ske9 smmetric matrices.

'$$itio#, multi6licatio# a#$ scalar multi6licatio# o matrices, sim6le6ro6erties o a$$itio#, multi6licatio# a#$ scalar multi6licatio#. No#:

commutatiit o multi6licatio# o matrices a#$ eiste#ce o #o#:8ero

matrices 9hose 6ro$uct is the 8ero matri !restrict to s4uare matrices o

or$er 2". )o#ce6t o eleme#tar ro9 a#$ colum# o6eratio#s. ;#ertible

matrices a#$ 6roo o the u#i4ue#ess o i#erse, i it eists< !=ere all

matrices 9ill hae real e#tries".

2. Dtr%inant! +25 riod!,

Determi#a#t o a s4uare matri !u6 to 3 > 3 matrices", 6ro6erties o

$etermi#a#ts, mi#ors, coactors a#$ a66licatio#s o $etermi#a#ts i# i#$i#g

the area o a tria#gle. a$?oi#t a#$ i#erse o a s4uare matri. )o#siste#c,

i#co#siste#c a#$ #umber o solutio#s o sstem o li#ear e4uatio#s b

eam6les, soli#g sstem o li#ear e4uatio#s i# t9o or three ariables

!hai#g u#i4ue solutio#" usi#g i#erse o a matri.

nit III : (A)()S

1. (ontin$it and Dirntiailit +20 riod!,

)o#ti#uit a#$ $iere#tiabilit, $eriatie o com6osite u#ctio#s, chai# rule,

$eriaties o i#erse trigo#ometric u#ctio#s, $eriatie o im6licit u#ctio#.

)o#ce6t o e6o#e#tial a#$ logarithmic u#ctio#s a#$ their $eriaties.0ogarithmic $iere#tiatio#. Deriatie o u#ctio#s e6resse$ i# 6arametric

orms. Seco#$ or$er $eriaties. Rolle@s a#$ 0agra#ge@s mea# Value

-heorems !9ithout 6roo" a#$ their geometric i#ter6retatio#s.

2. A''liation! o Dri6ati6! +10 riod!,

A''liation! o Dri6ati6! : Rate o cha#ge o bo$ies, i#creasi#gA

$ecreasi#g u#ctio#s, ta#ge#ts a#$ #ormals, use o $eriaties i#

a66roimatio#, maima a#$ mi#ima !irst $eriatie test motiate$

geometricall a#$ seco#$ $eriatie test gie# as a 6roable tool". Sam6le

6roblems !that illustrate basic 6ri#ci6les a#$ u#$ersta#$i#g o the sub?ect

as 9ell as real:lie situatio#s".

;. Intgral! +20 riod!,

;#tegratio# as i#erse 6rocess o $iere#tiatio#. ;#tegratio# o a ariet o

u#ctio#s b substitutio#, b 6artial ractio#s a#$ b 6arts, o#l sim6le

i#tegrals o the t6e to be ealuate$.

8;9 8(la!! *II : Math!9


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∫ dx

, ∫ dx , ∫ dx , ∫ dx ,

∫ dx

x

2

± a2 ax 2+ bx + c a

2 − x 2 ax 2 + bx + c

x2 ± a 2

( px + q ) ( px + q )dx ,dx , a

2 ± x 2 dx , x 2 − a 2 dx ,

∫ ax 2 + bx + c ∫ ∫ ∫

ax2 + bx + c

∫ ∫ ( px + q )ax 2 + bx + c dx a#$ ax 2 + bx + c dx Dei#ite i#tegrals as a limit o a sum, %u#$ame#tal -heorem o )alculus

!9ithout 6roo". Basic 6ro6erties o $ei#ite i#tegrals a#$ ealuatio# o

$ei#ite i#tegrals.

4. A''liation! o th Intgral! +15 riod!,

'66licatio#s i# i#$i#g the area u#$er sim6le cures, es6eciall li#es, area

o circlesA6arabolasAelli6ses !i# sta#$ar$ orm o#l", area bet9ee# a# o

the t9o aboe sai$ cures !the regio# shoul$ be clearl i$e#tiiable".

5. Dirntial E


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2. Thr-Di%n!ional &o%tr +15 riod!,

Directio# cosi#es a#$ $irectio# ratios o a li#e ?oi#i#g t9o 6oi#ts. )artesia#

a#$ ector e4uatio# o a li#e, co6la#ar a#$ ske9 li#es, shortest $ista#ce

bet9ee# t9o li#es. )artesia# a#$ ector e4uatio# o a 6la#e. '#gle

bet9ee# !i " t9o li#es, !ii " t9o 6la#es, !iii " a li#e a#$ a 6la#e. Dista#ce o a

6oi#t rom a 6la#e.

nit V : )I/EAR R&RAMMI/&

+20 riod!,

1. )inar rogra%%ing : ;#tro$uctio#, relate$ termi#olog such as

co#strai#ts, ob?ectie u#ctio#, o6timi8atio#. Diere#t t6es o li#ear 6rogrammi#g

!0.." 6roblems, mathematical ormulatio# o 0.. 6roblems, gra6hical metho$ o

solutio# or 6roblems i# t9o ariables, easible a#$ i#easible regio#s, easible a#$

i#easible solutio#s, o6timal easible solutio#s !u6 to three #o#:triial co#strai#ts".

nit VI : R7A7I)IT=

1. roailit +;0 riod!,

Multi6licatio# theorem o# 6robabilit. )o#$itio#al 6robabilit, i#$e6e#$e#t

ee#ts, total 6robabilit, Bae@s theorem, Ra#$om ariable a#$ its

6robabilit $istributio#, mea# a#$ aria#ce o a ra#$om ariable.

Re6eate$ i#$e6e#$e#t !Ber#oulli" trials a#$ Bi#omial $istributio#.

Marks per Question Total Number of Questions in

2013-14 2014-1

1 1&

+ 12 13

/ /

Total 2> 2?

859 8(la!! *II : Math!9


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MATEMATI(S +(DE /. 041,

Ti% : ; ho$r! Ma@. Mar#! 100

Sl.No. Typology of Questions Learning Outcomes Very Long Long Marks %

& Testing Short Anser Anser !eigh"

#ompetencies Anser $ $$ tage

M' ( M' ) M'

1. R%%ring C !K#o9le$ge base$

Sim6le recall 4uestio#s, to k#o9

s6eciic acts, terms, co#ce6ts,

6ri#ci6les, or theories< ;$e#ti, $ei#e,

or recite, i#ormatio#"

Reaso#i#g 2 3 1 2& 2&E

'#altical Skills

)ritical thi#ki#g

Deriatie

2. ndr!tanding C !)om6rehe#sio#: 2 2 1 1 1E

to be amiliar 9ith mea#i#g a#$ to

u#$ersta#$ co#ce6tuall, i#ter6ret,

com6are, co#trast, e6lai#,

6ara6hrase, i#ormatio#"

;. A''liation !Use abstract 1 3 2 2( 2(E

i#ormatio# i# co#crete situatio#, toa66l k#o9le$ge to #e9 situatio#s<

Use gie# co#te#t to i#ter6ret a

situatio#, 6roi$e a# eam6le, or

sole a 6roblem"

4. igh rdr Thin#ing S#ill! C 1 2 2 21 21E

!'#alsis F S#thesis:classi,

com6are, co#trast, or $iere#tiate

bet9ee# $iere#t 6ieces o

i#ormatio#< 5rga#ise a#$Aor

i#tegrate u#i4ue 6ieces o

i#ormatio# rom a ariet o

sources"

5. E6al$ation and M$lti-Di!i'linar

C !'66raise, ?u$ge, a#$Aor ?usti the

alue or 9orth o a $ecisio# or

outcome, or to 6re$ict outcomes

base$ o# alues"

21 1 1* 1*E !alues

base$"

?B1C 1;B4C B?C

TTA) ? 52 42 100 100

8(la!! *II : Math!9 8?9


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(/TE/TS

S.No. Chapter Page

1. Relatio#s a#$ %u#ctio#s H

2. ;#erse -rigo#ometric %u#ctio#s 1/

3 F +. Matrices a#$ Determi#a#ts 23

(. )o#ti#uit a#$ Diere#tiatio# 3H

. '66licatio#s o Deriaties +/

/. ;#tegrals 1

*. '66licatio#s o ;#tegrals *(

H. Diere#tial 74uatio#s H&

1&. Vectors 1&1

11. -hree:Dime#sio#al Geometr 111

12. 0i#ear rogrammi#g 122

13. robabilit 12/

Model !apers 1+1

89 8(la!! *II : Math!9


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5#to u#ctio# !sur?ectie" ' u#ctio# f # → $ is sai$ to be o#to i " f I $i)e) ∀ b ∈ $, there eists a ∈ # such that f !a" I b

' u#ctio# 9hich is #ot o#e:o#e is calle$ ma#:o#e u#ctio#.

' u#ctio# 9hich is #ot o#to is calle$ i#to u#ctio#.

7i"ti6 3$ntion : ' u#ctio# 9hich is both i#?ectie a#$ sur?ectie is

calle$ bi?ectie u#ctio#.

(o%'o!ition o TFo 3$ntion! : ; f # → $, * $ → + are t9ou#ctio#s, the# com6ositio# o f a#$ * $e#ote$ b * of is a u#ctio# rom #

to + gie# b, !* of " ! x " I * !f ! x "" ∀ x ∈ #

)learl * of is $ei#e$ i Ra#ge o f ⊂ $omai# o *) Similarl fo* ca# be$ei#e$.

In6rtil 3$ntion : ' u#ctio# f , → is i#ertible i it is bi?ectie.

; f , → is bi?ectie u#ctio#, the# u#ctio# * → , is sai$ to be

i#erse o f i fo* I . y a#$ *of I . x

9he# . x % . y are i$e#tit u#ctio#s.

g is i#erse o f a#$ is $e#ote$ b f C1

.

7inar 'ration : ' bi#ar o6eratio# LG@ $ei#e$ o# set # is a u#ctio#

rom # & # → #) G !a% b" is $e#ote$ b a G b)

Bi#ar o6eratio# G $ei#e$ o# set # is sai$ to be commutatie i

a G b ' b G a ∀ a% b ∈ #)

Bi#ar o6eratio# G $ei#e$ o# set # is calle$ associatie i a G !b G c " I !a

G b" G c ∀ a% b% c ∈ #

; G is Bi#ar o6eratio# o# #% the# a# eleme#t e ∈ # is sai$ to be the

i$e#tit eleme#t i a G e I e G a

∀ a ∈ #

;$e#tit eleme#t is u#i4ue.

; G is Bi#ar o6eratio# o# set #, the# a# eleme#t b is sai$ to be i#erse o

a ∈ # i a G b ' b G a ' e

;#erse o a# eleme#t, i it eists, is u#i4ue.

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1. ; ' is the set o stu$e#ts o a school the# 9rite, 9hich o ollo9i#g relatio#s

are U#iersal, 7m6t or #either o the t9o.

" 1 I J!a% b" a% b are ages o stu$e#ts a#$ a / b ≥ &

" 2 I J!a% b" a% b are 9eights o stu$e#ts, a#$ a / b &

" 3 I J!a% b" a% b are stu$e#ts stu$i#g i# same class

2. ;s the relatio# " i# the set # I J1, 2, 3, +, ( $ei#e$ as

" I J!a% b" b I a 1 releieO

3. ; " , is a relatio# i# set N gie# b

" ' J!a, b a ' b / 3, b P (,

the# $oes eleme#t !(, /" ∈ RO

+. ; f J1, 3 → J1, 2, ( a#$ * J1, 2, ( → J1, 2, 3, + be gie# b f I J!1,

2", !3, (", * I J!1, 3", !2, 3", !(, 1",

9rite go.

(. 0et *% f " → " be $ei#e$ b

( )=

x + 2 ( )9rite og x

( )* x 3 ,f x = 3 x − 2.

. ; f ( " → " $ei#e$ b

f ( x ) = 2 x − 1(

be a# i#ertible u#ctio#, 9rite f C1

! x ".

x ∀ x ≠ −1, 9rite fo f ( x )./. ; f ( x ) = x + 1

*. 0et be a Bi#ar o6eratio# $ei#e$ o# " , the# i

!i" a b I a b ab, 9rite 3 2

8119 8(la!! *II : Math!9


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( ) 2a + b ( )!ii" a b = +., 9ri te2 3

3

H. ; n! #" I n!$" I 3, the# ho9 ma# bi?ectie u#ctio#s rom # to $ ca# be

orme$O

1&. ; f ! x " I x 1, * ! x " I x C 1, the# !go" !3" I O

11. ;s f N → N gie# b f ! x " I x 2 o#e:o#eO Gie reaso#.

12. ; f ( " → #% gie# b

f ! x " I x 2 C 2 x 2 is o#to u#ctio#, i#$ set #.

13. ; f # → $ is bi?ectie u#ctio# such that n ! #" I 1&, the# n !$" I O

1+. ; f ( " → " $ei#e$ b f ( x ) = x

−

1

, i#$ fof x

2

1(. " I J!a% b" a% b ∈ N , a ≠ b a#$ a $ii$es b. ;s " releieO Gie reaso#

1. ;s f " → "% gie# b f ! x " I x C 1 o#e:o#eO Gie reaso#

1/. f " → $ gie# b f ! x " I si# x is o#to u#ctio#, the# 9rite set $.

( ) 1+ 2 ( )1*. ; = log , sho9 that = 2 .

1−

1+ 2

1H. ; LG@ is a bi#ar o6eratio# o# set Q o ratio#al #umbers gie# b

the# 9rite the i$e#tit eleme#t i# Q)

aba b =

(

2&. ; G is Bi#ar o6eratio# o# N $ei#e$ b a G b I a ab ∀ a, b ∈ N% 9rite thei$e#tit eleme#t i# N i it eists.

21. )heck the ollo9i#g u#ctio#s or o#e:o#e a#$ o#to.

!a" f " → " , f ! x " = 2 x − 3

/

!b" f " → " , f ! x " I x 1

!c" f " / J2 → "% f ( x ) =

3 x −1

x − 2

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!$" f ( " → QC1, 1, f ! x " I si#2 x

22. )o#si$er the bi#ar o6eratio# o# the set J1, 2, 3, +, ( $ei#e$ b a b I

=.).%. o a a#$ b. rite the o6eratio# table or the o6eratio# G.

23. 0et f B " − {−+ }→ " −{+} be a u#ctio# gie# b f ( x ) = + x . Sho93 3 + x 3 x + +

that f is i#ertible 9ith f −1 ( x ) = .+ − 3 x

2+. 0et " be the relatio# o# set # I J x x ∈ % & ≤ x ≤ 1& gie# b" I J!a% b" !a C b" is $iisible b +. Sho9 that R is a# e4uiale#ce

relatio#. 'lso, 9rite all eleme#ts relate$ to +.

2(. Sho9 that u#ctio#f # → $ $ei#e$ as f ( x ) =3 x + +

9here( x − /

/ 3 is i#ertible a#$ he#ce i#$ f

C1. # = " − , $ = " −

( ( 2. 0et G be a bi#ar o6eratio# o# Q such that a G b I a b C ab.

!i" roe that G is commutatie a#$ associatie.

!ii" %i#$ i$e#ti eleme#t o G i# T !i it eists".

2/. ; is a bi#ar o6eratio# $ei#e$ o# " C J& $ei#e$ b a b = 2a , the#b

2

check G or commutatiit a#$ associatiit.

2*. ; # I N > N a#$ bi#ar o6eratio# G is $ei#e$ o# # as

!a% b" G !c , d " I !ac% bd ".

!i" )heck G or commutatiit a#$ associatiit.

!ii" %i#$ the i$e#tit eleme#t or G i# # !; it eists".

2H. Sho9 that the relatio# " $ei#e$ b !a% b" " !c% d " ⇔ a d I b c o# theset N > N is a# e4uiale#ce relatio#.

3&. 0et be a bi#ar o6eratio# o# set T $ei#e$ b a b = ab

, so5 tat

+

!i" + is the i$e#tit eleme#t i# T.

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!ii" 7er #o# 8ero eleme#t o T is i#ertible 9ith

−1

=1

∈ Q −{ }

a , a & .a

31. Sho9 that f " → " $ei#e$ b f ( x ) =1

is bi?ectie 9here " is the2 x

set o all #o#:8ero 6ositie real #umbers.

32. 0et # I J1, 2, 3, ...., 12 a#$ R be a relatio# i# ' > ' $ei#e$ b !a, b" R

!c, $" i a$ I bc ∀ !a, b", !c, $", ∈ ' > '. roe that R is a# e4uiale#cerelatio#. 'lso obtai# the e4uiale#ce class Q!3, +".

33. ; LG@ is a bi#ar o6eratio# o# " $ei#e$ b a 6 b ' a b ab) roe that G iscommutatie a#$ associatie. %i#$ the i$e#ti eleme#t. 'lso sho9 that

eer eleme#t o " is i#ertible ece6t C1.

3+. ; f , * " → " $ei#e$ b f ! x " I x 2 C x a#$ * ! x " I x 1 i#$ !og" ! x " a#$!go" ! x ". 're the e4ualO

3(. f Q1, ∞" → Q2, ∞" is gie# b f ( x ) = x + 1 , i#$ f −1

( x ). x

3. f " → "% * " → " gie# b f ! x " I Q x , * ! x " I x the# i#$

− 2 (fo* ) a#$

3

− 2 (*of ) .

3

1. " 1 is u#iersal relatio#. " 2 is em6t relatio#.

" 3 is #either u#iersal #or em6t.

2. No, " is #ot releie.

3. !(, /"

∉ "

+. *of I J!1, 3", !3, 1"

(. !fo* "! x " I x ∀ x ∈ "

8(la!! *II : Math!9 8149


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. f −1 ( x ) = (

x

+

1

2

( )( x ) = x , x ≠ − 1/. fof

22 x + 1*. !i" 3 G 2 I 11

13H!ii" 2/

H.

1&. 3

11. es, f is o#e:o#e ∀ x

1, x 2 ∈N ⇒ x 12 = x 2

2 .

12. # ' Q1, ∞" because " f I Q1, ∞"

13. n!$" I 1&

1+. (fof ) ( x ) = x − 3

+

1(. No, " is #ot releie

(a, a) ∉" ∀ a ∈N

1. f is #ot o#e:o#eu#ctio#

f !3" I f !C1" I 2

3 ≠ C 1 i)e) $isti#ct eleme#tshae same images.

1/. $ ' QC1, 1

1H. e I (

2&. ;$e#tit eleme#t $oes#ot eists.

21.!a" Bi?ectie

!b" Neither o#e:o#e #or o#to.

!c" 5#e:o#e, but#ot o#to.

!$" Neither o#e:


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o#e #or o#to.

8159

8(la!! *II : Math!9


17/208

22.

1 2 3 + (

1 1 1 1 1 1

2 1 2 1 2 1

3 1 1 3 1 1

+ 1 2 1 + 1

( 1 1 1 1 (

2+. 7leme#ts relate$ to + are &, +, *.

2(. f−1

( x ) = / x + +

( x − 3

2. & is the i$e#tit eleme#t.

2/. Neither commutatie #or associatie.

2*. !i" )ommutatie a#$ associatie.

!ii" !1, 1" is i$e#tit i# N & N

32. (3, +)= {(3, +), (, *), (H, 12)}

33. 0 is the i$e#tit eleme#t.

3+. !fo* " ! x " I x 2 x

!*of " ! x " I x 2 C x 1

)learl, the are u#e4ual.

3(. f −1 ( x ) = x

+ x 2 − +

2

− 2 3. (fo* )= & 3

− 2 (*of ) = 1

3

8(la!! *II : Math!9 81?9


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(ATER 2

I/VERSE TRI&/METRI( 3/(TI/S

si# C1

x , cos C1

x , ... etc., are a#gles.

; si#θ = x a#$ θ ∈ −π

,π

the# θ I si# C1 x etc. 2 2

Fnction Domain !ange

(Principal Vale "ranch)

si# C1

x QC1, 1

−

π

,

π

2 2

cos C1

x QC1, 1 Q&, π

ta# C1

x

−

π

,

π

" 2 2

cot C1

x " !&, π"

C1 &, π −

π

sec x " C !C1, 1" [ ]

{2}

C1 π π { }cosec x " / !C1, 1" − , −2 2 &

si#

C1 !si# x " I x ∀ x ∈

−

π,π

2 2

cos C1

!cos x " I x ∀ x ∈ Q&, π etc.

si# !si# C1

x " I x ∀ x ∈ QC1, 1

cos !cos C1

x " I x ∀ ∈ QC1, 1 etc.

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si#−1

x = cosec C1 1

∀ x ∈[

−1, 1]

x

ta# C1

x I cot C1

!1A x " ∀ P & sec C

1 x I cos

C1 !1A x ", ∀ x ≥ 1

si# C1

!C x " I C si# C1

x ∀ x ∈ QC1, 1

ta# C1

!C x " I C ta# C1

x ∀ x ∈ "

cosec C1

!C x " I C cosec C1

x ∀ x ≥

1

cos C1

! /x " I π C cos C1 x ∀ > ∈ QC1,

1 cot C1

! /x " I π C cot C1 x ∀ x ∈ "

sec C1

! /x " I π C sec C1 x ∀ x ≥ 1

si#−1

x + cos −1 x = π , x ∈ [ −1, 1]2

ta# C1

x + cot C1 x =

π ∀ x ∈" 2

sec C1

x + cosec C1 x = π ∀ x ≥ 1

−1 −1

2

−1 x + y + ta# = ta#

−1. 1 + xy

C1 = ta# C1 2 x


20/208

8(la!! *II : Math!9 81H9

1. rite the 6ri#ci6alalue o

!i" si# C1

( C 32 )

!iii" ta# C1 C 1 3

!" cot C1 1

.

3

!ii" cos C1

( 32 ) .

!i" cosec C1

!C 2".

!i" sec C1

!C

2".

−1 − 3

−1 − 1 −1!ii"

(−1 3 )si# + cos + ta#

2 2 2. hat is the alue o the ollo9i#g u#ctio#s

!usi#g 6ri#ci6al alue".

!i" ta# C1 1

C sec

C1

3

!iii" ta# C1

!1" C cot C1

!C1".


21/208

C1 1 C1 3 !ii" si# C C cos .

2 2

!i" cosec C1( ) + sec C1 ( ) .2 2

!" ta# C1

!1" cot C1

!1" si# C1

!1".


22/208


23/208

+. roe that

−1 cos x −1 1 + cos x π∈ ( &, π 2) .ta# − cot =

1 − 1 − + si# x

cos x

ta# C1 x x

a 2 − x 2

(. roe that = si# C1 = cos C1 . a a

a2 C x

2

. roe that

C1 C1 * +

C1 C1 * =

C1 3&&

cot 2 ta# cos ta# 2 ta# si# ta# .

1/ 1/ 11

−1

1 + x 2 + 1 − x 2

π 1 −1 2

/. roe that ta#= + cos x .

1 + x 2

− 1 − x 2

+ 2

*. Sole cot C1

2 x + cot C1 3 x = π .+

H. roe that

−1 m

− ta#−1 m −n = π

, m, n >&

ta# +

n m +n

1 −1 2 x 1 −1 1− y 2

x + y

1&. roe t ha t ta# si# + cos

=

2

2

1+y

2

1+ x 2 1− xy

−1 x 2 −1 − 2 x 2π− 1 1

11. Sole or x , cos + ta# =2

2

x

2

1− x

3+ 1

12. roe that ta# −1

1

+ ta# −1 1

+ ta# −1 1

+ ta#−1 1

= π

3 ( / * +

13. Sole or x , ta#

(cos

− 1

x

)= si#

(ta#

−1 )> &2 < x

1+. roe that 2ta# C1 1 + ta# C1 1 = ta# C1 32


24/208

( + +3

8(la!! *II : Math!9 8209


25/208

1

C1

3

1(. 7aluate ta# cos

2

11

1. roe thatta#

C

1

a cos x − bsi# x

= ta#

C

1

a

− x

b cos x + a

si# x

b

1/. roe that −1 −1 1 −1 2 −1 2

cot ta# x + ta# + cos ( ) + cos ( 2 x ) = π, x > &

1− 2 x − 1

x

1*. roe that ta# C1 a − b + ta# C1 b − c + ta# C1 c − a = & 9here a% b% 1+ ab

1+ bc

1+ ca

c P &

1H. Sole or x , 2 ta# C1

!cos x " I ta# C1

!2 cosec x "

2&. 76ress si# C1

( x 1 − x − x 1− x 2 ) i# sim6lest orm.

21. ; ta# C1

a ta# C1

b ta# C1

c I π,

the# 6roe that a b c I abc

22. ; cos C1

x cos C1

y cos C1

7 I π, 6roe that x 2 y 2 7 2 2 xy7 I 1

Qint : 0et cos C1

x I #, cos C1

y I $, cos C1

7 I c the# # $ + ' π or

# $ ' π C c

-ake cos o# both the si$es.

1. !i" C

π

!ii"

π!iii"

Cπ!i"

Cπ3

!"π

!i"2π

!ii"π

.3 3

2. !i" & !ii"

−π

!iii" −

π !i" π

3 2 2


26/208

!" π !i"π

!ii"

Cπ!iii"

π.( +

8219 8(la!! *II : Math!9

*. 1

13.(

3

1H. x = π .+

21. Hint# 0et ta# C1

a I α

ta# C1

b I β

ta# C1

c I γ

the# gie#, α + β + γ = π

∴ α + β = π − γ

take ta#ge#t o# both si$es,

ta# !α + β" I ta# (π − γ)

π11. ta# = 2 − 3

12

11 − 31(.

3 + 11

2& sin C1

x C si# C1

√ x .


27/208

8(la!! *II : Math!9 8229


28/208

(ATER ; 4

MATRI(ES A/D DETERMI/A/TS

Matri@ : ' matri is a# or$ere$ recta#gular arra o #umbers or

u#ctio#s. -he #umbers or u#ctio#s are calle$ the eleme#ts o the

matri.

rdr o Matri@ : ' matri hai#g Lm@ ro9s a#$ Ln@ colum#s is calle$

the matri o or$er mxn)

S


29/208

Tran!'o! o a Matri@ : ; # I Qai $ m & n be a# m > n matri the# the

matri obtai#e$ b i#tercha#gi#g the ro9s a#$ colum#s o ' is calle$ the

tra#s6ose o the matri. -ra#s6ose o # is $e#ote$ b # or #T )

ro6erties o the tra#s6ose o a matri.

!i" ! #" I # !ii" ! # $" I # $

!iii" !k#" I k#, k is a scalar !i" ! #$" I $'

S%%tri Matri@ : ' s4uare matri #I Qai8 is smmetric i ai8 ' a

8i ∀i% 8 . 'lso a s4uare matri # is smmetric i #9 I #)

S#F S%%tri Matri@ : ' s4uare matri # I Qai8 is ske9:smmetric, i

ai8 ' / a 8i ∀ i% 8) 'lso a s4uare matri # is ske9 : smmetric, i # I C #)

Dtr%inant : -o eer s4uare matri # I Qai8 o or$er n > n, 9e ca#

associate a #umber !real or com6le" calle$ $etermi#a#t o #. ;t is$e#ote$ b $et # or #.

Properties

!i" #$ I # $

!ii" k#n > n I kn #n & n 9here k is a scalar.

'rea o tria#gle 9ith ertices ! x 1, y 1", ! x 2, y 2" a#$ ! x 3, y 3" is gie#

b

1 x 1 y 1 1

∆ = x 2 y 2 12

x 3 y 3 1

x 1 y 1 1

-he 6oi#ts ! x 1, y 1", ! x 2, y 2", ! x 3, y 3" are colli#ear ⇔ x 2 y 2 1 = & x 3 y 3 1

Ad"oint o a S n

ad8 # I Q # 8i n > n

8(la!! *II : Math!9 8249


30/208

Properties

!i" #!ad8 #" I !ad8 #" # I # I

!ii" ; # is a s4uare matri o or$er n the# ad8 # I #n C1

!iii" ad8 ! #$" I !ad8 $" !ad8 #".

QNote # )orrect#ess o ad8 ' ca# be checke$ b usi#g '.

!a$? '" I !a$? '" . ' I ' I

Sing$lar Matri@ : ' s4uare matri is calle$ si#gular i # I &, other9ise it

9ill be calle$ a #o#:si#gular matri.

In6r! o a Matri@ : ' s4uare matri 9hose i#erse eists, is calle$i#ertible matri. ;#erse o o#l a #o#:si#gular matri eists. ;#erse o a

matri # is $e#ote$ b # C1

a#$ is gie# b

#−1

= 1

. ad8 #

#

Properties

!i" ## C1

I # C1

# I .

!ii" ! # C1

" C1

I #

!iii" ! #$" C1

I $ C1

# /1

!i" ! #T " C1

I ! # C1

"-

!" #−1 = #

1 , # ≠ &

Sol$tion o !!t% o


31/208


32/208

11. %i#$ the alue o a + ib c + id

−c + id a − ib

12. ;2 x

+

( 3

= &, i#$ x .

( x + 2 H

13. %or 9hat alue o k , the matrik 2 3 + has #o i#erse.

1+. si# 3&° cos 3&° , 9hat is #.

; # =

− si# &° cos &°2 −3 (

1(. %i#$ the coactor o a12 i# & + .

1 ( −/

1 3 −2

1. %i#$ the mi#or o a23 i# + −( .3 ( 2

1/. %i#$ the alue o ! , such that the matri

−1 2

is si#gular . + !

1*. %i#$ the alue o x such that the 6oi#ts !&, 2", !1, x " a#$ !3, 1" arecolli#ear.

1H. 'rea o a tria#gle 9ith ertices !k , &", !1, 1" a#$ !&, 3" is ( u#it. %i#$ the

alue !s" o k .

2&. ; # is a s4uare matri o or$er 3 a#$ # I C 2, i#$ the alue o C3 #.

21. ; # I 2$ 9here # a#$ $ are s4uare matrices o or$er 3 > 3 a#$ $ I (,

9hat is #O

22. hat is the #umber o all 6ossible matrices o or$er 2 > 3 9ith each e#tr

&, 1 or 2.

23. %i#$ the area o the tria#gle 9ith ertices !&, &", !, &" a#$ !+, 3".

2+. ; 2 x + = −3 , i#$ x .−1 x 2 1

829 8(la!! *II : Math!9


33/208

x + y y + 7 7 + x 2(. ; # =

7 x y

, 9rite the alue o $et #)

1 1 1

2. rite the alue o the ollo9i#g $etermi#a#t

2 3 +

( *

x H x 12 x

2/. ; # is a #o#:si#gular matri o or$er 3 a#$ # I C 3 i#$ ad8 #.

2*. ; ( −3

i#$ (ad8 #) # = *

2H. Gie# a s4uare matri # o or$er 3 > 3 such that # I 12 i#$ the alue o

# ad8 #.

3&. ; # is a s4uare matri o or$er 3 such that ad8 # I *1 i#$ #. 31. 0et

# be a #o#:si#gular s4uare matri o or$er 3 > 3 i#$ ad8 # i

# I 1&.

32. ; 2 −1

i#$ ( #−1 ) −1 . # =

3

+ 3

33. ; # = [−1 2 3] a#$ $ =

−+

i#$ #$.

&

3+. %i#$ x% y% 7 a#$ 5 i

x − y 2 x + 7

=

C1 ( .

2 x − y 3 x

& 13

+ 5 3(. )o#struct a 3 > 3 matri # ' Qai8 9hose eleme#ts are gie# b

1+ i + 8 i i ≥ 8 a

i8I

i − 2 8 i i


34/208

3. %i#$ # a#$ $ i 2 # 3$ I 1

−2 3

3 & 1

2 & a#$ # − 2$ = −1 2 .

−1

− 1

3/. ; # =

a#$ $ = [−2 −1 −+], eri that ! #$" I $ #. 2

3 3 3 −1

3*. 76ress the matri

−2 −2

9here ! is a smmetric a#$ Q 1 = ! + Q

−+ −(

2a ske9:smmetric matri.

3H. cos θ si# θ

the# 6roe that #n =

cos n θ si#n θ ; # ' ,

− si# θ cos θ − si#n θ cosn θ9here n is a #atural #umber.

+&.2 −1

,( 2 + 2 ( , i#$ a matri : such that

0et # = $ = , =

3 + / + 3 *+: / #$ I ;)

1 3 2 1[1 x 1] 2 = &

+1. %i#$ the alue o x such that 2 ( 1 3 2 x

1(

+2. roe that the 6ro$uct o the matrices

cos2 θ cos θ si#θ cos2 φ cos φ si# φ

si#2 θ a#$

si#2 φ cos θ si#θ cos φ si#φ

is the #ull matri, 9he# θ a#$ φ $ier b a# o$$ multi6le o π

.2

+3. ; # = ( 3

, sho9 that #2 C 12 # C I I &. =e#ce i#$ #

C1.

/

12

82>9 8(la!! *II : Math!9

++. Sho9 that


35/208

# C1.

++(. ;

# =

2


36/208


37/208

3 1 −2the# sho9 that ! #$"

C1 I $

C1 #

C1.

−+ a#$ $ =

−1 3

+*. -est the co#siste#c o the ollo9i#g sstem o e4uatio#s b matri metho$

3 x C y I (< x C 2y I 3

+H. Usi#g eleme#tar ro9 tra#sormatio#s, i#$ the i#erse o the matri

−3, i 6ossible. # = −2

1

(&. B usi#g eleme#tar colum# tra#sormatio#, i#$ the i#erse o # =3 1 .

( 2

(1.

cos α − si# α; # = si# α a#$ # #9 ' I, the# i#$ the ge#eral alue o α.

cos αUsi#g 6ro6erties o $etermi#a#ts, 6roe the ollo9i#g Q (2 to Q (H.

a − b − c 2a 2a( 3

(2. 2b b − c − a 2b+ b + c

)

= a

2c 2c c − a − b

x + 2 x + 3 x + 2a

(3. x + 3 x + + x + 2b = & i a, b, c are i# #.! . x + + x + ( x + 2c

si# α cos α si#(α + δ)(+. si# β cos β si# (β + δ) = &

si# γ cos γ si#(γ + δ)

8(la!! *II : Math!9 8;09


38/208

b

2

+ c2

a 2 a2

((. b 2 c 2 + a2

b 2 = +a 2 b

2c

2.

c 2 c 2 a 2 + b2

b + c c + a a + b a b c

(. q + r r + p p + q = 2 p q r .y + 7 7 + x x + y x y 7

a2

bc ac + c2

(/. a

2

+ ab b2

ac = +a 2

b

2

c

2

.ab b

2 + bc c 2

x + a b c

= x 2 ( x + a + b + c ) .(*. a x + b c a b x + c

(H. Sho9 that

x y 7

x2

y2

72 = (y − 7 ) (7 − x ) ( x − y ) (y7 + 7x + xy ) .

y7 7x xy &. !i" ; the 6oi#ts !a% b" !a, b" a#$ !a C a9 , b C b" are colli#ear, sho9

that ab I a9b.

!ii" ; # =2 (

a#$ $ =+ −3

eri that #$ = # $ .2 1

2 (

& −1 2& 1

1. Gie# # = a#$ $ = 1 & . %i#$ the 6ro$uct #$ a#$2 −2 &

1 1

also i#$ ! #$" C1.

2. Sole the ollo9i#g e4uatio# or x .

a + x a − x a − x a − x a + x a − x = &.

a − x a − x a + x

8;19 8(la!! *II : Math!9


39/208

& − ta#α

23. ; # = α

a#$ I is the i$e#tit matri o or$er 2, sho9

&

ta# 2

that,

I + # = ( I − # )cos α − si# α

si# α cos α +. Use matri metho$ to sole the ollo9i#g sstem o e4uatio#s

( x C /y I 2, / x C (y I 3.

(. 5btai# the i#erse o the ollo9i#g matri usi#g eleme#tar o6eratio#s2 C1 +

&

+ 2.

C2

3 / 1 −1 & 2 2 C+

.

3

2 are t9o s4uare matrices, i#$ #$

; # = 2 + a#$ $ = C+ −+

1

2 C1

& 2 (

a#$ he#ce sole the sstem o li#ear e4uatio#s

x C y I 3, 2 x 3y +7 I 1/, y 27 I /.

/.

S olve the follow ing system of equations by m atrix m ethod, w here x ≠ &,y ≠ &, 7 ≠ &

2 − 3 + 3 = 1&, 1 + 1 + 1 = 1&, 3 − 1 + 2 = 13.y 7 x y 7 x x y 7

1 2 −3

*. %i#$ # C1

, 9here

2 3 2

# =

, he#ce sole the sstem o li#ear

3 −3 C+ e4uatio#s

x 2y C 37 I C +

2 x 3y 27 I 2 3 x

C 3y C +7 I 11

8(la!! *II : Math!9 8;29


40/208

H. -he sum o three #umbers is 2. ; 9e subtract the seco#$ #umber rom

t9ice the irst #umber, 9e get 3. B a$$i#g $ouble the seco#$ #umber a#$ the thir$ #umber 9e get &. Re6rese#t it algebraicall a#$ i#$ the

#umbers usi#g matri metho$.

/&. )om6ute the i#erse o the matri.

3 −1

# =

−1(

( −2

1 −( a#$ eri that # C1 # I . .

3(

1 1 2 1 2 & /1. ; the matri # = & 2 −3 a#$ $ C1 = & 3 the# C1 ,

3 −2 +

1 & 2

com6ute ! #$" C1

.

/2. Usi#g matri metho$, sole the ollo9i#g sstem o li#ear e4uatio#s

2 x C y I +, 2y 7 I (, 7 2 x I /.

&

11

# 2 − 3. −1 i # = −1/3. %i#$ # 1 & 1 . 'lso sho9 that # = .2

1 1

&

b 2 + c2

ab ac

/+. Sho9 that ba c2 + a 2 bc = +a

2 b

2c

2

ca cb a2 + b2

a b C c c

+ b = (a + b + c )

(a

2

+ b

2

+ c

2

)/(. Sho9 that a + c b c − aa − b b + a c

cos α − si# α &

/. ; # =

si# α cos α, eri that # . !ad8 #" I !ad8 #" . # I # . 3. &

& &

8;;9 8(la!! *II : Math!9


41/208

2 −1 1//. %or the matri # = −1 2 eri that #3 C #2 G H # C +. I &, he#ce −1 ,

1 −1 2

i#$ # C1

.

/*. %i#$ the matri , or 9hich

3 2 − 1 1 2 −1 / ( . , . −2 = & +

1

/H. B usi#g 6ro6erties o $etermi#a#ts 6roe the ollo9i#g

1+ a 2 − b 2 2ab −2b( 32 2 2

2ab 1− a + b 2a + b2 )

.= 1+ a

2b −2a 1− a 2 − b2

(y + 7 ) 2 xy 7x *&. xy ( x + 7 )

2

y7

x7 y7 2

( x + y )

3

= 2 xy7 ( x + y + 7 ) .

a a + b a + b + c

*1. 2a 3a + 2b +a + 3b + 2c = a3 .3a a + 3b 1&a + b + 3c

x x2

1+ x 3

*2. ; x% y% 7 are $iere#t a#$ y y2

1+ y 3 = &, sho9 that xy7 I C 1.

7 7 2 1+ 7 3

*3. ; x , y , 7 are the 1&th

, 13th

a#$ 1(th

terms o a G.. i#$ the alue o

log x 1& 1

∆ = log y 13 1 . log 7 1( 1

8(la!! *II : Math!9 8;49


42/208

*+. Usi#g the 6ro6erties o $etermi#a#ts, sho9 that

1+ a 1 1 1 1 1 1 1+ b 1 = abc 1+ + + = abc + bc + ca + ab

a b1 1 1+ c c

*(. Usi#g 6ro6erties o $etermi#a#ts 6roe that

−bc b 2 + bc c 2 + bc

a2 + ac −ac c

2 + ac ( )3

= ab + bc + ca

a2 + ab b 2 + ab −ab

3 2 1*. If # =

+ −1 2

, i#$ #

C1 a#$ he#ce sole the sstem o e4uatio#s

/ 3 −3

3 x +y /7 I 1+, 2 x C y 37 I +, x 2y C 37 I &.

& −11. x I 2, y I / 2. 1

& 3. 11. +. +

(.H −

. 3 −(

& 2H . −3 .

−1

/. #$ ' Q2. *. x I (

H. x I C ( 1&.

& 1 −1 .

& 11. a

2 b

2 c

2 d

2. 12. x I C 13

13. k = 3 1+. # I 1.2

1(. + 1. C+

8;59 8(la!! *II : Math!9


43/208

1/. ! I C *

1*.

x =

( .3

1H. k = C/ , 13 . 2&. (+.2 2

21. +&. 22. /2H

23. H s4. u#its 2+. x I W 2

2(. & 2. &

2/. H 2*. * 3

−

.

(2H. 1/2* 3&. # I W H

31. 1&& 32. 11

33. #$ I C 11 3+. x I 1, y I 2, 7 I 3, 5 I 1&

3 3 2 ( 2

+ ( 2

.3(. ( /

11 + H ( 2 1

−

− −

3. / / /

// / # = , $ =

1 1* + + 12 (

− −

/ / / / / /

+&. : = − 1H1 −11& . +1. x I C 2 or C 1+// ++

#−1

− / 3 # C1 1 + 3

+3. = . ++. = .12 −( C3

1/ 2

8(la!! *II : Math!9 8;?9


44/208

+(. x I H, y I 1+ +. 1

−2

x = .2

&

+*. ;#co#siste#t +H.;#erse $oes #oteist.

# C1 =

2 −1 α = 2 n π ±

π, n ∈7 (&.

− . (1.( 3

3

1.

#$

= 1 2

,

( #$

) C1 = 1 2 −2

2

.

−2 2

−1

2 &, 3a +. x = 11 , y = 1 .

2+2+

−2 1 12

(.

#−

1 =

. x I 2, y I C1, 7 I +

11 −1

− .

1

+ −−2

2

/.

x

=

1,

y =

1 ,

7

= 1

2 3 (

1 −1/ 13

*.

#

−1 =

1+ ( −*, x = 3, y = C2, 7 = 1

/

−1( H

H. x I 1, y I C 2, 7 I 2 1

1 12 1/1. ( #$ )−1 = 21 11 −/ .

1H −2 3 1&


45/208

2 & −1

/&. #−1

=

( 1 &

& 1 3

/2. x I 3, y I 2, 7 I 1.

8;9 8(la!! *II : Math!9

1− 1 1 1

−1 = −1 /3. # 1 1 .2

1 1

/*. , − 1 3= 2+ −(

.

*. x I 1, y I 1, 7 I 1.

1 3 1 −1

//. #−1

= 1 3 1 .

+ −1 1 3

*3. &


46/208

8(la!! *II : Math!9 8;H9


47/208

(ATER 5

(/TI/IT= A/D DI33ERE/TIATI/

' u#ctio#f ! x " is sai$ to be co#ti#uous at x Ic i limf ( x ) (c )= f

x →c

i)e)% lim f ( x ) = lim f ( x ) = f ( )c

x →c C x →c +

X X f ! x " is co#ti#uous i# !a% b X i it is co#ti#uous at X X x X = X c X ∀ X c X ∈(a, X b X ).

f ! x " is co#ti#uous i# Qa% b i

!i" ! x " is co#ti#uous i# !a% b"

!ii" lim f ( x ) = f (a),

x →a+

!iii" lim f ( x ) = f (b) x →b C

-rigo#ometric u#ctio#s are co#ti#uous i# their res6ectie $omai#s.

7er 6ol#omial u#ctio# is co#ti#uous o# R.

; f ! x " a#$ * ! x " are t9o co#ti#uous u#ctio#s a#$ c ∈ " the# at x I a

!i" f ! x " W * ! x " are also co#ti#uous u#ctio#s at x I a.

!ii" * ! x " . f ! x ", f ! x " c , cf ! x ", f ! x " are also co#ti#uous at x I a)

f ( x )

!iii" ( ) is co#ti#uous at x I a 6roi$e$ * !a" ≠ &. * x

f ! x " is $eriable at x I c i# its $omai# i

8;>9 8(la!! *II : Math!9


48/208

f ( x

) − f (c

)

= lim f

( x

)

− f

(c

)lim , a#$ is i#ite

x − c x − c x →c − x →c +

-he alue o aboe limit is $e#ote$ b f !c " a#$ is calle$ the $eriatie o

f ! x " at x I c)

d (u Y < ) =u Y

d


49/208


50/208

Man Val$ Thor% : ; f ! x " is co#ti#uous i# Qa, b a#$ $eriable i#

!a% b" the# there eists atleast o#e real #umber c ∈ !a% b" such that

f (c ) = f (b ) C f (a ) . b − a

f ! x " I loge x , ! x P &" is co#ti#uous u#ctio#.

1. %or 9hat alue o x% f ! x " I 2 x C / is #ot $eriable.

2. rite the set o 6oi#ts o co#ti#uit o * ! x " I x C 1 x 1.

3. hat is $eriatie o x C 3 at x I C 1.

( x

) + ( x )+. hat are the 6oi#ts o $isco#ti#uit o f

( x

) = − 1 + 1 .

( x − / ) ( x − )

(. rite the #umber o 6oi#ts o $isco#ti#uit o f ! x " I Q x i# Q3, /.

λ x − 3 i x 2 x ∈ "% i#$ λ.

ta#3 x , x ≠ &

/.

%or 9hat alue o = , f ( x ) = si#2 x

2= , x = &

co#ti#uous u#ctio# or all

is co#ti#uous ∀ x ∈ " .

*. rite $eriatie o si# x 5)r)t) cos x .

H. ; f ! x " I x 2* ! x " a#$ * !1" I , * ! x " I 3 i#$ alue o f 9 !1".

1&. rite the $eriatie o the ollo9i#g u#ctio#s

!i" log3 !3 x (" !ii" elog 2 x

loge( )

!iii" e x −1

, x > 1

8419 8(la!! *II : Math!9


51/208

!i" sec

−1

x + cosec −1

x , x ≥ 1.

!"

−1 (

x

/ 2

) !i" log x (, x P &.si#

11. Discuss the co#ti#uit o ollo9i#g u#ctio#s at the i#$icate$ 6oi#ts.

!i"f

( x

) =

x − x , x ≠ &

at x = &. x 2,

x = &

si# 2 x , x ≠ &

!ii" * ( x ) = 3 x at x = &. 3

x = &2

2 cos (1 x ) x ≠ &!iii" f ( x ) =

x

at x = &.& x = &

!i" f ! x " I x x C 1 at x I 1.

!" f ( x ) x −

[ x

], x ≠ 1 at x = 1.=

& x = 1

12. %or 9hat alue o k , f ( x ) =

∀ x ∈[&, 3].

13. %or 9hat alues o a a#$ b

x + 2 +

x + 2f ( x ) = a + b

x + 2 + 2 x + 2

3 x 2 − kx + (, & ≤ x < 2 is co#ti#uous1 − 3 x 2 ≤ x ≤ 3

i x C2

8(la!! *II : Math!9 8429


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1+. roe that f ! x " I x 1 is co#ti#uous at x I C1, but #ot $eriable at x I C1.

1(. %or 9hat alue o p%

p

si#

(1 x )

x

f ( x ) = &

= 1 −1

2 x +1. ; y ta#

1 − x 22

x ≠ &

x = & is $eriable at x I &.


53/208

2 ta#−1 1

, & < x


54/208

84;9 8(la!! *II : Math!9


55/208

2.;

x I aet !si#t C

cos t "

y I aet !si#t cost" the#

sho9 that

d y

at x

= π

is

1.

dx +

−1

d y

2

= si#

1− x −

1− x 2/.

; y

x x

the#i#$ C

.

d x

loge x x dy

2*.; y

= x

+ (loge x )

the#i#$

.

d x

2H.Diere#tiate

x x x

9.r.t. x .

3&.%i#$

d y ,

i

(cos x )y

= (cosy ) x

dx

− πd y

−1

1 +si# x

1 − si# x

31. ; y = ta#

9here cos for x ∈

, π .

2 2 2

1 32. ; x = si# log y the# sho9 that !1 C x 2" y C xy C a2y I &.

a

e

33.Diere#tiate

log x

, x > 15 .r .t . x

(log x )

dy =si#2 (a+


56/208

y

)

3+.; si# y I x si# !a y " the#sho9 that

.

dx si#

3(.; y I si#

C1 x ,

i#$

d2

y

i# terms o y .

dx 2

3.;

x2

+

y 2

= 1,

the# sho9that

d

2

y =−b

+.

a

2

b

2

dx2

a2 y

3

a cos −1 x ( 2 )d

2

y d y 2

3/.; y

= e

, − 1 ≤ x ≤ 1,

sho9that

1 − x

− x

− a y =&

dx 2

d x

d2

y =−2a2 x

2

3*.

; y 3 I

3ax 2

C x 3 the#

6roe that

.

dx 2

y (

8(la!! *II : Math!9 8449


57/208

3H. Veri RolleZs theorem or the u#ctio#, y I x 2 2 i# the i#teral Qa, b

9here a I C2, b I 2.

+&. Veri Mea# Value -heorem or the u#ctio#, f ! x " I x 2 i# Q2, +

1. x =/ . 2. " 2

3. C1 +. x I , /

(. oi#ts o $isco#ti#uit o f ! x " are +, (, , / i)e) our 6oi#ts.

Note # 't x I 3, f ! x " I Q x is co#ti#uous. becauselim f ( x ) = 3 = f (3).

x →3 +

. λ = / . /. k = 3.+2

*. Ccot x H. 1(

1&. !i"

3 log3 e !ii" e log2 x 1

.log2 e.3 x + ( x

!iii" ! x C 1"(

!i" &

/ x2

.

−loge (.!"

x

!i" 22

1− x / x (loge x )11. !i" Disco#ti#uous !ii" Disco#ti#uous

!iii" )o#ti#uous !i" co#ti#uous

!" Disco#ti#uous

12. k I 11 13. a I &, b I C1.

1(. p P 1. 1. &

1/. − x . 22. 11− x 2

23. Ccot2 x logea

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dy − x

x

(1

+ log x

) − yx

y −1 − y

x log y

2+.

= .

xy log x + xy x −1dx

2(.d

2 y

= 1

cosec θ sec +θ. dx 2 3a

dy 1 12/. = −

dx 1− x 2 2 x 1− x

log x 2log x ) x 12*. x + (log x

log x

+ log (log x ).

2H.dy x x x 1

= x . x log x 1+ log x + .dx x log x

3&.dy

=y ta# x + logcos y

dx x ta# y + logcos x

31.dy

= − 1

. dx 2

log x 1 log (log x )33. (log x ) + , x > 1

x x

3(. sec2y ta#.

8(la!! *II : Math!9 84?9


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(ATER ?

A)I(ATI/S 3 DERIVATIVES

Rat o (hang : 0et y I f ! x " be a u#ctio# the# the rate o cha#ge o

y 9ith res6ect to x is gie# b dy

= f ( x ) 9here a 4ua#tit y aries 9ith dx

a#other 4ua#tit x .

dy

or f ( x & ) re6rese#ts the rate o cha#ge o y 9.r.t. x at x I &. dx x = x & ; x I f !t " a#$ y I *

!t " B chai# rule

dy = dy dx i dx ≠ &.dx dt dt dt

!i" ' u#ctio# f ! x " is sai$ to be i#creasi#g !#o#:$ecreasi#g" o# a#

i#teral !a% b" i x 1


60/208


61/208

0et f be a u#ctio#. 0et 6oi#t c be i# the $omai# o the u#ctio# f at 9hich

either f 9 ! x " I & or f is #ot $eriable is calle$ a critical 6oi#t o f .

3ir!t Dri6ati6 T!t : 0et f be a u#ctio# $ei#e$ o# a# o6e# i#teral

I. 0et f be co#ti#uous at a critical 6oi#t c ∈ I) -he# i,

!i" f 9 ! x " cha#ges sig# rom 6ositie to #egatie as i#creases

through c , the# c is calle$ the 6oi#t o the local maima.

!ii" f 9 ! x " cha#ges sig# rom #egatie to 6ositie as x i#creases

through c , the# c is a 6oi#t o local minima.

!iii" f 9 ! x " $oes #ot cha#ge sig# as x i#creases through c , the# c is

#either a 6oi#t o local maxima #or a 6oi#t o local minima.

Such a 6oi#t is calle$ a 6oi#t o inflexion)

Sond Dri6ati6 T!t : 0et f be a u#ctio# $ei#e$ o# a# i#teral I

a#$ let c ∈ I. -he#

!i" x I c is a 6oi#t o local maima i f !c " I & a#$ f !c " &.

f !c " is local maimum alue o f .

!ii" x I c is a 6oi#t o local mi#ima i f !c " I & a#$ f [!c " P &. f !c " is

local mi#imum alue o f .

!iii" -he test ails i f9 !c " I & a#$ f99 !c " I &.

1. -he si$e o a s4uare is i#creasi#g at the rate o &.2 cmAsec. %i#$ the

rate o i#crease o 6erimeter o the s4uare.

2. -he ra$ius o the circle is i#creasi#g at the rate o &./ cmAsec. hat is

the rate o i#crease o its circumere#ceO

3. ; the ra$ius o a soa6 bubble is i#creasi#g at the rate o

1

cm sec. 't29hat rate its olume is i#creasi#g 9he# the ra$ius is 1 cm.

+. ' sto#e is $ro66e$ i#to a 4uiet lake a#$ 9aes moe i# circles at a

s6ee$ o + cmAsec. 't the i#sta#t 9he# the ra$ius o the circular 9ae

is 1& cm, ho9 ast is the e#close$ area i#creasi#gO

84>9 8(la!! *II : Math!9


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(. T he totalrevenue in rupees received from the sale of x u#its o a 6ro$uct

is gie# b

" ! x " I 13 x 2 2 x 1(. %i#$ the margi#al ree#ue 9he# x I /.

. %i#$ the maimum a#$ mi#imum alues o u#ctio# f ! x " I si# 2 x (.

/. %i#$ the maimum a#$ mi#imum alues !i a#" o the u#ctio#

f ! x " I C x C 1 / ∀ x ∈ " .

*. %i#$ the alue o La@ or 9hich the u#ctio# f ! x " I x 2 C 2ax , P & is

strictl i#creasi#g.

H. rite the i#teral or 9hich the u#ctio# f ! x " I cos x , & ≤ x ≤ 2π is$ecreasi#g.

1&. hat is the i#teral o# 9hich the u#ctio# f ( x ) = log x , x ∈(&, ∞) isi#creasi#gO

x

11. %or 9hich alues o x , the u#ctio# y = x + − + x 3 is i#creasi#gO

3

( ) 1

12. rite the i#teral or 9hich the u#ctio# f x = x is strictl $ecreasi#g.13. %i#$ the sub:i#teral o the i#teral !&, πA2" i# 9hich the u#ctio#

f ! x " I si# 3 x is i#creasi#g.

1+. ithout usi#g $eriaties, i#$ the maimum a#$ mi#imum alue o

y I 3 si# x 1.

1(. ; f ! x " I ax cos x is strictl i#creasi#g o# " , i#$ a.

1. rite the i#teral i# 9hich the u#ctio# f ! x " I x H 3 x

/ + is i#creasi#g.

1/. hat is the slo6e o the ta#ge#t to the cure f I x 3 C ( x 3 at the 6oi#t

9hose x co:or$i#ate is 2O

1*. 't 9hat 6oi#t o# the cure y I x 2 $oes the ta#ge#t make a# a#gle o +(\

9ith 6ositie $irectio# o the x :aisO

1H. %i#$ the 6oi#t o# the cure y I 3 x 2 C 12 x H at 9hich the ta#ge#t is

6arallel to x :ais.

8(la!! *II : Math!9 8509


63/208

2&. hat is the slo6e o the #ormal to the cure y I ( x 2 C + si# x at x I &.

21. %i#$ the 6oi#t o# the cure y I 3 x 2 + at 9hich the ta#ge#t is 6er6e#$icular

to the li#e 9ith slo6e − 1 .

22. %i#$ the 6oi#t o# the cure y I x 2 9here the slo6e o the ta#ge#t is

e4ual to the y C co:or$i#ate.

23. ; the cures y I 2e x

a#$ y I ae C x

i#tersect orthogo#all !cut at righta#gles", 9hat is the alue o aO

2+. %i#$ the slo6e o the #ormal to the cure y I * x 2 C 3 at x =

1 .

+2(. %i#$ the rate o cha#ge o the total surace area o a cli#$er o ra$ius r

a#$ height 9ith res6ect to ra$ius 9he# height is e4ual to the ra$ius o

the base o cli#$er.

2. %i#$ the rate o cha#ge o the area o a circle 9ith res6ect to its ra$ius.

=o9 ast is the area cha#gi#g 9.r.t. its ra$ius 9he# its ra$ius is 3 cmO

2/. %or the cure y I !2 x 1"3 i#$ the rate o cha#ge o slo6e o the

ta#ge#t at x I 1.

2*. %i#$ the slo6e o the #ormal to the cure

2 π x I 1 C a si# θ


64/208

13. ' balloo# 9hich al9as remai# s6herical is bei#g i#late$ b 6um6i#g i#

H&& cubic cm o a gas 6er seco#$. %i#$ the rate at 9hich the ra$ius o the balloo# i#creases 9he# the ra$ius is 1( cm.

1+. ' ma# 2 metres high 9alks at a u#iorm s6ee$ o metres 6er mi#ute

a9a rom a lam6 6ost ( metres high. %i#$ the rate at 9hich the le#gth

o his sha$o9 i#creases.

1(. ater is ru##i#g out o a co#ical u##el at the rate o ( cm3Asec. ; the

ra$ius o the base o the u##el is 1& cm a#$ altitu$e is 2& cm, i#$ therate at 9hich the 9ater leel is $ro66i#g 9he# it is ( cm rom the to6.

1. -he le#gth x o a recta#gle is $ecreasi#g at the rate o 2 cmAsec a#$ the

9i$th y is i#creasi#g as the rate o 2 cmAsec 9he# x I 12 cm a#$ y I (cm. %i#$ the rate o cha#ge o

!a" erimeter !b" 'rea o the recta#gle.

3/. Sa#$ is 6ouri#g rom a 6i6e at the rate o 12c.cAsec. -he alli#g sa#$

orms a co#e o# the grou#$ i# such a 9a that the height o the co#e is

al9as o#e:sith o the ra$ius o the base. =o9 ast is the height o the

sa#$ co#e i#creasi#g 9he# height is + cmO

3*. -he area o a# e6a#$i#g recta#gle is i#creasi#g at the rate o +*

cm2Asec. -he le#gth o the recta#gle is al9as e4ual to the s4uare o

the brea$th. 't 9hat rate is the le#gth i#creasi#g at the i#sta#t 9he# thebrea$th is +.( cmO

3H. %i#$ a 6oi#t o# the cure y I ! x C 3"2 9here the ta#ge#t is 6arallel to the

li#e ?oi#i#g the 6oi#ts !+, 1" a#$ !3, &".

+&. %i#$ the e4uatio# o all li#es hai#g slo6e 8ero 9hich are ta#ge#ts to the

cure y = 1 . x

2 − 2 x + 3

+1. roe that the cures x I y 2 a#$ xy I k cut at right a#gles i *k

2 I 1.

+2. %i#$ the e4uatio# o the #ormal at the 6oi#t !am2, am

3" or the cure

ay 2 I x 3.

+3. Sho9 that the cures + x I y 2 a#$ + xy I k cut as right a#gles i k

2 I (12.

++. %i#$ the e4uatio# o the ta#ge#t to the cure y = 3 x − 2 9hich is6arallel to the li#e + x C y ( I &.

8(la!! *II : Math!9 8529


65/208

+(. %i#$ the e4uatio# o the ta#ge#t to the cure x + y = a at the 6oi#t a 2 a 2

, .

+ + 1

+. %i#$ the 6oi#ts o# the cure +y I x 3 9here slo6e o the ta#ge#t is 3

.

+/. Sho9 that x + y = 1 touches the cure y I be C x Aa

at the 6oi#t 9here thea b

cure crosses the y-ais.

+*. %i#$ the e4uatio# o the ta#ge#t to the cure gie# b x I 1 C cos θ,

y I θ C si# θ at a 6oi#t 9here θ = π .+

+H. %i#$ the i#terals i# 9hich the u#ctio# f ! x " I log !1 x " C x , x > −11+ x

is i#creasi#g or $ecreasi#g.

(&. %i#$ the i#terals i# 9hich the u#ctio# f ! x " I x 3 C 12 x

2 3 x 1/ is

!a" ;#creasi#g !b" Decreasi#g.

(1. roe that the u#ctio# f ! x " I x 2 C x 1 is #either i#creasi#g #or $ecreasi#g i# Q&, 1.

%i#$ the i#terals o# 9hich the u#ctio# f ( x ) = x (2. is $ecreasi#g. x

2 + 1

(3. roe that f ( x ) = x3

− x 2 + H x , x ∈ [1, 2] is strictl i#creasi#g. =e#ce i#$3

the mi#imum alue o f ! x ".

π

(+. %i#$ the i#terals i# 9hich the u#ctio# f ! x " I si#+ x cos

+ x , & ≤ x ≤ 2 is

i#creasi#g or $ecreasi#g.

((. %i#$ the least alue o ZaZ such that the u#ctio# f ! x " I x 2 ax 1 is strictl

i#creasi#g o# !1, 2".

85;9 8(la!! *II : Math!9


66/208

3 (

(. %i#$ the i#teral i# 9hich the u#ctio# f ( x ) = ( x 2 − 3 x 2, x > & is strictl$ecreasi#g.

(/. Sho9 that the u#ctio# f ! x " I ta# C1

!si# x cos x ", is strictl i#creasi#g o#

π the i#teral &, .

+

π (*. Sho9 that the u#ctio# f ( x ) = cos 2 x + is strictl i#creasi#g o#

3π /π +

, .

* * f ( x ) =si# x π (H. Sho9 that the u#ctio# is strictl $ecreasi#g o# &, .

x 2 Usi#g $iere#tials, i#$ the a66roimate alue o !T. No. & to +".

1 1

&. (&.&&H ) . 1. (*& ) .3 +

1

2. (&.&&3/ ) . 3. &.&3/.2

+. 2(.&2 .

(. %i#$ the a66roimate alue o f !(.&&1" 9here f ! x " I x 3 C / x

2 1(.

. %i#$ the a66roimate alue o f !3.&2" 9here f ! x " I 3 x 2 ( x 3.

/. Sho9 that o all recta#gles i#scribe$ i# a gie# ie$ circle, the s4uare

has the maimum area.

*. %i#$ t9o 6ositie #umbers x a#$ y such that their sum is 3( a#$ the

6ro$uct x 2y ( is maimum.

H. Sho9 that o all the recta#gles o gie# area, the s4uare has the smallest

6erimeter.

/&. Sho9 that the right circular co#e o least cure$ surace area a#$ gie#

olume has a# altitu$e e4ual to 2 times the ra$ius o the base.

8(la!! *II : Math!9 8549


67/208

/1. Sho9 that the semi ertical a#gle o right circular co#e o gie# surace

area a#$ maimum olume is si# −1 1 .

3

/2. ' 6oi#t o# the h6ote#use o a tria#gle is at a $ista#ce a a#$ b rom thesi$es o the tria#gle. Sho9 that the mi#imum le#gth o the h6ote#use is

2 232

(a 3

+ b 3) .

/3. roe that the olume o the largest co#e that ca# be i#scribe$ i# a s6here

*

o ra$ius " is 2/ o the olume o the s6here.

/+. %i#$ the i#teral i# 9hich the u#ctio# f gie# b f ! x " I si# x cos x , & ≤ x ≤ 2π is strictl i#creasi#g or strictl $ecreasi#g.

/(. %i#$ the i#terals i# 9hich the u#ctio# f ! x " I ! x 1"3 ! x C 3"

3 is strictl

i#creasi#g or strictl $ecreasi#g.

/. %i#$ the local maimum a#$ local mi#imum o f ! x " I si# 2 x C x , − π


68/208

*2. %i#$ the e4uatio# o the ta#ge#ts at the 6oi#ts 9here the cure 2y I 3 x 2

C 2 x C * cuts the x :ais a#$ sho9 that the make su66leme#tar a#gles9ith the x-ais.

*3. %i#$ the e4uatio#s o the ta#ge#t a#$ #ormal to the h6erbola x

2

− y2

= 1a

2b

2

at the 6oi#t ! x &, y &".

*+. ' 9i#$o9 is i# the orm o a recta#gle surmou#te$ b a# e4uilateral

tria#gle. Gie# that the 6erimeter is 1 metres. %i#$ the 9i$th o the

9i#$o9 i# or$er that the maimum amou#t o light ma be a$mitte$.

*(. ' ?et o a# e#em is li#g alo#g the cure y I x 2 2. ' sol$ier is 6lace$

at the 6oi#t !3, 2". hat is the #earest $ista#ce bet9ee# the sol$ier a#$the ?etO

*. %i#$ a 6oi#t o# the 6arabola y 2 ' + x 9hich is #earest to the 6oi#t !2, C

*".

*/. ' s4uare 6iece o ti# o si$e 2+ cm is to be ma$e i#to a bo 9ithout to6

b cutti#g a s4uare rom each cor#er a#$ ol$i#g u6 the la6s to orm

the bo. hat shoul$ be the si$e o the s4uare to be cut o so that the

olume o the bo is the maimum.

**. ' 9i#$o9 i# the orm o a recta#gle is surmou#te$ b a semi circular

o6e#i#g. -he total 6erimeter o the 9i#$o9 is 3& metres. %i#$ the$ime#sio#s o the recta#gular 6art o the 9i#$o9 to a$mit maimum

light through the 9hole o6e#i#g.

*H. '# o6e# bo 9ith s4uare base is to be ma$e out o a gie# iro# sheet o

area 2/ s4. meter, sho9 that the maimum alue o the bo is 13.( cubic

metres.

H&. ' 9ire o le#gth 3 m is to be cut i#to t9o 6ieces. 5#e o the t9o 6ieces is

to be ma$e i#to a s4uare a#$ other i#to a circle. hat shoul$ be the

le#gth o t9o 6ieces so that the combi#e$ area o the s4uare a#$ the circle

is mi#imumO

H1. Sho9 that the height o the cli#$er o maimum olume 9hich ca# be

i#scribe$ i# a s6here o ra$ius " is2"

. 'lso i#$ the maimum olume. 3

H2. Sho9 that the altitu$e o the right circular co#e o maimum olume that

ca# be i#scribe$ is a s6here o ra$ius r is+r

. 3

8(la!! *II : Math!9 85?9


69/208

H3. roe that the surace area o soli$ cuboi$ o a s4uare base a#$ gie#

olume is mi#imum, 9he# it is a cube.

H+. Sho9 that the olume o the greatest cli#$er 9hich ca# be i#scribe$ i#

a right circular co#e o height a#$ semi:ertical a#gle α is+

π3

ta#2

α.

2/

H(. Sho9 that the right tria#gle o maimum area that ca# be i#scribe$ i# a

circle is a# isosceles tria#gle.

H. ' gie# 4ua#tit o metal is to be cast hal cli#$er 9ith a recta#gular bo

a#$ semicircular e#$s. Sho9 that the total surace area is mi#imum 9he#

the ratio o the le#gth o cli#$er to the $iameter o its semicircular e#$s is

π !π 2".

1. &.* cmAsec. 2. +.+ cmAsec.

3. 2π cm3Asec. +. *&π cm2Asec.

(. Rs. 2&*.

. Mi#imum alue I +, maimum alue I .

/. Maimum alue I /, mi#imum alue $oes #ot eist.

*. a ≤ &. H. Q&, π

1&. !&, e 11. x ≥ 1

π 12. !C ∞, &" U !&, ∞" 13. &, .

1+. Maimum alue I +, mi#imum ale I &. 1(. a P 1.

1. " 1/. /

1*. 1 , 1 . 1H. !2, C 3"

2 +

2&.1

21. !1, /"+

859 8(la!! *II : Math!9

22. !&, &", !2, +"

2+. C1

. +


70/208

2. 2πr cm2Acm, π cm2Acm

2*. − a .2b

3&. a P &.

(+, 11) a#$ −+, − 31 31. .

3

33.1

cm sec.

π

3(.+

cm sec.

+(π

3/.1

cm sec.

+*π

/ 1 3H. , .

2 +

+2. 2 x 3my I am2 !2 3m

2"

+(. 2 x 2y I a2

+*.( ) ( ) π

.2 C 1 x − y = 2 2 C 1 C +

23.1

.

2

2(. *πr

2/. /2

2H. Rs. *&.

32. − * cm sec.3

3+. + metresAmi#ute

3. !a" & cmAsec., !b" 1+ cm2Asec.

3*. /.11 cmAsec.

+&. y = 1

.

2

++. +* x / 2+y I 23


71/208

+. *

,12*

, −*

, −12*

. 3 2/ 3 2/

+H. ;#creasi#g i# !&, ∞", $ecreasi#g i# !C1, &".

(&. ;#creasi#g i# !C ∞, 2" ∪ !, ∞", Decreasi#g i# !2, ".

8(la!! *II : Math!9 85H9


72/208

(2. !C ∞, C1" a#$ !1, ∞". (3.2(

.3

π,

π &,

π (+. ;#creasi#g i# Decreasi#g i# .

+ + 2

((. a I C 2. (. Strictl $ecreasi#g i# !1, ∞".

&. &.2&*3 1. 2.HH&/

2. &.&&*3 3. &.1H2(

+. (.&&2 (. C3+.HH(

. +(.+

*. 2(, 1&

π (π /+. Strictl i#creasi#g i# &, ∪ , 2π

+ +

Strictl $ecreasi#g i# π (π

, .

+ + /(. Strictl i#creasi#g i# !1, 3"

∪ !3,

∞" Strictl

$ecreasi#g i# !C∞, C1" ∪ !C1, 1".

/. 0ocal maima at x = π

π0ocal ma. alue = 3 −2

0ocal mi#ima at x = − π

0ocal mi#imum alue =− 3

+π

2

//. Strictl i#creasi#g i# !C∞, 2 ∪ Q3, ∞"

Strictl $ecreasi#g i# !2, 3".

85>9 8(la!! *II : Math!9


73/208

oi#ts are !2, 2H" a#$ !3, 2*".

/*. − 3

metres mi#.

π

/H. x y ta#θ C a secθ I &.

*&. !&, &", !C1, C2" a#$ !1, 2".

*1. x y I 3

*2. ( x / y C 1& I & a#$ 1( x 3y 2& I &

*3. xx 2& −yy 2& =

1, a b

1

*+.

− 3

*. !+, C+"

**. & , 3&

π + + π ++π" 3

H1.

3 3

y

− y

& + x

− x & = &. a

2 y & b

2

x &

*(. (

*/. +cm

H&.1++

m,

3π m.

π + +π + +

8(la!! *II : Math!9 8?09


74/208

(ATER

I/TE&RA)S

;#tegratio# is the reerse 6rocess o Diere#tiatio#.

d ( ) ( ) the# 9e 9rite ∫ f ( x )dx = > ( )+ c .0et > x = f x x

dx

-hese i#tegrals are calle$ i#$ei#ite i#tegrals a#$ c is calle$ co#sta#t o

i#tegratio#.

%rom geometrical 6oi#t o ie9 a# i#$ei#ite i#tegral is collectio# o amil

o cures each o 9hich is obtai#e$ b tra#slati#g o#e o the cures

6arallel to itsel u69ar$s or $o9#9ar$s alo#g y :ais.

x

n +1

1. ∫ x n dx = + c n ≠ −1 n + 1

log x + c

n

= C1

n +1 (ax + b ) + c

n ≠ −1∫ n

( n + ) a(

)2. ax + b dx = 1 1

log ax + b + c n = −1 a

3. ∫ si# x dx = C cos x + c . +. ∫ cos x dx = si# x + c .

(. ∫ ta# x . dx = C log cos x + c = log sec x + c .8?19 8(la!! *II : Math!9


75/208

. ∫ cot x dx = log si# x + c . /. ∫

sec2 x . dx = ta# x + c .

*. ∫ cosec 2 x . dx = C cot x + c . H. ∫ sec x . ta# x . dx = sec x + c .

1&. ∫ cosec x cot x dx = C cosec x + c .

11. ∫ sec x dx = log sec x + ta# x + c .

12. ∫ cosec x dx = log cosec x C cot x + c . a x x x

∫ x

13. ∫ e dx = e + c .1+.

a dx = + c

loga

1(.∫

1

dx = si# C1

x + c , x 1.

∫ x x

2 − 1

1*. ∫ 1 dx =1 log a + x + c .

a2 − x 2 2a a − x

1H. ∫ 1 dx =

1 log x − a + c .

x 2 − a 2 2a x + a

2&. ∫ 1 dx = 1 ta# C1 x + c .a

2

+ x 2

a a

8(la!! *II : Math!9 8?29


76/208

21. ∫ 1 dx = si# C1 x + c .

a2 C x

2a

1 + c .22. dx = log x + a2 + x 2

∫ a 2 + x 2

23.

2+.

2(.

2

.

∫

∫

∫

∫


77/208

1 dx = log x + x2 − a 2 + c .

x2

C a2

x

+ a 2 si# C1

x

a

2

− x2

dx = a2

− x2

+ c .2 a2

x + a2

a 2 + x2

dx = a 2 + x2

log x + a 2 + x2 + c .

2 2

x a2

2 2 2 2 2 2 x − a dx = x − a − log x + x − a + c .

2 2

1. ∫ k .f ( x ) dx = k ∫ f ( x ) dx .

2. ∫ k {f ( x ) ± * ( x )}dx = k ∫ f ( x ) dx ± k ∫ * ( x ) dx .

2. ∫ e x { (

x

)+ f Z

(

x

)}dx = e

x

f

( x

)+ c .f

1. ∫ f V ( x ) ( x )dx = log f

+ c .f ( x )n +1

n

∫ [f ( x )]

2. [f ( x )] f ( x )dx = + c .

n + 1

8?;9 8(la!! *II : Math!9

3. ∫ f ( x )(f ( x

dx =n[f ( x )] C

∫ f

(

x

)

. *

(

x

)

dx

= f


78/208

x

)

dx

C

( )

.

∫ *

(

x

)

dx

dx .

∫ fV x

b

∫f

(

x

)

dx = >

(

b

)

− >

(

a

)

, 9here >

(

x

)

= ∫ f

(

x

)

dx .a

b f (a ) + f (a + ) + f (a + 2 )

∫ f ( x ) dx = lim

4 → & + ..... + f (a + n − 1 )

a

b − a b n

9here = . or

∫ f ( x ) dx = lim

∑f (a + r )

44 → &

a r =1

b a b b

1. ∫ f

(

x

)

dx = C ∫ f

(

x

)

dx . 2.

∫f

(

x

)

dx = ∫ f

( t ) dt .

a b a a

b c b

3. ∫ f

( )

∫ f

(

x

)

dx +

∫f

(

x

)

dx . x dx =

a a c

b b a a

+. !i"

∫f

( )

=

∫f

(

a + b − x

)

dx . !ii"

∫f

( )

= ∫ f

(

a − x

)

dx . x dx x dx a a & &

8(la!! *II : Math!9 8?49


79/208

a

(. ∫ f ( x

)= &< i f

( x

)is o$$ u#ctio#.

Ca

aa

. ∫ f ( x )dx = 2 ∫ f ( x ) dx , i f ! x " is ee# u#ctio#.&

−a

2a a ( ) ( ) ( )/. ∫ f ( x ) dx = 2∫

f x dx , i f 2a − x =f x

&( ) ( )

i f 2a − x = −f x &

&,

7aluate the ollo9i#g i#tegrals

1. ∫ (si#−1

+ cos −1 )dx . x x

3.

∫

1dx .

1 − si#2

x

1

(. ∫ x HH cos + x dx .−1

∫ π 2 + + 3si# x /. log dx .&

+ + 3cos x

2 H.

∫ cos2 x + 2si# x dx .

cos2 x

∫ 11. 1& C + x + x 2dx .


80/208

1

2. ∫ e x dx .−1

∫ x + x * +* + x +. * dx .

x *


81/208

. ∫ 1 dx .

x log x log(log x )

*. ∫

( a log x + e x log a ) dx .e

π 2

1&. ∫ si#/ x dx .− π

2

12.d (

x )dx

dx ∫

f .

8?59 8(la!! *II : Math!9

13.1 dx .

∫ si#

2 x cos

2 x

1(. ∫ e − loge x dx .

1/. ∫ 2 x e x dx .

1H. ∫ x dx .(

2

x +)

1

21.

∫ cos

2

α dx .

23. ∫ sec x .log ( sec x + ta# x ) dx .

2(. ∫ cot x .log si# x dx .

2/. ∫ 1 dx .

x (2 + 3 log x )

2H. ∫ 1 − cos x dx .

si# x

( )31. ∫

x + 1

( x + log x )dx . x

33. ∫ π cos x dx .&


82/208


83/208

1+. ∫ 1 dx .

x + x − 1

1.e x

dx .

∫ a x

1*. ∫ x dx . x + 1

e2&. ∫ x dx .

x

22. ∫ 1 dx . x cos α + 1

2+. ∫ 1 dx .

cos α + x si# α

∫ 1 3

2. x −

dx . x

2*. ∫ 1 − si# x dx . x + cos x

3&. ∫

x e −1 + e x −1dx .

xe + e x

∫ 1 2

32. ax −

ax

2

3+. ∫ [ x ] dx 9here Q is greatest i#teger u#ctio#.&

8(la!! *II : Math!9 8??9


84/208

3(. ∫ 1 dx

H − + x 2

3. ∫ b f ( x ) dx . 3/. ∫ 1 x dx .

f

(

x

)+ f

( ) −2 x aa + b − x

1

3*. ∫ −1 x

x dx . π

3H. ; ∫ a 1 = , the# 9hat is alue o a.& 1+ x 2 +

+&. ∫ b f ( x ) dx + ∫ a +1. ( )f ( x ) dx . ∫ e log x +1 −log x dx .a b

+2. ∫ si# x dx . +3. ∫ si# x si#2 x dx .si#2 x

π +

b

( )a

) x dx ( dx .

++. ∫ si# x dx . +(. ∫ f +∫ f a + b − x − π a b

+

+. ∫ 1 dx . +/. ∫ si#2 x dx .sec x + ta# x 1+ cos x

+*.

1− ta# x

dx . +H.

a x + b x

dx .∫ 1+ ta# x

∫ c x

x cosec (ta# C1 x 2)

!i" ∫ ∫ x + 1 − x − 1

(&. dx . !ii"

dx .

+1 + x x + 1 + x − 1

!iii"

1

dx . !i" ∫ cos

( x +a

)dx .

cos ( x − a


85/208

∫ si# ( x − a ) si# ( x − b ) )

8?9 8(la!! *II : Math!9


86/208

!" ∫ cos x cos 2 x cos 3 x dx . !i" ∫

cos( x dx .

!ii" ∫ si#2 x cos+ x dx . !iii" ∫ cot 3 x cosec + x dx .

!i"

∫ si# x cos x dx .

a2 si#

2 x + b 2 cos2 x

Qint : 6ut a2 si#

2 x b

2 cos

2 x I t or t

2

!"1 dx .

Qint : -ake sec2 x as #umerator∫ cos3 x cos ( x + a )

!i" ∫ si#

x + cos x dx . !ii" ∫ si# x + cos x dx .

2 2si# x cos x si# 2 x

(1. 7aluate

!i" ∫ x dx . Qint : 6ut x

2 I t

x + + x 2 + 1

!ii" ∫ 1 dx . Qint : 6ut log x I t x

(log x )2

+ / log x + 2

!iii" ∫ dx . !i" ∫ 1 dx .

1 + x − x 2 H + * x − x 2

!" 1 dx .

∫ ( x − a ) ( x − b )

∫ ( x − 2

∫

x 2

!i" dx . !ii" dx .

3 x

2 x 2 ++ 2 x + 1 x + 12

x + 2 ∫ !iii" dx . !i" x 1 + x C x 2 dx .∫ + x − x 2

8(la!! *II : Math!9 8?H9


87/208

( ) 2

!" ∫ 3 x − 2 x + x + 1 dx . !i" ∫ sec x C 1 dx .Qint : Multi6l a#$ $ii$e b sec x + 1

(2. 7aluate

!i" ∫ dx . x

(

x

/ )+ 1

!ii" si# x dx .

∫ (1 + cos x ) ( 2 + 3 cos x )!iii" ∫ si# θ cos θ d θ.

cos2

θ − cos θ − 2

!i" ∫ x − 1 dx .

( ) () (

x − 2 x + 3

) x + 1

!" ∫ x2

+ x + 2 dx .(

x − 2) (

x −)

1

(

x

2 ) ( x

2

+ 2

)

!i"+ 1 dx . Qint : x

2 I t

∫( x 3 + 3) ( x 2 + +)

!ii" ∫ dx .

(

2 x

) ( x 2 + + )+ 1

!iii" ∫ x

2 − 1dx .

x + + x 2 + 1

!i" ∫ dx .ta# x

8?>9 8(la!! *II : Math!9


88/208

(3. 7aluate

!i" ∫ x ( si# x 3dx .

!ii" ∫ sec 3 x dx .Qint : rite sec

3 x I sec x . sec

2 x a#$ take sec x as irst u#ctio#

!iii" ∫ eax

cos

(bx + c

!" ∫ cos x dx .!ii"

∫ e

2 x 1 + si# 2 x

1 + cos 2 x

∫ !i" 2ax − x 2dx .

!i" ∫ e x ( 2 + si# 2 x )

(1 + cos 2 x )

)dx .

dx .

dx .

!i" ∫ si# C1 x

dx .1 + H x 2

Qint : 6ut 3 x I ta# θ


89/208

!i" ∫ x 3 ta# C1 x dx .!iii"

∫

e x x − 1 dx . 2

2 x ( x

2 + )!" ∫ e

x 1

dx .( 2

x

)

+ 1

∫ 1 t

log (log x ) +

!ii" dx . Qint : 6ut log x I t ⇒ x I e (log x )2

!iii" ∫ ( x + () + x − x 2 dx .

!i" ∫ 1 dx . x

3

+ 1

!"∫ (2 x − () x 2 − + x + 3 dx .

!i"∫ x 2 − + x + * dx .

8(la!! *II : Math!9 809


90/208

(+. 7aluate the ollo9i#g $ei#ite i#tegrals

π

!i" +si# x + cos x

∫ H + 1 si# 2 x &

1 1 2

!iii" ∫ x − x dx2& 1 + x

Qint : 6ut x 2 I

π2

si# 2 x !" ∫ si#

+

x + cos&

π

!ii" 2 x + si# x dx∫ 1 + cos x &

int : rite

π2

!ii" ∫ cos 2 x log si# x dx .&

1 2

si#−1

x !i" ∫ dx .

(3 2

& 1 − x 2 )

2

( x 2!i" ∫ dx .

x

2

1+ + x + 3


91/208

x + si# x as

x + si# x 1 + cos x 1 + cos x 1 + cos x

((. 7aluate

3 π x

!i" ∫ { x − 1 + x − 2

+

x − 3 } dx . !ii" ∫ dx .1 + si# x 1 &

1 C1 x 1 + x +

x

2 π

!iii" ∫ e ta#

dx . !i" ∫ x si# x dx .2

1 + x

2

C1 1 + cos x

&

2 x − x

39he# − 2 ≤ x


92/208


93/208

!ii" ∫ 1 − x dx 1 + x ( 2 )

x 2

+ 1 x !iii" ∫ log + 1 − 2 log x

dx

x +

!i" ∫ x 2 dx ( x si# x

2

+ cos x )

∫ −1 x

!" si#dx

a + x

π

3 si# x + cos

!i" ∫ si# 2 x π

π

2

(si#!ii" ∫ x − C π

2

3

2

!iii" ∫ x si# π x −1

&. 7aluate theollo9i#gi#tegrals

∫ x ( + +

!i" dx x ( − x

!iii" ∫ 2 x

3

( x

)( x + 1


94/208

!ii"

∫ dx

dx

( x − 1 ) ( x 2 + +)

!i" ∫ x

+

dx

x

+

C 1

8;9 8(la!! *II : Math!9


95/208

π

21

!" ∫ ( ta# x + cot x ) dx . !i" ∫ dx . x + + 1&

x ta# C1

x

∫ 2 dx .

& (1 + x 2 ) 1. 7aluate the ollo9i#g i#tegrals as limit o sums

!ii"∞

+ 2

( )!i" ∫

(2 x

)!ii" ∫ x

2

+ 3 dx .+ 1 dx .2 &

3 +

!iii" ∫(3 x

2 − 2 x + +) dx . !i" ∫

(3 x

2 + e 2 x ) dx .

1 &

(

!" ∫ ( x 2 + 3 x ) dx .2

2. 7aluate

1

!i" ∫ cot−1 (1 − x + x 2 ) dx &

!ii" ∫ dx

(si# x − 2 cos x ) ( 2 si# x + cos x )

1 ( )π

log 1 + x 2

dx !iii" ∫ 1 + x 2

!i" ∫ (2 log si# x C log si# 2 x ) dx .& &

3. ∫ 1

dx . +. ∫ (3si#θ − 2)cos θ d θ.(− cos 2 θ − +si#θ

si# x + si#2 x

8(la!! *II : Math!9 849


96/208

(. ∫

1 (ta# C1 x ) 2 .

∫ e

2 x cos 3 x dx .

x dx &

/. ∫ πA2 log si# x dx &

1.π x + c . 2. 2e C 2

2

* x

xH

x2

3. ta# x c . +. + + *log x + + c .H 1log*

(. & . log log log x + c

x a +1 a x

/. & *. + + c a + 1 loga

H. ta# x c 1&. &

( ) x−

2 x 2 − + x + 1&11. + 3log x − 2 + x

2

− + x + 1& + c 2

12. f ! x " c

2 3 2 2 3 2

13. ta# x C cot x c 1+. x

−

( )+ c

3 3 x − 1

1(. log x c 1. e x log(e a ) + c

a 2 x e x 2 3 2 1 2

1/. + c 1*.(

x ) − 2 ( x + )

log ( 2e) 3 + 1 1 + c .

1

1H. log x + 1 + + c . 2&. 2e x + c x + 1

21. x cos2 α c 22. log x cos α + 1 + c .

cos α

859 8(la!! *II : Math!9


97/208

( log sec x + ta# x 2

23.)

2+ c

(logsi# x 22(.

)+ c

2

2/.1

log 2 + 3log x + c .3

2H. 2 log sec x A2 c)

2( x + log x )

31. + c 2

33. &

1si#

−1 2 x + c 3(.

2 3

3/. C1

3H. 1

+1. x log x c .

1 si#3 x +3. − − si# x + c or

32

2+.

log cos α + x si#α + c si#α

x + 1 3 x 22. + − + 3 log x + c .

2 x 2

+ 2

2*. log x cos x c

3&.1

log xe + e x + c .

e

x 2 log x 32. a + −2 x + c .

2 a

3+. 1

3.b − a

2

3*. &

+&. &

+2.1

log sec x + ta# x + c . 2

2 si#

3 x + c

3

++.2

− 2

+(. &

+. log 1 si# x c +/. x C si# x c

+*. log cos x si# x c

(a c ) x (b c ) x

+H. + + c 1.log a c log b c

8(la!! *II : Math!9 8?9


98/208

1 − 1 2 1 (&. !i" log cosec ( ta# x ) − + c .

2

x 2

1( 2 ) 12 2!ii" x − x x −1 + log x + x −1 + c .2 2

!iii"

1 log si# ( x − a ) + c

si# (a − b ) si# ( x − b )

!i" x cos 2a C si# 2a log sec ! x C a" c .

!"1

[12 x + si# 2 x + 3 si# + x + 2 si# x ] + c .+*

!i" si# x −2 si#

3 x + 1 si#

( x + c .

(

3

1 + 1−

1 si# + x − 1 + c .

!ii"2 x si# 2 x si# x

32 2

2

cot x cot + x

!iii" − + + c .

+

!i"1

a2 si#

2 x + b 2 cos2 x + c

a2 − b 2

!" C2 cosec a cos a − ta# x . si# a + c .

!i" ta# x C cot x C 3 x c.

!ii" si# C1

!si# x C cos x " c .

1 C1

2 x 2

+

(1. !i"ta# 1 + c .

3 3

!ii" log

2 log x + 1+�

12 Maths Good Notes

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