Problems Plus In Iit Mathematics. Part 2

332 Pages·2014·114.72 MB·english

by  A. Das Gupta

IT Mathematics
Problems Plus
in
IIT Mathematics

Dg of Ms
‘Science College, Patna

[@| Bharati Bhawan

Hl PUBLISHERS & DISTRIBUTORS
Preface

Protioms Plus in HT Mathematics is meant for students who aspire to take the entrance
‘csaminations of diferent technical institutions, espe

: ily LE wetting this book, T have
{raw ly rom my experience eaching stecent preparing fer JEFAITD, Many students
‘dowwellat heboand examinations andi theircoaching courses, ad yet aller al [EEE The
ress for this ate

They are unale to apply their knowledge to new situations
They often cannot detect the area(s) of mathematics on which a problem is based. At
BE UT, the student's ability to apply the concepts learnt to new situations is tested

‘umber ef concupts from different areas of mathematics are mixed, creating. a new

"rey fail to compete the paper within the given time

Thy are oon ie to dest the fof te problem, A problem may bestia
any one of the following forms: , , °

1 Direct 2.tliest 3. nverted

Ti illustrate the diferent forms, we take three prablems from conndinate geotnetry.

1 tmnt 9342 Nandan Bem

2 Find the value of « sich that the point (a,

2) is on the diameter of the circle
S24 as 2p 29-O through the pot (3), iret orm]
3. Find the point whose ordinate i -2 an which is collinear with the points (1-3)

and (2, 1, Uniertad form)
Basically, the theee problems test the same knowledge, namely the condition of
sollinearity nt thtee points.

‘You know that luce points (,y,): ¥=1.2,3 are collinear ifthe area of the triangle
with the points asits verticusis zem, i,

nowt
sy eo
sont
13
Problem 1 igeasiy solved by proving that F}-2 1 1! 0.
$2
i

For Problem 2, we have ta find the third point which is the contr ofthe circle for the
bother end of the diameter, because (1,3) is on the cite. So, the knowledge oF
finding the conte from the equation of «ciel is also requleed. Here, centre = (2,1)
and (1,3) satisfies the equation of the elle, So we have to find he value of a such,

wy
by.(o2, 1) and (1, -3) are collins be
that the points (2 -2.6-2 0 and

which given
2), So, we have to find the unknown a by using,
collgeaity the thee pois (3-2 and (2-2) Many eens Bind it
tnicult tavisuatige the baste af a problemin the fas

nt practice inamethodical manner tohelp studeny,
covers these ditticuties, It is assumed that they hy ay gone OMS tag
teathowhs ofthe 42 level anu Koo the different mathematical concepts sf this level, They
inset to solve problems up to a certain level of difficulty, and are now sendy to tackle
prublems which reguire abit mone application or in somie eases, additional Knowledge, Thy
fullectionof these problemshas, therefore, been dubbed Problems Plus. The salient Features of

the Book ate

or Prem &,ave take any point

Theaimot this book isto provide sufficien

+ At the beginning of every chapter, a summary of facts, formulae and working

techniques are given, with examples wherever necessary. Amumber of finer points,

A lange number of problems of higher-diffculty level have been worked out. These
include many problems already set in competitive examinations, and which illustrate
‘an important point or technique, and many newly framed ones. Since basic
‘knowledge of the chapter is assumed, the Selected Solved Examples soction starts with
problems with slightly higher level of difficulty.

+ Alarge number of the problems in the exercises are newly framed. They have been
‘put in short-answer,lang-answer and objective forms, Problems have been given in
Girect, indirect and inverted fo

‘+ At the end of each chapter there isa time-bound chapter test, Students should try to
finish the test within the time indicated.

‘+ At the end of the book, seven practice tests, based on the knowledge of all the
chapters, are given.

+ This is followed by Miscellaneous Exercises. This will help the reader in gaining
confidence in solving problems without prior knowledige of the chapter(s) to which
the problems belong.

The book contains about 1000 worked-out problems and about 4000 problems given as
exercise. After going through this book, Lam certain that the readers will find themselves more
‘comfortable and confident in facing new problems. Any suggestion to improve the book will
be thankfully received.

Asit Das Gupta

Note to the Third Edition

Tam grateful to the students and teachers, particularly Dr § § Chawla of FIITJFR for sending

invaluable suggestions. In this edition a number of new prot re bes pecially
invaluable supgeton Problems have been added, especially

Asit Das Gupta

oy
Contents

Algebra
1. Progressions, Relate Inequalit
2. Detoreninants and Cramer's Rule
2 Equations, Inequations and Expressions
44. Complex Nombers
5. Permtation and Combination

4, Binomial Theorem for Positive integral Index

AB
7. Principe of Mathematiea Induction (PMI) : ea
8 Infinite Series per
9. Matrices 285
Trigonometry
1. Cicwulae Fuetions,Ienides Ba
2, Solation of Equations B25
3 Inverse Cirle Functions 3a
4. Teigonomelrica Inequalities and Ineqatione Bw
5: Logasthin 357
6 Propertiog of Iiangle Be
7, Heights and Distances* Bs
Coordinate Geometry
1. Coordinate and Straight Lines ca
2 Pair of Straight Vines and Transformation of Axes : ca
3, Cites cas
4. Perabola cas
5. Flipse and Hypecbola 93
Calculus
1. Function Ds
2. Differentiation Das
3. Lint, Indeterminate Form De»
4. Continuity, Dilferentiability and Graph of Fonction Da
5. Application of dy/dx De?
6 Maxima and Minima Der,
7. Monotonic Function and Lagrange’s Theorem Das
8 Indefinite Integration of Blementary Functions 95

o
ad ;
jonal Functions
8. Indefinite tntegeation of Rational and Irrational F
10. Definite Integration
j 11. Properties and Application of Definite Integrals,
12. Area
13, Differontial Equation of the First Order

Vectors

2. Vectors and Addition of Vectors
2. Product of twa Vectors

3. Product of three or mote Vectors
4. Application of Vactors
Algebra
1. Progressions, Rei

Recap of Fact

sro pargiession is knwo by its fst sem

mminon slference d So waiie solving a
sotsgm ot AP, sve fait ae the fest teren anc
fi comanen dileranee. If given than take thent sy

to d

wp given, otherwise assume the fist oon 2
comune siflersner =

«Gesu sormetan APEa, edo Boe 43d

{lf chree or lous nomrbers in AP ane to be coneiseved,

they cam be taker, conveniently as «=, 4,4

o

irc progression is known by is ist toe
{he cermanan valio 1. $9 while solving a prablen

for GP te Ihave to lake the firs. sem and the
Chenmon ratis IF given then la them ae They are

# ard the

given, utherssise assume the fist term

fe

I vee or four numbers in SP ave ta he considered,
ney can be taken conveniently

+ Gan of a GP isa, a»

oe
ver

aaa

+ [fares or four nuunbers i Li! ave to be eisidered,
Vy can be taken as

~ respoctively

2. Classification of a

sen progression
When

ane required 16 verify whether a given
Sequence iin AP,GP ar 11" we proceeds follows

and Formulae

heat, ah

hen che saquunceis i At?

sa!
thes the sec

‘nial, For all ne

win Gr

Zp he= eonceant, for al

sham the sequen:

cisin HP,

etations between vatisbles and faets ale! AP
+ eth texan

Ph
+ By know.nyg he sur tn tormsef an AP wean kaa)
they sean using, = 5,

‘The terms of an AP ane either ine-tasing Gwen d Mt
for decreasing, (whi 20) 54,
+ inthe AP a

fetta. forall?

ag we have the toons

ip Hareateoin AP

levied er be
ayrlitg Bea Bn ot area NP

° leroy
te iy are a0 AP

Sy Bg Bp aye ae soit A.
= The arithmetic mean af v0 nim

4 Thearithentic mean A oft amber, 83,95
ao sel that

* Hetations Betwasn variables nd Fats whom CP
Sth Germ m5, ce

+ Suna term:

+ Sum of sini en “ Trt

provite

* GY towing the sum to terms of GP see ean know
ie atin tora using

in a Gt of positive terme
increasing (viens ty

IMIR tao,

he tenn ace
Secteasing (den

Ve hace the allow

a Sarak ine? geo,
Wels ts. ancalsoin Ge
fy pyro 888 se in GP

"he geometric mean G of 80 mnumbursw ane 66

umbors ya

+ The goometciemean G oti nis

+ Hes: aambers ane set between «and 6 sick thae

HOW Gay bare in GP Bien

5 Relations berween variables and facts ahout 11?

+ rh term

m of comrcspanding AP
9» Way 83 a3 gy atc iM HP then

hud

= constant.

1

+ Thehacmonie mean of two numbers ¢ and bis

«Fra nuabers ae set tweet anit sec) hay

My, Sy fig ar ts HIP en

whered

oan (ilies af positive numbers

ied, GH amin ce

oon

+ Yor tonsa
= For tons positive numbars,

equality hoki Forequal munbers.
+ fer nega positive nambers, A> £

equality holding when al! the nuenbers ae co

Summation of serve oF antral nailer,

+ 2 pou he ued py using the ident.ty

2865 1Be 0K" Ska

‘hich gives

foun of numbeas)?

= Sum of squares of marnbers 2 fem of
Produrts ef numbars taking tis sf
spin forwule fe esetal to fined Ose sum of

Fy eigersed numbers “aking Hw ata Sm Ure
eeeeitale nusvibers ad the sin. of Uheie

rams ein boobies

snatian of relate series

hr tersvof a series oF the fi

vk

Pe eur Tp"

Lined oy using she formulae for 24,

ne saves Be" being GP,
Spyies of following, faring admit of sumeation Ly
tiny she above methnd (enpysn by ¥ meshed)
Wg Pete dN ee) 1 G@= 2AM 1 214

Sehach is obtaivod by the sum of tke products cf

Chnssponding lerms of leo (or wore) APS,

ncgsample | 2207 5-1
ins which the first factors 1,3, 5
arein AP,

sas ovell ag the
second Faeiore 4 7.10
FHA +3e.0 1 a

Ii age ha 20 ee,
4 which te dillerances of eunsveutive terms are

and these ave m AP,

2 2n,1 3,

potas bile oeb

jn which the effferencos of consecutive terms are
ete OF ou and these are in GE.
tke E wethod is nt apoliable in a series whose nth
rf, canal be veritten as sun Of positive integral

powers of » aad powers of eomtunl, be,

ous Tf f, contains a texm of the

enh =e
forey wy or wip", exe, che E method cannot be
applied

Tnserivs ikea 1 (4 abv ~ (a 4 2)

whieh is obtained by the stim of the proftucts of
convespanding terms of an AP and a (3? we get a erm
fs mentioned) abave, Such series are Known a8
vitamelice-yeometric sevies, "he method of solving

Such serie as olTows

1 2+ Abr = (a4 243? +

baa Laie

abe a= dtr? fan Daw

s fan Layoe”

subtracting,

A= nS, 0b 4 Abr ae® 4

ce
evr mans nerie a ME Hern
Ned estbot the seine catoge tele

9, Method of dicference in
Sumrition uf secies

+ the mth term ofasetiescans

by fears 2

HED ORO punt
HWM
revere

rshon "sign ster
when sig een

J=Fy shen! “sgn ate
oF the wuziee Ye sue hat the eh sere and e = Tall
term can be related a3 follawshen sso the ees.

bead:

or By UY APNE EON
Gaim eR TI Bool
Lea, fn es
or Sy=q POM
6) tape hey FO
‘vhen .
Sten Stns tea BF
, or Bheah

= amy +f

FO) te. FDA
yor af

rp ODA