Preview Krishnas Chemistry For IIT JAM, 2nd Edition (2021)

https://1drv.ms/b/s!AmkCsf2WlV7n1WoyCT5nnvO18-29?e=EB8hdN

By

Alok Bariyar

Ph.D (CSIR fellow), M.Sc.(IIT Delhi), G.A.T.E., N.E.T.

Ex Nuclear Scientist

Bhabha Atomic Research Centre,

Syllabus

PHYSICAL CHEMISTRY

Basic Mathematical Concepts:

Atomic and Molecular Structure:

Basic Concepts in Organic Chemistry and Stereochemistry:

Organic Reaction Mechanism and Synthetic Applications:

Qualitative Organic Analysis:

Natural Products Chemistry:

1 Functions; maxima and minima; integrals; ordinary differential equations; vectors and matrices; determinants; elementary statistics and probability theory.

2 Fundamental particles; Bohr's theory of hydrogen-like atom; wave-particle duality; uncertainty principle; Schrodinger's wave equation; quantum numbers; shapes of orbitals; Hund's rule and Pauli's exclusion principle; electronic configuration of simple homonuclear diatomic molecules.

3 Equation of state for ideal and non-ideal (van der Waals) gases; Kinetic theory of gases; Boltzmann distribution law; equipartition of energy.

Maxwell-4 Crystals and crystal systems; X-rays; NaCl and KCl structures; close packing; atomic and ionic radii; radius ratio rules; lattice energy; Born-Haber cycle; isomorphism; heat capacity of solids.

5 Reversible and irreversible processes; first law and its application to ideal and nonideal gases; thermochemistry; second law; entropy and free energy; criteria for spontaneity.

6 Law of mass action; Kp, Kc, Kx and Kn; effect of temperature on K; ionic equilibria in solutions; pH and buffer solutions; hydrolysis; solubility product; phase equilibria-phase rule and its application to one- component and two-component systems; colligative properties.

7 Conductance and its applications; transport number; galvanic cells; EMF and free energy; concentration cells with and without transport; polarography; concentration cells with and without transport; Debey-Huckel-Onsagar theory of strong electrolytes.

8 Reactions of various order; Arrhenius equation; collision theory; transition state theory; chain reactions - normal and branched; enzyme kinetics; photochemical processes; catalysis.

9 Gibbs adsorption equation; adsorption isotherm; types of adsorption; surface area of adsorbents; surface films

on liquids.

10 Beer-Lambert law; fundamental concepts of rotational, vibrational, electronic and magnetic resonance spectroscopy.

and steric effects and its applications (acid/base property); optical isomerism in compounds with and without any stereocenters (allenes, biphenyls); conformation of acyclic systems (substituted ethane/n-propane/n-butane) and cyclic systems (mono- and di-substituted cyclohexanes).

free radicals, carbenes, nitrenes, benzynes etc .); Hofmann-Curtius-Lossen rearrangement, Wolff rearrangement, Simmons-Smith reaction, Reimer-Tiemann reaction, Michael reaction, Darzens reaction, Wittig reaction and McMurry reaction; Pinacol-pinacolone, Favorskii, benzilic acid rearrangement, dienone-phenol rearrangement, Baeyer-Villeger reaction; oxidation and reduction reactions in organic chemistry; organometallic reagents in organic synthesis (Grignard, organolithium and organocopper); Diels-Alder, electrocyclic and sigmatropic reactions; functional group inter-conversions and structural problems using chemical reactions.

20 Essentials and trace elements of life; basic reactions in the biological systems and the role of metal

ions, especially Fe , Fe , Cu and Zn ; structure and function of hemoglobin and myoglobin and carbonic anhydrase.

21 Basic principles; instrumentations and simple applications of conductometry, potentiometry and UV-vis spectrophotometry; analysis of water, air and soil samples.

22 Principles of qualitative and quantitative analysis; acid-base, oxidation-reduction and complexometric titrations using EDTA; precipitation reactions; use of indicators; use of organic reagents in inorganic analysis; radioactivity; nuclear reactions; applications of isotopes.

Aromatic and Heterocyclic Chemistry:

INORGANIC CHEMISTRY

Periodic Table:

Chemical Bonding and Shapes of Compounds

Main Group Elements (s and p blocks):

Transition Metals (d block):

Bioinorganic Chemistry:

Instrumental Methods of Analysis:

Analytical Chemistry:

Weightage of Inorganic, Organic & Physical Chemistry

Questions Asked in Last Three Years

INORGANIC

JAM 2016

Physical Chemistry : 23 Questions

Organic Chemistry : 20 Questions

Inorganic Chemistry : 17 Questions

JAM 2017

Physical Chemistry : 27 Questions Organic Chemistry : 20 Questions Inorganic Chemistry : 13 Questions

JAM 2018

Physical Chemistry : 28 Questions Organic Chemistry : 22 Questions Inorganic Chemistry : 10 Questions

INORGANIC

PHYSICAL

ORGANIC

PHYSICAL INORGANIC

ORGANIC PHYSICAL

Preface

I am happy to present this book entitled "Chemistry for IIT JAM" It has been written according to the Latest syllabus

to fulfil the requirement of students.

The book is written with the following special features:

1 It is written in a simple language so that all the students may understand it easily.

2 It has an extensive and intensive coverage of all topics.

3 The complete syllabus has been divided into 3 sections having 22 Chapters.

4 Sufficient Numerical Problems, Subjective Questions and Objective type questions with Hints & Solutions given at the end of

each chapter will enable students to understand the concept.

5 Sufficient Mock Test Papers have also been added for the practice of students

First of all I want to express my sincere gratitude to Dr S.B.P Sinha, Prof J.C Ahluwalia, Prof N.K Jha for their

invaluable guidance, immense interest and constant encouragement for the successful completion of the work I am

also thankful to Bandana Bariyar, Abhishek Bariyar, Purnima Sinha & family for their kind help at many occasions.

I am extremely grateful to my respected and beloved parents whose incessant inspiration guided us to accomplish

this work I also express gratitude to my respected family members for their moral support.

I am immensely thankful to Managing Director, Executive Director,

getting the book published.

The originality of the ideas is not claimed and criticism and suggestions are invited from the students, teaching

community and other readers.

Author

one or more than one

3 contains a total of 10 Each MSQ type question is similar to MCQ but with a difference that there may be choice(s) that are correct out of the four given choices The candidate gets full credit if he/she selects all the correct answers only and no wrong answers Questions Q.31-Q.40 belong to this section and carry 2 marks each with a total of 20 marks

4 contains a total of 20 questions For these NAT type questions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor No choices will be shown for these type of questions Questions Q.41-Q.60 belong to this section and carry a total of 30 marks Q.41-Q.50 carry 1 mark each and Questions Q.51-Q.60 carry 2 marks each

5 In all sections, questions not attempted will result in zero mark In Section-A (MCQ), wrong answer will result in

marks For all 1 mark questions, 1/3 marks will be deducted for each wrong answer For all 2 marks questions, 2/3 marks will

be deducted for each wrong answer In (MSQ), there is and marking provisions There is marking in Sections-C (NAT) as well

1 Students generally start their study with reference books and other advanced study materials They forget that to fight a battle in the field it's important to learn basic concepts of the fight, which is applicable

to IIT JAM as well Not working on conceptual terms tend to make students puzzled when the question is slightly twisted.

2 It's a natural tendency of humans to say, “I'll do it from tomorrow onwards” Making a schedule alone is not enough to make student win this battle, you must work on it with full dedication To create an effective time- schedule, save the important dates for the exam and plan up your activities accordingly

3 Self confidence is the key to success, you can lose even a small fight if you have doubts about yourself

On the other hand, self-confidence is the only key that can lead you to win over the battles Giving a little space to doubts leads to hesitation and lack of confidence Have trust in yourself and make a positive mind set about whatever you have prepared is more than enough for exams This will increase your confidence and you will perform well in the exam.

4 Mock Tests are practice tests used as a mirror to your skill sets They help you in finding out the true analysis of your performance & knowledge level Helping you in increasing your speed, they also improve your answer giving skills with greater accuracy One should never skip these tests as they are the most important part of preparation.

5 Taking too much stress or no stress, both can be harmful to a student Many students panic a lot which worsens their performance in the exams and many of them are so relaxed that they do not bother even about the revision Students must understand that the preparation they have done might be enough but revision is of equal importance

Avoiding the Basic Concepts :

Tomorrow Never Comes :

2 By making summary notes students can focus on key points of important topics.

3 During revision take short breaks after every hour not every 10 minutes so that you feel energetic.

4 Having less sleep could impact your next day revision and could lead to less concentration for revision Research has shown that lack of sleep leads to clogging of brain and makes us forget sooner.

5 Revise in a room that is quiet and you don't get distracted from outside noises Use proper chair and table for revision Don't try to revise in bed Wash your face and eyes with water to avoid sleeping state Lastly, be confident before exams with all the revision you have done Remember revision is a skill that leads to improvement of memory and enahnces learning ability of students Practice makes one perfect but effective Revision Techniques put excellence in ones perfectness.

Start Revising Early:

Make Summary Notes:

Take Short Breaks:

Have a Proper Sleep:

Revision Room and Sitting Conditions:

Mock Test Papers

Latest Examination Paper

Section

Chapter 1: Basic Mathematical Concepts

Chapter 4: Solid State

Chapter 7: Electrochemistry

Chapter 10: Spectroscopy

Chapter 2: Atomic and Molecular Structure

Chapter 5: Chemical Thermodynamics

Chapter 8: Chemical Kinetics

Chapter 3: Theory of Gases

Chapter 6: Chemical and Phase Equilibria

Chapter 9: Adsorption

• Functions • Maxima and minima • Integrals • Ordinary differential equations • Vectors and matrices • Determinants • Elementary statistics

and probability theory.

INTRODUCTION TO FUNCTION

Let A & B be two sets & let there exist a rule or manner or correspondence ’f ’ which associates to each element of A, a unique element in B Then f is called a function or mapping from A to B It is denoted by the symbol f A : → or A f B which reads “f ” is a function from A to B or f maps ′ B A to B.

A relation R from a set A to a set B is called a function if

(i) Each element of A is associated with some element of B.

(ii) Each element of A has unique image in B.

Some important points :

To every x1 there is one & only one y i.e.

So it can be concluded that One many ⇒ Not a function Many one ⇒ Function.

INTERVALS

The set of numbers between any two real numbers is called interval Types of intervals :

1 Closed intervals : Representation : [ , ] , a b i e a ≤ x ≤ b

2 Open intervals : Representation : ] , [ a b or ( , ) , a b i e a < x < b

3 Semiclosed or semi open : [ , ] a b → a ≤ x < b

12

123

a b c

a b c d

Not a function

Is a function

Is a function

EXERCISE A

1 Solve − ≤ 6 2 1 ( − x ) ≤ 8

2 Solve − ≤ 3 3 − 2 x ≤ 7 & − ≤ − 4 1 5 x < 6

WAVY CURVE METHOD

If a b , = 0 ⇒ either a = 0 or b = 0 But if a b ⋅ > 0 ⇒ if we conclude

Step I : First arrange all values of x at which either numerator or

denominator becomes zero.

Step II : Values of x at which numerator becomes zero should be

marked with dark circle.

Step III : Values of x at which denominator becomes zero should

be marked with empty circle Check the value of f x ( ) for any real

number greater than the right most marked number or the number

line.

Step IV : From right to left, draw a wavy curve (Beginning above

the number line in case of value of f x ( ) is positive in step III otherwise

form below the number line) Passing throughly all the marked

points so that when passes through a points (exponent or power

whose factor is odd) intersect the number line & when passing

thoroughly a point (exponent whose corresponds factor is even) the

curve doesn’t intersect the real line & remain on the same side of real

line.

Step V : The appropriate intervals are chosen in accordance with

the sign of inequality (the function f x ( ) is positive wherever the

curve is above the number line It is negative if the curve is found

below the number line) Their union represents the solution of

Check f x ( ) for x > 7, f( )8>0

∵ Exponents of factor of −1 3, , & are even, 4

So wave will not change the direction at these points

So lu tion : Don’t cancel common factor, instead, add their power Here

for x − 1, 3 5+ = i.e., even wave will not change the direction.8

Range → Set of images { , , , 1 4 9 16 }

Range is subset of codomain

i.e., Range ≤ Codomain

Domain : Set of values of x for which the (input) function is defined

(Not zero in the denominator) & real (Not negative in square root).

Range : Set of collection of values of y (output) for which f x ( ) is

defined & real.

Ex am ple 1: Find the do main of fol low ing func tions

So lu tion : In given question

x

x

x x

0 0 Now in case (I) | | x ≥ 0

in case (II) | | x ≥ x

But | | x ≠ less than x.

Example 1: Solve for x : | x −1|+2(x −2)=x+4

149162536

EXERCISE D SOLVE FOR x

2x − ≤ − or 21 2 x − ≥1 2

x ≤ −1

2 or x ≥

32Using property I

− ≤4 2x − ≤1 4 or −3≤ ≤

2

52

12

32

52

= | |

| |

3 | x ± y | | | | | ≤ x + y

GREATEST INTEGER FUNCTION OR STEP UP FUNCTION

The function y = f x ( ) = [ ] is called the greatest integer function x

where [ ] x denotes the greatest integer less than or equal to x.

f x ( ) : [ ] = Integer part of x just less than or equal to x x

if if

2 f x ( ) = ln ( [ 2 x + 2 14 ] − )

3 f x

( ) [ ] [ ]

=

1 6

∵ 2 3 = 2 + 0 3

x = [ ] x + { } x Definition f x ( ) = { } x = x − [ ] x gives the fractional part of x &

x ∈( , ) 0 1

e.g., { } 27 = 07 { } 3 = 0 { − 3 } = 0

2 7

2 7

= − 1 2 =

7

5 7 { } ∩ = ∩ − ∩ = ∩ − 3 [ ]

5 { x ± I } = { } x i.e., { x + 2 } = { } x

–2–3

–1–2–3

+1+2+3

Ex am ple 2: Find do main for f x

4 or { }x ≠

34

x≠n+ 1

4 or { }x ≠n+

34

14

The function f x ( ) = ax = ex where

a > 1 & a ≠ 1 is called an exponential function The inverse of this

function is called logarithmic function i.e., f x a a

if a > 1 i.e., 2x, 3x then graph

if 0 < a < 1 then graph

4 log (a m × n ) ⇔ logam + logan

5 logam loga loga

Ex am ple 1: Find the value of :

(i) 0 8 1 [ +9log3 8]log5 5

(ii) log23log34log45log56log67log78

So lu tion : For do main log1 2/

12

So common value D f = −∞( , )1 ∪[ , )0 5

EXERCISE J

Ex am ple 1: Find do main of

(i) f x( )=log (1−log (x2−5x+16)

(ii) f x( )=sin log x

3(iii) f x( )=log sin (10 x−3)+ 16−x2

Set of collection of values of y for which f x ( ) is defined & real.

Method of find range :

Method I: When Df = Finite element

= { , a a a2, 3 an} then Rf = { ( ), ( ), ( )} f a1 f a2 f an

x x x

Express x in terms of y i.e., x= ( ).g y

Find domain of g y ( ) It will be range for f x( )

Ex am ple 1: Find range of func tion:

(i) f x x

( ): : ( { }

2 2

x y2 = +1 x2 or x2(y−1)=1

x y

= ±

−

11for this to be defined 1

1 1

11

yx2+yx+y=x2−x+1

x2(y−1)+(y+1)x +y−1=0

Taking y as constant

y − ≠1 0 or y ≠ 1 Also D ≥ 0 or ( y+1)2−4(y−1) (y−1)≥0

−1x cot−1x, cosec−1x

CLASSIFICATION OF FUNCTION

I One-one function (injective):

It can be defined as different x ⇒ diferently.

II Many one function:

It can be defined as more than one value of x = same y

HOW TO IDENTIFY ONE-ONE & MANY ONE FUNCTION

Method I: By inspection :

If domain R → , many one R

If domain ∈ N one-one

Method II: By graphical method :

If a line parallel to x-axis cut the graph exactly at one point, then

function is one-one otherwise many one if Df→ R → R

Method III: By increasing or decreasing function for one one

function ′ f x ( ) ≥ 0 or ′ f x ( ) ≤ 0 i.e., function must be strictly increasing

or decreasing.

Method IV: ONTO function (surjective)

Here in this function, no extra element in co-domain i.e., Rf =

Co-domain.

Method V: INTO function

Here in this function, at least one element extra in the co-domain.

Method VI : If a function is injective & surjective then that function

is called bijective.

How to test into function.

First of all find range.

x

x x x x

21







log

32010

ψ2

y

x x

32012

−x=Using previous result

2012

20112012

22012

So lu tion : (i) fog f g x: ( ( )) : (sinf x+ cos )x

: (sinx+cos )x2−1= +1 2sinxcosx −1=sin2 x

(ii) gof g f x: ( ( )) : (g x2−1)=sin(x2−1)+cos (x2−1)

(iii) gog g g x: ( ( )) : (sing x+ cos )x =sin (sinx +cos )x

(iv) fof f f x: ( ) : (f x2−1) : (x2−1)2−1:x4−2x2

Ex am ple 2: Let g x( ) : 2x + and h x1 ( ) : 4x2+4x+ then find a7

func tion f x ( ) such that fog= h

Ex am ple 1: Let f x ( ) : 2 and g x x ( )= 1/x and h x( ) :e x then

prove that (gog oh) =fo goh( )

Ex am ple 2: If the do main of f x( ) is [ / , ],1 2 1 then find the

Def II: If f g x ( ( )) = g f x ( ( )) = x = each one is inverse of other I

Inverse of a function is unique.

If we are given whether a function is invertibile or not, we will check

that y = ( ) is one-one and onto f x

Ex am ple 1: If f : ( ,2 ∞ → and f x) R ( ) :− +1 ln x( −2 Find f) −1( ).x

So lu tion : ∵ f x( ) :− +1 ln x( −2 is bijective)Let y= f x( )= − +1 ln x( −2)

Ex am ple 3: Let f : [ / ,1 2 ∞ →) [ / ,3 4 ∞ & f x) ( ) :x2−x+ then1solve

2

34

So lu tion : x = 1

ODD FUNCTION AND EVEN FUNCTION

If f ( − x ) = − f x ( ) ⇒ f x ( ) + f ( − x ) = 0 then called as odd function.

e g , xodd power, sin , tan cot , cos x x x ec x

odd functions are symmetric about the origin or opposite

quadrant and f o ( ) = 0 (if defined)

A-12

1234

14916

14916

1234

f : Y → X

z g (y) g f(x)

g of

y g (x) x

f : X → Y g : Y → Z

EVEN FUNCTION

If f ( − x ) = f x ( ) i.e., f x ( ) − f ( − x ) = 0 then called as even function.

e.g., xeven power, cos , sec ,| | x x x

Even functions are symmetrical about y axis y =| | x

f x ( ) = 0 is even as well as odd function.

f x( ) g x( ) f x( )+g x( ) f x( )−g x( ) f x g x f x

g x

( ) ( )

gof x( )

PERIODIC FUNCTION

A function f x ( ) is called periodic if there exist a positive number

T T ( > 0 called the period of the function such that: )

f x ( + T ) = f x ( ) for all values of x within the domain of x.

Note : { } x is a periodic function with the periodic = 1 sin x is a

periodic function with the period = 2π.

LIST OF SOME STANDARD PERIODIC FUNCTION

Periodic function Fundamental period

sin , cosx ecx, cos , secx x 2π

|sin |,|cos |,|cot |x x x

|tan |,|sec |,|cosx x ecx| π

sinn x, cosn x, cosecn x, secn x n (even) = 17

n (other than even) = 2π

periodsinx2, cos x, sinx3 Not a periodic function

HOW TO FIND PERIOD

Rule I: If f x ( ) → then f ax T b T

a

| | . + = But f x ( ) ± a a f x , ( ) and

f x a

( )

has period T only.

Rule II: gof x ( ) = g f x ( ( )) is a periodic if f x ( ) is a periodic function.

And period of gof x ( ) = period of f x ( ) but it may give fundamental period.

Rule III: If f x ( ) = f x1( ) − f x2( ) + f x3( ) ( ) + f xn then period of

f x ( ) = LCM ( , T T T1 2, 3 Tn) Again it may not give always fundamental period.

Ex am ple 1: Find pe riod of f x( )=2| cos (πx +3) |+7

So lu tion : T = π =

Ex am ple 2: Let f x( )=e3x+ −1 3[ x+1] has a pe riod T1 &

g x( )=e3(x−[ ])x has a pe riod T2 Find re la tion between T1 & T2

So lu tion : As f x ( ) = e3x+ −1 [3x−1]= e{3x+1} is a pe ri odic func tion with pe riod 1 3 /

Ex am ple 3: Find pe riod for f x( +4)=f x( +6)

2017 2

++

So lu tion : (i) Odd

−+

sin

sin

coscot

t t

t

t dt

−+

( )11

1

tan−+

1 If m → even and n → odd; put sin x = t

2 If m → odd and n → even; put cos x = t

3 m n1 is odd, put either sin x or cos x = t

4 m n1 is even, then simplify

5 If m n1 is fraction and m + n = negative even integer put

± + or Algebric twins

Divide numerator and denomenator by x2 and use a2+ b2

= ( a ± b )2 ∓ 2ab as per required.

Ex am ple 1 In te grate the fol low ing:

1 1

+ +

1 1

+ +

∫ t / / t t dt or 1 1

3 2

+

∫ ( t / ) t / t dt Put t − 1 / t = Z

12

12

tantan

1

2 2

12

a sin x + b cos x ± a sin x cos x

Divide numerator and denomenator by cos2x A-16

Type II:

a ± b sin x , a ± b cos x , a sin x ± b cos x a , sin x ± b cos x + c

tan / , cos

tan / tan /

x x

=

− +

2

2 or convert into polar form.

Type III : ± sin ± cos

(sin )

Use sin 2 x = (sin x + cos ) x 2− 1

or sin 2 x = 1 − (sin x − cos ) x2

∫

So lu tion : (i) 1

2 3

33

23

sin x lnx , sin x will be v.

Ex am ple 1: In te grate the fol low ing :

Two classical integerals :

2 2

2sin (log )−cos (log )+

3 sin−1xsec sin−1x−log | sec sin−1x+ tan−1x|+ c

7 xsin (log ) +x c

INTEGRATION BY USING PARTIAL FRACTION

Type I: When denominator has only linear factor (non-reperted).

How to partiate the fraction.

1 4

1

1 2

5 4 3

Q x

R x

∫

1 2sin (x − cos )x dx

11

12

11

34

Bx C x

+

++

4

2 2

12

514

72

52

1

2

tan−

++







+

DIMENSION GEOMETRY

DIRECTION COSINE OF LINE

Just like in 2D, slope is basic for a line In same, 3D, for a directed line segment or vector, direction cosine is basic to represent it Let →a = x^i + y^j + z^k

Then angle in cosine made by x-axis, y-axis and z-axis is known as

direction cosine of a vector or line.

m b

n c

a

→

DC

α

Direction ratio of line through two points give

Note : Dr of any line may be many but 2 DC of any line in unique.

Angle B/W two lines (in terms of dr/dc)

c c

1

2

1 2 1 2

= = or l

l

m m

n n

1 2 1 2 1 2

Ex am ple 1: The pro jec tion of a vec tor on the co or di nate axis

are 2, 3, 6 re spec tively Find its di rec tion co sine

67

23

BC = −2^i+ 2^j+^k So DC of BC = <−2 >

3

23

13

Ex am ple 1: A line in x y, plane makes an an gle of 30° with

the x-axis Find its di rec tion co sine.

Ex am ple 2: A line makes an an gle of α β γ, , with x y, and

z-axis re spec tively then find

1 sin2α+ sin2β+sin2γ

2 Σ cos2x

di ag o nal of a cube

Ex am ple 4: A line makes an an gle α β γ, , and δ with the four

di ag o nals of a cube Then prove thatcos2α+cos2β+cos2γ+cos2δ=4 3/

di rec tion co sine are con nected by the re la tion

STRAIGHT LINE IN SPACE

Type I: One point form:

Equation of line passing through a point A ( )→a and parallel to vector

1 1 1

→

u< l i + m

, j + n k

1 1 1

→

V< l

i + m , j +

n k

2 2 2

→

V< a

i + b , j +

c k

2 2 2

A(x , y , z )1 1 1 B(x , y , z )2 2 2

Symmetric form/cartesian form

Type II: Two point form

Note : Two lines are parallel if distance between

them is always same and both line in the same plane.

Note : Coplaner = Intersecting

SKEW LINES

Such pair of lines which are neither parallel nor intersecting i.e.,

non-coplaner formula for shortest distance: SD

2

35

35

23

12

3

2 at a

dis tance of 5 units from point P( , , ).1 3 3

So lu tion : So ran dom point

Q= x+2=y+ = z− =

3

12

Ex am ple 3: If the lines x=ay+b , z=cy +d and ′ = ′ +x a y b , ′

=

P(1,3,3)

Q (3

,–1,–2,2λ–1,2λ+

XY plane P(x,y,o)

EXERCISE B

Ex am ple 1: Find the equa tion of the line pass ing through a

point with the po si tion vec tor 2i^−3^j+5k^ and ⊥ to each of

the lines Whose equa tion is

If they intersect then find the point of intersection

Ex am ple 3: Find the co or di nates of the foot of ⊥ r from the

1

12

23

dis tance of point from line and im age of the point w.r.t to

Ex am ple 5: Find the length of short est dis tance between the

3

81

72

64Also find the equation of line of shortest distance

52

6

815

33

where a b c , , are dr of normal to the plane.

Type II: Point and normal vector to the plane.

(→r −→ →a n ) = 0

or a x ( − x1) + b y ( − y1) + c z ( − z1) = 0

Type III: If three points are collinear, then infinite planes will pass

through three collinear points.

But if three non-collinear point then only one and only the plane is possible

y b

c z

C (o,o,c)

A (a,o,o)

B (o,b,o)

ANGLE BETWEEN TWO PLANES OR ANGLE BETWEEN THEIR NORMALS

Corollary : If two parallel if normals are parallel i.e.,

d r of both planes are proportional and distance

between two parallel planes.

Ex am ple 1: Find the dis tance of plane −3x+ 4y−12+39=0

from the or i gin

Ex am ple 2: Find the equa tion of plane through the mid point

of line seg ment A ( , , ) −1 2 3 and B ( ,1−1 2, ) and per pen dic u lar

52

2

52

^ ^ ^ and n→=2^i −3^j−^k

So equation of plane (→r −→ →a n) = 0 or r n→.→=→ →a n,

Ex am ple 3: Find the equa tion of plane which makes the

in ter cept twice, thrice and four times on the co or di nate axis

Ex am ple 5: Find the equa tion of line through ( ,2 −1 1, )

par al lel to the plane 4x +y+ z+2= and per pen dic u lar to0

51

=

So lu tion : Let dr of the line < a b c, , >

i.e., line is perpendicular to normal of plane.

Also line is perpendicular to another line

51

3 13

Ex am ple 6: Find the dis tance of point ( ,1−2 3, ) from the

23

117

157

Ex am ple 2: A vec tor n→ of mag ni tude 8 units makes an an gle

of 30° with x-axis, 60° with y-axis and an acute an gle with

z-axis If the plane passes through the point ( ,2 − 2 2, ) and

nor mal to the n→ vec tor, then find the equa tion of plane

in ter sec tion x+y+ z= 6 and 2x =3y+4z+5= and the0point ( , , ).4 4 4

32

3

So lu tion : 12x+13y+10z−210=0

A-25

Ex am ple 9: A line through the point P ( ,2 −1 2, ) with the

pos i tive di rec tion co sine makes equal an gle with the

co or di nate axis If the line in ter sect the planes 2x+y+z=a

at a point Q then find the length of PQ.

So lu tion : PQ = 3

Ex am ple 10: A vari able plane re mains at con stant dis tance

3p from the or i gin cuts the co or di nate axis at A, B and C

re spec tively Then find the lo cus of cen troid of ∆ABC.

Vector is a directed line segment

2 If a b r→ → →, , are coplanar then

(i) Triple product is zero

AREA OF PARALLELOGRAM AND THE TRIANGLE

1 Area of the with parallelogram with a→ and b→ as adjacent side

PRODUCT OF THREE VECTORS

1 Scalar triple product:

= 1 → → →

6 [ a b c ]

= 1 → → ×→

6 [ ( a b c )]

Where a b→ →, and c→ are coinitial vectors.

PROPERTIES OF SCALAR TRIPLE PRODUCT:

B

Ex am ple 3: Let A( , , ),1 2 3 B( , , )2 1 2 C ( , , ), 2 3 1 D ( , ,1 2 4 Find the ).

pro jec tion of AB on CD

Ex am ple 5: Find the ara of par al lel o gram con structed on the

vec tor a→ =→p + 2 and b→q → =2→p+→q where p→ and q→ are any two

unit vec tors form ing an actute an gle of 30°

So lu tion : Area of par al lel o gram =|→a×→b| |(= →p+2→q)×(2→p+→q) |

p q sin π

Ex am ple 6: The num ber of dis tinct real value of λ for which

the vec tors −λ2i +i^+^j+k^

or (1 2) (1 2) 0

Hence ± 2 is distinct real values of λ

Ex am ple 7: If a b c→,→, are unit coplaner vec tors then find the→sca lar tri ple prod uct of [2→a−b→,2→b−→c,2→c−→a]:

So lu tion : Given | | | | | |→a = b→ = →c = 1Also [→ → →a b c]= 0 (Coplaner)Now [2→a−→b,2b→ −→c,2→c−→a]

Then find a vector which is coplaner to a→ and b→ Also has a

projection along c→ of magnitude 1

3

So lu tion : r→ =2^i−^j+ 2k^

Ex am ple 4: Let a→=2^i+^j−2k^

and b→ = +^i ^j If c→ is a vec tor such

that a c c→.→|→| , |→c−→a|= 2 2 and the an gle be tween (→a×b→) and

Ex am ple 5: If a b c→,→, are any three vec → tors such that

Ex am ple 7: Find the vol ume of parallelopiped whose edges

are rep re sented by the vec tors

So lu tion : (i) 1 (ii) 3

Ex am ple 11: Let a→= − −^j k^, b→ = − +i^ ^j , c→= +^i 2^j+3 and if r k^ → is

a vec tor such that r→×b→ =→c × and r b→ →.→a=0 then find r b→.→

So lu tion : 9

MATRIX AND DETERMINANT

Regular array of numbers or letters is called Matrix.

Trace of matrix ( AB = Sum of diagonal elements Also trace of ( ) AB )

may or may not be equal to trace of ( BA ).

Interchanging of rows and columns is called the transpose.

= − , it is called skew symmetric matrix.

SOME SPECIAL TYPE OF MATRICES

ADJOINT OF A MATRIX (ADJ A)

It is the transpose of matrix form by the cofactors of matrix A Cofactors is calculated as :

Cij (cofactors of i, j row and colom) = − +

A-29

Method to solve it : use cramer’s rule

Also remember if D ≠ 0, unique solution Further if D ≠ 0 but

Dx = Dy= Dz= 0 : zero solution or trivial solution.

and if D ≠ 0 and atleast D D D1, 2, 3≠ 0 : Non trivial solution.

But if D = 0 and D1= D2= D3= 0 : Infinite solution.

If D = 0 and atleast D D D1, 2, 3≠ 0 ; no solution.

Also D ≠ 0 ; Trivial or zero solution.

D = 0 ; Non-trivial or non-zero solution.

Determinant of skew symmetric matrix of odd order = 0.

PROPERTIES OF DETERMINANT

I : Interchange of all rows and columns in a determinants doesn’t

affect its value.

This property is also known as switching property.

Note : For triangular determinant like

a b c

0 0 0 α

P-V : If any row/column is added or substracted by the scalar.

Multiple of any other row/column doesn’t affect the determinant (atleast on row/column remaining as it is).

REMEMBER THESE AS A RESULT

(i) Symmetric det :

So lu tion : ∵ Equa tion has atleast one so lu tion

Ex am ple 6: If the equa tion

ax+4y+z=0

bx+3y+ z=0

cx+2y+z=0 have non-trival solution, then a, b, c are in which progression?

So lu tion : For non-triv ial so lu tion, D = 0 a

b c

4 1

3 1

2 10

=

a(3−2)−4(b c− )+1 2( b−3c)= or a0 + c= 2b Hence a b c , , are in AP.

Ex am ple 7: Find the value of de ter mi nant :

n n

K n

4 Fill in the blanks :

(i) If A is involuntary matrix, then inverse of A

has no solution Find α :

Question related to determinant :

Ex am ple 1: If [x r,y r]=r= 1 2 3 are any three points where , ,

x r (r = 1 2 3 i.e., ( ,, , ) x x1 2,x3) and y r (r = 1 2 3 i.e., ( ,, , ) y y y1 2, 3)

both are on a aP with same com mon ra tio, then find the ara

+++

3 1 2

Ex am ple 3: If a b c, , are the p th q th

, and r th term of HP, then

So lu tion : (i) 2A (ii) A T

Ex am ple 5: If a a a1, 2, 3 .a a n n+1,a n+2 . are in GP, then find

the value of

=

∑1

DEPENDENT AND INDEPENDENT EVENT

If P A P B ( ) ( ) = P A ( ∩ B ) then events are independent, else

dependent.

MULTIPLICATION THEOREM OF PROBABILITY

Using conditional probability we know that

i i i

n

i i

Binomial distribution generally deals with the dischotomous

situation i.e., Head or Tail, agree or disagree, success or failure etc.

Let n → number of trials (Binomial trial)

P → probability of success

q → probability of failure then p + q = 1 or q = 1 − P Now let X = Random variable of Binomial distribution

X r

P X=( )

Theorem of Total Probability

Let E E E1, 2, 3 En are mutually exclusive i.e., Ei∩ Ej= φ ∀ i j ,

and are exhaustive i.e., E1∪ E2∪ E3 ∪ En= 5

And A be any event which is associated with E E1, 2 En Clearly A = A ∩ S = A ∩ ( E1 ∪ E2∪ E3 ∪ En) = ( A ∩ E1) ∪ ( A ∩ E2) ∪ ( A ∩ E3) ∪ ( A ∩ En)

P A ( ) = P A ( ∩ E1) + P A ( ∩ E2) + + P A ( ∩ En) = 

n n

i

n i i

Ex am ple 1: 3 Nat u ral num ber are cho sen at ran dom from

first hun dred nat u ral num ber What is the chance that these

num bers are di vis i ble by 2 and 3

C C

prob a bil ity that the num ber ap peared on them make a sum

13

13

34

23

112

Ex am ple 3: As sum ing that each born child is equally likely

to be a boy or a girl A fam ily has two chil dren Find the

prob a bil ity that both chil dren are girls that atleast one is

the girl in the fam ily

So lu tion : Here sam ple space : {BB GG BG GB, , , }

A : Both are girls : { GG}=n A( )= 1

B : At least one girl { GG GB BG, , }

Ex am ple 4: A box of 15 or anges con tain 3 de fec tive or anges

3 or ange are drawn one af ter one with out re place ment

Then find the prob a bil ity that all three or ange are de fec tive

C

( 1, 2, 3)

3 3 15 3

1455

Ex am ple 5: A speaks truth in 60% of cases while B speaks

truth in 75% of cases Then find the prob a bil ity that theycon tra dict in stat ing the same fact

So lu tion : P (A speaks truth) = 60 =

100

35

P A( ) ⇒ speak lie = 40 =

100

25

Similarly P B( )= 3, ( )P B =

4

14

So required P=P AB( )+P AB( ) =3× + × =

5

14

25

35

920

Ex am ple 6: A coin is tossed three times Find the prob a bil ity

vari ance and stan dard de vi a tion

38

18

=

∑x P i i i

32 Variance =E X( 2)−( ( ))E X 2= 3

4Standard deviation = Variance=0 86

Ex am ple 7: The prob a bil ity of hit ting the tar get is 1

4 Thenfind the prob a bil ity of two hits in five shots

So lu tion : P = Prob a bil ity of hit ting tar get =1

1352

C q p

Ex am ple 8: Three groups A B C, , are con test ing for the board

of di rec tor of a com pany And the prob a bil i ties of their

win ning are 0.7, 0.5 and 0.6 re spec tively If group A wins,

then prob a bil ity of in tro duc ing the new prod uct is 0.1 while

prob a bil ity of launch ing the new prod uct

So lu tion : Here E1 = Win ning group A P E( 1)=07

A-34

SOLVED EXAMPLES

C