kvbilaspurlibrary.files.wordpress.com · 2 श्री संतोष कु ा ल्ल,...

1

REFERENCE MATERIAL

2

श्री संतोष कुमार मल्ल, भा.प्र.से. माननीय आयुक्त

के. वि. सं. नई दिल्ली Mr.Santosh Kumar Mall, IAS

Hon’ble Commissioner

KVS New Delhi

श्री जी. के. श्रीिास्ति

अपर आयुक्त (प्रशा.) के. वि. सं. नई दिल्ली

Mr.G.K.Srivastava, IAS

Additional Commissioner (Admin.)

KVS New Delhi

श्री यू. एन. खिारे

अपर आयुक्त (शैक्षणिक)

के. वि. सं. नई दिल्ली Mr.U.N.Khaware

Additional Commissioner (Acad.)

KVS New Delhi

डॉ. शची कांत

संयुक्त आयुक्त (प्रशश.)

के. वि. सं. नई दिल्ली Dr.Shachi kant

Joint Commissioner (Trg.)

KVS New Delhi

डा.(श्रीमती) िी विजयलक्ष्मी संयुक्त आयुक्त (शैक्षक्षक) के. वि. सं. नई दिल्ली

Dr.(Smt.) V Vijayalakshmi

Joint Commissioner (Acad)

KVS New Delhi

संरक्षक /Patrons

3

OUR TEAM

COURSE DIRECTOR

Ms USHA ASWATH IYER

DIRECTOR & DEPUTY COMMISSIONER

ZIET MUMBAI

RESOURCE PERSONS

Sh M SRINIVASAN

PGT (MATHS)

ZIET MUMBAI

Sh K SELVARAJU

PGT(MATHS)

K.V. GANESHKHIND, PUNE

SUPPORTED BY ZIET FACULTY

Mrs.Radha Subramaniam, PGT(Bio)

Mrs.Pushpa Verma, PGT(Eco)

Sh M Gopala Reddy, PGT(Phy)

Sh Eugin D Leen, PGT(Eng)

Sh S Singhal, PGT(Eco)

Mrs.Kanta Bara, Librarian

Mrs.R Jayalakshmi, HDM

Sh Herman Churra, HDM

4

INDEX

S.No. PARTICULARS PAGE NO.

1 From Director’s Desk 5

2 List of Groups formed 6

3 Daily schedule 7

4 Report of the workshop 8

Chapter wise reference material

5 Relations & Functions 10

6 Matrices and Determinants 23

7 Differential Calculus 43

8 Integral Calculus 68

9 Differential Equations 88

10 Vectors and Three Diminension 99

11 Linear Programming 114

12 Probability 124

13 Activities at the Workshop 139

14 Contact Details of the participants 141

5

From the Director’s Desk

The three day workshop for PGT Mathematics teachers covered areas of difficulty as well as Error Analysis in detail. The participants first listed the topics which are considered difficult either for the teacher regarding ‘how’ to teach or for the student on ‘how’ to learn. The learned professors from IIT Powai, Mumbai were a source of information and enrichment for the teachers of Mathematics.

The first thing that teachers of Maths must keep in mind is how to remove the fear in the minds of students. To learn Maths, help students to enjoy Maths. Encourage them, analyze the errors they commit, and try out new methods to overcome those errors.

Revising topics taught earlier and establishing links between old topics and new ones should be the aim of the teacher.

It is also important to encourage students to set their own targets for self- improvement; this will lead to the deep involvement of the students in the teaching-learning process.

USHA ASWATH IYER

DIRECTOR

ZIET MUMBAI

6

KENDRIYA VIDYALAYA SANGATHAN, NEW DELHI

ZONAL INSTITUTE OF EDUCATION AND TRAINING, MUMBAI

3 DAY WORKSHOP ON CONTENT ENRICHMENT IN MATHEMATICS : 22.8.16 TO 24.8.16

GROUPS FORMED

GROUP I

S.NO. Name K.V Region Topics Allotted

1 Sh V K Mishra NO.2 Ahmedabad AHMEDABAD

1. Relation and Functions 2. Inverse Trigonometric Functions 3. Matrices 4. Determinants

2 Sh N M Giri NO.1 Baroda AHMEDABAD

3 Sh.SP Mathur KV 5 Jaipur JAIPUR

4 Sh.Narendra Kumar KV Jalipa Cantt JAIPUR

5 Sh Bhola Singh Dhanapur Cantt PATNA

6 Sh Jintendra Kumar Jawaharnagar PATNA

7 Dr.D S Rai Dhanpuri RAIPUR

GROUP II

1 Sh Santosh B Silvasa AHMEDABAD

1. Continuity 2. Differentation 3. Application of Derivates

2 Sh.S.L.Mehta No.1 Udaipur JAIPUR

3 Sh U N Singh Kankarbagh II Shit PATNA

4 Sh S P Thakur Muzaffarpur I shit PATNA

5 Sh M N Nandanwar Durg RAIPUR

6 Sh Praveen Kumar Khairagarh RAIPUR

GROUP III

1 Sh Kapil Kumar Soni Dantiwada AHMEDABAD

1. Integral Calculus 2. Differential Equations 3. Vectors

2 Sh Prasant Tiwari NO.2 Jamnagar AHMEDABAD

3 Sh.Mohd.Rafique Lalgarh Jattan JAIPUR

4 Mrs.Suman Singh HFC Baruani PATNA

5 Sh J S Pandit Mokamaghat PATNA

6 Sh U N Kurrey NO.1 Raipur RAIPUR

7 Sh D P Chaubey Jhagrakhand RAIPUR

GROUP IV

1 Sh AK Chaudhary Bhavnagar Para AHMEDABAD

1.Three dimension Geometry 2. LPP 3. Probability

2 Ms Seema Rajput ONGC SURAT AHMEDABAD

3 Sh.RPS Rathore KV Jhunjhunu JAIPUR

4 Sh Awadesh Prasad Bailey Road, I Shit PATNA

5 Sh E T Babu Kashmunda RAIPUR

6 Sh Vivekanand Pradhan NTPC Korba RAIPUR

7


3 DAY WORKSHOP ON CONTENT ENRICHMENT IN MATHEMATICS FOR PGT(MATHS) : 22.8.16 TO 24.8.16

DAILY SCHEDULE

DAY 09.30 A.M. TO 09.30 A.M. 09.30 A.M. TO 11.00 A.M 11.15 A.M. TO 01.00 P.M. 02.00 P.M. TO 03.30 P.M. 03.45 P.M. TO 05.30 P.M.

22.8.16 MODAY

Registration and

Inauguration

Relations, Functions Properties of Determinants, Types of Differentiation GROUP WORK

Sh K Selvarju, RP Sh M Srinivasan, RP

23.8.16 TUESDAY

PRAYER GROUP WISE TOPIC

DISCUSSION

GROUPWISE TOPIC

DISCUSSION

Types of Integration,

Differential

Equations

Session on

‘Application of Derivatives’

Sh M Srinivasan, RP Sh K Selvaruju, RP

Prof I K Rana, IIT Mumbai

24.8.16 WEDNESDAY

PRAYER

Session on

‘Probability’

Session on

Vectors and Three Dimension Geometry

GROUP WISE TOPIC

DISCUSSION

TOPIC BY RP & VALEDICTORY

FUNCTION

Prof.S.Sivaramakrishnan IIT, Mumbai Prof.Ananthanarayan

IIT, Mumbai

11.00 A.M. TO 11.15 A.M. & 03.30 P.M. TO 03.45 P.M. – Tea Break

01.00 P.M. to 02.00 P.M. – Lunch Break

8

ZONAL INSTITUTE OF EDUCATION & TRANING, MUMBAI

3 DAY WORKSHOP ON CONTENT ENRICHMENT FOR PGT(MATHS)

DAILY REPORT

DAY 1 : 22.8.16

The 3 day workshop on ‘Content Enrichment for PGT(Maths) started with

Prayer and was inaugurated by lighting of lamp by Ms Usha Aswath Iyer,

Director, ZIET, Mumbai.

After self-introduction by the participants, Ms Usha Aswath Iyer, Director, ZIET,

Mumbai welcomed all the participants and emphasized the need of clarity in

teaching Mathematics and the need of enriching the content by teaches. She

also listed out the difficulties faced by the students and teachers during the

teaching and learning process of Mathematics. She instructed the participants

to compile material which can be used by all teachers handing class XII

Mathematics.

The first session was taken by Sh K Selvaraju, Resource Person on Relations and

Functions. He explained the topics with typical problems.

After tea break, Sh M Srinivasan, Resource Person took a session on

‘Determinants and Differential Calculus’. He discussed typical problems in

properties of determinants and differential calculus.

Sh M Srinivasan, Resource person divided the participants into 4 groups and

explained the group works to be done.

The groups were given different topics and were asked to prepare concrete

and objective action plan under the heads

Important concepts in the chapter

The concepts in the chapter which students find difficult to understand

Common errors by students in the chapter

Tips and techniques for scoring high

After the lunch break, the participants did group work in the topics allotted to

them.

9

DAY 2 : 23.8.16

After Morning Prayer, group wise discussion on topics was done. Each group

submitted the concrete and objective action plan in the topic allotted to them.

The discussion was made effective by the Resource Persons who discussed

some important questions in the topic under discussion.

The discussion continued after tea break.

After lunch break, Sh K Selvaraju, Resource Person discussed some important

problems in indefinite integrals and properties of definite integral.

Sh M Srinivasan, Resource Person discussed some typical problems in

Differential Equations.

After tea break, Prof I K Rana, Department of Mathematics, IIT, Mumbai took a

session on ‘Application of Derivatives’. He explained the concept of Limits and

how differentiation is used to get the rate of change of quantities. He also

explained the nature of graph of functions by giving typical examples.

DAY 3 : 24.8.16

The day started with the prayer and recollection of previous days’ work.

Prof S Sivaramakrishnan, Department of Mathematics, IIT Powai took a session

on ‘Probability’. He explained the basic concept of probability and the idea of

conditional probability. He did some typical problems under Baye’s theorem.

After tea break, Prof Ananathanarayan, Department of Mathematics, IIT,

Powai, took a session on vector algebra and three dimension geometry. He

explained how the idea of two dimension geometry is extended for three

dimension geometry. He also explained the properties of lines and planes in

three dimension geometry.

After lunch break, Sh M Srinivasan, explained some problems in Linear

Programming. He explained how to draw the flow chart in transportation

problems.

The valedictory function was presided over by the Director, ZIET, Mumbai. She

emphasized the importance of error analysis in teaching learning process and

gave some tips to carry out the task.

10

CHAPTER I : RELATIONS & FUNCTIONS

CONCEPT MAPPING

1. RELATIONS AND FUNCTIONS

Cartesian Product:

The Cartesian product of two sets A and B = A B = {(a, b): a A, b B}

Relation: – A relation R is a subset of the Cartesian product A B. Domain : – The set of all first elements of the ordered pairs in a relation R from a set A to set B is called the domain of the relation R. Range : – The set of all second elements in a relation R from a set A to set B is called the range of the relation R. The set B is called the co domain of the relation R.

If (a, b) R we say that ‘a is related to b’ under the relation R and we write as a R b.

Ex. :– Let A={3,4} and B={1,6}. Find R such that {(a, b): a <b, a A, b B} and also its domain and range. Solution: R= {(3, 1),(4,1)}, Domain of R= {3,4}, Range of R={1} NOTE: Number of relations from set A to set B having n and m elements = nm2 Types of Relations

1. Empty Relation :– A relation R in a set A is called empty relation if R=

2. Universal Relation:– A relation R is universal relation, if R= A A 3. Reflexive Relation:– A relation R in a set A is called reflexive,

If (a,a) R, for every aA. 4. Symmetric Relation:– A relation R in a set A is called symmetric,

if(a,b) R R (b,a) R , (a,b) R A. 5. Transitive Relation:– A relation R is called transitive,

if (a,b) R , (b,c) R implies that(a,c) R, a,b,c A. 6. Equivalence Relation:– A relation R is said to be an equivalence relation , if R is reflexive, symmetric and transitive. Example:– Let P be the set of all lines in a plane, and the relation R in set P is

given by R= {( a,b) P P: line a is parallel to line b} show that R is an equivalence relation. Example:– Let T be the set of all triangles in a plane with R is a relation in T is given by R={T1,T2): T1 is congurent to T2} show that R is an equivalence relation.

11

Example : In N N, show that the relation defined by (a,b)R(c,d) iff ad=bc is an equivalence relation

FUNCTIONS Function:– A relation f from a set A to set B is said to be a function if every element of Set A has one and only one image in Set B. The function f from A to B is denoted by f: A→B Types of Functions. One–One (or injective) function A function f:X→Y is defined to be one–one (or

injective) if x1,x2 X, f(x 1)=f(x2) x1=x2 or If x1 ≠ x2f(x1) ≠ f(x2) x1, x2 X

Onto (or surjective)function: A function ƒ: X → Y is said to be onto, if every

element of Y is the image of some element of X under f, i.e for every y Y, there exists an element x in X s.t f(x)=y see fig 1 (iii) and (iv)

f1 f 2 f3 (ii) f4 (iii) (iv) Fig 1 (i) to (iv)

Composition of functions:– let f : A B and g : BC be two functions. Then the composition of f and g, denoted by gof, is defined as the function gof :

AC given by gof (x) = g( f (x)) x A

1

2

3

4

a

b

c

d

e

f

2

3

4

1

2

3

4

a

b

c

d

e

f

2

3

4

1

2

3

4

a

b

d

2

3

4

1

2

3

4

a

b

c

d

12

Note: gof: A→C is one–one and f: A→B is ontog: B→C is one –one. Invertible function Def: – Let f: A→B be a bijection, Then a function g: B→A which associates each

element yB to a unique element xA s.t. f(x) = y is called the Inverse of f i.e

f(x)= y g(y) =x. The Inverse of f is generally denoted by f–1

A f(x) =2x B B f–1(x) = 1

2x A

Binary Operations

Binary Operation:– A Binary operation * on a set A is a function * : A×A A. we denote * (a, b) by a * b (an operation is a process which produces a new element from two given elements) OR

A Binary operation * on a set A associates only two elements a,b A to a unique element

a*bA. Thus it is a function from A×A to A

Closure Property: For every a, b A, a *b A

Commutative Law: For every a, b A, a *b = b *a

Associative Law: For every a, b, c A, a*(b*c)= (a*b)*c)

Identity Law: For every a A , a * e = e * a = a, Identity element = e = a

Inverse Law: For every aA, a *b = b*a = a, Inverse of element a = b Composition Table: A table showing all the results w. r. to a binary operation * on a set A is called a Composition table.

Example: Discuss the commutavity and associativity of the binary operation on R defined

by a b =ab/4 a,b R

1

-1

2

3

2

-2

4

6

2

3

4

2

-2

4

6

1

-

1

2

3

2

3

4

x

A

B

C f g

gof

13

INVERSE TRIGONOMETRIC FUNCTIONS

BASIC CONCEPTS 𝐈𝐧 𝐠𝐞𝐧𝐞𝐫𝐚𝐥 𝐭𝐫𝐢𝐠𝐨𝐧𝐨𝐦𝐞𝐭𝐫𝐢𝐜 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧𝐬 𝐚𝐫𝐞 𝐧𝐨𝐭 𝐨𝐧𝐞 – 𝐨𝐧𝐞 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧𝐬

sin 30 o =1/2 , sin150o = 1/2

𝐓𝐫𝐢𝐠𝐨𝐧𝐨𝐦𝐞𝐭𝐫𝐢𝐜 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧𝐬 𝐚𝐫𝐞 𝐧𝐨𝐭 𝐨𝐧𝐞 – 𝐨𝐧𝐞 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧𝐬

𝐁𝐮𝐭 𝐛𝐲 𝐫𝐞𝐬𝐭𝐫𝐢𝐜𝐭𝐢𝐧𝐠 𝐭𝐡𝐞 𝐝𝐨𝐦𝐚𝐢𝐧 𝐢𝐧𝐯𝐞𝐫𝐬𝐞 𝐭𝐫𝐢𝐠𝐨𝐧𝐨𝐦𝐞𝐭𝐫𝐢𝐜 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧𝐬 𝐜𝐚𝐧 𝐛𝐞 𝐝𝐞𝐟𝐢𝐧𝐞𝐝

𝐈 − 𝐃𝐨𝐦𝐚𝐢𝐧 𝐚𝐧𝐝 𝐑𝐚𝐧𝐠𝐞 𝐨𝐟 𝐈𝐧𝐯𝐞𝐫𝐬𝐞 𝐓𝐫𝐢𝐠𝐧𝐨𝐦𝐞𝐭𝐫𝐢𝐜 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧𝐬:

The value of an Inverse trignometric function which lies in the range of Principal

branch is called the Principal value of the Inverse trignometric function.

Example:

1. The Principal value of sin-1 (3/2) = /3

2. The Principal value of tan-1 (–1)= – /4

𝑷𝒓𝒐𝒑𝒆𝒓𝒕𝒊𝒆𝒔 𝒐𝒇 𝑰𝒏𝒗𝒆𝒓𝒔𝒆 𝑻𝒓𝒊𝒈𝒐𝒏𝒎𝒆𝒕𝒓𝒊𝒄 𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔

Function Domain Range

(Principal Value Branch)

1 𝒔𝒊𝒏−𝟏𝒙 [ – 1 , 1 ] [ −𝝅

𝟐 ,𝝅

𝟐 ]

2 𝒄𝒐𝒔−𝟏𝒙 [ – 1 , 1 ] [ 𝟎 ,𝝅 ]

3 𝒄𝒐𝒔𝒆𝒄−𝟏𝒙 R – ( – 1 , 1 ) [ −𝝅

𝟐 ,𝝅

𝟐 ] − { 0 }

4 𝒔𝒆𝒄−𝟏𝒙 R – ( – 1 , 1 ) [ 𝟎 , 𝝅 ] − { 𝟎 }

5 𝒕𝒂𝒏−𝟏𝒙 R (− 𝝅

𝟐 ,𝝅

𝟐 )

6 𝒄𝒐𝒕−𝟏𝒙 R ( 𝟎 , 𝝅 )

II.

𝒔𝒊𝒏−𝟏 ( 𝒔𝒊𝒏 𝒙 ) = 𝒙

14

𝒄𝒐𝒔−𝟏 ( 𝒄𝒐𝒔 𝒙 ) = 𝒙

𝒕𝒂𝒏−𝟏 ( 𝒕𝒂𝒏 𝒙 ) = 𝒙

𝒄𝒐𝒔𝒆𝒄−𝟏 ( 𝒄𝒐𝒔𝒆𝒄 𝒙 ) = 𝒙

𝒔𝒆𝒄−𝟏 ( 𝒔𝒆𝒄 𝒙 ) = 𝒙

𝒄𝒐𝒕−𝟏 ( 𝒄𝒐𝒕 𝒙 ) = 𝒙

III.

𝒔𝒊𝒏−𝟏 (−𝒙 ) = − 𝒔𝒊𝒏−𝟏 (𝒙 )

𝒄𝒐𝒔𝒆𝒄−𝟏 (−𝒙 ) = − 𝒄𝒐𝒔𝒆𝒄−𝟏 (𝒙 )

𝒕𝒂𝒏−𝟏 (−𝒙 ) = − 𝒕𝒂𝒏−𝟏 (𝒙 )

𝒄𝒐𝒔−𝟏 (−𝒙 ) = 𝝅 − 𝒄𝒐𝒔−𝟏 (𝒙 )

( 𝒔𝒆𝒄 𝒙 ) = 𝝅 − 𝒔𝒆𝒄−𝟏 (𝒙 )

𝒄𝒐𝒕−𝟏 ( 𝒄𝒐𝒕 𝒙 ) = 𝝅 − 𝒄𝒐𝒕−𝟏 (𝒙 )

IV.

𝒔𝒊𝒏−𝟏 ( 𝟏

𝒙 ) = 𝒄𝒐𝒔𝒆𝒄 −𝟏 (𝒙 )

𝒄𝒐𝒔𝒆𝒄−𝟏 ( 𝟏

𝒙 ) = 𝒔𝒊𝒏 −𝟏 (𝒙 )

𝒄𝒐𝒔−𝟏 ( 𝟏

𝒙 ) = 𝒔𝒆𝒄−𝟏 (𝒙 )

𝒔𝒆𝒄−𝟏 ( 𝟏

𝒙 ) = 𝒄𝒐𝒔−𝟏 (𝒙 )

𝒕𝒂𝒏−𝟏 ( 𝟏

𝒙 ) = 𝒄𝒐𝒕 −𝟏 (𝒙 )

15

𝒄𝒐𝒕 −𝟏 ( 𝟏

𝒙 ) = 𝒕𝒂𝒏 −𝟏 (𝒙 )

V.

𝒔𝒊𝒏−𝟏 𝒙 + 𝒄𝒐𝒔−𝟏 𝒙 =𝝅

𝟐

𝒕𝒂𝒏−𝟏 𝒙 + 𝒄𝒐𝒕−𝟏 𝒙 =𝝅

𝟐

𝒔𝒆𝒄 −𝟏 𝒙 + 𝒄𝒐𝒔𝒆𝒄−𝟏 𝒙 =𝝅

𝟐

VI.

𝒔𝒊𝒏−𝟏 𝒙 = 𝒄𝒐𝒔−𝟏 √𝟏 − 𝒙𝟐

𝒄𝒐𝒔−𝟏 𝒙 = 𝒔𝒊𝒏−𝟏 √𝟏 − 𝒙𝟐

𝟑 𝒔𝒊𝒏−𝟏 𝒙 = 𝒔𝒊𝒏−𝟏 (𝟑𝒙 − 𝟒𝒙𝟑)

𝟑 𝒄𝒐𝒔−𝟏 𝒙 = 𝒄𝒐𝒔−𝟏 (𝟒𝒙𝟑 − 𝟑𝒙)

VII.

𝒔𝒊𝒏−𝟏 𝒙 = 𝒄𝒐𝒔−𝟏 √𝟏 − 𝒙𝟐

𝒄𝒐𝒔−𝟏 𝒙 = 𝒔𝒊𝒏−𝟏 √𝟏 − 𝒙𝟐

𝟑 𝒔𝒊𝒏−𝟏 𝒙 = 𝒔𝒊𝒏−𝟏 (𝟑𝒙 − 𝟒𝒙𝟑)

𝟑 𝒄𝒐𝒔−𝟏 𝒙 = 𝒄𝒐𝒔−𝟏 (𝟒𝒙𝟑 − 𝟑𝒙)

VIII.

𝟐 𝒕𝒂𝒏−𝟏 𝒙 = 𝒕𝒂𝒏−𝟏 ( 𝟐𝒙

𝟏 − 𝒙𝟐 )

𝟐 𝒕𝒂𝒏−𝟏 𝒙 = 𝒄𝒐𝒔−𝟏 ( 𝟏 − 𝒙𝟐

𝟏 + 𝒙𝟐 )

16

𝟐 𝒕𝒂𝒏−𝟏 𝒙 = 𝒔𝒊𝒏−𝟏 ( 𝟐𝒙

𝟏 + 𝒙𝟐 )

VIII.

𝒕𝒂𝒏−𝟏 𝒙 + 𝒕𝒂𝒏−𝟏 𝒚 = 𝒕𝒂𝒏−𝟏 (𝒙 + 𝒚

𝟏 − 𝒙𝒚 )

𝒔𝒊𝒏−𝟏 𝒙 + 𝒔𝒊𝒏−𝟏 𝒚 = 𝒔𝒊𝒏−𝟏 (𝒙√𝟏 −𝒚𝟐 + 𝒚√𝟏− 𝒙𝟐 )

𝒄𝒐𝒔−𝟏 𝒙 + 𝒄𝒐𝒔−𝟏 𝒚 = 𝒄𝒐𝒔−𝟏 (𝒙 𝒚 − √𝟏− 𝒙𝟐 √𝟏 − 𝒚𝟐 )

QUESTION BANK ON RELATIONS , FUNCTIONS &

INVERSE TRIGONOMETRIC FUNCTIONS

ONE – MARK QUESTIONS: -

LEVEL-1

1. How many relations are possible from a set A of n elements to another set of m

elements?

2. If )(xf = x+7 and )(xg = x -7, x R find )( fog (7)

3. Let * be a binary operation on N given by NbabaHCFba ,),,(* write the value

of22*4.

4. Let * be a binary operation defined by a* b=2a+b-3 find 3*4

5. Let * be a binary operation defined by x * y = x – y find (5 * 3)* 1

6. Find the Principal value of sin–1 3/2

7. Find the value of

4

3tantan 1 .

8. Evaluate –1cos 3 cos 1 2

LEVEL - 2

9. Write the domain and Principal branch of x1tan .

10. Let f(x)= x2 and g(x) = Sin3x find fog and gof

11. If A={3,4,5,6} and B={1,3,6} and R= { (a,b) A B:a > b} Find R.

12. Let *be a binary operation on N given by a * b = L.C.M (a, b) find 2 * 3 .

13. Find the principal value of :

2

1sin 1

14. Let f : RR such that f(x) = Sinx and g :R>R s.t g(x) = x2 find f o g

17

LEVEL-3

15. Give an example of a relation on A={1,2,3,4}, which is Symmetric but neither reflexive nor transitive

16. Let the function ),3(),,2(),,1( bcaf find 1f .

17. Find the Identity element in the set Q+ of all positive rational numbers for the

operation* defined by a * b = ab/2 a,b Q+


2

1sin2

2

1cos 11 .

19. Find the value of )2(sec3tan 11 .


6

7coscos 1 .

ANSWERS:

LEVEL-1

1. nm2

2. 7

3. 2

4. 7

5. 1

6. /3

7. –/4

8. 1

LEVEL-2

9. Domain = R, Range = (−𝜋

2 ,𝜋

2)

10. fog = Sin2 3x, gof= Sin 3x2

11. R= {(a,b) A xB: a > b}={(3,1), (4,1), (4,3), (5,1), (5,3), (6,1), (6,3)}.

12. (i) 6

13. – /6

14. Sin x2

LEVEL-3

15. R={(1,2)(2,1),(1,4),(4,1)}

18

16. )3,(),2,(),1,( bca

17. 2

18. 2/3

19. –/3

20. 5/6

Four Marks Questions:

Level – 1:

1. Show that the Relation R in R defined as R={(a,b):a b} is reflexive and transitive but not

symmetric.

2. Show that the relation R in the Set {1,2,3} given by R={(1,2),(2,1)} is symmetric but neither

reflexive nor transitive.

3. If f(x) is an invertible function, find the inverse of f(x) = 5

23 x .

4. Discuss the commutavity and associativity of the binary operation on R defined by a

b =ab/4 a,b R

5. Prove that 48

1tan

5

1tan

2

1tan 111


2

3cot

5

3sintan 11

7. Prove that 7

24tan

5

3sin2 11

8. Prove that 36

77tan

5

3sin

17

8sin 111

9. Prove that 65

33cos

13

12cos

5

4cos 111

10. Prove that 65

56sin

5

3sin

13

12cos 111

11. Prove that 419

8tan

5

3tan

4

3tan 111

Answer: 3. 5𝑥+2

3 6. 17/6

Level -2:

1. Show that the relation R defined in the set A of all triangles as R={T1,T2): T1 is similar to T2}is

equivalence relation. Consider three right angled triangles T1 with sides 3,4,5, T2 with sides

5,12,13 and T3 with sides 6,8,10. Which triangles among T1,T2 and T3 are related.

19

2. Prove that the relation R in the set A = }5,4,3,2,1{ given by R = }:,{ evenisbaba , is an

equivalence relation.

3. Show that the relation R defined by babaR :),{( is divisible by },:3 Nba is an

equivalence relation.

4. Let P be the set of all point in a plane and the relation R in set P be defined as

PPBAR )( : distance between points A and B is less than 3 units . Show that the

relation R is not an equivalence relation.

5. Show that the function f : Ro Ro, defined as f(x) = 1/x, is one–one onto where Ro is the set of

all non zero real numbers.

6. Check the surjectivity of the function f : Z Z given by f(x) =|x|.

7. Show that the function RRf : defined by 34)( xxf is Invertible. Find the Inverse of

f .

8. Find the identity element for the binary operation in set NNA given by

),(),(*),( dbcadcba for NNdcba ),(),,( . Show that * is commutative and

associative.

9. Show that binary operation 4-bab * a , a, b R is associative. Also find its Identity

element and inverse of an element.


2

1sin

2

1cos)1(tan 111 .

11. Show that85

84cos

17

8sin

5

3sin 111 .

Answer : 6. Not surjective 8. Identity does not exist 10 . 3𝜋

4

Level – 3:

1. Show that the relation R on NN defined by(a,b) R(c,d) a+d=b+c is an equivalence relation.

2. In N N, show that the relation defined by (a,b) R (c,d)<=> ad = bc is an equivalence relation

3. Show that the relation R on the set R of all real numbers defined as

R={(a,b):a ≤ b2 } is neither reflexive, nor symmetric nor transitive.

4. If 42x

1xtan

2x

1xtan 11

then find the value of x.

5. If 2x 1

f (x)2x 3

,x –1/2 then show that

2x 1f (f (x))

2x 3

provided x – 3/2

6. Let A= R–{2} and B=R–{1} If f: AB is a function defined by ƒ(x)= (x–1 / x-2)

show that f is a bijection.

20

7. . Let NNf : be defined by

evenisnifn

oddisnifn

xf

,2

,2

1

)(

Examine whether f is one to one, onto or bijective, Justify your answer.

8. Let A and B be sets. Show that ABBAf : , Such that ),(),( abbaf is

bijective function.

9. State whether the function RRf : defined by 21)( xxf is one-one, onto or

bijective .

10. Prove that the function f : N × N defined by f(x) = x2+x+1 is one one but not onto.

11. Consider ),5[: Rf given by 569)( 2 xxxf show that f is invertible.

Find the inverse of f .

12. Let f : NR be a function defined as f(x)= 4x2 + 12x +15, show that f: NS, where

S is the range of f , is invertible. Find the inverse of f.

13. Prove that 5

3cos

13

5sin

16

63tan 111

14. Prove that

15. a

b

b

a

b

a 2cos

2

1

4tancos

2

1

4tan 11

16. Prove that 3

22sin

4

9

3

1sin

4

9

8

9 11

17. Prove that 𝒕𝒂𝒏−𝟏 (√𝟏+𝒙−√𝟏−𝒙

√𝟏+𝒙+√𝟏−𝒙) =

4,0,cos

2

1

4

1 xx

18. Solve the equation 3

2tan)2(tan)2(tan 111 xx

19. Prove that 3

22sin

4

9

3

1sin

4

9

8

9 11

Answer: 4. ±1

√2 7. F is not one one , so not onto. 9. F is not one one and not onto.

11. 3

16)(1 y

yf 12. f is invertible with f–1

=f 18. x = ±𝟑

21

Common Mistakes

1. Relations and functions

1. Let A = { 1,2,3}. Check whether R = { (1,2),(2,1), (1,1), (1,3)} is symmetric or not.

Mistake done : Here, (1,2)∈ 𝑅, (2,1) )∈ 𝑅. So, it is symmetric.

Correction : The student thinks that only an ordered pair is to check for Symmetric.

So, he forgets to check for (1,3).

2. If A = {1,2,3}, check whether R = {(1,1),(1,2),(2,1)} is transitive or not.

Mistake done: (1,2) )∈ 𝑅 , (2,1) )∈ 𝑅 implies that (1,1) )∈ 𝑅. So it is transitive.

Correction: Here the student forgets to see for (2,1) )∈ 𝑅, (1,2) )∈ 𝑅 implies (2,2)

)∈ 𝑅or not.

3. Find the domain and range of the function f(x) = √𝑥 − 3.

Mistake done: Domain = { 3,4,5,6,…….}, Range = { 0,1,2,…………}.

Correction : The student considers only the natural numbers. So, he forgets for real

number. The correct answer is Domain= [ 3, ∞) and Range = [ 0, ∞)

4. Which of the following are functions from A = {1,2} to B = { a,b} Find f-1

(if exists).

(i). f = { (1,a), (2,a)}

(ii) f= { (1,a),(2,b),(1,b)}

(i). Mistake done: f-1

= { (a,1),(a,2)}.

Correction : The child got confusion for finding the inverse of a relation and that of

a function. He forgot to see for one-one onto.

(ii). Mistake done: f-1

= { (a,1),(b,2),(b,1)}.

Correction : The child got confusion for finding the inverse of a relation and that of

a function. He forgot to see for one-one onto.

5. Compute fog(4) if f(4) = -4, g(4) = -2 ,f(-2) = -1

Mistake done : fog(4) =f(4)g(4) = (-4)(-2) = 8

Correction : fog(4) =f(g(4)) = f(-2) = -1.

6.If f(x) = 3x ,g(x) = sinx find fog(x).

22

Mistake done : fog(x) = sin(3x).

Correction : fog(x) = 3sinx

7. Mistake done : cos-1

x = 1

𝑐𝑜𝑠𝑥

Correction: cos-1

x is inverse function of cosx. It is not reciprocal of cosx.

8.Mistake done : Students use example to prove a relation which is reflexive or symmetric or

transitive.

Correction : When we prove any result it must be generally but when we disprove any

result then give example.

9. R= {(1,1),(2,2),(3,3)} Is R transitive relation?

Mistake done: R is not transitive.

Correction: R is transitive since we do not have any two elements (a,b) , (b,c) ∈ R such that

(a,c)∈ R.

TIPS AND TECHNIQUES RELATIONS AND FUNCTIONS – CHAPTER 1

INVERSE TRIGONOMETRIC FUNCTIONS – CHPATER-2

1. In Function every element of x A must have a UNIQUE but not in the case of

Relation

2. The Domain of the Relation is the set of all first entries of the ordered pairs of the

relation.

3. After finding the f-1 it can verified with (fo f-1)(x ) = x

4. Principle value branch must be known before solving Inverse Trigonometric

Functions related problems

5. Properties of Rationals (or Reals) must be discussed before studying properties of a

Binary Operation[ mainly for identity element and inverse of an element]

6. While solving the inverse trigonometric functions both sides must be taken ( some

students take LHS and prove RHS

7. While using the formula namely 𝟐 𝒕𝒂𝒏−𝟏 𝒙 = 𝒕𝒂𝒏−𝟏 ( 𝟐𝒙

𝟏− 𝒙𝟐 ) the formula for

tan2 can be recalled 8. LHS and RHS should be considered for proving COMMUTATIVE Law or Associative

Law .

9. While using the formula namely 𝒕𝒂𝒏−𝟏 𝒙 + 𝒕𝒂𝒏−𝟏 𝒚 = 𝒕𝒂𝒏−𝟏 (𝒙 + 𝒚

𝟏 −𝒙𝒚 ) the

formula for tan (A + B) can be recalled 10. gof (x ) = g [f(x)] and fog (x ) =f[g(x)]

23

MATRICES AND DETERMINANTS

CONCEPT MAPPING

1. What is matrix?

* Matrix is a rectangular array of objects.

*[3 4 61 0 7

] ←Rows

↑ Columns

2. Order of matrix.

* If a matrix has ‘m’ number of rows and ‘n’ number of columns then order of matrix is mxn.

* [6 94 3

] order -2x2 i.e. A square matrix of order 2

*[7 1 35 7 9

] order 2x3 i. e. A rectangular matrix of order 2x3.

3. Types of matrices.

* Row matrix--- [1 2 3]

* Column matrix ---[456

]

* Square matrix of order 2 ---[1 24 3

] , of order 3----[9 7 21 6 35 0 6

]

*Diagonal matrix --[6 0 00 8 00 0 7

] ,in which all the nom diagonal elements of a square matrix

are all zero

*Scalar matrix --[7 0 00 7 00 0 7

]( it is a diagonal matrix whose all diagonal elements are equal)

*Unit/Identity matrix—I2= [1 00 1

] , 𝐼3 = [1 0 00 1 00 0 1

]

* Null/Zero matrix --[0 00 0

] ,[0 0 00 0 00 0 0

]

4. Equality of matrices.

24

5. Operation on matrices

(a) Addition/subtraction of matrices.

* Same order matrices can be added as well as subtracted.

* Addition as well as subtraction of matrices are binary operations.

(b) Multiplication of matrices.

*Two matrices are compatible for multiplication if number of columns of first matrix is equal

to the number of rows of second matrix.

*𝐴 = [1 23 4

] , 𝐵 = [5 7 42 5 2

] , AB exists but BA does not exist.

* Multiplication of matrices is also binary operation under certain conditions.

6.. Transpose of matrix.

* By interchanging rows to columns or vice versa transpose of matrix is obtained.

*𝐴 = [2 5 81 7 0

] , AT=[2 15 78 0

]

7. Symmetric and skew symmetric matrices.

(a) Symmetric matrix if A=AT,(where A is a square matrix)

(b) Skew symmetric matrix if AT= -AT

(c) Every square matrix can be expressed as the sum of a symmetric and skew symmetric

matrices.

i.e. 𝐴 =1

2(𝐴 + 𝐴𝑇) +

1

2(𝐴 − 𝐴𝑇)

=(Symmetric matrix)+ (Skew symmetric matrix)

8. Elementary transformations for finding inverse of a matrix.

(a) Elementary row transformations A=IA

(b )Elementary column transformations A=AI

Note: Elementary row transformations and Elementary column transformations should not

be used simultaneously to find the inverse of a matrix.

9.What is determinant?

25

*Every square matrix can be associated with a unique number that is called its determinant.

i.e.|𝑎 𝑏𝑐 𝑑

|=ad-bc

10. Minors and co factors of the elements of a determinant.

* Aij=(-1)i+jMij ,where Aij is co factor and Mij is minor.

11. Evaluation of a determinant.

* The sum of the products of elements and its corresponding co factors of a row or column.

12. Properties of determinants.

* The value of the determinant remains unchanged if its rows and columns are

interchanged.

*If any two rows(or columns) of a determinant are interchanged the sign of determinant

changes.

* If any two rows (or columns) of a determinant are identical(all corresponding elements are

same) then value of determinant is zero.

* If each element of a row (or a column) of a determinant is multiplied by a constant ‘k’ then

its value gets multiplied by’ k’

*If some or all elements of a row or column of a determinant are expressed as sum of two

(or more) terms, then the determinant can be expressed as sum of two (or more)

determinants.

*If ,to each element of any row or column of a determinant , the equimultiples of

corresponding elements of other row (or column) are added, then value of determinant

remains the same.

* Use of appropriate property to evaluate the determinant easily.

13. Adjoint and inverse of a matrix.

*adjA=[𝐴11 𝐴12𝐴21 𝐴22

]𝑇

=[𝐴11 𝐴21𝐴12 𝐴22

] ,where Aij is a co factor of aij.

* A(adj A)=I |A|=(adjA)A

∗ 𝐴 (𝑎𝑑𝑗𝐴)

|𝐴| =I=

(𝑎𝑑𝑗)𝐴

|𝐴|

*A-1=𝑎𝑑𝑗𝐴

|𝐴|

26

14. Conditions for a system of linear equations to be consistent/inconsistent

* A system of linear equations is called consistent if it has at least one solution.

i) |A|≠ 0 (System has unique solution)

ii) |A|=0≠ (𝑎𝑑𝑗 𝐴 )B (System has no solution)

iii) |A|=0=(adj A)B ( System has infinitely many solutions)

15. Solving a system of linear equations using matrix method.

3x+2y-5z=10

2x+7y-4z =20

8x+6y-5z=25

A =[3 2 −52 7 −48 6 −5

], X=[𝑥𝑦𝑧], B=[

102025

]

AX=B, X=A-1B

16.To form linear equations from given word problems and their solution using matrix

method.

QUESTION BANK

OPERATION ON MATRICES AND TRANSPOSE OF MATRIX LEVEL-1(1 marks)

1. if A=[1 2 31 4 5

] and B=[−1 2 43 −2 6

] .Find the value of 2A-B.

2. If [𝑎 + 𝑏 25 𝑎𝑏

] = [6 25 8

] .Find the value of a and b.

3. find x,y,a and b if [2𝑥 − 3𝑦 𝑎 − 𝑏 3

1 𝑥 + 4𝑦 3𝑎 + 4𝑏]= [

1 −2 31 6 29

]

4. find x and y if [𝑥 + 𝑦𝑥 − 𝑦]= [

84]

5. if A=[𝑥 − 1 𝑦 + 22 𝑧 + 1

]= ,B=[1 22 4

]. Find x,y and z if A=B

6. Find x,y,z,w if

[𝑥 6−1 2𝑤

]+[4 𝑥 + 𝑦

𝑧 +𝑤 3] = 3 [

𝑥 𝑦𝑧 𝑤

]

7. if 2[3 45 𝑥

] + [1 𝑦0 1

] = [7 010 5

] ,find x and y.

27

8. Find X and Y if X+Y=[7 02 5

] and X-Y=[3 00 3

]

9. Find X and Y if X+y=[5 20 9

], X-Y=[3 60 −1

]

10.Find matix B such that

[2 5−3 7

]B=[17 −147 −13

]

11.if A=[−123]and B=[−2 −1 −4] ,verify that(𝐴𝐵)𝑇 = 𝐵𝑇𝐴𝑇

LEVEL-2(4 marks)

1. if A= [3 2 01 4 00 0 5

] , show that 𝐴2 −7𝐴 + 10𝐼 = 0

2. if A=[1 2 33 −2 14 2 1

], then show that 𝐴3 − 23𝐴 − 40𝐼 = 0

3.Find the number x and y such that 𝐴2 + 𝑥𝐴 + 𝑦𝐼 = 0. Where A=[1 54 3

]

4.For A=[1 𝑎0 1

], show that 𝐴𝑚 = [1 𝑚𝑎0 1

] , for all positive integers m.

5. if 𝐴 = [1 1 11 1 11 1 1

] , 𝑡ℎ𝑒𝑛 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝐴𝑛 = [3𝑛−1 3𝑛−1 3𝑛−1

3𝑛−1 3𝑛−1 3𝑛−1

3𝑛−1 3𝑛−1 3𝑛−1], for all positive integers n.

6.If A=[𝑐𝑜𝑠𝜃 𝑠𝑖𝑛 𝜃−𝑠𝑖𝑛𝜃 𝑐𝑜𝑠 𝜃

] , thenprove by principle of mathematical induction that 𝐴𝑛 =

[𝑐𝑜𝑠𝑛𝜃 𝑠𝑖𝑛𝑛𝜃−𝑠𝑖𝑛𝑛𝜃 𝑐𝑜𝑠𝑛𝜃

] for

all 𝑛 ∈ 𝑁.

7. if A=[2 3 45 7 9−2 1 1

], B= [4 0 51 2 00 3 1

], verify that (𝐴𝐵)𝑇 = 𝐵𝑇𝐴𝑇 .

8. Express the matrix A=[4 2 −13 5 71 −2 1

] as the sum of a symmetric and skew –symmetric matrix.

9. Express [1 3 5−6 8 3−4 6 5

] as a sum of symmetric and skew –symmetric matrices.

10.if A=[0 6 7−6 0 87 −8 0

] , B=[0 1 11 0 21 2 0

] , C=[2−23], verify (𝐴 +𝐵)𝐶 = 𝐴𝐶 + 𝐵𝐶 .

11.find the matrix X so that 𝑋 [1 2 34 5 6

] = [−7 −8 −92 4 6

]

28

ELEMENTARY ROW AND COLUMN OPERATIONS ON A MATRIX

1. Find the inverse of the following matrix by using elementary row operation.

Level 1

(i) [7 14 −3

] (ii) [3 12 7

] (iii) [2 35 7

] (iv) [5 22 1

] (v) [1 6−3 5

]

Level 2

(i) [2 0 −15 1 00 1 3

] (ii) [2 3 12 4 13 7 2

] (iii)[1 1 23 1 12 3 1

] (iv)

[1 3 −2−3 0 12 1 0

]

(v) [2 −1 31 2 43 1 1

]

Level 3

(i) [3 −1 −22 0 −13 −5 0

] (ii) [1 3 −2−3 0 −12 1 0

] (iii) [1 2 −2−1 3 00 −2 1

]

(iv) [2 −1 44 0 23 −2 7

] (v) [−1 1 21 2 33 1 1

]

2. Find the inverse of the following matrix by using elementary column

transformation

Level 1

(i)[2 31 2

] (ii) [1 32 7

] (iii) [1 −12 3

] (iv) [3 −1−4 2

] (v) [−10 −2−5 1

]

Level 2

(i) [−1 1 21 2 33 1 1

] (ii) [1 2 02 3 −11 −1 3

] (iii) [0 1 21 2 33 1 1

]

(iv) [3 −3 42 −3 40 −1 1

] (v) [1 2 −2−1 3 00 −2 1

]

Level 3

(i) [1 3 −2−3 0 −52 5 0

] (ii) [3 0 −12 3 00 4 1

] (iii) [1 2 32 5 7−2 −4 −1

]

30

|𝑎 + 𝑥 𝑎 − 𝑥 𝑎 − 𝑥𝑎 − 𝑥 𝑎 + 𝑥 𝑎 − 𝑥𝑎 − 𝑥 𝑎 − 𝑥 𝑎 + 𝑥

| = 0

5.Without expanding the determinant, evaluate: ||

1

𝑎𝑎 𝑏𝑐

1

𝑏𝑏 𝑐𝑎

1

𝑐𝑐 𝑎𝑏

||

6. Prove that |𝑏 + 𝑐 𝑎 𝑎𝑏 𝑐 + 𝑎 𝑏𝑐 𝑐 𝑎 + 𝑏

| = 4𝑎𝑏𝑐

7. Prove that | 𝑎 𝑎 + 𝑏 𝑎 + 2𝑏

𝑎 + 2𝑏 𝑎 𝑎 + 𝑏𝑎 + 𝑏 𝑎 + 2𝑏 𝑎

| = 9𝑏2 (𝑎 + 𝑏)

8.If X= -4 is a root of |𝑥 2 31 𝑥 13 2 𝑥

| = 0 then find other two roots.

9. Show that |1 𝑎 𝑎2

1 𝑏 𝑏2

1 𝑐 𝑐2|=|1 𝑎 𝑏𝑐1 𝑏 𝑐𝑎1 𝑐 𝑎𝑏

|

10.If a+b+c=0 and |𝑎 𝑏 𝑐𝑏 𝑐 𝑎𝑐 𝑎 𝑏

| = 0 then prove that a=b=c

Level 3 (six marks)

1. Using properties of determinants, prove that

|1 + 𝑎2 − 𝑏2 2𝑎𝑏 −2𝑏

2𝑎𝑏 1 − 𝑎2 + 𝑏2 2𝑎2𝑏 −2𝑎 1 − 𝑎2 − 𝑏2

| = (1 + 𝑎2 + 𝑏2)3

2.Using the properties of determinants , prove that:

|

𝑥 𝑥2 1 + 𝑝𝑥3

𝑦 𝑦2 1+ 𝑝𝑦3

𝑧 𝑧2 1+ 𝑝𝑧3| = (1 + 𝑝𝑥𝑦𝑧)(𝑥 − 𝑦)(𝑦 − 𝑧)(𝑧 − 𝑥)

3. Solve for x: |15 − 2𝑥 11− 3𝑥 7− 𝑥11 17 1410 16 14

| = 0

31

4. Solve for x: |

(𝑎2 + 𝑏2)/𝑐 𝑐 𝑐

𝑎 (𝑏2 + 𝑐2)/𝑎 𝑎

𝑏 𝑏 (𝑐2 + 𝑎2)/𝑏

| = 4𝑎𝑏𝑐

5.Prove that |𝑎2 𝑏2 𝑐2

(𝑎 + 1)2 (𝑏 + 1)2 (𝑐 + 1)2

(𝑎 − 1)2 (𝑏 − 1)2 (𝑐 − 1)2| = 4 |

𝑎2 𝑏2 𝑐2

𝑎 𝑏 𝑐1 1 1

|

6. Prove that: |

1 cos(𝛽 − 𝛼) cos(𝛾 − 𝛼)cos(𝛼 − 𝛽) 1 cos(𝛾 − 𝛽)cos(𝛼 − 𝛾) cos(𝛽 − 𝛾) 1

| = 0

7.In a triangle ABC,if|1 1 1

1+ sin𝐴 1 + sin𝐵 1 + sin𝐶sin𝐴 + sin2 𝐴 sin𝐵 + sin2 𝐵 sin𝐶 + sin2 𝐶

| = 0 then prove that ∆ABC is

an isosceles triangle.

8.In a triangle ABC,if|1 1 1

1 + cos𝐴 1 + cos𝐵 1 + cos𝐶cos𝐴 + cos2 𝐴 cos𝐵 + cos2 𝐵 cos𝐶 + cos2 𝐶

| = 0 then prove that ∆ABC

is an isosceles triangle.

9.If |𝑝 𝑏 𝑐𝑎 𝑞 𝑐𝑎 𝑏 𝑟

| = 0 then find the value of p/(p-a) + q/(q-b) + r/(r-c)

10.Prove that |1 𝑎2 + 𝑏𝑐 𝑎3

1 𝑏2 + 𝑐𝑎 𝑏3

1 𝑐2 + 𝑎𝑏 𝑐3| = −(𝑎 − 𝑏)(𝑏 − 𝑐)(𝑐 − 𝑎)(𝑎2 + 𝑏2 + 𝑐2)

ADJOINT AND INVERSE OF MATRIX LEVEL-1

1.Find the minors of (1,2)th entry in the given matrix [2 −3 56 0 41 5 −7

]

2. Find the minor of (2,2)th entry in the given matrix [4 12 3

]

3.Find the cofactor of (1,3)th entry in the following : [1 2 32 3 01 5 4

]

4.Find the value of x, if |𝑥 + 2 3𝑥 + 5 4

| = 3

32

5. Find x , if |2𝑥 + 5 35𝑥 + 2 9

| = 0

6. For what value of x the given matrix[3 − 2𝑥 𝑥 + 12 4

] is singular ?

7. Evaluate |𝑠𝑖𝑛30° 𝑐𝑜𝑠30°−𝑠𝑖𝑛60° 𝑐𝑜𝑠60°

|

8. If A is a square matrix of order 3 such that |𝑎𝑑𝑗 𝐴| = 25, find |𝐴| 9. If A is a square matrix of order 3 and |𝐴| = 5 find A(adjA).

LEVEL-2(4-MARKS)

1. If A=[−3 −21 −4

], find A.(Adj A)

2. if A= [1 3−1 4

], find |𝑎𝑑𝑗 𝐴|

3. If A= [−2 0 03 4 010 −7 3


4. Given that A=[𝑐𝑜𝑠𝑥 𝑠𝑖𝑛𝑥−𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥

] and A(adjA)= k[1 00 1

], find the value of k.

5.If A=[𝑥 0 00 𝑦 00 0 𝑧

], find adjA.

6. Find the adjoint of the matrix [1 −23 −5

]

7. Find the inverse of the matrix [𝑎 −𝑏𝑐 −𝑑

]

8.For the matrix A=[5 2−3 −1

], verify that adj A’= (𝐴𝑑𝑗𝐴)′

9. If A=[1 −10 3

] and B=[2 4−3 0

] ,then 𝑎𝑑𝑗(𝐴𝐵) = (𝑎𝑑𝑗𝐴)(𝑎𝑑𝑗𝐵)

10.Find the inverse of the matrix A= [3 −12 6

].Also verify that 𝐴−1𝐴 = 𝐼2 = 𝐴−1𝐴

11.Given A=[2 −3−4 7

], compute 𝐴−1 and show that 2𝐴−1 = 9𝐼 −𝐴

12.If A=[4 52 1

], show that A-3I= 2(I+3A-1)

13.If A=[2 𝑥4 2

],x≠ 1,calculate (i) A2 (ii) (A2)-1

33

14.Find the inverse of the matrix A=[𝑎 𝑏

𝑐1+𝑏𝑐

𝑎

] and show that: aA-1=(a2+bc+1)I-aI.

!5.Find the condition that the matrix [𝑎 𝑏𝑐 𝑑

]

May be invertible .If the condition is satisfied find the inverse of the matrix.

16.Find the co factors of the elements of the third row of the determinant|2 −3 56 0 41 5 −7

| and verify

that A31 a11+A32a12+A33a13=0.

17.Using cofactors of elements of third column evaluate the determinant|

1 𝑥 𝑦𝑧1 𝑦 𝑧𝑥1 𝑧 𝑥𝑦

|.

18. If [1 −𝑡𝑎𝑛𝑥

𝑡𝑎𝑛𝑥 1][

1 𝑡𝑎𝑛𝑥−𝑡𝑎𝑛𝑥 1

]−1

= [𝑎 −𝑏𝑏 𝑎

],then find the values of a and b.

19. Find the adjoint of the matrix A=[1 23 4

], also verify that 𝐴(𝑎𝑑𝑗𝐴) = |𝐴|𝐼2 = (𝑎𝑑𝑗𝐴)𝐴

20. If A= [3 1 22 −3 −11 2 1

], find A(adjA) without calculating adjA.

LEVEL-3(6-MARKS)

1.If A=[0 1 21 2 33 𝑎 1

] and 𝐴−1 = [

1

2

−1

2

1

2

−4 3 𝑏5

2

−3

2

1

2


2.Find the adjoint of the matrix A=[1 −1 22 3 5−2 0 1

],also verify that A(adjA)= |𝐴|𝐼3=(adjA)A

3. Find the inverse of the matrix A=[2 1 34 −1 0−7 2 1

], also verify that 𝐴−1𝐴 = 𝐼3

4.If A=[1 3 41 4 31 3 4

],then verify that𝐴(𝑎𝑑𝑗𝐴) = |𝐴|𝐼. Hence find 𝐴−1.

5. Given A=[5 0 42 3 21 2 1

], 𝐵−1 = [1 3 31 4 31 3 4

],compute (𝐴𝐵)−1

6.If A=1

9[−8 1 44 4 71 −8 4

],prove that 𝐴−1 = 𝐴𝑇 .

7. If A=[1 −1 12 −1 01 0 0

],prove that 𝐴−1 = 𝐴2

34

8.If 𝐴−1 = [3 −1 1−15 6 −55 −2 2

] and B=[1 2 −2−1 3 00 −2 1

], find (𝐴𝐵)−1

9. If A=[1 −2 1−2 3 11 1 5

], verify that (i) (𝑎𝑑𝑗𝐴)−1 = 𝑎𝑑𝑗(𝐴−1)

(ii) (𝐴−1)−1 = 𝐴

10. If A=[1 2 22 1 22 2 1

], prove that 𝐴2 − 4𝐴− 5𝐼 = 0. Hence find 𝐴−1

ADJOINT AND INVERSE OF MATRIX LEVEL-1

1.Find the minors of (1,2)th entry in the given matrix [2 −3 56 0 41 5 −7

]

2. Find the minor of (2,2)th entry in the given matrix [4 12 3

]

3.Find the cofactor of (1,3)th entry in the following : [1 2 32 3 01 5 4

]

4.Find the value of x, if |𝑥 + 2 3𝑥 + 5 4

| = 3

5. Find x , if |2𝑥 + 5 35𝑥 + 2 9

| = 0

6. For what value of x the given matrix[3 − 2𝑥 𝑥 + 12 4

] is singular ?

7. Evaluate |𝑠𝑖𝑛30° 𝑐𝑜𝑠30°−𝑠𝑖𝑛60° 𝑐𝑜𝑠60°

|

8. If A is a square matrix of order 3 such that |𝑎𝑑𝑗 𝐴| = 25, find |𝐴| 9. If A is a square matrix of order 3 and |𝐴| = 5 find A(adjA).

LEVEL-2(4-MARKS)

1. If A=[−3 −21 −4

], find A.(Adj A)

2. if A= [1 3−1 4


3. If A= [−2 0 03 4 010 −7 3


35

4. Given that A=[𝑐𝑜𝑠𝑥 𝑠𝑖𝑛𝑥−𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥

] and A(adjA)= k[1 00 1

], find the value of k.

5.If A=[𝑥 0 00 𝑦 00 0 𝑧

], find adjA.

6. Find the adjoint of the matrix [1 −23 −5

]

7. Find the inverse of the matrix [𝑎 −𝑏𝑐 −𝑑

]

8.For the matrix A=[5 2−3 −1

], verify that adj A’= (𝐴𝑑𝑗𝐴)′

9. If A=[1 −10 3

] and B=[2 4−3 0

] ,then 𝑎𝑑𝑗(𝐴𝐵) = (𝑎𝑑𝑗𝐴)(𝑎𝑑𝑗𝐵)

10.Find the inverse of the matrix A= [3 −12 6

].Also verify that 𝐴−1𝐴 = 𝐼2 = 𝐴−1𝐴

11.Given A=[2 −3−4 7

], compute 𝐴−1 and show that 2𝐴−1 = 9𝐼 −𝐴

12.If A=[4 52 1

], show that A-3I= 2(I+3A-1)

13.If A=[2 𝑥4 2

],x≠ 1,calculate (i) A2 (ii) (A2)-1

14.Find the inverse of the matrix A=[𝑎 𝑏

𝑐1+𝑏𝑐

𝑎

] and show that: aA-1=(a2+bc+1)I-aI.

!5.Find the condition that the matrix [𝑎 𝑏𝑐 𝑑

]

May be invertible .If the condition is satisfied find the inverse of the matrix.

16.Find the co factors of the elements of the third row of the determinant|2 −3 56 0 41 5 −7

| and verify

that A31 a11+A32a12+A33a13=0.

17.Using cofactors of elements of third column evaluate the determinant|

1 𝑥 𝑦𝑧1 𝑦 𝑧𝑥1 𝑧 𝑥𝑦

|.

18. If [1 −𝑡𝑎𝑛𝑥

𝑡𝑎𝑛𝑥 1][

1 𝑡𝑎𝑛𝑥−𝑡𝑎𝑛𝑥 1

]−1

= [𝑎 −𝑏𝑏 𝑎


19. Find the adjoint of the matrix A=[1 23 4

], also verify that 𝐴(𝑎𝑑𝑗𝐴) = |𝐴|𝐼2 = (𝑎𝑑𝑗𝐴)𝐴

20. If A= [3 1 22 −3 −11 2 1

], find A(adjA) without calculating adjA.

36

LEVEL-3(6-MARKS)

1.If A=[0 1 21 2 33 𝑎 1

] and 𝐴−1 = [

1

2

−1

2

1

2

−4 3 𝑏5

2

−3

2

1

2


2.Find the adjoint of the matrix A=[1 −1 22 3 5−2 0 1

],also verify that A(adjA)= |𝐴|𝐼3=(adjA)A

3. Find the inverse of the matrix A=[2 1 34 −1 0−7 2 1

], also verify that 𝐴−1𝐴 = 𝐼3

4.If A=[1 3 41 4 31 3 4

],then verify that𝐴(𝑎𝑑𝑗𝐴) = |𝐴|𝐼. Hence find 𝐴−1.

5. Given A=[5 0 42 3 21 2 1

], 𝐵−1 = [1 3 31 4 31 3 4

],compute (𝐴𝐵)−1

6.If A=1

9[−8 1 44 4 71 −8 4

],prove that 𝐴−1 = 𝐴𝑇 .

7. If A=[1 −1 12 −1 01 0 0

],prove that 𝐴−1 = 𝐴2

8.If 𝐴−1 = [3 −1 1−15 6 −55 −2 2

] and B=[1 2 −2−1 3 00 −2 1

], find (𝐴𝐵)−1

9. If A=[1 −2 1−2 3 11 1 5

], verify that (i) (𝑎𝑑𝑗𝐴)−1 = 𝑎𝑑𝑗(𝐴−1)

(ii) (𝐴−1)−1 = 𝐴

10. If A=[1 2 22 1 22 2 1

], prove that 𝐴2 − 4𝐴− 5𝐼 = 0. Hence find 𝐴−1

SOLVING SYSTEM OF LINEAR EQUATIONS USING MATRIX METHOD

BASIC LEVEL QUESTIONS – 6 MARKS

Solve the system of equations using matrix method

1. 3x-2y+3z=8, 2x+y-z=1, 4x-3y+2z=4

2. X+y+z=3, 2x-y+z=-1, 2x+y-3z=-9

3. 3x+4y+7z=14, 2x-y+3z=4, x+2y-3z=0

4. 2x-y-z=7, 3x+y-z=7, x+y-z=3

37

5. X+y-5z=26, x+2y+z=-4, x+3y+6z=-29

6. 2x-3y+5z=11, 3x+2y-4z=-5, x+y-2z=-3

7. X+y+z=6, x+2z=7, 3x+y+z=12

8. X+2y-3z=-4, 2x+3y+2z=2, 3x-3y-4z=11

9. X+2y+z=7, x+3z=11, 2x-3y=1

10. 2x-y+3z=13, x+3y+2z=1, 3x-4y-z=8

AVERAGE LEVEL QUESTIONS – 6 MARKS

1. If a A= [2 −3 53 2 −41 1 −2

] find A-1.. Using A-1 solve the system of equations :- 2x-3y+5z=11,

3x+2y-4z=-5 and x+ y – 2z =-3

2. Solve the system of equations:2

𝑥+

3

𝑦+10

𝑧= 4 ,

4

𝑥−6

𝑦+5

𝑧= 1 ,

6

𝑥+9

𝑦−20

𝑧= 2

3. The sum of three numbers is 6.If we multiply third number by 3 and add second number to

it ,we get 11.By adding first and third numbers we get double of the second number.

Represent it algebraically and find the numbers using matrix method.

4. The cost of 4 chocolates ,3 samosas and 2 apples is Rs 60.The cost of 2 chocolates ,4

samosas and 6 apples is Rs.90.The cost of 6 chocolates ,2 samosas and 3 apples is Rs. 70.Find

the cost of each item by matrix method. What do you think is the healthiest diet? Suggest an

item that could replace samosa to make the diet healthier.

5. Using matrix method ,solve the system of equation x+2y+z,x-y+z=4,x+3y+2z=10.If x

represents the number of persons who take food at home, y represents the number of

persons who take junk food in market and z represent the number of persons who take food

at hotel. Which way of eating food you prefer and why?

6. There are three families, first family consists of 2 male members,4 female members and 3

children. Second family consists of 3 male members,3 female members and 2 children. Third

family consists of 2 male members, 2 female members and 5 children. Male member earns

Rs 500 per day and spends Rs 300 per day. Female member earns earns Rs 400 per day and

spends Rs 250 per day. Child member spends Rs 40 per day. Find the money each family

saves per day using matrices. What is the necessity of saving in the family?

7. Three shopkeepers A, B and C are using polythene, handmade bags (prepared by persons)

and newspapers envelopes as carry bags. It is found that the shopkeepers A,B ,C are

using(20,30,40) ,(30,40,20) ,(40,20,30) polythene ,handmade bags and newspapers

envelopes respectively. The shopkeepers A, B,C spent Rs 250,Rs 220 and Rs 200 on these

carry bags respectively. Find the cost of each carry bags using matrices. Keeping in mind the

social and environmental conditions, which shopkeeper is better and why?

8. Ina legislative assembly election , a political party hired a public relation firm to promote its

candidate in three ways; telephone, house calls and letters. The numbers of contacts of each

type in three cities A,B and C are (500,1000,5000) ,(3000,1000,10000) and (2000,1500,4000)

38

respectively. The party paid Rs.3700 ,Rs 7200 and Rs. 4300 in cities A,B and C respectively.

Find the costs per contact using matrix method keeping in mind the economic condition of

the country.

9. In a survey of 25 richest person of three localities x, y ,z. It is found that in locality A 7

believes in honesty,12 in hard work and 6 in unfair means. While in Y 10 believes in honesty

11 in hard work , 4 in unfair means and in Z, 8 believes in honesty ,12 in hard work 5 in unfair

means. If the income of 25 richest persons of locality x, y and z are respectively

Rs.1,19,000,Rs.1,18,0001,18,000per day then find income of each type of person by matrix

method. In your view which type of person is better for society and for nation.

10. Three friends A,B and C visited the reliance fresh to purchase fruits. A purchased 2kg

apples,1 kg grapes and 3 kg of oranges; B purchased 3kg apples,2 kg grapes and 1kg oranges;

C purchased 4 kg apples, 3 kg grapes and 2kg oranges, the amount paid by them are

respectively Rs.440,Rs.410and Rs.620. Find the cost of 1 kg of each fruit using matrix

multiplication. Why we take fruits in our diet.

ABOVE AVERAGE LEVEL QUESTIONS – 6 MARKS

1. Given that A=[1 −1 02 3 40 1 2

] and B=[2 2 −4−4 2 −42 −1 5

] find AB.Use this to solve the following

system of equations.x-y=3,2x+3y+4z=17,y+2z+7.

2. Given that A=[−4 4 4−7 1 35 −3 −1

] and B= [1 −1 11 −2 −22 1 3

] Find AB. Use this to solve the following

system of equations.

3. If A= [1 −1 12 1 −31 1 1

] find A-1 and hence solve the system of equations x+2y+z=4, -x+y+z=0, x-

3y+z =2.

4. An amount of Rs.5000 is put into three investments at the rate of interest of 6%,7% and 8%

per annum respectively. The total annual income is Rs.358. If the combined income from

the first two investments is Rs.70 more than the income from the third, Find the the amount

of each investment by matrix method.

5. A school wants to award its students for the values of honesty,regularity and hardwork with

a total cash award of Rs.6000. Three times the award money for hard work with a total cash

award of Rs.6000.Three times the award money for hardwork added to that given for

honesty amounts to Rs.11000.The award money given for honesty and hard work together

is double the one given for regularity. Represent the above situation algebraically and find

the award for each value using matrix method . Apart from these values namely honesty

,regularity and hard work suggest one more value which the school must include for awards.

6. For keeping fit X people believes in morning walk, Y people believe in Yoga and Z people join

Gym. Total number of people is 70, further 20%, 30% and 40% people are suffering from any

disease who believe in morning walk, Yoga and Gym respectively. Total number of such

people is 21. If morning walk costs Rs.0, Yoga costs Rs.500 per month and gym cost Rs.400

per month and total expenditure is Rs.23,000. (1) Formulate a matrix problem . (2) Calculate

the number of each type of people. (3) Why exercise is important for health.

39

7. An amount of Rs.600 crores is spend by the government in three schemes. Scheme A is for

saving girl child from the cruel parents who don’t want girl child and get the abortion before

her birth. Scheme B is for saving of newly wed girls from death due to dowry. Scheme C is

planning for good health for senior citizen. Now twice the amount spent on scheme C

together with amount spent on scheme A is Rs.700 Crores .And three times the amount

spent on scheme A together with amount spent on scheme B and scheme C is Rs.1200

Crores. Find the amount spent on each scheme using matrices. What is importance of saving

girl child from the cruel parents who don’t want girl child and get the abortion before her

birth.

8. Two schools A & B want to award their selected teachers on the values of honesty ,

hardwork and regularity. The school A wants to award Rs. x each, Rs.y each and Rs. Z each

for three respective values to 3,2and 1 teachers with a total award money of Rs.1.28 lakhs.

School B wants to spend Rs.1.54 lakhs lakhs to award its 4, 1 and 3 teachers on the

respective values (by giving the same award money for the three values as before). If the

total amount of award for one prize on each value is Rs.57,000 using matrices find the award

money for each value.

9. A School wants to reward the students participating in co-curricular activities (category I)

and with 100% attendance (category II) brave students (category III) in a function.The sum

of the numbers of all the three category student is 6. If we multiply the number of category

III by 2 and added to the number of category I to the result , we get 7. By adding II and III

category would to three times the first category we get 12. Form the matrix equation and

solve it.

10. Use the product [1 −1 20 2 −33 −2 4

] [−2 0 19 2 −36 1 −2

]to solve the system of equations x-y+2z=1,2y-

3z=1 and 3x-2y+4z=2.

THE COMMON MISTAKES COMMITTED BY THE STUDENTS IN

MATRICES AND DETERMINANTS:

1. While solving the questions based on equality of determinants students equate the corresponding

elements whereas the question must be solved by equating the values of determinants.

2. For finding the inverse of a matrix using elementary transformations

In the case of Elementary Row Transformations one should start with A=IA, while in the case of

finding inverse using Elementary Column Transformations one should use A=AI, the students make

mistake in it.Although the answer comes out to be correct but the in between steps are not

equivalent.

40

1-Mistakes of signs in calculating the cofactors

For example-

Evaluate

1 2 34 5 67 8 9

= 1|5 68 9

| + 2|4 67 9

| + 3|4 57 8

|

Rectification-

Minors and cofactors should be explained properly before expansion of

determinants

2-Mistake

students generaly using expansion method in place of using properties of

determinants.

Rectification – practice should be done so that students can apply the

properties of determinants.

3- Mistake

Students write the operation but the same is not implemented during solving

the problem.

Evaluate

1 𝑤 𝑤2𝑤 𝑤2 1𝑤2 1 𝑤

= 0 where w is the cube root of unity

Applying R1 -> R1+R2+R3

1 + 𝑤 + 𝑤2 𝑤 𝑤21 + 𝑤 + 𝑤2 𝑤2 11 + 𝑤 + 𝑤2 1 𝑤

Rectification - difference between Row and Column operation should be

explained clearly and more practice is required.

41

4- Mistake

Mistakes found in finding the common in determinants.

For example-

2 4 65 2 78 1 2

= 2 1 4 65 1 78 1 1

Rectification –

Common should be taken either from row or column , it should not be taken

from the diagonal elements.

5- Mistake

Mistakes found in breaking the determinants.

For example-

𝑎 + 1 𝑏 + 1 𝑐 + 1𝑎 + 2 𝑏 + 2 𝑐 + 2𝑎 + 3 𝑏 + 3 𝑐 + 3

= 𝑎 𝑏 𝑐𝑎 𝑏 𝑐𝑎 𝑏 𝑐

+ 1 2 31 2 31 2 3

Rectification –

Breaking of determinants should be explained clearly with help of different

type of examples.

TIPS AND TECHNIQUES:

1.While finding the inverse of a square matrix of order 3 the course of action

should be like :

*First try make a11 =1, then a21=0 ,a31=0 in case of Elementary Row

Transformations.

*First try to make a11=1,then a12=0,a13=0 in case of Elementary Column

Trasformations.

42

*For finding inverse of a matrix using Elementary Operations the operations

allowed are CIM (C= Clubbing i.e. R1→ 𝑅1 + 𝑘𝑅2, I= Interchanging i.e.𝑅1 ↔ 𝑅2

and M=Multiplying i.e.𝑅1 →1

𝑘𝑅2 ) This is also applicable for column

operations.

*If in any determinant all the elements of a particular row are all unity(i.e. 1)

then apply 𝐶2 → 𝐶2-𝐶1,𝐶3 → 𝐶3 − 𝐶1.

* If in any determinant all the elements of a particular column are all unity(i.e.

1) then apply 𝑅2 → 𝑅2-𝑅1,𝑅3 → 𝑅3 − 𝑅1.

*If the elements of a particular row or column are not all unity try to make

them unity if possible by applying proper property .

*Before expanding a determinant try to make maximum number of zeroes in a

row or column.

* In the particular problems regarding proving the determinant equal to the

given algebraic expression by using Properties of Determinant before

expansion two entries of any row or column must be zero.

*In multiplication of matrices the compatibility of multiplication must be

ascertained and the order of resulting matrix must be written properly.

43

CONTINUITY AND DIFFERENTIABILITY

1. Knowledge of functions :

(i) Polynomial functions: e.g. f(x) = x2 + 2 x + 5

(ii) Modulus function : f(x) = |𝑥|

(iii) Greatest Integer Function : f(x) = [x]

(iv) Signum function : The signum function, denoted 𝑠𝑔𝑛, is defined as

follows: 𝑠𝑔𝑛(𝑥) = { 1, 𝑥 > 0−1, 𝑥 < 0 0, 𝑥 = 0

(v) Trigonometric functions : sin x, cos x etc.

(vi) Inverse Trigonometric functions : sin- 1

x, cos- 1

x etc.

(vii) Logarithmic functions : f(x) = log x

(viii) Exponential functions : f(x) = ex

2. Continuity & Discontinuity :

Continuous at a Point

The function f is continuous at the point a in its domain if:

1. 𝑓(𝑥)𝑥→𝑎𝑙𝑖𝑚 ,

2. 𝑓(𝑥)𝑥→𝑎𝑙𝑖𝑚 = 𝑓(𝑎)

If f is not continuous at a, we say that f is discontinuous at a.

Note

If the point a is not in the domain of f, we do not talk about whether or not f is continuous at

a.

Continuous on a Subset of the Domain

The function f is continuous on the subset S of its domain if it continuous at each point of

S.

Points of Discontinuity in a Graph

Let f have the graph shown below.

44

Looking at the figure, we see that the possible points of discontinuity at x = -1, 0, 1, and 2.

Points of Discontinuity in a Graph

Let f have the graph shown below.

function is discontinuous at x = - 1 and 1.

Note: 1. All polynomial functions are continuous.

2. All rational functions are continuous provided denominator does not vanish.

3. All trigonometric functions are continuous.

4. All exponential functions are continuous.

Derivative; Differentiability

1) The derivative of the function f at the point a in its domain is given by 𝑓′(𝑎) =𝑓(𝑎+ℎ)−𝑓(𝑎)

ℎℎ→0

𝑙𝑖𝑚

2) The function f is differentiable at the point a in its domain if f'(a) exists.

3) The function f is differentiable on the subset S of its domain if it differentiable at each

point of S.

45

4)A function can fail to be differentiable at a point a if either 𝑓(𝑎+ℎ)−𝑓(𝑎)

ℎℎ→0

𝑙𝑖𝑚 does not exist, or is infinite

Note In the former case, we sometimes have a cusp on the graph, and in the latter case, we

get a point of vertical tangency.

As we see in the above graph on the right, there are no points of vertical tangency or cusps.

As we can see, the graphs provide immediate information as to where to look for a point of

non-differentiability: a point where there appears to be a cusp or a vertical tangent.

(a) Not all continuous functions are differentiable. For instance, the closed-form function f(x)

= |x| is continuous at every real number (including x = 0), but not differentiable at x = 0.

(b) However, every differentiable function is continuous.

3. Derivatives of different types of functions :

(i) Direct formulae based

Y = sin x dy

dx = cos x , etc.

(ii) Chain Rule :

Y = f (u)

dy

dx= f ′(u)

du

dx

(iii) By taking Logarithmic

Y = xx

log y = x log x 1

y dy

dx = x .

1

x + log x

dy

dx = y (1 + log x)

dy

dx = xx (1 + log x)

46

4. Rolle’s Theorem: Let f be continuous on a closed interval [a, b] and differentiable on

the open interval (a, b). If f(a) = f(b), then there is at least one point c in (a, b) where

f '(c) = 0

5. Mean Value Theorem: Let f be continuous on a closed interval [a, b] and

differentiable on the open interval (a, b). Then there is at least one point c in (a, b)

where f ′(c) =f(b)−f(a)

b−a.

(The Mean Value Theorem claims the existence of a point at which the tangent is parallel to

the secant joining (a, f(a)) and (b, f(b)). Rolle's theorem is clearly a particular case of the

MVT in which f satisfies an additional condition, f(a) = f(b).)

Question Bank on Continuity & Differentiability

Basic Level Questions:

1. At what points is the function given by 𝑓(𝑥) = 𝑥+1

(𝑥−2)(𝑥−3) is continuous?

2. Find f(0), so that 𝑓(𝑥) = 𝑥

1−√1−𝑥 becomes continuous. (Ans. F(0) = 2)

3. If the function 𝑓(𝑥) = 𝑆𝑖𝑛 10𝑥

𝑥, 𝑥 ≠ 0 is continuous at 𝑥 = 0, find f(0). (Ans. 10)

4. Show that the exponential function ex is continuous at each point of its domain.

5. Check the continuity of f(x) = 2x + 3 at x = 1. Discuss the continuity of the function f(x) = |x| at

x = 0.

6. Examine the continuity of the function at x = 2 if 𝑓(𝑥) = {1 + 𝑥 , 𝑖𝑓 𝑥 ≤ 25 − 𝑥 , 𝑖𝑓 𝑥 > 2

.

7. Discuss the continuity of the function f at x = 0 if; (𝑥) = {2𝑥 − 1, 𝑥 < 02𝑥 + 1, 𝑥 ≥ 0

.

8. Show that f(x) = 2x - |x| is continuous at x = 0.

9. Find the relation between a and b so that given function is continuous at x = 3, if 𝑓(𝑥) =

{𝑎𝑥 + 1, 𝑥 ≤ 3𝑏𝑥 + 3, 𝑥 > 3

. (Ans. a = b + 2/3)

10. Show that the function f(x) = |x + 2| is continuous at every 𝑥 ∈ 𝑅, but fails to be differentiate

at

x = 2.

Average Level Questions:

1. Find the value of a so that function 𝑓(𝑥) defined by 𝑓(𝑥) = {𝑆𝑖𝑛2𝑎𝑥

𝑥2, 𝑥 ≠ 0

1 , 𝑥 = 0 may be continuous at

𝑥 = 0. (Ans. 𝑎 = ±1)

http://www.cut-the-knot.org/do_you_know/few_words.shtml#continuity

http://www.cut-the-knot.org/do_you_know/few_words.shtml#cinterval

http://www.cut-the-knot.org/do_you_know/few_words.shtml#ointerval

http://www.cut-the-knot.org/do_you_know/few_words.shtml#continuity

http://www.cut-the-knot.org/do_you_know/few_words.shtml#cinterval

http://www.cut-the-knot.org/do_you_know/few_words.shtml#ointerval

47

2. Determine the value of k so that the function 𝑓(𝑥) = {𝑘𝑥2 , 𝑖𝑓 𝑥 ≤ 23 , 𝑖𝑓 𝑥 > 2

is continuous. (Ans. k = ¾)

3. Show that the function given by 𝑓(𝑥) = {𝑥 , 𝑖𝑓 𝑥 ≥ 1

𝑥2 , 𝑖𝑓 𝑥 < 1 is continuous everywhere on R.

4. Show that the function: 𝑓(𝑥) = {𝑥, 𝑖𝑓 𝑥 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟

0, 𝑖𝑓 𝑥 𝑖𝑠 𝑛𝑜𝑡 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 is discontinuous at each integral value

of x.

5. Show that the function 𝑓(𝑥) = {𝑒1𝑥−1

𝑒1𝑥+1

, 𝑤ℎ𝑒𝑛 𝑥 ≠ 0

0, 𝑤ℎ𝑒𝑛 𝑥 = 0

is discontinuous at x = 0.

6. If the function defined by 𝑓(𝑥) = {log(1+𝑎𝑥)−log (1−𝑏𝑥)

𝑥 , 𝑥 ≠ 0

𝑘 , 𝑥 = 0 is continuous at x = 0, find k.

7. If the function 𝑓(𝑥) = {3𝑎𝑥 + 𝑏 𝑖𝑓 𝑥 > 1

11 𝑖𝑓 𝑥 = 15𝑎𝑥 − 2𝑏 𝑖𝑓 𝑥 < 1

is continuous at 𝑥 = 1, find a and b. (Ans. A = 3, b = 2)

8. Find all the points of discontinuity of the function given by 𝑓(𝑥) = {𝑥 + 2 , 𝑥 ≤ 1

𝑥 − 2, 1 < 𝑥 < 20 , 𝑥 ≥ 2

. (Ans. x = 1)

9. Find the value of a and b so that given function is continuous; 𝑓(𝑥) = {5 , 𝑥 ≤ 2

𝑎𝑥 + 𝑏, 2 < 𝑥 < 1021 , 𝑥 ≥ 10

.

(Ans. a = 2, b = 1).

10. Locate the points of discontinuity of the function 𝑓(𝑥) = {𝑥4−16

𝑥−2, 𝑖𝑓 𝑥 ≠ 2

16, 𝑖𝑓 𝑥 = 2

Above Average Level Questions:

1. If 𝑓(𝑥) =

{

𝑥−5

|𝑥−5|+𝑎

𝑎 + 𝑏𝑥−5

|𝑥−5|+ 𝑏

is continuous function. Find a and b. (Ans. a=1, b = -1)

2. Show that function f defined as follows, is continuous at x = 2, but not differentiable at x = 2.

𝑓(𝑥) = {3𝑥 − 2, 0 < 𝑥 ≤ 12𝑥2− 𝑥, 1 < 𝑥 ≤ 25𝑥 − 4 , 𝑥 > 2

.

3. Discuss the continuity of the function 𝑓(𝑥) = {|𝑥| + 3, 𝑥 ≤ −3−2𝑥 , −3 < 𝑥 < 36𝑥 + 2 , 𝑥 ≥ 3

.

4. Let𝑓(𝑥) = {

1−𝑠𝑖𝑛3𝑥

3𝑐𝑜𝑠2𝑥𝑎

𝑏(1−sin 𝑥)

(𝜋−2𝑥)2

is continuous function at 𝑥 =𝜋

2 , find a and b. (Ans. a = ½ , b = 4)

48

5. Find all the points of discontinuity of the function f defined by 𝑓(𝑥) = {𝑥 + 2 , 𝑥 ≤ 1

𝑥 − 2, 1 < 𝑥 < 20 , 𝑥 ≥ 2

6. For what value of ‘k’ is the following function continuous at x = 2? 𝑓(𝑥) = {2𝑥 + 1 , 𝑥 < 2𝑘 , 𝑥 = 2

3𝑥 − 1 , 𝑥 > 2

7. Discuss the continuity of the following function at x = 0. 𝑓(𝑥) = {𝑥4+2𝑥3+𝑥2

tan−1𝑥 , 𝑥 ≠ 0

0 , 𝑥 = 0.

8. Show that the function ‘f’ defined by 𝑓(𝑥) = |1 – 𝑥 + |𝑥|| , 𝑥 ∈ 𝑅 is continuous.

9. Show that the function 𝑓(𝑥) = {𝑥 𝑆𝑖𝑛

1

𝑥, 𝑖𝑓 𝑥 ≠ 0

0, 𝑖𝑓 𝑥 = 0 is continuous at x = 0.

10. For what value of ‘k’, the function 𝑓(𝑥) = {√5𝑥+2−√4𝑥+4

𝑥−2, 𝑖𝑓 𝑥 ≠ 2

𝑘, 𝑖𝑓 𝑥 = 2 is continuous at x = 2.

Topic: Chain rule and differentiation of inverse trigonometric functions


Differentiate the following.

1. y = (2x3 - 7)

3

2. y = (x2 -8x + 9)

4

3. y = (x4– 9x3

+ 3x - 2)2

4. y=𝑒√cot𝑥

5. y=2xe

6. y = cos (x2)

7. y = cos2(x)

8. y = sin-1(

2x).

9. y = tan-1

(2 − x2)

10. y = sin-1

(ex)

49



1. y = ex + e2x

+ e3x

2. y = sin (ex)

3. y = xe2x

4. y = cos (3x2 + ex

)

5. y= tan−1 (sin𝑥

1+cos𝑥)

6. y = 4 tan-1

(2x4)

7. y = (x2 + 1) sin

-1(3x)

8. y = 𝑒𝑠𝑖𝑛− 1 𝑥

9. y= sin−1 (2𝑥

1+ 𝑥2)

10. y= sec−1 (1

2𝑥2−1)



1. y = sin3(8x)

2. y = log(cot(ex+ 1))

3. y = sin(sin(sin(x)))

4. y = cosec (log (2x4))

5. y = 𝑠𝑖𝑛−1 [5𝑥+12√1−𝑥2

13]

6. y = sin−1(𝑥2 √1 − 𝑥2 + 𝑥 √1 − 𝑥4)

50

7. If 𝑦 = 𝑒𝑎𝑐𝑜𝑠−1𝑥 , −1 ≤ 𝑥 ≤ 1 then show that: (1 − 𝑥2)

𝑑2𝑦

𝑑𝑥2− 𝑥

𝑑𝑦

𝑑𝑥− 𝑎2𝑦 = 0

8. If y = 1

2 2

5tan

6

ax

a x

then show that

2 2 2 2

3 2

9 4

dy a a

dx a x a x

9. If y = (tan-1

x)2, show that 212)1( 2

2

222

dx

dyxx

dx

ydx

10. If y = tan−1 (2𝑥

1− 𝑥2) + sec−1 (

1+ 𝑥2

1−𝑥2), then prove that

𝑑𝑦

𝑑𝑥=

4

1+ 𝑥2

Topic: Logarithmic and parametric differentiation


1. Find dx

dywhen x = 2at2, y = at4

2. Find dx

dywhen x = acos3t, y = bsin3t

3. Find dx

dywhen x = a (Ɵ + sin Ɵ ), y = a(1 + cos Ɵ )

4. Find dx

dywhen x = et(sin t + cos t), y = et (sint – cos t)

5. Find dx

dywhen x = cos Ɵ – cos 2 Ɵ, y = sin Ɵ – sin 2 Ɵ

6. Find dx

dyif y = e3logx

7. Find dx

dyif y = 2log2 𝑥

2

8. Find dx

dyif y = 5𝑥

5

9. Find dx

dyif y = xx

10. Find dx

dyif y = esinx


1. Find dx

dy when x =

tt

2

2

1

2

, y =

t

t2

1

2

2. Find dx

dy when x = a sin pt and y = b cos pt at t = π/4.

3. Find dx

dy when y = a cos3θ, x = b sin3θ

51

4. Find dx

dy when x = eθ (1 +

1) and y = e-θ (1 –

1)

5. Find dx

dy when x = eθ( 2sin θ + sin2 θ) , y = eθ( 2cos θ + cos2 θ)

6. Find dx

dy when xy yx = ab

7. Find dx

dy when y = (logx)x + xlogx

8. Find dx

dy when y = xx + x1/x

9. Find dx

dy when y = sin(xx)

10.Find dx

dy when y = xx + yx = 1


1. Find dx

dy when x = a

tt

2

2

1

1

, y =

t

bt2

1

2

2. If x = atsin 1

and y = atcos 1

, show that dx

dy= -

x

y

3. If x = sec Ɵ – cos Ɵ and y = sec n Ɵ - cos

n Ɵ ,

then show that (x 2 + 4) (

dx

dy)

2 = n

2 (y

2 + 4)

4. If x = log t and y = t

1, prove that y2 + y1 = 0.

5. If y = sin pt and x = sin t , then prove that (1 – x2) y2 – x y1 + p

2 y =0.

6. Differentiate w.r.t. x, esin x

+ (tan x ) x

7. Differentiate w.r.t. x, xcos x

+ (sin x ) tan x

8. Differentiate w.r.t. x, xlog x

+ (log ) x

9. If xy = e

x – y , prove that

dx

dy=

)log1(2

log

x

x

10. Find dx

dy when x

y + y

x = (x + y)

x + y

52

Topic: Mean Value theorem


1. Find the slope of chord A ( 0, 1) B ( 2,5)

2. Is f(x) = tanx on [ -1 , 1] is continuous ?

3. Is f(x) = |x| on [ -1 , 1] is continuous and differential ?

4. Is logx is continuous on for all x>0

5. Find the derivative of function f(x) = (x-2) (x-3) (x-4)

6. Find the derivative of function cos 2( x- π/4)

7. If (cosx –sinx) =0 then what is x ?

8. If sin2x = 0 then what is x ?

9. Is f(x) = 2+(x-1)2/3 on (0,2) is differentiable ?

10. Find the derivative of 𝑠𝑖𝑛𝑥

𝑒𝑥


Verify mean value theorem for the following function

1. F(x) = x (x-2) on [ 1,3]

2. F(x) = x3-2x2-x+3 on [0,1)

3. F(x) = (x-3) (x-6) (x-9) on [3,5]

4. F(x) = 𝑒1−𝑥2 0n [-1,1]

5. F(x) = 1

4𝑥−1 on [1,4]

6. F(x) = cosx on [0,π/2 ]

7. F(x) = √25− 𝑥2 0n [1,5]

8. F(x) = 𝑥1/3 on [-1,1]

9. F(x) =𝑒𝑥 on [0,1 ]

10. F(x) = x (x+4)2 on [0,4]


Verify mean value theorem for the following function

1) F(x) = log𝑒 𝑥 on [ -1 ,1]

2) F(x) = x-2sinx on [ -π,-π]

3) F(x) = 2sinx +sin2x on [0,π]

4) Find c of mean value theorem of the function

F(x) = x (x-1) (x-2) on [0,1/2]

53

5) Using mean value theorem find a point p on the curve y=√𝑥2 − 4

defined in the interval [2,4] where tangent is parallel to the chord joining

the points on the curve .

6) Find a point on the on y= x3-3x where the tangent is parallel to the

chord joining (1,-2) and (2,2) .

7) Discuss the applicability of mean value theorem for the function Is f(x)

= |x| on [ -1 , 1]

8) find c so that f’(c) = 𝑓(6)−𝑓(4)

6−4 where f(x) = √𝑥 + 2 and

c € (4,6)

9) Find a point on the on y= (x-3)2 where the tangent is

parallel to the chord joining (3,0) and (4,1) .

1o) Find a point on the on y= 12(X+1) (X-2) in the interval

[-1,2] where the tangent is parallel to x-axis.

Topic: Second order derivatives


1. Find second order derivative of x3.

2. If y = cot x, find 𝑑2𝑦

𝑑𝑥2 at =

𝜋

2 .

3. If y = 5 cosx -3 sinx , prove that 𝑑2𝑦

𝑑𝑥2 + y = 0.

4. If √𝑥 +√𝑦 = √𝑎, find 𝑑2𝑦

𝑑𝑥2 at x=a.

5. If y=500 𝑒7𝑥 + 600 𝑒−7𝑥 , prove that 𝑑2𝑦

𝑑𝑥2 = 49y.

6. If 𝑥

𝑎 + 𝑦

𝑏 =1, find

𝑑2𝑦

𝑑𝑥2.

7. If 𝑥 = 𝑎 𝑠𝑖𝑛 𝑝𝑡 and 𝑦 = 𝑏 𝑐𝑜𝑠 𝑝𝑡, find the value of 𝑑2𝑦

𝑑𝑥2 at x=0.

8. If 𝑥 = 𝑎𝑡2, 𝑦 = 2𝑎𝑡 find 𝑑2𝑦

𝑑𝑥2.

9. If 𝑦 = 𝑡𝑎𝑛𝑥, prove that 𝑑2𝑦

𝑑𝑥2= 2y

𝑑𝑦

𝑑𝑥.

10. Find 𝑑2𝑦

𝑑𝑥2 when 𝑦 = 𝑡𝑎𝑛𝑥 + 𝑠𝑒𝑐𝑥


.If y= (cos−1 𝑥)2 , prove that (1-x2)y2 –xy1=2.

2.If y=log(x+√𝑥2 + 𝑏2) prove that ( x2+ b2) y2 + xy1 =0.

54

3.If y=𝑒tan−1 𝑥 prove that (1+x2) y2 + (2x-1)y1 = 0.

4. If x = 2𝑎𝑡2

1+𝑡 and y =

3𝑎𝑡

1+𝑡 find

𝑑2𝑦

𝑑𝑥2.

5.If 𝑥𝑦 = 𝑠𝑖𝑛𝑥 , prove that 𝑑2𝑦

𝑑𝑥2 + 2

𝑥 𝑑𝑦

𝑑𝑥 + y =0.

6. If x = asin3t , y = bcos3t , find 𝑑2𝑦

𝑑𝑥2 at t=

𝜋

4.

7. If y = 3 cos (logx) + 4 cos (logx) prove that

x2 𝑑2𝑦

𝑑𝑥2 + x

𝑑𝑦

𝑑𝑥 + y =0

8. If y =𝑒𝑥( sinx + cosx ), prove that 𝑑2𝑦

𝑑𝑥2 -2

𝑑𝑦

𝑑𝑥 + 2y =0

9. If y=A𝑒𝑚𝑥+ B 𝑒𝑛𝑥, show that 𝑑2𝑦

𝑑𝑥2- (m+n)

𝑑𝑦

𝑑𝑥 mny=0.

10.If 𝑒𝑦(x+1) =1 ,show that𝑑2𝑦

𝑑𝑥2 = (

𝑑𝑦

𝑑𝑥)2


1.If y= (1+√𝑥2 − 1)m , prove that (𝑥2 − 1) 𝑦2 + x y1 = m2 y

2. y = 𝑠𝑖𝑛− 1𝑥

√1− 𝑥2 , show that (1-x2)y2 -3xy1 –y =0.

3.If x= asin2t (1+cos2t) and y= bcos2t( 1-cos2t) , find 𝑑2𝑦

𝑑𝑥2 at t=

𝜋

4.

4.If y=x log(𝑎

𝑎+𝑏𝑥),prove that

𝑑2𝑦

𝑑𝑥2=

1

𝑋 (

𝑎

𝑎+𝑏𝑥)2

5. If y = [𝑙𝑜𝑔(𝑥 + √𝑥2 + 1)]2

, Show that (1+x2) y2+xy1-2=0

6. If y= A 𝑒−𝑘𝑥 cos( pt+c) prove that 𝑑2𝑦

𝑑𝑥2 + 2 k

𝑑𝑦

𝑑𝑥 + ( p2 +k2)y =0.

7.If y = sin(𝑚 sin−1 𝑥), prove that (1-x2) y2-xy1 + m2y =0

8.If 𝑥 = 𝑎( 𝑐𝑜𝑠𝑡 + 𝑡𝑠𝑖𝑛𝑡), 𝑦 = 𝑎 (𝑠𝑖𝑛𝑡 − 𝑡𝑐𝑜𝑠𝑡) , 𝑓𝑖𝑛𝑑 𝑑2𝑦

𝑑𝑥2.

9.If 𝑠𝑖𝑛(𝑥 + 𝑦) = 𝑘𝑦 ,prove that y2 +y(1+y1)3 =0.

10.If y= 𝑒𝑎 sin−1 𝑥, prove that (1-x2) y2 –xy1-a2y =0

55

COMMON ERRORS COMMITTED BY STUDENTS IN CONTINUITY & DIFFERENTIABILITY

S. No. Errors Correction Remarks 1. 𝑦 = 𝑥𝑦 + 𝑦𝑥

𝑙𝑜𝑔𝑦 = 𝑙𝑜𝑔𝑥𝑦 + 𝑙𝑜𝑔𝑦𝑥

𝑦 = 𝑥𝑦 + 𝑦𝑥 𝑦 = 𝑢 + 𝑣

𝑢 = 𝑥𝑦 ,𝑙𝑜𝑔 𝑢 = 𝑦 𝑙𝑜𝑔 𝑥 & 𝑣 = 𝑦𝑥,𝑙𝑜𝑔 𝑣 = 𝑥 𝑙𝑜𝑔 𝑦

Proper application of logarithmic properties

2. 𝑑(𝑙𝑜𝑔𝑒𝑒)

𝑑𝑥=1

𝑒

𝑙𝑜𝑔𝑒𝑒 = 1 𝑑(1)

𝑑𝑥= 0

Proper application of logarithmic properties

3. 𝑑

𝑑𝑥(𝑥𝑥) = 𝑥𝑥

𝑑

𝑑𝑥(𝑥𝑥) = 𝑥𝑥(1+ 𝑙𝑜𝑔𝑥) Proper application of

logarithmic properties

4. 𝑑

𝑑𝑥(𝑥𝑥) = 𝑥. 𝑥𝑥−1

𝑑

𝑑𝑥(𝑥𝑥) = 𝑥𝑥(1+ 𝑙𝑜𝑔𝑥) Proper application of


5. 𝑑(sin2𝑥)

𝑑𝑥= cos 2𝑥

𝑑(sin2𝑥)

𝑑𝑥= 2cos2𝑥 Application of chain

rule 6. 𝑑

𝑑𝑥(𝑎𝑥) = 𝑥. 𝑎𝑥−1

𝑑

𝑑𝑥(𝑎𝑥) = 𝑎𝑥𝑙𝑜𝑔𝑒𝑎 Proper application of


7. 𝑑

𝑑𝑥(𝑎𝑥) = 𝑎𝑥

𝑑

𝑑𝑥(𝑎𝑥) = 𝑎𝑥𝑙𝑜𝑔𝑒𝑎 Proper application of


8.

dx

dy= sin t

tdx

ydcos

2

2

dx

dy= sin t

dx

dtt

dx

yd.cos

2

2

Application of chain rule

9. 𝐼𝑓 𝑥 = 𝑓(𝑡)𝑎𝑛𝑑 𝑦 = 𝑔(𝑡),

𝑑2𝑦

𝑑𝑥2=

𝑑2𝑦𝑑𝑡2

𝑑2𝑥𝑑𝑡2

𝑑2𝑦

𝑑𝑥2=𝑑(𝑓(𝑡))

𝑑𝑡.𝑑𝑡

𝑑𝑥

Application of chain rule

10. Differentiate 𝑓(𝑥) = (𝑥 − 1)2

3 on [0,2].

Answer: 𝑓′(𝑥) =2

3(𝑥 − 1)

−1

3

f′(x) =2

3(x − 1)

−1

3 , but Left hand

derivative and right hand derivative at x = 1 are not equal. So at x = 1 it is not derivable and hence not differentiable in (0, 2)

Check LHD and RHD.

11. 𝑓(𝑥) =1

𝑥 then

𝑓′(𝑥) =1𝑑𝑑𝑥𝑥 − 𝑥

𝑑𝑑𝑥1

𝑥2

𝑓′(𝑥) =𝑥𝑑𝑑𝑥1 − 1

𝑑𝑑𝑥𝑥

𝑥2 =

1

𝑥2

𝑜𝑟 𝑓′(𝑥) =𝑑(𝑥−1)

𝑑𝑥= −1 𝑥−2

=−1

𝑥2

Apply correct way of quotient rule.

56

Tips and techniques for scoring good marks

1) To show the function is continuous such as

lim𝑥→𝑎 𝑓(𝑥) is continuous at x = a then find lim𝑥→𝑎 𝑓(𝑥) and f(a) if

both are equal then it is continuous.

2) To show the function is continuous such as

f(x) = { 𝑓(𝑥) 𝑖𝑓 𝑥 ≠ 𝑎

𝑔(𝑥) 𝑖𝑓 𝑥 = 𝑎 then find RHL ,LHL and f(a) if all are

equal

then f (x) is continuous.

3) To find the value of k if it is given that the function is continuous. Then

find only RHL or LHL and f(a) and equate.

4) For logarithmic differentiation, students must be aware of all the rules

of logarithmic & exponential.

5) To find second order derivative in parametric differentiation, proper

use of chain rule should be followed.

6) To verify Role’s theorem & Mean Value theorem, there is no need to

prove continuity & differentiability for certain functions

(i) Polynomial functions, sin x , cos x, exponential functions etc. are

always cont. for all x ε R.

(ii) tan x, cot x, sec x, & cosec x are in certain intervals.

(iii) log x is always cont. for all x > 0

(iv) |𝑥| is continuous but not differentiable at 𝑥 = 0.

(v) [𝑥]𝑖𝑠 neither differentiable nor cont.

57

APPLICATION OF DERIVATIVES

1. RATE OF CHANGE OF QUANTITY:

1) 𝑑𝑦

𝑑𝑥 represents the rate of change of y with respect to x.

Here y is the depending variable and x is the independent variable.

2) 𝑑𝑦

𝑑𝑥 at a particular point x0 represents the rate of change of y w.r.t. x at x = x0.

3) If x = f(t) and y = g(t), then 𝑑𝑦

𝑑𝑥=

𝑑𝑦

𝑑𝑡/𝑑𝑥

𝑑𝑡 .

4) Marginal cost represents the instantaneous rate of change of the total cost at any level

of output. If C(x) represents the cost function for x units produced, then marginal cost,

denoted by MC, is given by MC = 𝑑

𝑑𝑥{𝐶(𝑥)}.

5) Marginal revenue represents the rate of change of total re4venue with respect to the

number of items sold at an instant. If R(x) is the revenue function for x units sold ,

then marginal revenue , denoted by MR, is given by MR = 𝑑

𝑑𝑥{𝑅(𝑥)}.

6) The profit function ( )P x is given by the formula: ( ) ( ) ( )P x R x C x .

7) The rate of change of the profit function ( )P x is '( )P x and is called the marginal profit

function for the product. 8) The point at which the profit is zero is called the break-even point.

Increasing / Decreasing Functions

1) Let & 'x x be any two points taken from an interval ( , )a b .

2) A real function ( )y f x is increasing on the (a, b), if ( ) ( ')f x f x whenever 'x x .

3) The function is said to be decreasing on ( , )a b , if ( ) ( '),f x f x whenever 'x x .

Increasing / decreasing test:

The following result, called Increasing/decreasing test, is very useful in applications:

(i) If '( ) 0f x for all ( , )x a b , then f is increasing on ( , )a b

(ii) If '( ) 0f x for all ( , )x a b , then f is decreasing on ( , )a b

(iii) If '( ) 0f x for all ( , )x a b , then f is constant on ( , )a b .

***: A function f is said to be monotonic on an interval, if it is either increasing or decreasing on

that interval.

5) A function ( )y f x is said to have a critical point at x c , if any one of the following

conditions is satisfied:

(a) '( ) 0f c (b) '( )f c is undefined, but ( )f x is continuous at x c .

58

6) Let ( )y f x be a given function. The points where '( ) 0f x are called stationary points of the

function. So, we can find the stationary points of a function ( )y f x by solving the equation

'( ) 0f x for x .

TANGENTS AND NORMALS

1)For the curve y = f(x), 𝑑𝑦

𝑑𝑥 , represents the slope of the tangent to the curve

2) Slope of the tangent to the curve at (x1, y1) is 𝑑𝑦

𝑑𝑥 ]( x1, y1).

3) Equation to the tangent at (x1, y1) to a curve y = f(x) with slope m is y – y1 = m(x – x1)

4) If m is the slope of the tangent to the curve y = f(x) at( x1, y1 ), then slope of the normal at (x1, y1)

is -1/m.

5) Equation of normal at (x1, y1) is 𝑦 − 𝑦1 = −1

𝑚(𝑥 − 𝑥1)

Approximation

1) Let 𝑓:𝐷 ⊆ 𝑅 → 𝑅 be a function y = f(x). Δx is small increment in x and Δy is increment in y

corresponding to increment Δx in x. Then ∆𝑦 = 𝑑𝑦

𝑑𝑥. ∆𝑥.

2) In approximation, 𝑓(𝑥 + ∆𝑥) = 𝑓(𝑥) + 𝑓′(𝑥). Δx

𝑦 + ∆𝑦 = 𝑦 + 𝑑𝑦

𝑑𝑥∆𝑥

Maxima & Minima

1)There are two types of extreme positions: local (relative) and global (absolute).

2)A function ( )f x defined on an interval [ , ]a b is said to have a local (or relative) maximum at a

point x c , if ( ) ( )f c f c h for all sufficiently small negative as well as positive values of h . The

function is said to have a local (or relative) minimum at x c , if ( ) ( ).f c f c h

3)The point x c , where '( ) 0f c or '( )f c does not exist, is called a critical point of the function

( )f x .

4)A maximum or a minimum value of a function is also termed as extremum or extreme value of the

function.

5) Let f be function defined in the closed interval I. If there exist a point ‘a’ in the interval I such that

f(a) ≥ f(x) for every x є I, then the function is said to attain absolute maximum at x = a, and f(a) is

absolute maximum value.

59

6) Let f be function defined in the closed interval I. If there exist a point ‘a’ in the interval I such that

f(a) ≤f(x) for every x є I, then the function is said to attain absolute minimum at x = a, and f(a) is

absolute minimum value.

7) To find the absolute maxima or minima in [a, b] we have to find out the value at the end point of

interval [a, b] i.e. f(a) and f(b) along with local maxim or minima.

Test for maximum or minimum:

1) First Derivative Test : If a function ( )f x has either local maxima or minima at a point x c ,

then either '( ) 0f c or '( )f c does not exist, i.e, x c is a critical point of the function. Of course,

there may be functions for which ( )f c is not a local extremum, even when x c is a critical point

of the function.

2) If f’(x) does not change its sign in the neighbourhood of ‘x1’, then ‘x1’ is neither point of local

maxima nor local minima, then x1 is called the point of inflexion.

Second Derivative Test: Let '( ) 0f c for a given function ( )f x defined on ( , )a b . Then

(i) ''( ) 0 ( )f c f c is a local maximum of ( )f x

(ii) ''( ) 0 ( )f c f c is a local minimum of ( )f x .

NOTE: If second derivative test fails, then apply first derivative test. ( In case of linear

polynomial second derivative test fails, as the first derivative becomes constant

and hence it’s second derivative become zero.

MATHEMATICAL MODELLING OF THE WORD PROBLEM

(MAXIMA AND MINIMA)

1) Draw the labeled figure wherever it is required. 2) Write the maximizing or minimizing function (correct formula). 3) Convert the function in a single independent variable using either by previous knowledge or

given data. 4) Differentiate the function got in step (3) w.r.t. that independent variable. 5) Find the critical points by equating first derivative to zero. 6) Find the second derivative got in step (3). 7) Find the value of second derivative at critical points. 8) If the value is less than zero then the function has maxima and the maximum is the value of

the function( in step 3) at the critical point.

9) If the value is greater than zero then the function has minima and the minimum is the value of the function(in step 3) at the critical point.

60

QUESTION BANK ON APPLICATION OF DERIVATIVE

Sub Topic : Quantity as a rate measurer

Level 1

Q1 Find the rate of change of volume of the cone of constant height, w.r.t. radius of the cone.

(1) Q2 Find the rate of change of area of a circle w.r.t. the radius x. (1) Q3 The side of an equilateral triangle is increasing at the rate of 0.5cm/sec. Find the rate of

increase of its perimeter. (1)

Q4 An edge of a variable cube is increasing at the rate of 5 cm/sec. How fast is the

volume of the

cube is increasing when edge is 10cm long? (4) Ans: 1500

cm3/sec.

Q5 A balloon which always remains spherical is being inflated by pumping in gas at the

rate of 900cm3/sec. Find the rate at which the radius of the balloon is

increasing when the radius of the

balloon is 15 cm. (4)

Q6 The volume of cube is increasing at the rate of 6cm3/sec. How fast is the surface area

increasing when the length of an edge is 15cm. (4)

Q7 A particle moves along the curve 6y = x3 + 2. Find the points on the curve at which

the y-coordinate is changing 5 times as fast as x-coordinate. (4) Q8 A kite is moving horizontally at the height of 15.1 metres. If the speed of the kite is 10m/sec,

how fast is the string being let out, when the kite is 250 m away from the boy who is flying the kite? The height of the boy is 1.5m. (6)

Q9 At what point of the ellipse 16x2 + 9y2 = 400, does the ordinate decrease at the same rate at which the abscissa increases?

Q10

Level 2 Q1 The side of a square is increasing at the rate of 0.2 cm/sec. Find the rate of increase of

perimeter of the square. (1) Q2 The radius of the circle is increasing at the rate of 0.7 cm/sec. What is the rate of increase of

its circumference? (1) Q3 If the radius of a soap bubble is increasing at the rate of ½ cm sec. At what rate its volume

is increasing when the radius is 1 cm. (1) Q4 The total revenue received from the sale of x units of a product is given by R(x) = 3x3 + 12x

+10. Find the marginal revenue when x = 3. (1)

Q5 x and y are the sides of two squares such that y = x – x2. Find the rate of change of the

area of Second Square with respect to the area of the first square.

Q6 The volume of a spherical balloon is increasing at the rate of 25 cm3/sec. Find the rate

of change

of its surface area at the instant when the radius is 5 cm. (4) Ans: 10

cm2/sec.

Q7 The surface area of a spherical bubble is increasing at the rate of 2 cm2/sec. Find the

rate at which the volume of the bubble is increasing at the instant if its radius is 6 cm.

(4) Ans:80πcm2/s

Q8 A spherical balloon is being inflated by pumping in 16cm3/sec of gas. At the instant

when balloon contain 36π cm3of gas, how fast is its radius increasing? (4)

61

Q9 The bottom of the rectangular tank is 50cm x 20cm. Water is pumped into the tank at

the rate of 500c.c./min. Find the rate at which the level of water in the tank is rising.

(4)

Q10 Water is running out of a conical funnel at the rate of 5 cm3/sec. If the radius of the

base of the funnel is 10 cm and altitude is 20 cm, find the rate at which the water level

is dropping when it is 5 cm from the top. (6)

Level 3

Q1 A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/sec. At the instant when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing? (1)

Q2 For the curve y = 5x – 2x3, if x increases at the rate of 3 unit/sec, then how fast is the

slope of curve changing when x = 4. (1)

Q3 Find an angle α, which increases twice fast as its sine. (1)

Q4 Sand is pouring from a pipe at the rate of 12cm3/sec. The falling sand forms a cone

on the ground in such a way that the height of the cone is always one-sixth of the

radius of the base. How fast is the height of the sand cone in increasing when the

height is 4 cm? (4) Ans : 1

48𝜋cm/ sec (4)

Q5 A man 160 cm tall walks away from a source of light situated at the top of the pole

6m high at the rate of 1.1 m/sec. How fast is the length of the shadow increasing when

he is 1m away from the pole? (4)

Q6 Gas is escaping from a spherical balloon at the rate of 900 cm3/sec. How fast is the

surface area ,

radius of the balloon shrinking when the radius of the balloon is 30cm? (4)

𝑑𝐴

𝑑𝑡= 60𝑐𝑚2/s,

𝑑𝑟

𝑑𝑡=

1

4𝜋𝑐𝑚/𝑠

Q7 Water is passed into an inverted cone of base radius 5 cm and depth 10 cm at the rate

of 3

2 c.c /sec. Find the rate at which the level of water is rising when depth is 4 cm.

(4) Ans: 3

8𝜋cm/ sec

Q8 A ladder 13m long is leaning against a vertical wall. The bottom of the ladder is

dragged away from the wall along the ground at the rate of 2cm/sec. How fast is its

height on the wall decresing when the foot of the ladder is 5m away from the wall?

(4)

Q9 Water is dropping out from a conical funnel of semi-vertical angle π/4 at the uniform

rate of 2cm2/sec in its surface area through a tiny hole at the vertex in the bottom.

When the slant height of the water is 4 cm, find the rate of decrease of the slant height

of the water. (6)

Q10 A solid is formed by a cylinder of radius r and height h together with two hemisphere

of radius r attached at each end. It the volume of the solid is constant but radius r is

increasing at the rate of 1

2𝜋metre/ min ? How fast must h (height) be

changing when r and h are 10 metres. (6)

62

SUB TOPIC: INCREASING AND DECREASING Level 1

Q1. Find the value of a for which the function f(x) = x2

– 2ax + 6, x > 0 is strictly

increasing.

Q2. Write the interval for which the function f(x) = cos x, 0 ≤ 𝑥 ≤ 2𝜋 is decreasing.

Q3. Write the interval for which the function f(x)= 1/x is strictly decreasing.

Q4. If f (x) = ax + cos x is strictly increasing on R, find a.

Q5. Show that the function sin 𝑥

𝑥 is strictly decreasing in (0,

𝜋

2)

Q6. Find the interval in which f(x)= log 𝑥

𝑥 , 𝑥 ∈ (0,∞) is increasing ?

Q7. For which values of x, the functions y =𝑥4−4

3𝑥3 is increasing?

Q8. Find the least value of 'a' such that the function f(x) = x2 + ax + 1 is strictly increasing

on (1, 2).

Q9. Find the interval in which function is increasing and decreasing:

f(x) = – 2x3 – 9x

2 – 12x + 1

Q10. For what value of a, the function f(x) =a(x + sin x) + a ,is increasing or R.

Level 2 Q 1 Show that f(x) = x

3 –6x

2 + 18x + 5 is an increasing function for all x ∈ R

Q2 Write the interval in which the function f(x) = x9 + 3x

7 + 64 is increasing

Q3 Prove that the function f(x) = x2 – x + 1 is neither increasing nor decreasing in [0, 1].

Q4 Find the intervals on which the function 𝑥

𝑥2+1 is decreasing.

Q5 Find the intervals in which the function f(x)

=𝑠𝑖𝑛2𝑥 𝑖𝑛[0,𝜋]𝑖𝑠 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑜𝑟 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔

Q6 Find ‘a’ for which f (x) = a (x + sin x) is strictly increasing on R.

Q7 Find the intervals in which the function f(x) = (x + 1)3 (x – 3)

3 is strictly increasing or

strictly decreasing.

Q8 Find the interval in which the function f(x)=5𝑥3

2 −3𝑥5

2, 𝑥 > 0 is strictly decreasing.

Ans: Inc (0,1) and dec (1,∞)

Q9 Prove that the function f(x) =𝑥3

3− 𝑥2 + 9𝑥 is strictly increasing. Hence find the

minimum value of f(x)

Q10 Find the intervals in which the function f(x) = x3 – 12x

2 + 36x + 17 is (a) Increasing

(b) Decreasing.

Level 3 Q1 Find the intervals in which the function f(x) = sin

4x + cos

4x, 0≤ 𝑥 ≤

𝜋

2 is increasing

or decreasing. Ans: 𝑖𝑛𝑐. (𝜋

4,𝜋

2)𝑑𝑒𝑐 (0,

𝜋

4)

Q2 Find the intervals in which the function f(x) = log (1 + x) – 𝑥

1+𝑥 𝑤ℎ𝑒𝑟𝑒 𝑥 > −1is

increasing or decreasing Ans: 𝑖𝑛𝑐 (0,∞) 𝑑𝑒𝑐(−1,0)

Q3 Show that the function f(x) = tan–1

(sin x + cos x), is strictly increasing on the interval

(0,𝜋

4)

Q 4 Find the interval in which the function f given by f(x) = sin x - cos x, 0 <x < 2𝜋 is

increasing or decreasing. Ans :𝑖𝑛𝑐 (0,3𝜋

4) ∪ (

7𝜋

4)𝑎𝑛𝑑 𝑑𝑒𝑐 (

3𝜋

4,7𝜋

4)

Q5 Prove that the function y = 4 𝑠𝑖𝑛𝜃

2+𝑐𝑜𝑠𝜃− 𝜃is an increasing function of 𝜃in [0,

𝜋

2 ]

Q6 Find the interval of monotonicity of the function f(x) = 2x2 – log x , x ≠ 0

Q7 Find the intervals for which the function f(x) =log(2 +x) − 2𝑥

2+𝑥 is increasing or

decreasing Ans: 𝑖𝑛𝑐.(2,∞),𝑑𝑒𝑐(−2,2)

63

Q8 Find the interval in which the given function is increasing or decreasing:

f(x)=4 𝑠𝑖𝑛𝑥−2𝑥−𝑥 𝑐𝑜𝑠𝑥

2+cos 𝑥 Ans: 𝑖𝑛𝑐 (0,

𝜋

2) ∪ (

3𝜋

2, 2𝜋) 𝑑𝑒𝑐 (

𝜋

2,3𝜋

2)

Q9 Show that the function cos (2x+𝜋

4 ) is strictly increasing on(

3𝜋

8,7𝜋

8)

Q10 Find the sub-interval of the interval (0, 𝜋/2) in which the function f(x) = sin 3x

is increasing.

Sub Topic : Tangents and Normal

Level 1

Q1 Find the slope of the tangent to the curve y = 3x4 – 4x at x = 4. (1)

Q2 Find the slope of tangent to the curve y = 𝑥−1

𝑥−2 , x ≠ 2 at x = 10. (1)

Q3 Find the slope of the tangent to the curve y = x3 – x + 1 at the point whose

x –coordinate is 2. (1)

Q4 Find the equations of tangent and normal to the curve 𝑦2 = 𝑥2

4−𝑥 at (3, - 3) (4)

Q5 Find the equations of the normal lines to the curve y = 4x3 + 3x + 5 which are parallel

to the line 9y + x + 3 = 0. (4)

Q6 Find the slope of the normal at the point (am3, am

2) to the curve ax

2 = y

3. (4)

Q7 At what points on the curve x2 + y

2 – 2x – 4y +1 = 0, the tangents are parallel to x –

axis? (4)

Q8 Find the equations of normal lines to the curve y = x3 – 3x which are parallel to the

line

x + 9y = 14. (6)

Q9 Find the equation of the tangent and normal to the curve 3x2 – y

2 = 3, which are

perpendicular to the line x + 3y = 2. (6)

Q10 Find the points on the curve y = x3 at which the slope of the tangents is equal to

ordinate of that point. (6)

Level 2

Q1 Find the equation of the tangent to the curve y = x3 at (1, 1). (1)

Q2 Find the equation of the normal to the curve y = x2 at (0,0). (1)

Q3 Show that the tangents to the curve y = 2x3 – 4 at the points where x = 2 and x = - 2

are parallel. (1)

Q4 Show that the tangents to the curve y = x2 – 5x + 6 at the points where x = 2 and x = 3

are at right angles. (1)

Q5 Find the equation of tangent at the point ‘t’ on the curve x = asin3t, y = bcos

3t. (4)

Q6 Find the equation of tangents to the curve y = (x3 – 1)(x – 2) where the curve cuts x-

axis. (4)

Q7 Find the points on the curve 4x2 + 9y

2 = 1, where the tangents are perpendicular to

the line

2y + x = 0. (4)

Q8 Prove that the curves y2 = 4ax and xy = c

2 cut at right angles, if c

4 = 32a

4. (6)

Q9 Show that the curves x = y2 and xy = k cut orthogonally if 8k

2 = 1. (6)

Q10 Show that the curves 2x = y2 and 2xy = k cut at right angles if k

2 = 8. (6)

64

Level 3

Q1 Find the slope of the normal to the curve x = 1 – asinθ, y = bcos2θ at θ = π/2. (1)

Q2 Find the angle between the curves xy=6 and x2y=12. (1)

Q3 At what point(s) on the curve y = x2 does the tangent make an angle of 45

ᵒ. (1)

Q4 Show that the equation of tangent to the ellipse 𝑥2

𝑎2+

𝑦2

𝑏2= 1 𝑎𝑡 (x1 , y1) is

𝑥𝑥1

𝑎2+

𝑦𝑦1

𝑏2= 1.

(4)

Q5 Prove that the curves xy = 4 and x2 + y

2 = 8 touch each other. (4)

Q6 Find the equation of tangent to the curve x = sin3t, y = cos2t at t = π/4. (4)

Q7 For the curve y = 4x3 – 2x

5, find all the points at which the tangents passes through

the origin. (6)

Q8 If the tangent to the curve y = x3 + ax + b at point

(1, - 6 )is parallel to the line y – x =

5. Find the value of a and b. (6)

Q9 Find the equation of tangents and normal to the curve x1/2

+ y1/2

= at the point (a

2/4,

a2/4). (6)

Q10 The curve y = ax3 + bx

2 + cx + 5 touches

the x-axis at the point ( -2, 0) and cuts the y

–axis at a point where its gradient is 3. Find a, b and c. (6)

Sub Topic : Approximation by differentiation Level 1

Q1 For finding the approximate value of (82)1/2

, write the function f and the value of x

and Δx you will start with. (1)

Q2 If the error in the side of a square is 0.5%, find the percentage error in its area. (1)

Q3 For the function y = x3, if x = 5 and ∆x = 0.01, find ∆y

Q4 Using differentials find the appropriate value of (4 marks each)

(i) (82)1

4 (ii) (26)1/2

(iii) (127)1/3

(iv) (37)1/2

(v) (401)1/2

Level 2 Q1 Find the appropriate change in volume of a cube when side increases by 1%. (1)

Q2 Find the percentage error in calculating the volume of a cubical box if an error of

0.1% is made in measuring the length of edges of the cube. (1)

Q3 Find the approximate value of (1.995)5. (4)

Q4 If the radius of the sphere is measured as 9 cm with an error of 0.03 cm, then find the

approximate error in calculating in the volume. (4)

Q5 Use differentials, to calculate approximate value of log 9.01. Given that log 3 =

1.0986.

Level 3 Q1 For the function y = x

3, if x = 5 and Δx = 0.01, find Δy. (1)

Q2 Find the approximate change in volume of a be when side increases by 1%. (1)

Q3 Find the approximate value of (by differentiation) (4 marks each)

(i) tan-1

(0.999) (ii) 1

(2.002)2 (iii) √0.009

3

Q4 If 2x4 – 160 = 0, find the approximate value of x. (4)

Q5. Find the approximate value of f (5.001) where f(x) = x3 – 7x

2 + 15. (4)

Q6. Find the approximate value of f (3.02) where f (x) = 3x2 + 5x + 3. (4)

65

Sub Topic :Maxima and Mininma Level 1

Q1 What is the maximum and minimum value of the function f(x) = x? (1) (Ans: f(x)

has no extreme value)

Q2 At what value of x is the local maxima for the function f(x) = x3

– 3x +3? (1)

(Ans: x = 1)

Q3 What is the minimum value of x2+(250/x )(x>0)? (1) (Ans: 75)

Q4 Maximum value of f(x) = sinxcosx is _______ (1) (Ans: 1/2)

Q5 Find the point on the curve y2=4x which is the nearest to the point (2,-8). (4)

Q6 Find the maximum and minimum value of the function f(x) = 3-2sinx. (4)

Q7 Find two positive numbers whose sum is 24 and whose product is maximum. (4)

Q8 Show that of all the rectangle of given area, the square has the smallest perimeter. (4)

Q9 Show that the function f(x) = x3+x

2+x+1 has neither a maximum value nor a

minimum value. (4)

Q10 A window is in the form of a rectangle surmounted by a semicircular opening. The

total perimeter of the window is 10m. Find the diameter of the window to get

maximum light through the whole opening. (6)

Level 2

Q1 Check whether the function f(x) = 2x3-6x

2+6x+5 has a local maximum or local

minimum at

x = 1? (1) (Ans: Neither minima or maxima at x = 1)

Q2 Let f(x) = { |x| for 0<|x|<=1 then f has a _____ at x=0 1 for x = 0. (1) (Ans:

Local Maxima)

Q3 At x = 0 the function f(x) = x3 has a ______ (1) (Ans: Point of inflection)

Q4 What is the maximum value of f(x) = 1/(4x2+2x+1) , xεR ? (1) (Ans: 4/3)

Q5 Prove the area of a right angled triangle of given hypotenuse is maximum when the

triangle is isosceles. (4)

Q6 Find the area of the greatest isosceles triangle that can be inscribed in a given ellipse

x2/a

2 + y

2/b

2=1 with its vertex coinciding with one extremity of the major axis. (4)

Q7 Show that the height of a closed right circular cylinder of given surface and maximum

volume is equal to the diameter of the base. (4)

Q8 A piece of wire 28 units long is cut into two pieces. One piece is bent into the shape

of a circle and other into the shape of a square. How the wire should be cut so that the

combined area of the two figures is as small as possible. (6)

Q9 A farmer wants to construct a circular well and a square garden in his field. He wants

to keep sum of their perimeters fixed. Then prove that the sum of their areas is least

when the side of square garden is double the radius of the circular well. Do you think

good planning can save energy, time and money? (6)

Q10 If length of 3 sides of trapezium other than base are equal to 10cm each, then find the

area of trapezium when it is maximum?(ans 75√3𝑐𝑚2) (6)

Level 3 Q1 A manufacturer can sell x items at a price of Rs 5 –( x/100) each. The cost price of x

pens is Rs (x/5) + 500. What is the number of items, the manufacturer should sell to

earn maximum profit? (1) (Ans: 240)

Q2 Find the maximum and minimum value of the function without using derivative

f(x) = -|x+2|+3. (1)

66

Q3 Show that y=ex has no local maxima or local minima. (1)

Q4 A jet of an enemy is flying along the curve y=x2+2. A soldier is placed at the point

(3,2). What is the nearest distance between the soldier and the jet. (4)

Q5 Find the altitude of a right circular cone of maximum curved surface which can be

inscribed in a sphere of radius r. (4)

Q6 A car parking company has 500 subscribers and collects fixed charges of Rs 300 per

subscriber per month. The company proposes to increases the monthly subscription

and it is believed that for every increase of Rs 1, One subscriber will discontinue the

service . What increase will bring maximum income of the company? What values are

driven by this problem? (6)

Q7 If the performance of the student ‘y’ depends on the number of hours ’x ’ of hard

work done per day is given by the relation y = 4x – (x2/2). Find the number of hours,

the student work to have the best performance. “Hours of hard work are necessary for

success” Justify. (6)

Q8 Prove that the volume of the largest cone that can be inscribed in a sphere of radius R

is 8

27 of the volume of sphere. (6)

Q9 Show that the semi vertical angle of the cone of the maximum volume & of given

slant height is 𝑡𝑎𝑛−1√2 . (6)

Q10 Profit function of a company is given as P(x) =( 24x/5) –( x2/100) – 500 where x is

the number of units produced. What is the maximum profit of the company?

Company feels its social responsibility and decided to donate 10% of his profit for the

orphanage. What is amount contributed by the company for the charity? Justify that

every company should do it. (6)

Q11 An expensive square piece of golden color board of side 24cm is to be made into a

box without top by cutting each corner and folding the flaps to form a box. What

should the side of the square piece to be cut from each corner of the board to hold

maximum volume and minimum wastage? What is the importance of minimizing the

wastage in utilizing the resources? (6)

Tips and Techniques to score high (Group IV)

General

1. Student must know about the Time management so that He/She can attempt the entire

question paper.

2. Student must not write the unwanted steps in a solution for time saving purpose.

3. Student must utilize the reading time in planning to attempt the question paper.

4. 01 Mark questions should be dealt with straight answer without the solving steps.

5. Question carrying 06 marks may be attempted first.

Sub Topic related

6. In case of finding the equation of Tangent and Normal it is better to draw the rough sketches

of the tangent and Normal to the curve.

7. In maxima minima problems student at least should write the respective mensuration

formula and should draw the appropriate figure of it.

67

8. In case of Increasing Decreasing function student should draw the number line to represent

intervals in which the function is increasing or decreasing.

9. While differentiating a function w.r.t.any variable for finding increasing or decreasing

domain of the function should always kept in mind. i.e. for log function ,inverse function etc.

10. In case of Approximation, value of x and ∆𝑥 should be chosen as per the requirement of the

question. I.e. for √25.1 𝑎𝑛𝑑 ∛26.8, x=5 and 3 respectively.𝑎𝑛𝑑 𝑓𝑜𝑟 ∆𝑥 =0.1 and

-0.2resp ectively.

11. In maxima minima problem always write the inter relation of the variables contained in the

maximizing/ minimizing problems.

Common errors and mistakes committed by Students during exam

Application of Derivatives

1) Rate of Measure

1) Students differentiate w.r.t. ‘r’ instead of ‘t’ while finding rate of change of volume

of cylinders, spheres etc. w.r.t. ‘t’.

2) Got confused in the formulae of volumes, surface area of different 3-D objects.

2) Increasing and decreasing function

1) Strictly increasing in open interval but student generally writes in close interval i.e

[1,2] instead of (1,2).

2) Instead of finding the sign of f ‘ (x) students find the sign of f (x) in between the

critical points.

3) Approximation

1) Sign. of ∆𝑥 wrongly interpreted while finding the approximate value . For example

In finding approximate value of √25.1 and √24.9 .

3) Tangent and Normal

1) Student usually finds slope of tangent and normal to the curve at a general point

instead of Slope of the tangent and normal at a given point for finding the

equation of the tangent and normal at a given point.

2) Children get confused while finding the tangent to the curve at a point on the

curve and tangent from an external point to the curve.

4) Maxima and Minima

1) After finding the critical point student generally do not finds the sign. Of second

derivative to decide the Maxima or Minima.

2) Sometimes they differentiates the constants also w.r.t. the independent variable.

i.e.

V=𝜋𝑟2h (h is a constant )

dv/dr = 2𝜋r or 𝜋 (𝑟2 +2rh)

68

INTEGRALS

CONCEPT MAPPING

(1) ,xfxFdx

d Then we write cxFdxxf . These integrals are called

indefinite integrals; C is called a constant of integration.

(2) Some properties of indefinite integrals

(a) The process of differentiation and integration are inverse of each

other, xfdxxfdx

d and Cxfdxxf ' , where C is any

arbitrary constant

(b) Two indefinite integrals with the same derivatives lead to the same

family of curves and so they are equivalent. So it f and g are two

functions such that dxxgdx

ddxxf

dx

d .Then dxxf and dxxg

are equivalent.

(c) The integral of the sum of two functions equal the sum of the

integrals of the functions i.e., dxxgdxxfdxxgxf

(d) A constant factor may be written either before or after the integrals

sign i.e., dxxfadxxaf , where a is a constant.

(3) Methods of integrations .

(a) Integration by substitution

(b) Integration using partial fractions

(c) Integration by parts

(4) Definite integrals –

The definite integral is denoted by b

a

dxxf , where a is the lower

limit of the integral and b is the upper limit of the integral .The definite

integral is evaluated in the following two steps

(a) The definite integral as the limit of a sum

69

hnafhafhafafhdxxf

or

hnafhafhafafn

abdxxf

h

b

a

n

b

a

1..........2lim

1..........21

lim

0

Where 0

n

abh as 0n

(5) Fundamental Theorem of Calculus.

(a) Area function : The function b

a

dxxfxA

(b) Fundamental theorem of integral Calculus – Let f be a Continuous

function defined on the closed interval [a ,b] and F be an

antiderivative of f .

aFbFxFdxxfb

a

b

a

(6) Some properties of Definite Integrals.

,:

:

1

0

a

b

b

a

b

a

b

a

dxxfdxxfp

dttfdxxfP

In particular 0a

a

dxxf

aa

a

aa

aaa

aa

b

a

b

a

b

c

c

a

b

a

dxxfdxxfiP

xfxafif

xfxafifdxxfdxxfP

dxxafdxxfdxxfP

dxxafdxxfP

dxxbafdxxfP

dxxfdxxfdxxfP

0

7

0

2

0

6

00

2

0

5

00

4

3

2

2)(:

.2,0

2,2:

.2:

.:

.:

.:

If f is a even function i.e., xfxf

70

0)(

a

a

dxxfii If f is a odd function i.e., xfxf

QUESTION BANK

Part A( 1 MARK)

Evaluate: (LEVEL 1)

1. ∫log𝑥

𝑥 dx.

2. ∫ 𝑡𝑎𝑛2(7 − 4𝑥) dx

3. ∫(𝑎𝑥 + 𝑏)3dx

4. ∫𝒅𝒙

√𝟏−𝒙𝟐

5. ∫ 𝑒3 log𝑥 dx

6. ∫1+cot𝑥

𝑥+log sin𝑥 dx

7. ∫𝑠𝑒𝑐2 𝑥

3+tan𝑥dx.

8. xdxx log

9. xdx1tan

10. dxex x2

(LEVEL 2)

11. dxxxe x )cos(sin

12. dxx

xe x

2

1

1

1tan

13. dxxe x

sin

14. xdxlog

71

15.

dx

x

xxe x

2cos

cossin1

16. dxx 21

17.∫ √𝟒 − 𝒙𝟐𝟐

𝟎dx.

𝟏8. Given ∫ ex(tanx + 1)secx dx = ex f(x) +c, write f(x) satisfying the

above.

(LEVEL 3)

𝟏9. ∫ ex(1

x

2

1 −

1

x2) dx.

20. ∫1

1+ex dx.

21. ∫ √1 − sin 2𝑥 𝜋

2𝜋

4

dx.

𝟐2. ∫ 𝑠𝑖𝑛5𝑥𝜋

2−𝜋

2

dx.

23. ∫ [x]1.5

0dx.

24∫ 𝑙𝑜𝑔 [3+5 cos𝑥

3+5 sin𝑥]

𝜋

20

𝐝𝐱.

25. If ∫ 3x21

0+2x+k dx = 0, find k.

PART B( 4 MARKS)

(LEVEL 1)

Evaluate the following integrals

1. ∫𝒆𝒙

√𝟓−𝟒𝒆𝒙−𝒆𝟐𝒙 dx. Discuss the importance of integration (unity) in life.

2. ∫𝟓𝒙𝟐

𝒙𝟐+𝟒𝒙+𝟑 dx .

3. ∫𝟏−𝒙𝟐

𝒙(𝟏−𝟐𝒙) dx

72

4. ∫𝟔𝒙+𝟕

√(𝒙−𝟓)(𝒙−𝟒) dx.

5. ∫𝒙𝟐+𝟏

(𝒙−𝟏)𝟐(𝒙+𝟑) dx

6. ∫(𝒙𝟐+𝟏)(𝒙𝟐+𝟒)

(𝒙𝟐+𝟑)(𝒙𝟐−𝟓) dx.

(LEVEL 2)

7. ∫𝒙+𝐬𝐢𝐧𝒙

𝟏+𝐜𝐨𝐬𝒙. dx .

8. ∫𝟐+𝐬𝐢𝐧𝒙

𝟏+𝐜𝐨𝐬 𝒙𝒆𝒙/𝟐 dx.

9. ∫𝒆𝒙(𝐬𝐢𝐧𝟒𝒙−𝟒

𝟏−𝐜𝐨𝐬𝟒𝒙) dx.

10. ∫𝒙−𝟒

(𝒙−𝟐)𝟑𝒆𝒙 dx.

11. ∫𝒙𝒔𝒊𝒏−𝟏𝒙

√𝟏−𝒙𝟐 dx.

12. ∫ log (log x) + 1

(log x)2 dx.

13. dxxx 1log .

14. xdxx 12 tan .

(LEVEL 3)

15. dxxlogsin .

16.

dx

x

xe x

cos1

sin1 .

17. xdxx 2sec .

18. dxx

21sin .

19. dxxx tan1log2cos .

73

20.

dx

x

xx

sin1

cos .

21. xdxe x 3sin2

22. ∫𝑑𝑥

𝑥(𝑥8+1) .

23. ∫1

1+𝑒𝑠𝑖𝑛 𝑥

2𝜋

0 dx.

24. ∫𝑑𝑥

1+√tan𝑥

𝜋

3𝜋

6

25 Evaluate as limit of sum ∫ 3𝑥23

1 + 2x dx.

PART C( 6 MARKS)

(LEVEL 1)

1. Find ∫ 𝑥1

0(𝑡𝑎𝑛−1𝑥)2 𝑑𝑥

2.Evaluate ∶ ∫ log 𝑠𝑖𝑛𝑥 𝑑𝑥.𝜋

20

3. Show that ∫sin2x

sin x+cos x

π

20

= 1

√2 𝑙𝑜𝑔 (√2 + 1)

4. Evaluate ∶ ∫𝑥 tan𝑥

sec𝑥+tan𝑥

𝜋

0 dx.

5. Evaluate ∶ ∫sin2𝑥

𝑠𝑖𝑛4𝑥+𝑐𝑜𝑠4𝑥

𝜋

20

dx.

(LEVEL 2)

6. Evaluate: ∫(√𝑐𝑜𝑡 𝑥 + √𝑡𝑎𝑛 𝑥 ) 𝑑𝑥 .

7. Evaluate: ∫1

𝑠𝑖𝑛4𝑥+ 𝑠𝑖𝑛2 𝑥 𝑐𝑜𝑠2𝑥+𝑐𝑜𝑠4𝑥 dx.

8. Evaluate: ∫1

𝑐𝑜𝑠4𝑥 + 𝑠𝑖𝑛4 𝑥 dx.

9. Prove that ∫ 𝑒𝑥 𝑐𝑜𝑠2𝑥 dx = 1

2𝑒𝑥 +

𝑒𝑥

10(𝑐𝑜𝑠2𝑥 + 2𝑠𝑖𝑛2𝑥) + 𝒄

10. Evaluate: ∫𝑡𝑎𝑛 𝜃+ 𝑡𝑎𝑛3𝜃

1+ 𝑡𝑎𝑛3𝜃 d𝜃 .

74

11. Prove that ∫ |𝑥 𝑐𝑜𝑠𝜋𝑥| 𝑑𝑥2

−2 = 8

𝜋 .

(LEVEL 3)

Evaluate the following

12. dx

x

xxx

4

22 log21log1 .

13.

dxx

x

1

1tan 1 .

14.

dx

xx

xx

11

11

cossin

cossin

15. xdx3sec .

16.

dxxax 22log .

17.

dxxa

x1sin .

18.

dx

xxx

x2

2

cossin.

19.

dxx

xxe x

2

2tan

1

11

20. ∫𝐬𝐢𝐧𝒙+𝐜𝐨𝐬𝒙

√𝐬𝐢𝐧𝟐𝒙

𝝅

𝟑𝝅

𝟔

dx.

COMMON ERRORS

1.∫𝒔𝒊𝒏𝒙 𝒅𝒙 = − 𝒄𝒐𝒔𝒙. Instead of – 𝒄𝒐𝒔𝒙 + c.

Tips: Give more examples, like ,cossin xxdx

d also

xcxdx

dcossin where C is any arbitrary constant

2. Considering ∫𝟏

𝒙 𝒅𝒙 as ∫𝒙−𝟏 𝒅𝒙

Tips:- x

cxdx

d 1log

75

3. Degree of the numerator must be less than the degree

of the denominator. (e.g) ∫𝟓𝒙𝟐

𝒙𝟐+𝟒𝒙+𝟑 dx ,

Divide and reduce the degree of the numerator.

4. Applying wrong formulae.

(e.g.) ∫√𝒙𝟐 − 𝒂𝟐 dx and ∫√𝒂𝟐 − 𝒙𝟐 dx etc.

5. In applying ∫𝒙𝟐 − 𝒂𝟐 𝒅𝒙 , ∫ √𝒂𝟐 − 𝒙𝟐 dx,

∫𝒂𝒙𝟐 + 𝒃𝒙+ 𝒄 𝒅𝒙 etc.,

Tips: the co-efficient of 𝒙𝟐 must be unity.

6. They are not able to select which one is a function and

which one is its derivatives.

dxxxe x seclogtan Errors xxfxxf seclog,tan '

7. They integrate Sinbx instead of differentiating it

dxa

e

b

bx

a

ebxbxdxe

axaxax cos

.sinsin

8. They are not able to reduce inverse trigonometric

function by using substitution.

dx

x

xx

6

2312

1

sin Incorrect substitutions sinx or sin3 x

Correct substitution tx 31sin

9. They are applying a lengthy process and take more time.

dx

xxdx

xx

xdx

xx

xdx

xx

xx

93

4

939393

4222

2

2

2

10. They are not able to identify first function and second

function in integration by parts .

76

Use the Conviction ILATE

11. When to apply properties of Definite Integrals (e.g.)

∫ 𝒙𝟏

𝟎(𝒕𝒂𝒏−𝟏𝒙)𝟐 𝒅𝒙 , ∫

𝒙 𝐭𝐚𝐧𝒙

𝐬𝐞𝐜 𝒙+𝐭𝐚𝐧𝒙

𝝅

𝟎 dx. , ∫

𝟏

𝟏+𝒆𝒔𝒊𝒏 𝒙𝟐𝝅

𝟎 dx.

TIPS AND TECHNIQUES ON INTEGRATION :-

(i) When the differentiation of one function is another try to do by

substation method.

(ii) When the degree of numerator is greater or equal to the degree

of the denominator divide and reduce the degree of the

numerator less than the degree of the denominator.

(iii) When the denominator of integration is factorizable try to do by

partial fraction.

(iv) When an integration involves two different functions, try to do

by parts

(v) In definite integrals, try to apply the properties of the definite

integral first.

(vi)

4

0

321 dxxxx

TIPS :- Open intervals 633214,3

3213,2

43212,1

633211,0

xxxxxf

xxxxxf

xxxxxf

xxxxxf

dxxdxxdxxdxxI

4

3

3

2

2

1

1

0

63463

(vii)

4

02sin169

sincos

dxx

xx

TIPS :- 222 cossin2sin1cossin2sincos xxxxxxx

Hence, convert the Denominator

77

2cossin1625cossin2111692sin169 xxxxx

Put txx cossin then dtdxxx sincos

(vii) 41 x

dx

TIPS :- Numerator and Denominator are divided by 2x

dx

xx

xdx

xx

x

xx

dxxx

dx

xx

x

2

2

2

2

2

2

2

2

22

2

2

2

1

11

2

1

1

11

2

1

1

111

1

2

1

1

1

If numerator

2

11

x then put dtdxx

tx

x

2

11

1 and

convert the denominator 211

2

2

2

xx

xx

If numerator

2

11

x then put dtdxx

tx

x

2

11

1 and

convert the denominator 211

2

2

2

xx

xx

(ix) 3

1

2

1

xx

dx

TIPS:- dttdxtxtxtx 523

1

32

1

6

1

6,

(x) 1xe

dx

TIPS :-Multiply the numerator and denominator by xe and put te x

(xi) 1nxx

dx

TIPS :- Multiply the numerator and denominator by 1nx and put txn

78

CONCEPT MAPPING APPLICATION OF INTEGRATION

The area of the region bounded by the curve y=f(x),x-axis and the lines x=a,x=b is given by

• A=∫ 𝒚𝒅𝒙𝒃

𝒂

The area of the region bounded by the curve x=f(y),y-axis and the lines y=a, y=b is given by

A=∫ 𝒙 𝒅𝒙𝒃

𝒂

Steps to find area

CURVE Parabola 𝑦2 =x ,x=a, x=4

STEPS DRAW THE DIAGRAM MAKE A SHADED REGION FIND INTERSECTION POINTS IDENTIFY THE LIMITS WRITE THE INTEGRAL(S) FOR THE REGION EVALUATE THE INTEGRAL

THE VALUE SHOULD BE POSITIVE

Y=f(x)

a b

x=f(y)

a

a

79

Two curves are given

A=∫ 𝒇(𝒙 )–𝒈(𝒙)𝒅𝒙𝒃

𝒂

Examples Circle and line

80

Parabola and line

Parabola modulus

81

Parabola and y=a

Ellipse and line

82

Inequality problem

Exterior area problem

Find the area of the circle x2 + y2 = 16 exterior to the parabola y2 = 6x

83

Area of triangle

Using the method of integration find the area of the triangle ABC, coordinates of whose vertices are A (2, 0), B (4, 5) and C (6, 3

TOPIC-Application of Integrals

Sub topic-Area included by two curves

1-Mark Questions:-

(A)Basic Level Questions:

(1) Find the area enclosed by the given curves 𝑦 = 𝑥2, 𝑥 = −1, 𝑥 = 2.

(2) Find the area enclosed by the given curves x= 𝑦2 , 𝑥 = 1, 𝑥 = 4.(Ans:14/3 sq.units.)

(3) Find the area enclosed by the given curves 𝑦 = |𝑥|, 𝑥 = −1, 𝑥 = 2.

(4) Find the area enclosed by the given curves 𝑦 = 𝑥, 𝑥 = −1, 𝑥 = 3.

(B)Average Level Questions:

(5) Find the area enclosed by the given curves 𝑦 = 𝑥2, 𝑥 = 𝑦 .(Ans:1/6 sq.units.)

(6) Find the area enclosed by the given curves 𝑦 = |𝑥 + 3| 𝑥 = −6, 𝑥 = 0.

(7) Find the area enclosed by the given curves 𝑦 = sin 𝑥 , 𝑥 = 0, 𝑥 =𝜋

4.

(8)Find the area enclosed by the curves y = cos2x , x = 0 and x =π

2 and x −

axis. (𝐴𝑛𝑠:𝜋

4 𝑠𝑞. 𝑢𝑛𝑖𝑡𝑠. )

84

(C)Above Average Level Questions:

(9)Find the area enclosed by the given curves 𝑦 = 𝑥2, 𝑦 = √𝑥

(10)Find the area enclosed by the given curves 𝑦2 = 4𝑎𝑥 and its letus rectum

4-Marks Questions:-


(1)Using integration find the area enclosed by a circle of radius 𝑟.(Ans: 𝜋r2)

(2)Find the area of the region bounded by the curves 𝑦2 = 2𝑦 − 𝑥 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑦 −

𝑎𝑥𝑖𝑠.(Ans: 4/3 sq.units.)

(3) Using integration find the area enclosed by the curves y = x, and the curves y =

x3. (𝐴𝑛𝑠: 1

2𝑠𝑞. 𝑢𝑛𝑖𝑡𝑠)


(4)Find the area of the region bounded by the curves 𝑦2 = 4𝑎𝑥and x2 =

4ay. (𝐴𝑛𝑠: 16𝑎2/3 sq.units).

(5) Using integration find the area of the region in the first quadrant enclosed by x-

axis, 𝑥 = √3𝑦 and the circle𝑥2 + 𝑦2 = 4,(Ans: 𝜋


(6) Draw the rough sketch of the curves 𝑦 =

sin x , y = cos x as x varies from 0 toπ

2 and find the area enclosed by them and

x − axis. (𝐴𝑛𝑠. 2 − √2 )

(7)Find the area enclosed by the ellipse 𝑥2

𝑎2+𝑦2

𝑏2= 1, 𝑎 > 𝑏.(Ans: 𝜋ab sq.units.)


(8)Find the area enclosed by the given curves |𝑥| + |𝑦| = 1.(Ans: 2 sq.units.)

(9)Find the area bounded by the curves 𝑥 = 𝑦2 and 𝑥 = 3 − 2𝑦2. (𝐴𝑛𝑠: 4 𝑠𝑞. 𝑢𝑛𝑖𝑡𝑠. )

(10)Find the area of the region bounded by the curves y = x2 + 2, and the lines y =

x, x = 0 and x = 3.

6-Marks Questions:-


(1) Using integration find the area of the triangle ABC,where Ais (2,1), B is(3,4) and C is

(5,2). (Ans. 4 sq.units.)

85

(2)Find the area of the region{(𝑥, 𝑦): 𝑥2 + 𝑦2 ≤ 1 ≤ 𝑥 + 𝑦 }.(Ans: 𝜋

4−

1


(3)Find the area of the region bounded by the curves 𝑥2 = 4𝑦 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑥 = 4𝑦 −

2. (𝐴𝑛𝑠:9

8𝑠𝑞. 𝑢𝑛𝑖𝑡𝑠)


(4) Compute the area bounded by the linesx + 2y = 2, y − x = 1 and 2x + y = 7.

(Ans: 28/3 sq.units)

(5)Prove that the curves 𝑦 = 𝑥2 and 𝑥 = 𝑦2divide the square bounded by 𝑥 = 0, 𝑦 =

0, 𝑥 = 1, 𝑦 = 1 in to three equal parts.

(6) Find the area of the region{(𝑥, 𝑦): 𝑥2 ≤ 𝑦 ≤ |𝑥|}. (𝐴𝑛𝑠: 1

3𝑠𝑞. 𝑢𝑛𝑖𝑡𝑠. )

(7)Find the area of the smaller region bounded by the ellipse x2

a2+

y2

b2= 1 and the

straight line x

a+

y

b= 1. (𝐴𝑛𝑠:

1

3𝑠𝑞. 𝑢𝑛𝑖𝑡𝑠. )


(8) Find the area of the region enclosed between the two circle x2 + y2 = 1 and (x - 1)2 +

y2 = 1. (Ans: 2𝜋

3−√3

2 sq. units)

(9) Find the area of the region{(x, y): y2 ≤ 4x, 4x2 + 4y2 ≥ 9}

(𝐴𝑛𝑠: 2√2

3−1

2+9𝜋

8−9

4𝑠𝑖𝑛−1

1


(10) Sketch the graph 𝑦 = √𝑥 + 1 in (0,4) and determine the area of the region

enclosed by the curve, the axis of x and the lines 𝑥 = 0, 𝑥 = 4 (Ans: 28/3 sq.units.)

SUB-TOPIC:- Area of simple curves & area of triangle

Q1. Find the area enclosed between the given curves x=y 2 and y=x2.

Q2. Using application of integration, find the area of triangle whose vertices are (0,2),(2,4

)and (5,1).

Q3.Find the area of region bounded by y2=9x, x=2, x=4 and the x-axis in first quadrant.

Q4. Find the area of region { (x, y): y2 ≤ 4x , 4x2+4y 2≤ 9 }.

Q5.Find the area of region enclosed by the parabola x2=y, the line y=x+2 and the x-axis.

Q5.Find the area of the circle 4x2+4y2=9 which is interior to the parabola y2=4x.

86

Q6.Find the area of region bounded two parabolas 4y=x2 and 4x=y2.

Q7.Find the area of region in the first quadrant enclosed by the x-axis , the line x=√3y and

the circle x2+y2 = 4.

Q8.Find the area of the smaller part of the circle x2+y2=a2 cut off by the line x=a/√2.

Q9.Find the area of region enclosed between two circles x2+y2=1 and(x-1)2+y2=1.

Q10.Using application of integration, find the area of region bounded by lines x=y, x=5 and

x-axis.

Q11.Find the area of region bounded by the parabola y=x2 and y = IxI .

Common Errors Committed by the students:-

1. Incorrect drawing of the graph for the given curves.

2. Incorrect locating of the required enclosed region.

3. Finding point of intersection incorrectly.

4. Basic integration errors.

5. Area of the region below x-axis.

6. Unit of the area.

Measures to reduce the errors:-

1. Explain graph of the different types of curves like straight line, parabola, circle,

ellipse etc. correctly.

2. Explain how to find point of intersection i.e. to find common points of the given

curves.

3. To shade the correct region by giving many examples of standard curves.

4. When area is below x-axis, area should be added with the correct sign.

87

Common Errors Committed by the students:-

S.no. topic Errors committed by the students

remedial

1 Finding area through integration

Incorrect drawing of the graph for the given curves.

Explain graph of the different types of curves like straight line, parabola, circle, ellipse etc. correctly.


Finding point of intersection incorrectly.

Explain how to find point of intersection i.e. to find common points of the given curves.


Basic errors in process of integration

Basics of integration should be discussed


Area of the region below x-axis Negative sign should be given before integrand


Unit of the area. Area of unit like sq. unit should be highlighted

88

DIFFERENTIAL EQUATION

CONCEPT MAPPING:

Definition of differential equation. Order of a differential equation

Without radical sign: Order is the order of the highest order derivative appearing in the equation

Example: 𝑑2𝑦

𝑑𝑥2− 2

𝑑𝑦

𝑑𝑥+ 5𝑦 = 0.

With radical sign : like √ , √3

, 𝑒𝑡𝑐. remove the radical sign by squaring, cubing etc. then Order is the order of the highest order derivative appearing in the equation.

Example: √1 − (𝑑𝑦

𝑑𝑥)2= (

𝑑2𝑦

𝑑𝑥2)

1

3

Degree of a differential equation

The degree of a differential equation is the degree of the highest order derivative occurring in the equation, when the differential coefficients are made free from radicals if each term involving derivatives of differential equation is polynomial(can be expressed as a polynomial).

Example: (𝑑𝑦

𝑑𝑥)2+𝑑𝑦

𝑑𝑥− 𝑠𝑖𝑛𝑌′ = 0

Verifying the given function as a solution of the given differential

equation. Formation of differential equation

Steps involved

Write down the given equation of family of curves

Differentiate the given equation as many times as the number of arbitrary constants.

Eliminate the arbitrary constants from the given equation and the equation obtained by differentiation.

HOTS

The equation representing the parabola having vertex at the origin and axis along the positive direction of X-axis- 𝑦2 = 4𝑎𝑥.

The family of circles touching y-axis at the orgin-(𝑥 − 𝑟)2 + 𝑦2 = 𝑟2.

89

The family of circles touching the x-axis: 𝑥2 + (𝑦 − 𝑟)2 = 𝑟2

Solving a differential equation:

Variable separable

Type-1: 𝑑𝑦

𝑑𝑥=

𝑓(𝑥)

𝑔(𝑦), 𝑜𝑟

𝑔(𝑦)

𝑓(𝑥) then separate the variables and integrate.

Type-2: 𝑓1(𝑥). 𝑔1(𝑦)𝑑𝑦 + 𝑓2(𝑥). 𝑔2(𝑦)𝑑𝑥 = 0, then divide by 𝑓1(𝑥). 𝑔2(𝑦) and integrate.

Type-3: 𝑑𝑦

𝑑𝑥= 𝑓(𝑎𝑥 + 𝑏𝑦 + 𝑐), 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 𝑡 𝑎𝑛𝑑 𝑠𝑜𝑙𝑣𝑒.

Homogeneous

If the question is asked to prove homogeneous find 𝑓(𝜆𝑥, 𝜆𝑦) =𝜆𝑛𝑓(𝑥, 𝑦)then “f” is homogeneous of degree “n”.

Step 1: put y= vx and 𝑑𝑦

𝑑𝑥= 𝑣 + 𝑥.

𝑑𝑣

𝑑𝑥 where “v” is a function of “x” alone.

Step 2: The equation reduce to variable separable and solve.

Some homogeneous equation can be solved by substituting 𝑥 = 𝑣𝑦.

Linear Differential equation

Type1 : 𝑑𝑦

𝑑𝑥+𝑃𝑦 = 𝑄,

where P and Q are functions of “x” alone or constants.

Step 1: Identify P and Q.

Step 2: Find ∫𝑃 𝑑𝑥 and find 𝑒∫𝑃𝑑𝑥(Note: If ∫𝑃 𝑑𝑥 = log(𝑓(𝑥)), then

𝑒∫𝑃 𝑑𝑥 = 𝑓(𝑥)).

Step 3: Solution 𝑦. 𝑒∫𝑃𝑑𝑥 = ∫𝑄. 𝑒∫𝑃𝑑𝑥 𝑑𝑥 + 𝐶.

Type2 : 𝑑𝑥

𝑑𝑦+𝑃𝑥 = 𝑄,

where P and Q are functions of “y” alone or constants.

Step 1: Identify P and Q.

Step 2: Find ∫𝑃 𝑑𝑦 and find 𝑒∫𝑃 𝑑𝑦(Note: If ∫𝑃 𝑑𝑦 = log(𝑓(𝑦)), then

𝑒∫𝑃 𝑑𝑦 = 𝑓(𝑦)).

90

Step 3: Solution 𝑥. 𝑒∫𝑃 𝑑𝑦 = ∫𝑄. 𝑒∫𝑃 𝑑𝑦 𝑑𝑦 + 𝐶.

Find the type of the following differential equations:

(𝑥2 + 1)𝑑𝑦

𝑑𝑥+ 2𝑥𝑦 = √𝑥2 + 4 .

sec 𝑥 𝑑𝑦

𝑑𝑥− 𝑦 = sin 𝑥.

𝑥2𝑑𝑦

𝑑𝑥= 𝑦2 + 2𝑥𝑦

𝑥𝑑𝑦

𝑑𝑥= 𝑦 − 𝑥 𝑡𝑎𝑛 (

𝑦

𝑥)

𝑐𝑜𝑠2 𝑥𝑑𝑦

𝑑𝑥+ 𝑦 = tan 𝑥

(𝑥 cos𝑦

𝑥+ 𝑦 sin

𝑦

𝑥)𝑦 − (𝑦 sin

𝑦

𝑥− 𝑥 cos

𝑦

𝑥)𝑥

𝑑𝑦

𝑑𝑥= 0

DIFFERENTIAL EQUATIONS

BASIC QUESTIONS

ONE MARKS QUESTIONS:-

Q.1 Form the differential equation representing the family of curves.

Y = A cosx + B sinx ans- [y’’ + y=0]

Q2.Form the differential equation representing the family of curves.

x²-y² =a² ans- [x-yy’=0]


x²-y² =a² ans- [x-yy’=0]


y² =4ax ans- [y-2xy’=0]


91

y=𝑒𝑎𝑥 ans- [xy’=ycosx]

Average level questions

Q.6. Form the differential equation for the family of curves

Y=Aeᵡ +Beˉᵡ ans.[y’’- y=0]


x²-y² =a² ans- [x-yy’=0]

Q.8. 𝑑𝑦

𝑑𝑥 = x² +sin 3x ans- [y=1/3x³ -1/3cos 3x +c]

Q.9 Solve 𝑑𝑦

𝑑𝑥 = sec y ans- [x=siny +c]

Above average level questions

Q.10. Solve 𝑒𝑑𝑦

𝑑𝑥= x+1 given that x=0,y=3 ans- [y=xlog (x+1) + log(x+1) –

x+3 ]

Q.11. Solve 𝑑𝑦

𝑑𝑥 = sec y ans- [x=siny +c]

Q.12 Solve . 𝑑𝑦

𝑑𝑥 = (1+x²) (1+y²) ans- [tanˉ¹y = x+x³/3 +c]

Q13. Determine the order and degree of the equation.

03

2

2

22

dt

sds

dt

ds, ans Order -2, Degree – 2

Q14. Show that differential equation is homogeneous

FOUR MARKS QUESTIONS:-

BASIC LEVEL

Q.15. Find the diffrential equation form.

y = k𝑒sin−1 𝑥+3 Ans.[y’√1-x² - y+3=0]

Q16..√1 + 𝑥2 + 𝑦2 + 𝑥2𝑦2 +xy𝑑𝑦

𝑑𝑥 =0

22 943 yxdx

dyxy

92

ABOVE AVERAGE LEVEL QUESTIONS

Q17. obtain the differential equation by elemineting a and b from the

equation y=eᵡ(a cosx +b sinx) ans.[y’’-2y’+2y=0]a

Q18. Find the differential equation of all straight lines which are at fixed

distance p from origin. ans.[p² (1+y’²) =(y-xy’)² ]b

Q.19 Find the general solution of differential equation.

(yᶟ+1)(eᵡ+xeᵡ)dx-xeᵡy²dy=0 ans. [logx + x2/2= 1/2log|𝑦3 + 1|+c]

Q.20. solve dy/dx =eᵡˉᵞ+x²eˉᵞ ans.[eᵞ=x³/3+eᵡ+c]

(ans. [√1+y²+1/2log|√1 + 𝑥² − 1 /+√1 + x2 + 1|+√1+y²=c)

AVERAGE LEVEL QUESTIONS

21. Show that (x-y) dy

dx= (x + y) is a homogeneous differential equation and

solve it.

22. Solve the Differential Equation by using appropriate substitution

23. Find Solution of the differential equationxcos𝑦

𝑥

𝑑𝑦

𝑑𝑥= 𝑦𝑐𝑜𝑠 (

𝑦

𝑥) + 𝑥, given

that when x = 1 y = 𝜋

4

24. Solve: dx

dy+ y cot x = sec x.

SIX MARKS QUESTIONS

BASIC LEVEL QUESTIONS

Q.25. 𝑑𝑦

𝑑𝑥+𝑦

𝑥= 0 x denotes the % population living in city and y denotes the

area for living healthy life of population find the particular solutionwhen x =

100,y=1

0222 dyxdxxxyy

93

AVERAGE LEVEL

Q26. Find the particular solution of differential equation log dy/dx=3x+4y give that y=0

when x=0 ans. [4e³ᵡ + 3eˉ⁴ᵞ=7]

27. The rate of increase of bacteria in a certain culture is proportional to the number of

bacteria present .if it is found that the number doubles in 5 hrs . prove that the bacteria

becomes 8 time at the end of 15 hrs

28. 9 Solve. dy/dx= x (2logx +1)/siny +ycosy ans.[ysiny = x²logx +c

ABOVE AVERAGE

29A wet porous substance in the open air loses its moisture at the rate proportional to the

moisture content . If a sheet hung in the wind loses half its moisture during the first hrs.

when will it has lost 99% when after condition remaining the same? ans.[ log100/log2

hrs.]

Q30. In the college hostel accommodating 1000 students one of them came in carrying a

flue virus then the hostel was isolated if the rate at which virus spreads is assumed to be

proportional to the product of number n of infected students and the number of non –

infected student and if the number of infected student is 50 after 4 days then 1.Find the %

of infected student after 10 days. (use195/2.999−3/2=0.05) comment on the

administration of hostel authorities. ans. [95.2%]

S.no. topic Errors committed by the

students

remedial

1 Differential

equations

Writing incorrect order of

differential equations

order is the order of highest order

derivative

2 Differential equations

Writing incorrect degree of

differential equation

Write degree of highest derivative

after removing all radical signs


Wrong identification of

form of Differential

equations

Form of Differential equations

like variable separable

,homogeneous and linear should

be explained in comparative

manner with three examples.


Incorrect integration of the

given expression

Basics of integration should be

discussed


Constant of integration It should be used with every

solution

94

VALUE-BASED QUESTIONS

Q.1.If in a triangular field is bounded by the lines x+2y=2, y-x=1 and 2x+y=7. Using

integration compute the area of the field

(i)if in each square unit area, 4 trees may be planted. Find the number of trees can be

planted in the field.

(ii)why plantation of trees is necessary?

Q2.A Farmer has a piece of land .He wishes to divide equally in his two sons to maintain

peace and harmony in the family .If his land is denoted by area bounded by the curvey2=4x

and x=4 and to divide the area equally he draws a line x=a. What is the value of a ?

What is the importance of equally among the people?

Q3.A poor deceased farmer has agriculture land bounded by the curve y= cosx between x=0

and x=2π.He has two sons .Now they want to distribute this land in three parts. Find the

area of each part.

Which parts should be given to the farmer& why ? Justify your answer.

DIFFERENTIAL EQUATION

Q4.Solve the differential equation (x+2y2)dy/dx=y

Given that when x=2,y=1.If x denotes the % of people who are polite and y denotes the % of

people who are intelligent ,find x when y=2%.

A polite child is always liked by all in society. Do you agree ? Justify.

Q5. dy/dx=y/x, where x denotes the percentage population living in a city & y denotes the

area for living a healthy life of population. Find the particular solution when x=100, y=1.Is

higher density of population harmful ? Justify your answer?

TIPS FOR THE CHILDREN

APPLICATION OF INTEGRATION

When two vertices of a triangle is given to find equation use determinant formula(x,y any

point on the line and they are

𝒙 𝒚 𝟏𝒙𝟏 𝒚𝟏 𝟏𝒙𝟐 𝒚𝟐 𝟏

=0

95

collinear)

Example A(1,2) B(3,5) 𝑿 𝒀 𝟏 𝟏 𝟐 𝟏𝟑 𝟓 𝟏

=0 IS

3X-2Y+1=0 To save time be thorough with

standard formulae and diagrams (𝒙 − 𝒂)𝟐 + (𝒚 − 𝒃)𝟐 = 𝒓𝟐 circle

𝒚𝟐 = 𝟒𝒂𝒙 𝒐𝒓 𝒙𝟐 =4ay parabola 𝒙

𝒂𝟐

𝟐+

𝒚

𝒃𝟐

𝟐 =1 ellipse

ax+by=c linear Learn formulae without mistakes Very important formulae

√𝒙𝟐 − 𝒂𝟐=𝒙

𝟐 √𝒙𝟐 −𝒂𝟐 ∓

𝒂𝟐

𝟐 𝐬𝐢𝐧−𝟏

𝒙

𝒂

Draw digrams with proper scale To avoid confusion

Practice all text book problems only.

TIPS FOR THE CHILDREN

Differential equations

To find degree remove radical √𝟏 + 𝒅𝒚

𝒅𝒙 =5 degree is 2(square

on both sides to remove radical) To form differential equation

identify number of arbitrary constants General rule

Y=kcos3x+5 k is arbitrary

constants not 5.So differentiate once.

96

If one parameter family of curves it represents by a first degree DE So differentiate once. If two parameter family of curves it represents by a second degree DE So differentiate twice.

Y=sinax+cosbx

Differentiate twice

Flow chart

Types of differential equations

If it is the above form proceed otherwise

Find dy/dx

1

97

Then it is said to be homogenous equation Proceed following

method Otherwise

98

In particular solution no arbitrary constant

In general solution arbitrary constant exists.

99

CONCEPT MAPPING OF VECTOR & 3-D

1. Definition of a vector → Vector is a physical quantity having

magnitude and direction, a directed line segment is used to represent it,

length of the line segment gives magnitude and arrow mark gives

direction.

𝐴𝐵⃗⃗⃗⃗ ⃗ , A is the Initial point and B is the terminal point.

Example–velocity, acceleration, force, momentum etc

2. Type of vectors →

Zero vector → magnitude zero and arbitrary direction or initial and terminal

points coincides in a vector.

a) Unit vector → magnitude one unit and having a definite direction or

particular direction.

b) Equal vectors → having equal magnitude and same direction

c) Like vectors→ having same direction. Unlike vectors → having opposite direction.

d) Co - initial vectors → having same initial point. Collinear vectors → which can be represented on a same line.

e) Coplanar vectors → vectors lying on the same plane.

f) Negative of a vector → a vector having same magnitude and opposite direction as that of a given vector.

g) Position vector of a point → position vector of a point P is 𝑂𝑃⃗⃗⃗⃗ ⃗ where O is the origin.

h) Any vector 𝑃𝑄⃗⃗⃗⃗ ⃗ = 𝑂𝑄⃗⃗⃗⃗⃗⃗ -𝑂𝑃⃗⃗⃗⃗ ⃗ .

i) If P is any point (x, y, z) then 𝑂𝑃⃗⃗⃗⃗ ⃗ = x�̂� +y�̂� +z�̂� . 3) Operation on vectors; addition of vectors → having same direction,

opposite directions and different directions.

Triangle law of vector addition →if two vectors are represented by two

sides of a triangle taken in order then their resultant is represented by

the third side taken in the opposite direction.

Parallelogram law of vector addition → if two vectors are represented

by two adjacent sides of a parallelogram, then their resultant is

represented by the diagonal of the parallelogram passing through the

common vertex of the adjacent sides.

100

4) Properties of vector addition → Commutative property → 𝑎 +�⃗� = �⃗� +

𝑎 .

Associative property → (𝑎 + �⃗� ) + 𝑐 = 𝑎 + ( �⃗� + 𝑐 )

Identity property → 𝑎 +𝑜 = 0⃗ + 𝑎 .

Inverse property → 𝑎 + (-𝑎 ) = 𝑜

5) Subtraction of vectors. 𝑎 -�⃗� = 𝑎 +(-�⃗� ).

6) Multiplication of a vector by a scalar → if k is a scalar and 𝑎 is a

vector then k𝑎 is a vector whose magnitude is k times that of 𝑎 and

direction is same or opposite according as k is positive or negative.

Distributive property → k ( 𝑎 + �⃗� ) = k𝑎 + k�⃗� .

7) Definition of dot (scalar) product of vectors— 𝑎 .�⃗� = abcosθ.

Angle between two vectors is given by cos θ = �⃗� .�⃗�

𝑎𝑏 .

Condition for perpendicularity of two vectors as 𝑎 .�⃗� = 0.

Geometrical meaning → 𝑎 .�⃗� = (Magnitude of 𝑎 ) (Projection of �⃗� on 𝑎 )

= (Magnitude of�⃗� ) (Projection of 𝑎 on �⃗� )

Orthonormal triads; �̂� ,�̂� and �̂� → �̂� . �̂� = �̂�. �̂� = �̂� . �̂� =1 and �̂� . �̂� = �̂� . �̂� = �̂� . �̂� =0.

POINTS TO REMEMBER.

1) 𝑎 .�⃗� = abcosθ.

2) cos θ = �⃗� .�⃗�

𝑎𝑏 .

3) Projection of 𝑎 on �⃗� = �⃗� .�⃗�

|�⃗� | .

4) If 𝑎 = a1�̂� +a2�̂� +a3�̂� and �⃗� = b1�̂� +b2�̂� +b3�̂� then 𝑎 .�⃗� = 𝑎1𝑏1 + 𝑎2𝑏2 +

𝑎3𝑏3

4) If 𝑎 and �⃗� are perpendicular then 𝑎 .�⃗� = 0.

5) |𝑎 + �⃗� |2 = |𝑎 |2 + |�⃗� |

2 +2(𝑎 .�⃗� )

6) |𝑎 − �⃗� |2 = |𝑎 |2 + |�⃗� |

2 - 2(𝑎 .�⃗� ) .

7) (𝑎 +�⃗� ) . (𝑎 -�⃗� ) = |𝑎 |2 - |�⃗� |

2.

101

8) If 𝑎 = a1�̂� +a2�̂� +a3�̂� then |𝑎 |2 =a12 +a2

2 +a3

2.

9) |𝑎 + �⃗� |2 +|𝑎 − �⃗� |2 = 2(|𝑎 |2 +|�⃗� |2).

10) |𝑎 + �⃗� + 𝑐 |2 =|𝑎 |2 +|�⃗� |2 + |𝑐 |2 + 2 𝑎⃗⃗⃗ .�⃗� +2�⃗� .𝑐 +2𝑐 .𝑎 .

11) If �̂� is a unit vector and 𝑎 is any vector then �̂� = �⃗�

|�⃗� | .

TOPIC: CROSS PRODUCT AND SCALAR TRIPLE PRODUCT OF VECTORS

I. CONCEPT MAPPING

1. Definition

2. Types

3. Formula

4. Concept of related terms

5. Problems

6. Example.

105

CONCEPT MAPPING OF THREE –DIMENSIONAL GEOMETRY

1. Distance formula: Distance between two points A( x1 ,y1, z1 ) and B(x2,

y2,z2) is

AB=√(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)

2 + (𝑧2 − 𝑧1)2

2. Section formula: Coordinates of a point P, which divides the join of two

given points

A( x1 ,y1, z1 ) and B(x2, y2,z2) in the ratio l:m

(i). internally,

are P((𝑙𝑥2+𝑚𝑥1

𝑙+𝑚,𝑙𝑦2+𝑚𝑦1

𝑙+𝑚,𝑙𝑧2+𝑚𝑧1

𝑙+𝑚) ,

the Coordinates of a point Q dividing the join in the ratio l:m

(ii). externally are Q(𝑙𝑥2−𝑚𝑥1

𝑙−𝑚,𝑙𝑦2−𝑚𝑦1

𝑙−𝑚,𝑙𝑧2−𝑚𝑧1

𝑙−𝑚)

(iii).coordinate of mid-point are R((𝑥1+𝑥2

2,𝑦1+𝑦2

2,𝑧1+𝑧2

2)

3. Direction cosines of a line :

(i).The direction of a line OP is determined by the angles 𝛼, 𝛽, 𝛾 which

makes with OX, OY,OZ respectively. These angles are called the direction

angles and their cosines are called the direction cosines.

(ii).Direction cosines of a line are denoted by l,m,n . l=cos 𝛼

,m=cos 𝛽, 𝑛 = cos 𝛾

(iii).Sum of the squares of direction cosines of a line is always 1.

l2+m2+n2=1 i.e cos2𝛼 + cos2𝛽 +cos2𝛾 =1

4. Direction ratio of a line :(i)Numbers proportional to the direction

cosines of a line are called direction ratios of

a line .If a ,b ,c, are , direction ratios of a line, then

𝑙

𝑎=

𝑚

𝑏=

𝑛

𝑐.

(ii). If a ,b ,c, are , direction ratios of a line , then the direction cosines

are

±𝑎

√𝑎2+𝑏2+𝑐2 , ±

𝑏

√𝑎2+𝑏2+𝑐2 ±

𝑐

√𝑎2+𝑏2+𝑐2

(𝑖𝑖𝑖). Direction ratio of a line AB passing through the points

A(x1,y1,z1)and B (x2,y2,z2) are

𝑥2 −x1, y2-y1, z2-z1

106

4. STRAIGHT LINE:. (i). Vector equation of a Line passing through a point 𝑎

and along the direction�⃗⃗� , : 𝑟 =𝑎 + 𝝁�⃗⃗� ,

(ii).Cartesian equation of a Line:𝑥−𝑥1

𝑎=

𝑦−𝑦1

𝑏=

𝑧−𝑧1

𝑐. Where (x1,y1,z1 is

the passing through a point and along the direction ratios are a,b,c

6. (i). Vector equation of a Line passing through two points, with

position vectors 𝑎 𝑎𝑛𝑑 �⃗� 𝑟 =𝑎 + 𝝁(�⃗⃗� - 𝑎 )

(ii). ).Cartesian equation of a Line:𝑥−𝑥1

𝑥2−𝑥1=

𝑦−𝑦1

𝑦2−𝑦1=

𝑧−𝑧1

𝑧2−𝑧1, two points are

(x1,y1) and (x2,y2).

7. ANGLE between two lines (i). Vector equations: 𝑟 =𝑎1⃗⃗⃗⃗ + 𝝀𝑏1⃗⃗ ⃗and:

𝑟 =𝑎2⃗⃗⃗⃗ + 𝝁𝑏2⃗⃗⃗⃗ ,

cos 𝜃 =𝑏1⃗⃗⃗⃗ ⃗.𝑏2⃗⃗⃗⃗ ⃗

|𝑏1⃗⃗⃗⃗ ⃗|.|𝑏2⃗⃗⃗⃗ ⃗|

(ii) ).Cartesian equations: 𝑥−𝑥1

𝑎1=

𝑦−𝑦1

𝑏1=

𝑧−𝑧1

𝑐1,𝑥−𝑥2

𝑎2=

𝑦−𝑦2

𝑏2=

𝑧−𝑧2

𝑐2

cos𝜃 =𝑎1.𝑎2+𝑏1.𝑏2+𝑐1.𝑐2

√𝑎2+𝑏2+𝑐2 √𝑎2+𝑏2+𝑐2

(iii). If two lines are perpendicular, then 𝑏1⃗⃗⃗⃗ . 𝑏2⃗⃗⃗⃗ =0, a1a2+b1b2+c1c2=0

(iv) . If two lines are parallel, then 𝑏1⃗⃗⃗⃗ = 𝑡 𝑏2⃗⃗⃗⃗ , where t is a scalar. OR

𝑏1⃗⃗⃗⃗ × 𝑏2⃗⃗⃗⃗ =0, 𝑎1

𝑎2=

𝑏1

𝑏2=

𝑐1

𝑐2

(v).If 𝜃 𝑖𝑠 𝑡ℎ𝑒 angle between two lines with direction cosines ,l1,m1,n1

and l2,m2,n2 then

(a).cos 𝜃 =l1l2+m1m2+n1n2 (b). if the lines are parallel, then 𝑙1

𝑙2=

𝑚1

𝑚2=

𝑛1

𝑛2

(c). if the lines are perpendicular, then l1l2+m1m2+n1n2=0

8.(a).Shortest distance between two skew- lines:

(i).Vector equations: 𝑟 =𝑎1⃗⃗⃗⃗ + 𝝀𝑏1⃗⃗ ⃗, and: : 𝑟 =𝑎2⃗⃗⃗⃗ + 𝝁𝑏2⃗⃗⃗⃗ ,

d=|(𝑎2⃗⃗⃗⃗ ⃗−𝑎1⃗⃗⃗⃗ ⃗).(𝑏1⃗⃗⃗⃗ ⃗×𝑏2⃗⃗⃗⃗ ⃗)

|𝑏1⃗⃗⃗⃗ ⃗×𝑏2⃗⃗⃗⃗ ⃗||.

If shortest distance is zero, then lines intersect and line intersects in

space if they are coplanar. Hence if above lines are coplanar

If (𝑎2⃗⃗⃗⃗ − 𝑎1⃗⃗⃗⃗ ). (𝑏1⃗⃗⃗⃗ × 𝑏2⃗⃗⃗⃗ ) = 0

(ii). Cartesian equations: 𝑥−𝑥1

𝑎1=

𝑦−𝑦1

𝑏1=

𝑧−𝑧1

𝑐1,𝑥−𝑥2

𝑎2=

𝑦−𝑦2

𝑏2=

𝑧−𝑧2

𝑐2

107

D=

|𝑥2−𝑥1 𝑦2−𝑦1 𝑧2−𝑧1𝑎1 𝑏1 𝑐1𝑎2 𝑏2 𝑐2

|

√(𝑏1𝑐2−𝑏2𝑐1)2+(𝑐1𝑎2−𝑐2𝑎1)2+(𝑎1𝑏2−𝑎2𝑏1)2

9.If shortest distance is zero, then lines intersect and line intersects in

space if they are

coplanar. Hence if above lines are coplanar

|

𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1𝑎1 𝑏1 𝑐1𝑎2 𝑏2 𝑐2

| = 0

(b).Shortest distance between two parallel lines: If two lines are parallel,

then they are coplanar.

Let the lines be : 𝑟 =𝑎1⃗⃗⃗⃗ + 𝝀�⃗� , and: : 𝑟 =𝑎2⃗⃗⃗⃗ + 𝝁�⃗� ,

D=|�⃗� ×(𝑎2⃗⃗⃗⃗ ⃗−𝑎1⃗⃗⃗⃗ ⃗)

|�⃗� ||

10.General equation of a plane in vector form :- It is given by 𝑟 . �⃗� + 𝑑 = 0 ,

�⃗� is a vector normal to plane.

11.General equation of a plane in Cartesian form :- 𝒂𝒙 + 𝒃𝒚+ 𝒄𝒛 + 𝒅 = 𝟎 ,

Where a,b,c are direction ratios of normal to the plane.

12.General equation of a plane passing through a point :- if position vector of

given point is 𝑎 then equation is given by ( 𝑟 − 𝑎 ). �⃗� = 0, �⃗� is a vector

perpendicular tothe plane.

13.General equation of a plane passing through a point :- if position vector of

given point is 𝑎 then equation is given by ( 𝑟 − 𝑎 ). �⃗� = 0, �⃗� is a vector

perpendicular tothe plane.

14.General equation of a plane passing through a point :- if coordinates of

point are(𝑥, 𝑦, 𝑧) then equation is𝑎(𝑥 − 𝑥1) + 𝑏(𝑦 − 𝑦1 ) + 𝑐(𝑧 − 𝑧1) = 0,

a,b,care direction ratios of a line perpendicular to the plane.

15.Intercept form of equation of a plane :-General equation of a plane which

cuts off intercepts a, b and c on x-axis, y-axis, z-axis respectively is 𝑥

𝑎+𝑦

𝑏+𝑧

𝑐=

1.

108

16. Equation of a plane in normal form:- 𝑟 . �̂� = p,where �̂� is a unit vector

along perpendicular from origin and ‛p’ is distance of plane from origin.p is

always positive.

17.Equation of a plane in normal form :- It is given by 𝑙𝑥 + 𝑚𝑦 + 𝑛𝑧 = 𝑝,

where 𝑙,𝑚, 𝑛 are direction cosines of perpendicular from origin and ‛p’ is

distance of plane from origin. p is always positive.

18.Equation of a plane passing through three non-collinear points :- If 𝑎 , �⃗� , 𝑐

are the position vectors of three non-collinear points, then equation of a plane

through three points is given by –

(𝑟 − 𝑎 ). {(�⃗� − 𝑎 ) × (𝑐 − 𝑎 )} = 0.

19. Equation of a plane passing through three non-collinear points(Cartesian

system) :- If plane passing through points (𝒙𝟏, 𝒚𝟏, 𝒛𝟏 ) , (𝒙𝟐, 𝒚𝟐, 𝒛𝟐) and

(𝒙𝟑, 𝒚𝟑, 𝒛𝟑 ) then equation is-

|

(𝑥 − 𝑥1) (𝑦 − 𝑦1) (𝑧 − 𝑧1)(𝑥2 − 𝑥1) (𝑦2 − 𝑦1) (𝑧2 − 𝑧1)(𝑥3 − 𝑥1) (𝑦3 − 𝑦1) (𝑧3 − 𝑧1)

| = 0

20.If 𝜽 is angle between two planes �⃗� . 𝒏𝟏⃗⃗ ⃗⃗ + 𝒅𝟏 = 𝟎 and �⃗� . 𝒏𝟐⃗⃗ ⃗⃗ + 𝒅𝟐 = 𝟎 then

cos 𝜃 = 𝑛1⃗⃗⃗⃗ ⃗.𝑛2⃗⃗⃗⃗ ⃗

|𝑛1⃗⃗⃗⃗ ⃗||𝑛2⃗⃗⃗⃗ ⃗|

(i) If planes are perpendicular, then 𝑛1⃗⃗⃗⃗ . 𝑛2⃗⃗⃗⃗ =0

(ii) If planes are parallel, then 𝑛1⃗⃗⃗⃗ × 𝑛2⃗⃗⃗⃗ = 0 or 𝑛1⃗⃗⃗⃗ = 𝑡𝑛2⃗⃗⃗⃗ ,t is a scalar.

21. If 𝜽 is angle between two planes 𝒂𝟏 𝒙+ 𝒃𝟏𝒚+ 𝒄𝟏𝒛 + 𝒅𝟏 = 𝟎 𝒂𝒏𝒅 𝒂𝟐𝒙+

𝒃𝟐𝒚+ 𝒄𝟐𝒛 + 𝒅𝟐 = 𝟎

Then 𝐜𝐨𝐬𝜽 =𝒂𝟏𝒂𝟐+𝒃𝟏𝒃𝟐+𝒄𝟏𝒄𝟐

√(𝒂𝟏𝟐+𝒃𝟏𝟐+𝒄𝟏𝟐)( 𝒂𝟐𝟐+𝒃𝟐

𝟐+𝒄𝟐𝟐 )

(i) If planes are perpendicular ,then 𝑎1𝑎2 + 𝑏1𝑏2 + 𝑐1𝑐2 = 0

(ii) If planes are parallel , then 𝑎1

𝑎2=

𝑏1

𝑏2=

𝑐1

𝑐2

109

22. If 𝜽 is angle between line �⃗� = �⃗⃗� + 𝝀�⃗⃗⃗� and the plane �⃗� . �⃗⃗� + 𝒅 = 𝟎

,then 𝐬𝐢𝐧 𝜽 = �⃗⃗⃗� .�⃗⃗�

|�⃗⃗⃗� |.|�⃗⃗� |

(i)If line is parallel to plane ,then �⃗⃗� . �⃗� =0 and

(ii)If line is perpendicular to plane , then �⃗⃗� × �⃗� = 0 or �⃗⃗� = 𝑡�⃗� ,t is a

scalar.

23. . If 𝜽 is angle between line 𝒙−𝒙𝟏

𝒂𝟏=

𝒚−𝒚𝟏

𝒃𝟏=

𝒛−𝒛𝟏

𝒄𝟏 and the plane 𝒂𝒙 + 𝒃𝒚 +

𝒄𝒛 + 𝒅 = 𝟎 ,then

𝐬𝐢𝐧 𝜽 =𝒂𝒂𝟏+𝒃𝒃𝟏+𝒄𝒄𝟏

√(𝒂𝟏𝟐+𝒃𝟏𝟐+𝒄𝟏𝟐)( 𝒂𝟐+𝒃𝟐+𝒄𝟐 )

(i)If line is parallel to the plane ,then 𝑎𝑎1 + 𝑏𝑏1 + 𝑐𝑐1 = 0

(ii)If line is perpendicular to the plane, then 𝑎

𝑎1=

𝑏

𝑏1=

𝑐

𝑐1

24. General equation of a plane parallel to the plane 𝑟 . �⃗� + 𝑑 = 0 𝑖𝑠

𝑟 . �⃗� + 𝜆 = 0, where 𝜆 is a constant and can be calculated from given condition.

25. General equation of a plane parallel to the plane ax+by+cz+d=0 is

ax+by+cz+ 𝜆 = 0, where 𝜆 is a constant and can be calculated from given

condition.

26. General equation of a plane (vector form) passing through the line of the

intersection of planes

𝑟 . 𝑛1⃗⃗⃗⃗ + 𝑑1 = 0 and 𝑟 . 𝑛2⃗⃗⃗⃗ + 𝜆𝑑2 = 0 is 𝑟 . (�⃗� 1 + 𝜆�⃗� 2) + (𝑑1 + 𝜆𝑑2) = 0 ,

where 𝜆 is a constant and can be calculated from given condition.

27. General equation of a plane(Cartesian form) passing through the line of the

intersection of planes a1x+b1y+c1z+d1=0 and a2x+b2y+c2z+d2=0 is(

a1x+b1y+c1z+d1)+ 𝜆 (a2x+b2y+c2z+d2 )=0 , where 𝜆 is a constant and can be

calculated from given condition.

28. Distance of a plane(vector form) 𝑟 . �⃗� + 𝑑 = 0

, from a point with position vector 𝑎 , is|�⃗� .�⃗� +𝑑

|�⃗� ||.

110

29. Distance of a plane(Cartesian form) ax+by+cz+d=0, , from a point

(x1,y1,z1) is|𝑎𝑥1+𝑏𝑦1+𝑐𝑧1+𝑑

√𝑎2+𝑏2+𝑐2|.

QUESTIONS ON VECTORS & 3-D.

LEVEL-1

1) Find the projection of 𝑖̂ - 𝑗 ̂on 𝑖̂ + 𝑗 ̂.

(1 Mark)

2) If |𝑎 | =2, |�⃗� | =√3 and 𝑎 .�⃗� =√3.Find the angle between 𝑎 and �⃗� .

(1 Mark)

3) Find the value of λ when the projection of 𝑎 =λ𝑖̂ + 𝑗 ̂+4�̂� on �⃗� =2𝑖̂ +6𝑗+̂3�̂� is 4 units.

(1 Mark)

4) If 𝑎 .𝑎 =0 and 𝑎 .�⃗� = 0 Then what can be concluded about the vector �⃗� ?

(1 Mark)

5) Show that angle in a semicircle is a right angle using vectors.

(4 Marks)

6) Show that diagonals of a rhombus are perpendicular to each other using vectors.

(4 Marks)

7) Show that the vectors 1

7 (2𝑖 ̂+3𝑗+̂6�̂�),

1

7 (3𝑖 ̂- 6𝑗+̂2�̂�) and

1

7 (6𝑖̂ +2𝑗-̂3�̂�) are mutually

perpendicular unit vectors.

(4 Marks)

8) 3 Find the Co ordinate of foot of the perpendicular from origin to the plane 3x+4y-5z=7

(4 Marks)

9) Q 1: Prove that if a plane has the intercepts a, b, c and is at a distance of p units from the origin,

then 2 2 2 2

1 1 1 1

a b c p

(6 Marks)

10) Q 2: Find the equation of the plane passing through the point (-1,3, 2) and perpendicular to the

planes x +2y +3z = 5 and3x +3y + z = 0.

(6 Marks)

11) Q 3: Find the co-ordinates of the point where the line through (3, 4, 1) and (5, 1, 6) crosses the

xy-plane . (6 marks)

LEVEL-2

1. Find the angle between the vectors 𝑖̂ -2𝑗+̂3�̂� and 3𝑖̂ -2𝑗+̂�̂� .

(1 Mark)

2. Find |𝑎 − �⃗� | if |𝑎 | =2, |�⃗� | =3 and 𝑎 .�⃗� = 4.

(1 Mark)

111

3. If 𝑎 is a unit vector and (𝑥 - 𝑎 ) . (𝑥 + 𝑎 ) =15. Find |𝑥 | .

(1 Mark)

4. If 𝑎 and �⃗� are two vectors such that |𝑎 + �⃗� | =|𝑎 | then prove that the vector 2𝑎 + �⃗� is perpendicular to

�⃗� . (4 Marks)

5. Show that the angle between the diagonals of a cube is cos−11

3 .

(4 Marks)

6. If vectors 𝑎 =2𝑖̂ +2𝑗+̂3�̂�, �⃗� =𝑖̂ +𝑗+̂�̂� and 𝑐 =3𝑖̂ +𝑗 ̂are such that 𝑎 + λ �⃗⃗� is perpendicular

to �⃗⃗� .Find the value of λ.

(4 Marks)

7. If 𝑎 and �̂� are two unit vectors and θ is the angle between them ,then prove that

a. sin θ /2 =1

2 |𝑎 − �⃗� |.

(4 Marks)

8. If 𝑎 and �̂� are unit vectors and θ is the angle between them ,then prove that

9. tan θ/2 =⌊ �⃗� −�⃗�

�⃗� +�⃗� ⌋ .

(4 Marks)

10. 7. Find the distance of the point (−1, −5, −10) from the point of intersection of the

line 𝑟 = (2𝑖̂ − 𝑗̂ + 2�̂�) + 𝜆(3 𝑖̂ + 4𝑗̂ + 2�̂� ) and the plane 𝑟 . (𝑖̂ − 𝑗̂ + �̂�) =

5. 𝐴𝑁𝑆: 13 𝑢𝑛𝑖𝑡𝑠. (6 Marks)

11. 8. A line makes angles 𝛼, 𝛽, 𝛾 𝑎𝑛𝑑 𝛿 with the four diagonals of a cube.

12. Prove that:cos2α + cos2β + cos2γ+cos2𝛿 =4

3.

(6 Marks)

13. Find the distance between the point P(6,5,9) and the plane determined by the

points A(3,−1,2),B(5,2,4) and C(−1,−1,6). ANS:6

√34units.

(6 Marks)

14. 10. Find the equation of the plane which is perpendicular to the plane 5x+3y+6z+8=0

and which contains the line of intersection of the planes x+2y+3z-4=0 and 2x+y-

z+5=0. (6 Marks)

𝐴𝑁𝑆:51x+15y-50z+173=0

LEVEL-3

1. Find the value of p so that 𝑎 =2𝑖̂ + p𝑗 ̂+�̂� and �⃗� =𝑖̂ -2𝑗+̂3�̂� are perpendicular to each

other. (1 Mark)

2. Find |𝑎 | if |𝑎 |=2|�⃗� | and (𝑎 +�⃗� ) . (𝑎 -�⃗� ) = 12.

(1 Mark)

3. If |𝑎 + �⃗� | =60, |𝑎 − �⃗� | =40 and |�⃗� | =46. Find |𝑎 | .

(1 Mark)

112

4. IF �⃗⃗� =(3i+2j-3k) and �⃗� =4i+7j-3k Find vector projection of �⃗⃗� in the direction of �⃗� .

(1 Mark)

5. The two adjacent sides of a parallelogram are )ˆ32ˆ(&)ˆ5ˆ4ˆ2( kjikji .Find the unit

vectors parallel to its diagonals. Also find its area.

(4 Marks)

6. If )ˆ6ˆ(&)ˆ2ˆ2ˆ3(),ˆ3ˆ5ˆ2(),ˆˆˆ( kjikjikjikji are the position vectors of

points A, B, C & D respectively, then find the angle between AB & CD. Deduce that AB & CD

are parallel. (4 Mark)

7. Find the value of such that the vectors )ˆˆ2(&)ˆ3ˆ2ˆ(),ˆ5ˆˆ3( kjikjikji are

coplanar.(4 Marks)

8. Find the magnitude and equation of the line of the shortest distance between the lines:

kjirandkjir )2()12()1()1()1()1(

(6 Marks) i. Ans.:-

)(2;2

25

kiir

9. Find the equation of the plane which is perpendicular to the plane 08635 zyx

and which contains the line of intersection of the planes

0520432 zyxandzyx . (6 Marks)

i. Ans.:-

0173501551 zyx 10. 12.Find the equation of a plane which is at a distance of 7 units from the origin and which is

normal to the vector .653

kji Ans.:-

0707)653.(

kjir (6 Marks)

11. . Find the distance of the point (1,−2,3)𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 plane x-y+z=5 measured ‖ to the

line𝑥+1

2=

𝑦+3

3=

𝑧+1

−6. ANS: other point (

9

7, −

11

7,15

7) , 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 1 𝑢𝑛𝑖𝑡

(6 Marks)

113

Common Errors

1. Condition for parallelism and perpendicularity of 2 vectors

a. If 𝑎 = k �⃗� or 𝑎1

𝑏1 =

𝑎2

𝑏2 =

𝑎3

𝑏3 (parallel)

b. 𝑎1𝑏1 + 𝑎2𝑏2 + 𝑎3𝑏3 =0 (perpendicular)

2. Applying correct identity.

3. Dot and cross product of 𝑖̂ ,𝑗 ̂and �̂� .

4. Taking general vector 𝑎 = a1𝑖̂ +a2𝑗 ̂+a3�̂� .

5. Not understanding difference between lines and planes in vector and Cartesian form

of the equations.

6. Not defining direction cosine and direction ratios between lines and planes.

7. Not understanding the condition for parallel and perpendicular lines and planes.

8. Not using proper formula according to the question of lines and planes.

TIPS AND TECHNIQUES IN SCORING MARKS IN

VECTORS AND THREE DIMENSIONAL GEOMETRY.

1) Identify the correct identity or formulas.

2) Condition for parallel and perpendicularity in vectors and 3D.

3) Understanding the correct formula in vector form and Cartesian form.

4) In shortest distance questions check whether the lines are parallel or not.

5) In shortest distance questions take 𝑎1⃗⃗⃗⃗ and 𝑏1⃗⃗ ⃗ from line I and 𝑎2⃗⃗⃗⃗ and 𝑏2⃗⃗⃗⃗ from line II.

6) Angle between two lines or planes better use cosine formulae whereas angle between a line and a

plane use cos (90-𝜃)

114

LINER PROGRAMMING

CONCEPT MAPPING

Basic Concepts And Formulas:-

1. Definition:- Linear programming (LP) is an optimization technique in which a linear function is optimised (i.e. minimized or maximized) subject to certain constraints which are in the form of linear inequalities or/and equations. The function to be optimised is called objective function.

2. Application of linear programming:- Linear programming (LP) is used in determining optimum combination of several variables subject to certain constraints or restrictions.

3. Formation of linear programming problem (LPP): The basic problem in the formulation of a linear programming problem is to set-up some mathematical model. This can be done by asking the following question:

(a) What are the unknowns ( variables)? (b) What is the objectives? (c) What are the restrictions?

For this, let x1,x2,x3,………………xn be the variables. Let the objective function to be optimized

(i.e. minimized or maximized) be given by z.

(i) Z=C1x1+c2x2+………………+cnxn ,where 𝑐𝑖𝑥𝑖(i=1,2,3,….n) are constraints.

(ii) Let there be mn constants and let bi be a set of constants such that

a11x1 + a12 x2+……..+a1nxn(≤,=,≥)𝑏1

a21x1 + a22 x2+……..+a2nxn(≤,=,≥)𝑏2

……… ……. ……….

……… …….. ……….

Am1x1 + am2 x2+……..+amnxn(≤,=,≥)𝑏1𝑚

(iii) Finally, let x1 ≥0 ,x2 ≥0,x3 ≥0 ,……………,xn ≥ 0, called non –negative constraints.

The problem of determining the values of x1,x2,x3,………………xn which makes Z, a minimum or maximum and which satisfies(ii) and (iii) is called the general linear programming problem.

4.General LPP:

(a) Decision variables: The variables x1,x2,x3,………………xn whose values are to be decided, are called decision variables.

(b)Objective function: The linear function Z=c1x1+c2x2+………………+cnxn, which is to be optimized (i.e. minimized or maximized)is called the objective function or preference function of the general linear programming problem.

115

(c) Structural constraints: The inequalities given in (ii), are called the structural constraints of the general linear programming problem. The structural constraint are generally in the form of inequalities of ≥ 𝑡𝑦𝑝𝑒 𝑜𝑟 ≤ type,but occasionally, a structural constraint may be in the form of an equation.

(d) Non-negative constraints: The set of inequalities(iii) is usually known as the set of non-negative constraints of the general LPP. These constraints imply that the variables x1,x2,x3,………………xn , Cannot take negative values.

( e )Feasible solution: Any solution of a general LPP which satisfies all the constraints, structural and non-negative, of the problem, is called a feasible solution to the general LPP.

(f)Optimum solution: Any feasible solution which optimizes(i.e. minimized or maximized) the objective function of the LPP is called Optimum solution.

5.Requirement for Mathematical formulation of LPP:-Before getting the mathematical form of a linear programming problem, it is important to recognize the problem which can be handled by linear programming problem. For the formulation of a linear programming problem, the problem must satisfy the following requirements:

(i) There must be an objective to minimise or maximize something. The objective must be capable of being clearly defined mathematically as a linear function.

(ii)The resources must be in economically quantifiable limited supply .This gives the constraints to LPP.

(iii) The constraints(restrictions) must be capable of being expressed in the form of linear equations or inequalities.

6.Solving Linear programming problem: To solve linear programming problem .Corner Point Method is adopted, Under this method following steps are performed:

Step 1. At first , feasible region is obtained by plotting the graph of given linear constraints and its corner points are obtained by solving the two equations of the lines intersecting at that point.

Step 2. The value of objective function z = ax + by is obtained for each corner point by putting its x and y- coordinate in place of x and y in Z= ax + by .Let M and m be largest and smallest value of Z respectively.

Case-I: If the feasible region is bounded , then M and m are the maximum and minimum values of Z.

Case-II: If the feasible region is unbounded, then we proceed as follows:

Step-3: The open half plane determined by ax + by > 𝑀 and ax + by < 𝑚 are obtained.

Case I:If there is no common point in the half plane determined by ax + by > 𝑀 and feasible region, then M is maximum value of Z otherwise Z has no maximum value.

Case II: If there is no common point in the half plane determined by ax + by < 𝑚 and feasible region , then m is minimum value of Z otherwise Z has no minimum value.

116

Above facts can be represented by arrow diagram as:-

Feasible region (having largest and smallest values M and m of Z=ax+by at corner point)

Bounded Unbounded

(M is maximum and m is minimum Value of Z)

TYPES OF LINEAR PROGRAMMING PROBLEMS :-

(I)Manufacturing Problems (profit always maximize/manufacturing cost always minimize)

(ii)Diet Problems (Cost Always Minimize)

(iii)Transportation Problems(Transportation Cost Always Minimize)

Note:- The feasible region of transportation problem has 2 pair of parallel lines.

LEVEL-1

MANUFACTURING PROBLEMS

Question 1: A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftsman’s time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftsman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman’s time.

(ii) What number of rackets and bats must be made if the factory is to work at full capacity?

If no common point in the half plane determined by ax + by > 𝑀 and feasible region, then M is maximum value of Z otherwise Z has no maximum value.

If there is no common point in the half plane determined by ax + by < 𝑚 and feasible region , then m is minimum value of Z otherwise Z has no minimum value.

117

(ii) If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find the maximum profit of the factory when it works at full capacity.

Ans:- (max. profit Rs.200 when factory makes 4 tennis rackets and 12 cricket bats)

Question 2: A manufacturer produces nuts and bolts. It takes 1 hour of work on machine

A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A

and 1 hour on machine B to produce a package of bolts. He earns a profit, of Rs 17.50

per package on nuts and Rs. 7.00 per package on bolts. How many packages of each

should be produced each day so as to maximize his profit, if he operates his machines

for at the most 12 hours a day?

Ans:- (max. profit Rs.73.5 when 3 package nuts &3

package bolt are produce)

Question 3: A factory manufactures two types of screws, A and B. Each type of screw requires the use of two machines, an automatic and a hand operated. It takes 4 minutes on the automatic and 6 minutes on hand operated machines to manufacture a package of screws A, while it takes 6 minutes on automatic and 3 minutes on the hand operated machines to manufacture a package of screws B. Each machine is available for at the most 4 hours on any day. The manufacturer can sell a package of screws A at a profit of Rs 7 and screws B at a profit of Rs10. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce in a day in order to maximize his profit? Determine the maximum profit.

Question 4: A cottage industry manufactures pedestal lamps and wooden shades,

each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on

grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It

takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to

manufacture a shade. On any day, the sprayer is available for at the most 20 hours and

the grinding/cutting machine for at the most 12 hours. The profit from the sale of a

lamp is Rs. 5 and that from a shade is Rs. 3. Assuming that the manufacturer can sell all

the lamps and shades that he produces, how should he schedule his daily production in

order to maximize his profit?

Ans:-(manufacturer should produce 4 lamps & 4 shades to

get maximum profiy of Rs. 32.)

Question 5: A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours of assembling. The profit is Rs. 5 each for type A and Rs. 6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the profit?

118

LEVEL-2 Question 1:Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs. 60/kg and Food Q costs Rs. 80/kg. Food P contains 3 units /kg of vitamin A and 5 units /kg of vitamin B while food Q contains 4 units /kg of vitamin A and 2 units /kg of vitamin B. Determine the minimum cost of the mixture? Question 2 : One kind of cake requires 200g flour and 25g of fat, and another kind of cake requires 100g of flour and 50g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes? (Ans :- 20 cakes Ist type & 10 cakes IInd type) Question 3: A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F1and F2 are available. Food F1costs Rs. 4 per unit food and F2 costs Rs. 6 per unit. One unit of food F1 contains 3 units of vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements?

(Ans:- Minimum cost 104 Rs., 24 units of food F1 and 4

3 units of food F2 is required.)

Question 4: A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs. 250 per bag contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing Rs. 200 per bag contains 1.5 units of nutritional elements A, 11.25 units of element B, and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

Question 5: A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours of assembling. The profit is Rs. 5 each for type A and Rs. 6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the profit?

Question6: A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of Rs. 12 and Rs. 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximize the profit?

119

Question 7: A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid at least 270 kg of potash and at most 310 kg of chlorine. If the grower wants to minimize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?

LEVEL-3

Question 1: Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops, D, E and F whose requirements are 60, 50 and 40 quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table:

Transportation cost per quintal (in Rs)

From/To A B

D

E

F

6

3

2.50

4

2

3

How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost?Write precaution while transporting the things?

Question 2: An oil company has two depots A and B with capacities of 7000 L and 4000 L respectively. The company is to supply oil to three petrol pumps, D, E and F whose requirements are 4500L, 3000L and 3500L respectively. The distance (in km) between the depots and the petrol pumps is given in the following table:

kg per bag

Brand P Brand Q

Nitrogen

Phosphoric acid

Potash

Chlorine

3

1

3

1.5

3.5

2

1.5

2

120

Distance in (km)

From/To A B

D

E

F

7

6

3

3

4

2

Assuming that the transportation cost of 10 litres of oil is Re 1 per km, how should the delivery be scheduled in order that the transportation cost is minimum? What is the minimum cost? What is the use of Oil in our daily life & How to save petrol and diesel for future generation?

Question 7: A merchant plans to sell two types of personal computers − a desktop model and a portable model that will cost Rs. 25000 and Rs. 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs. 70 lakhs and if his profit on the desktop model is Rs. 4500 and on portable model is Rs. 5000. What is the advantage of computer in daily life?

Ans:- (Maximum profit of Rs. 11,50,000 is obtained when he stocks 200 desktop & 50 portable computer.)

Question 8: There are two types of fertilizers F1 and F2. F1consists of 10% nitrogen and 6% phosphoric acid and F2consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F1cost Rs. 6/kg and F2 costs Rs. 5/kg, determine how much of each type of fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost? Why natural fertilizers are better than chemical fertilizers?

121

COMMON MISTAKE(LPP)

(1) SOME CASES STUDENTS ARE NOT CLERIFY ABOUT EXACT SHADED REGION .

ANS:- first we check the inequalities taking origine whether it satisfies or not , if it satisfies

then the open half plane containing origin otherwise another half plane.

EXP-

2 X + 3y >6

Mistake graph Correct graph

(0,2) (3,0)

(0,2) (3,0)

(2) If at least & at most is comes then student doing mistake in symbols (≤,≥,

Ans:-if at least comes in the question then corresponding inequalities contain ≥ , if

at most is comes in the question then corresponding inequalities contain ≤ .

EXP:- A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F1and

F2 are available. Food F1 costs Rs 4 per unit food and F2 costs Rs 6 per unit. One unit of food

F1 contains 3 units of vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of

vitamin A and 3 units of minerals. Formulate this as a linear programming problem.

SOLUTION:-

Let the diet contain x units of food F1 and y units of food F2. Therefore,

x ≥ 0 and y ≥ 0

The given information can be complied in a table as follows.

122

Vitamin A (units) Mineral (units) Cost per unit

(Rs)

Food F1 (x) 3 4 4

Food F2 (y) 6 3 6

Requirement 80 100

MISTAKES CORRECTION

3x + 6y ≤ 80

3x + 6y ≥ 80

4x + 3y ≤ 100

4x + 3y ≥ 100

(3) When region is unbounded then some student do not clarify about

max./min. Value is possible & not possible.

: The open half plane determined by ax + by > 𝑴 and ax + by < 𝒎 are obtained.

Case I:If there is no common point in the half plane determined by ax + by > 𝑴 and feasible region, then M is maximum value of Z otherwise Z has no maximum value.

Case II: If there is no common point in the half plane determined by ax + by <m and

feasible region , then m is minimum value of Z otherwise Z has no minimum value.

MISTAKE CORRECTION

123

(4) STUDENTS ARE DOING MISTAKE NOT CONVERTING IN SAME UNIT.

EXP-Question 5:

A factory manufactures two types of screws, A and B. Each type of screw requires the use of two

machines, an automatic and a hand operated. It takes 4 minutes on the automatic and 6 minutes

on hand operated machines to manufacture a package of screws A, while it takes 6 minutes on

automatic and 3 minutes on the hand operated machines to manufacture a package of screws B.

Each machine is available for at the most 4 hours on any day. The manufacturer can sell a

package of screws A at a profit of Rs 7 and screws B at a profit of Rs10. Assuming that he can

sell all the screws he manufactures, how many packages of each type should the factory owner

produce in a day in order to maximize his profit? Determine the maximum profit.

Answer :

Let the factory manufacture x screws of type A and y screws of type B on each day. Therefore,

x ≥ 0 and y ≥

.

MISTAKE CORRECT

Screw

A

Screw

B

Availability

Automatic

Machine

(min)

4 6 4

Hand

Operated

Machine

(min)

6 3 4

Screw

A

Screw

B

Availability

Automatic

Machine

(min)

4 6 4 × 60 =240

Hand

Operated

Machine

(min)

6 3 4 × 60 =240

TIPS & TRICKS :-

(1) if at least comes in the question then corresponding inequalities contain ≥ , if at

most comes in the question then corresponding inequalities contain ≤. (2) Taking same units of time ,weight & rupees,etc. (3) If inequality contains ≤ ,then open half will be the plane containing origin. (4) If inequality contains ≥ ,then open half will be the plane not containing origin.

124

PROBABILITY

CONCEPT MAPPING

1 .Definition of probability ,its formula .

2. Random Experiment ,outcomes ,sample space,events .

3. simple event ,compound events .

4.Mutually exclusive events : - Two events are said to be mutually exclusive if the

occurrence of one prevents the occurrence of the other . in other words ,if A and B have no

common elements , then they are said to be mutually exclusive events .

i.e. A ∩ B = ∅ .

5.Conditional Probability :- If E and F are two events associated with the same sample

space of a random experiment, the conditional probability of the event E given that F has

occurred,

i.e. P (E|F) is given by

P(E|F) = 𝑃(𝐸 ∩𝐹)

𝑃(𝐹) ,provided P(F) ≠ 0 .

6. Properties of conditional probability

Let E and F be events of a sample space S of an experiment, then we have

Property 1: P(S|F) = P(F|F) = 1

Property 2: If A and B are any two events of a sample space S and F is an event

of S such that P(F) ≠ 0, then P((A ∪ B)|F) = P(A|F) + P(B|F) – P((A ∩ B)|F)

Property 3: P(E′|F) = 1 − P(E|F)

7. Multiplication Theorem on Probability

P(E ∩ F) = P(E) P(F|E) = P(F) P(E|F) , provided P(E) ≠ 0 and P(F) ≠ 0.

The above result is known as the multiplication rule of probability.

8. Multiplication rule of probability for more than two events :- If E, F and G are three

events of sample space, we have P(E ∩ F ∩ G) = P(E) P(F|E) P(G|(E ∩ F)) = P(E) P(F|E)

P(G|EF) Similarly, the multiplication rule of probability can be extended for four

or more events.

125

9. Independent Events :- E and F are two events such that the probability of occurrence of

one of them is not affected by occurrence of the other. Such events are called independent

events .

OR

Let E and F be two events associated with the same random experiment,

then E and F are said to be independent if

P(E ∩ F) = P(E) . P (F) (it is known as multiplication theorem also)

Remarks

(i) Two events E and F are said to be dependent if they are not independent, i.e. if

P(E ∩ F ) ≠ P(E) . P (F)

(ii) Mutually exclusive events never have an outcome common, but independent events,

may have common outcome. Clearly, ‘independent’ and ‘mutually exclusive’ do not have

the same meaning.

In other words, two independent events having nonzero probabilities of occurrence can not

be mutually exclusive, and conversely, i.e. two mutually exclusive events having nonzero

probabilities of occurrence can not be independent .

(iii) if the events E and F are independent, then

(a) E′ and F are independent,

(b) E′ and F′ are independent

(c) If A and B are two independent events, then the probability of occurrence

of at least one of A and B is given by 1– P(A′) P(B′) .

10.Addition Theorem :- (a) When the events are not mutually exclusive :

𝑃 (𝐴 ∪ 𝐵) = P(𝐴) + P (B) - P(A∩ B)

(b) If Aand B are mutually exclusive i.e. (𝐴 ∩ 𝐵) = ∅ ,then 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) .

11. Partition of a sample space :- A set of events E1, E2, ..., En is said to represent a

partition of the sample space S if

(a) Ei ∩ Ej = φ, i ≠ j, i, j = 1, 2, 3, ..., n

(b) E1 ∪ Ε2∪ ... ∪ En= S and

(c) P(Ei)> 0 for all i = 1, 2, ..., n

126

.In other words, the events E1, E2, ..., En represent a partition of the sample spaceS if they

are pairwise disjoint, exhaustive and have nonzero probabilities.

12. Theorem of total probability

Let {E1, E2,...,En} be a partition of the sample space S, and suppose that each of the

events E1, E2,..., En has nonzero probability of occurrence. Let A be any event

associatedwith S, then

P(A) = P(E1) P(A|E1) + P(E2) P(A|E2) + ... + P(En) P(A|En) .

13. Bayes’ Theorem :- If E1, E2 ,..., En are n non empty events which constitute a partitionof

sample space S, i.e. E1, E2 ,..., En are pairwise disjoint and E1∪ E2∪ ... ∪ En = S andA is any

event of nonzero probability, then

𝑃 (𝐸𝑖

𝐴) =

𝑃(𝐸𝑖)𝑃(𝐴

𝐸𝑖)

∑ 𝑃(𝐸𝑖).𝑃(𝐴

𝐸𝑖)𝑛

𝑖=1

, for any i = 1, 2, 3, ..., n.

14. Random Variable :- A random variable is a real valued function whose domain is the

sample

space of a random experiment.

15. Probability Distribution : - The probability distribution of a random variable X is the

system of numbers

X 𝑥1 𝑥2 𝒙𝟑 …………… 𝑥𝑖 ……………………………………. 𝑥𝑛

P(X) 𝑝1 𝑝2 𝑝3 …………… 𝑝𝑖 …………………………………… 𝑝𝑛

where, 𝑝𝑖 > 0 , ∑𝑝𝑖 = 1, i = 1, 2,..., n .

16. Mean of a random variable: -Let X be a random variable whose possible values x1, x2,

x3, ..., xn occur with probabilities p1, p2, p3,..., pn, respectively. The mean of X, denoted by

μ,

𝜇 = 𝐸(𝑋) = ∑ 𝑥𝑖𝑝𝑖𝑛𝑖=1

And variance denoted by 𝜎2 = ∑ 𝑥𝑖2𝑛

𝑖=1 𝑝𝑖 − 𝜇2

Standard deviation = √𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒

16. Bernoulli trials :- Trials of a random experiment are called Bernoulli trials, if they satisfythe

following conditions :

(i) There should be a finite number of trials.

127

(ii) The trials should be independent.

(iii) Each trial has exactly two outcomes : success or failure.

(iv) The probability of success remains the same in each trial.

17.Binomial Distribution : - , the probability distribution of number of successes in an experiment

consisting of n Bernoulli trials may be obtained by the binomial expansion of (𝑝 + 𝑞)𝑛.

. Hence, this PROBABILITY distribution of number of successes X can be written as

X 1 2 3 ……….. x ………. n

P(X) 𝑛𝐶0𝑞𝑛 𝑛𝐶1𝑞

𝑛−1𝑝 𝑛𝐶2𝑞𝑛−2𝑝2 …………. 𝑛𝐶𝑥𝑞𝑛−𝑥𝑝𝑥 …………. 𝑛𝐶𝑛𝑝𝑛

The above probability distribution is known as binomial distribution with parameters

n and p, because for given values of n and p, we can find the complete probability

distribution.

The probability of x successes P(X = x) is also denoted by P(x) and is given by

P(x) = 𝑛𝐶𝑋𝑞𝑛−𝑥𝑝𝑥

, x = 0, 1,..., n. (q = 1 – p)

This P(x) is called the probability function of the binomial distribution .A binomial distribution

with n-Bernoulli trials and probability of success in each trial as p, is denoted by B(n, p).

128

QUESTION BANK

CONDITIONAL PROBABILITY

LEVEL 1

1. Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P(E|F)

2. Compute P(A|B), if P(B) = 0.5 and P (A ∩ B) = 0.32

3. If P(A) = 0.8, P (B) = 0.5 and P(B|A) = 0.4, find P(A ∩ B) .

4. Evaluate P(A ∪ B), if 2P(A) = P(B) = 5/13 and p(A/B) =2/5 .

5 . Given that E and F are events such that P(E) = 0.4, P(F) = 0.5 and P(E ∩ F) = 0.2, find

P(F|E) .

6. . If P(A) = 0.8, P (B) = 0.5 and P(B|A) = 0.4, find (i) P(A|B) .

7. . If P(A) = 0.7, P (B) = 0.3 and P(B|A) = 0.4, find (i) P(A ∪ B).

8. If P(A) = 6/11 , P(B) = 5/11 , P(A ∪ B) = 7/11, find P(A∩B) .

9. . If P(A) = 4/11 , P(B) = 3/11 , P(A ∪ B) = 5/11, find P(A|B) .

10. . If P(A) =4/7 , P(B) = 2/7 , P(A ∪ B) =5/7, find P(B|A) .

LEVEL 2

Question 1: find P (E|F) A coin is tossed three times, where

(i) E: head on third toss, F: heads on first two tosses

(ii) E: at least two heads, F: at most two heads

(iii) E: at most two tails, F: at least one tail

Question 2: find P (E|F) A die is thrown three times,

E: 4 appears on the third toss, F: 6 and 5 appears respectively on first two tosses

Question 3: find P (F|E) Mother, father and son line up at random for a family picture

E: son on one end, F: father in middle

Question 4: Assume that each born child is equally likely to be a boy or a girl. If a family has two

children, what is the conditional probability that both are girls given that (i) the youngest is a

girl, (ii) at least one is a girl?

129

LEVEL-3

Question 1: An instructor has a question bank consisting of 300 easy True/False questions, 200 difficult

True/False questions, 500 easy multiple choice questions and 400 difficult multiple choice

questions. If a question is selected at random from the question bank, what is the probability

that it will be an easy question given that it is a multiple choice question?

Question 2: Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die again

and if any other number comes, toss a coin. Find the conditional probability of the event ‘the

coin shows a tail’, given that ‘at least one die shows a 3’.

INDEPENDENT EVENT & BAYE’S THEOREM

LEVEL 1

1. If P(A) =3/5 ,and P(B) =1/5 ,find P(A ∩ B) if A and B are independent events.

2. Let E and F be events with P(E) =3/5 ,P(F) =3/10 ,P(E ∩ F) =1/5 .Are E and F independent ?

3. Given that the events A and B are such that P(A) =1/2 ,P(A ∪ B) =3/5 and P(B)=K .Find K if

they are independent.

4. Let A and B be independent events with P(A) =0.3 and P(B)= 0.4 .Find P(A ∩ B) .

5. Let A and B be independent events with P(A) =0.3 and P(B)= 0.4 .Find P(A ∪ B) .

6. Let A and B be independent events with P(A) =0.3 and P(B)= 0.4 .Find P(A / B) .

7. Let A and B be independent events with P(A) =0.3 and P(B)= 0.4 .Find P(B / A) .

8.If A and B are two events such that P(A) =1/4 ,P(B) =1/2 and P(A ∩ B) =1/8 .

Find P(not A and not B) .

9.Events A and B are such that P(A) =1/2 ,P(B) =7/12 and P(A 𝑛𝑜𝑡 𝐴 𝑜𝑟 𝑛𝑜𝑡 𝐵 ) =1/4 .State

whether A and B are independent .

10.Given two independent events A and B such that P(A) =0.3 ,P(B)= 0.6 .find P(A and not B).

11. Bag I contains 3 red and 4 black balls while another Bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is

found to be red. Find the probability that it was drawn from Bag II.

130

LEVEL 2

11.A die is tossed thrice .find the probability of getting an odd number at least once.

12.Two balls are drawn at random with replacement from a box containing 10 black and 8 red

balls .find the probability that one of them is black and other is red.

13. Probability of solving specific problem independently by A and B are 1/2 and 1/3

respectively. If both try to solve the problem independently. Find the probability that (I )

the problem is solved. (ii) exactly one of them solves the problem.

14.one card is drawn at random from a well shuffled deck of 52 cards .Check whether the

following events E and F are independent .

E: the card drawn is a spade , F :the card drawn is an ace.

15.In a hostel , 60% of the students read Hindi news paper and 40% read English newspaper

and 20% read both Hindi and English newspaper. A student is selected at random.

(a)Find the probability that the student reads neither Hindi nor English newspaper.(b)if

she reads English newspaper ,find the probability that she reads Hindi newspaper.

16.The probabilities that a husband and wife will be alive 20 years from now are 0.8 and 0.9

respectively .find the probability that in 20 years (a)both (b) neither (c) atleast one will

be alive .

17.The odds against A solving a certain problem are 4 to 3 and the odds in favour of solving

the same problem are 7 to 5 .find the probability that the problem will be solved.

18. Two cards are drawn at random from a pack of 52 cards .find the probability that the cards

are either both red or both aces .

19. A class consists of 80 students; 25 of them are girls and 55 boys ; 10 of them are rich and

the remaining poor ;20 of them are fair complexioned .what is the probability of selecting

a fair complexioned rich girl .

20. A scientist has to make a decision on each of the two independent events I and II .suppose

the probability of error in making decision on event I is 0.02 and that on event II is 0.05

.find the probability that the scientist will make the correct decision on (i) both the

events (ii) only one event .

21. In a factory which manufactures bolts, machines A, B and C manufacture Respectively 25%, 35% and 40% of the bolts. Of their outputs, 5, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it is manufactured by the machine B?

131

LEVEL 3

21. A and B throw a pair of dice turn by turn .The first to throw 9 is awarded a prize .if A starts

the game ,show that the probability of A getting the prize is 9/17 .

22. A has two fire extinguishing engines functioning independently .The probability of

availability of each engine , when needed , is 0.95. What is the probability that

(i) neither of them is available when needed ?

(ii) an engine is available when needed ?

(iii) exactly one engine is available when needed ?

23. A speaks truth in 55 percent cases and B speaks truth in 75 percent cases .Determine the

percentage of cases in which they are likely to contradict each other in stating the same

fact .

24. The probability that a teacher will give an unannounced test during any class meeting is

1/5 .if a

Student absent twice, find the probability that the student will miss at least one test.

25. A machine operates if all its three components function .The probability that the first

component

Fails during the year is 0.14, the second component fails is 0.10 and the third component

fails is 0.05.

What is the probability that the machine will fail during the year?

26. Three critics receive a book. Odds in favour of the book are5:2, 4:3 and 3:4 respectively

for three

critics. Find the probability that the majority are in favour of the book.

27. The probability of the student A passing an examination is 3/5 and of the student B

passing is 4/5. Assuming the two events: ‘A passes’, ‘B passes’ as independent find the

probability of:

(a) Both students passing the examination.

(b) Only A passing the examination.

(c) Only one the two passing the examination.

(d) Neither of the two passing the examination.

28. A and B decide to meet at Hanuman Temple between 5 to 6 p.m. with the condition that

no one .Would wait for the other for more than 15 minutes. What is the probability that

they meet?

132

29. Three persons A, B, C throw a die is successful till one gets a ‘six’ and win the

game.Find their the respective probabilities of the winning ,if A begin.

30. A and B take turn in throwing two twice dice. The first to throw the 9 being awarded.

Show that If A has the first throw, their chances of winning are in the ratio 9:8.

31. In a bulb factory, machines A,B and C manufacture 60% , 30% and 10% bulbs

Respectively 1% , 2% and 3% of the bulbs produced respectively by A,B and C are found

to be defective. A bulb is picked up at random from the total production and found to be

defective . Find the probability that the bulb was produced by the machine A

32. Coloured balls are distributed in three bags as shown in the following table

Bag Colour of the ball

Black White Red

I 1 2 3

II 2 4 1

III 4 5 3

A bag is selected at random and then two balls are

randomly drawn from the selected bag. They happen to

be black and red. What is the probability that they came

from bag I?

33. A fair die is rolled. If 1 turns up, a ball is picked up at random from bag A, if 2 or 3

turns up ,a ball is picked up at random from bag B, otherwise a ball is picked up from bag

C. Bag A contains 3 red and 2 white balls, bag B contains 3 red and 4 white balls and bag

C contains 4 red and 5 white balls. The die is rolled, a bag is picked up and a ball is drawn

from it. If the ball drawn is red, what is the probability that bag B was picked up?

Probability Distribution & Binomial Distribution

LEVEL 1

1. Two cards are drawn successively with replacement from a well shuffled pack of 52

cards . Find the (distribution) probability distribution of the number of aces.

2. Find the probability distribution mean and variance of the number of kings drawn

when two cards are drawn one by one without replacement from a pack of 52 cards.

3. Find mean 𝜇 and variance for the probability distribution.

X 0 1 2 3

P(X) 1/8 3/8 3/8 1/8

4. A pair of dice is thrown four times . getting a doublet is considered as a success.

Find the probability distribution of number of success.

5. An urn contains 4 white and 3 red balls. Let X be the number of in red balls in a

random draw of three balls . Find the mean and variance of X.

6. On a multiple choice examination with three possible answers (out of which one is

correct) for each of the five question, what is the probability of the candidate would

get four or more correct answers just by guessing.

133

7. A random variable X has the following probability distribution

8.

X 0 1 2 3 4 5 6 7

P(X) 0 k 2k 2k 3k 𝑘2 2𝑘2 7𝑘2 + 𝑘

Find i) k ii) P(X<3) iii) P(X>6) iv) P(0<X<3)

9. A die is thrown 10 times. If getting an even number is success , find the probability of

getting at least 9 success.

10. Find the probability of getting atmost two sixes in 6 throws of a single die.

11. Does the following tables represent probability distribution.

X -1 0 1

P(X) 1/3 ½ 1/6

X 0 1 2 3 4

P(X) 0.1 0.3 -0.2 0.4 0.4

LEVEL-2

1. A die is rolled once, find the probability distribution for the number obtain on it and

find its mean and variance

2. A die is thrown again and again until three sixes are obtained. Find the probability of

obtaining a third six in the sixth throw of die.

3. From a lot of 30 bulbs , which includes 6 defective bulbs, a sample of three bulbs is

drawn at random with replacement . Find the probability distribution of number of the

defective bulbs.

4. A random variable X has a probability distribution P(X) of the following form where

k is some number.

k , if X=0

P(X)= 2k, if X=1

3k, if X=2

0 , otherwise

Determine i) k ii) P(X<2) iii) P(X<= 2) iv) P(X>= 2)

5. A bag contains 10 balls each marked with one of the digits 0 to 9 . If four balls are

drawn successively with replacement from bag , find the probability that none is

marked with digit 6.

6. In a meeting 65% of the members favour and 35% oppose a certain proposal . A

member is selected at random and we take X=0 if he oppose and X=1 if he favours.

Find the mean and variance.

7. Find the mean variance and standard deviation of the number of heads in a

simultaneous loss of three coins.

134

8. Let X denote number of hours you study during a randomly selected school day. The

probability that the X can take the value X has the forms, where k is some unknown

constant.

0.1 , if X=0

P(X)= kx, if X=1 Or 2

k(5-X), if k=3 Or 4

0 , otherwise

i) Find the values of k.

ii) What is the probability that you study atleast 2 hours ?

iii) Exactly 2 hours.

iv) Less than 2 hours.

9. Let x represent the difference between the number of heads and number of tails when

a coin is tossed 6 times. What are the probable values of x ?

10. In a 20 questions true-false examination, suppose a student tosses a fair coin to

determine the answers to each questions if the coins fall heads he answers “True”, If it

falls tails he answers “False”. Find the probability that he answers at least 12

questions correctly.

LEVEL-3

1. The items produced by company contains 10% defective items. Show that the

probability of getting two defective items in a sample of 8 items is 28x96

108.

2. There is a group of 100 people who are patriotic out of which 70 believe in non

violence. Two persons are selected at random . Write the probability distribution for

selected persons who are non –violent also find the mean of the distribution.

Explain the importance of non- violent in patriotism.

3. The probability that a bulb produced by a factory will face after 160 days of use is

0.06. Find the probability that out of 5 such bulbs atmost one bulb will fuse after 160

days of use . Do you satisfy the importance of “quantity over quality” ? Comment.

4. Past experience shows that 80% of operation performed by a doctor are successful. If

he performs 4 operation in a day , what is the probability that atleast three operation

will be successful ? Which values are reflected by the doctor.

5. In a hurdle race , a player has to cross 10 hurdles . The probability that he will clear

each hurdle is 5/6 . What is the probability that he will knock down fever than two

hurdles? What is the importance of sports in ones life.

135

6. The probability that a student entering a college will graduate is 0.6. Find the

probability that out a group of 6 students i) none ii) at least one iii) at most three ,

will graduate.

7. How many times must a man toss a fair coin so that the probability of having one

head is more than 80% ?

8. Two cards are drawn at random from a pack of 52 cards and kept out . Then one card

is drawn from the remaining 50 cards . Find the probability that it is an ace.

9. In a game , a person is paid Rs. 5 if he gets all heads or all tails when three coins are

tossed and he will pay Rs. 3 it either one or two heads occour . What he expect to win

on the average per game.

10. Six dice are thrown 729 times, How many times do you expect atleast three dice to

show five or six.

Answers :

LEVEL-2

Q1. Mean=21/6,Variance=(91/6)-(21/6)2

Q2. 0.0267

Q3.

X 0 1 2 3

P(x) 64/125 48/125 12/125 1/125

Q4. K=1/6,1/2,1,1/2

Q5. 4C6(9/10)

4(1/10)

0

Q6. Mean =0.65,Variance=0.23

Q7.

X 0 1 2 3

P(x) 1/8 3/8 3/8 1/8

Mean = 3/2,Variance = ¾, Standard Deviation =0.87

Q8.

X 0 1 2 3

P(x) 0.1 2k 2k k

136

i) K=0.15

ii) 5k=0.75

iii) 0.3

iv) 0.55

Q9. -6,-4,-2, 0,2,4,6

Q10. (20

C12 + 20

C 13 + 20

C 14…....20

C20) / 220

LEVEL-3

Q2.

X 0 1 2

P(x) 29/330 140/330 161/330

Q3. The quantity should not be given more importance over the quality. By not

keeping the quality we are losing people’s trust.

Q4. (8/5)(4/3)3

The values reflected are responsibility, dedication towards work

Q5. (5/2)(5/6)9

Sports form an integral part of life, it helps to be responsible, friendly and to be

mutual.

Q6. 1424/3125

Q7. 3 times

Q8. 1/13

Q9.

X 5 -3

P(x) 2/8 6/8

Mean = -1

137

COMMON MISTAKES AND REMEDIES

1.In question , with replacement is mentioned , then it is independent event. 2. In question , without replacement is mentioned , then it is dependent event. 3. if number of trials are more than three and the trial has only two outcomes ,then is the question of binomial distribution.

S.NO. COMMON MISTAKES REMEDIES 1. Confusion in independent and

Mutually exclusive events Independent events have common outcome and probability of one does not affect probability of other events .then P(𝐴 ∩ 𝐵) = 𝑃(𝐴). 𝑃(𝐵) Mutually exclusive events have no outcomes. then 𝑃(𝐴 ∪𝐵) = 𝑃(𝐴) + 𝑃(𝐵)

2. Difference in dependent and independent events

Total outcomes for 2𝑛𝑑 , 3𝑟𝑑 ….out comes reduce by one. Ex:- two kings are drawn one by one from a deck of 52 cards.

𝑃(𝐾1) =4

52 and P(𝐾2) =3/51

Total outcomes for ,2𝑛𝑑3𝑟𝑑 ….out comes will remain same. Ex:- two kings are drawn one by one from a deck of 52 cards. P(𝐾1) =4/52 and P(𝐾2) =4/52

3 Conditional probability 𝑃 (𝐴

𝐵) =

𝑃(𝐴∩𝐵)

𝑃(𝐵)=

𝑛(𝐴∩𝐵)

𝑛(𝐵) i.e. selection of the element

of A from B 4 Students are un able to understand

the question based on Bays Theorem

Formula :-

𝑃 (𝐸1

𝐴) =

𝑃(𝐸1)𝑃(𝐴

𝐸1)

𝑃(𝐸1 )𝑃(𝐴

𝐸1)+𝑃(𝐸2)𝑃(

𝐴

𝐸2)+⋯…….+𝑃(𝐸𝑛)𝑃(

𝐴

𝐸𝑛)

=𝑃(𝐸1∩𝐴)

𝑃(𝐴) . selection of the element of 𝐸1 from A.

In question highlight on the last sentence . Three urns A ,B and C contain 6 red and 4 white ball; 2 red and 6 white balls,1 red and 5 white balls respectively .An urn is chosen at random and a ball is drawn is found to be red, find the probability that the ball was drawn from urn A .

5 Students are un able to understand the question based on Binomial distribution.

Formula (𝑃 + 𝑞)𝑛 The probability distribution is given by P(𝑋 = 𝑟) = 𝑛𝐶𝑟𝑞

𝑛−𝑟𝑝𝑟

1.trials are independent . 2. each trial has exactly two outcomes. 3.there should be finite number of trials.

138

General tips for the children

Don't study too much! Spending too much time going over and over the

same stuff won't help and will just overwhelm you.

Ask for help. There are lots of people around who can help if you just ask

or ask your teacher.

Work a variety of problems. Learn how to answer each kind of question - - multiple choice, gridded

response, shortanswer, and extended response.

Draw pictures to help solve math problems

Use the correct formulas

Read each problem carefully.

Think about what is being asked.

Choose an appropriate strategy.

Solve the problem using your strategy.

Go to class regularly

Do your homework without fail.

Ask questions in the class

Pay attention to neatness.

Do not rush! If you make a mistake, erase the problem completely.

Never give up on yourself. As long as you live always try your very best.

http://www.tc3.edu/instruct/sbrown/math/test.htm#Variety

http://www.wikihow.com/Do-Homework

http://www.wikihow.com/Write-Neatly

139

ACTIVITIES AT THE WORKSHOP

Inaugural speech by Director Madam

Session by Sh M Srinivasan, Resource Person

Session by Sh K Selvaraju, Resource Person

Session by Prof. I K Rana, IIT,Mumbai Session by Prof.S.Sivaramakrishnan, IIT, Mumbai

Session by Prof.Ananathanarayanan, IIT, Mumbai

140

ACTIVITIES AT THE WORKSHOP

141

KENDRIYA VIDYALAYA SANGATHAN, NEW DELHI


3 DAY WORKSHOP ON CONTENT ENRICHMENT IN MATHEMATICS : 22.8.16 TO 24.8.16

CONTACT DETAILS OF PARTICIPANTS

S.NO. Name K.V Region CONATACT NUMBER E MAIL ADDRESS

1 Sh M.Srinivasan ZIET MUMBAI ZIET MUM 9757440353 [email protected]

2 Sh K Selvarju Ganeshkhind MUMBAI 9423246976 [email protected]

3 Sh N M Giri NO.1 Baroda AHMEDABAD 8401221960 [email protected]

4 Sh Santosh B Silvasa AHMEDABAD 9408738976 [email protected]

5 Sh Kapil Kumar Soni Dantiwada AHMEDABAD 9812309192 [email protected]

6 Sh Prasant Tiwari NO.2 Jamnagar AHMEDABAD 9455650002 [email protected]

7 Ms Seema Rajput ONGC SURAT AHMEDABAD 7698386422 [email protected]

8 Sh.SP Mathur KV 5 (2nd Shift) Jaipur JAIPUR 9161673712 [email protected]

9 Sh.Narendra Kumar KV Jalipa Cantt

JAIPUR 9560955001 [email protected]

10 Sh.S.L.Mehta No.1 Udaipur


11 Sh.Mohd.Rafique Lalgarh Jattan


12 Sh.RPS Rathore KV Jhunjhunu


13 Sh Bhola Singh Dhanapur Cantt PATNA 9431618786 [email protected]

14 Sh Jintendra Kumar Jawaharnagar PATNA 9708726044 [email protected]

15 Sh U N Singh Kankarbagh II Shit PATNA 9162592402 [email protected]

142

S.NO. Name K.V Region CONATACT NUMBER E MAIL ADDRESS

16 Sh S P Thakur Muzaffarpur I shit PATNA 9661391896 [email protected]

17 Mrs.Suman Singh HFC Baruani PATNA 7549127464 [email protected]

18 Sh J S Pandit Mokamaghat PATNA 9905050934 [email protected]

19 Sh Awadesh Prasad Bailey Road, I Shit PATNA 9430078392 [email protected]

20 Dr.D S Rai Dhanpuri RAIPUR 9425960078 [email protected]

21 Sh M N Nandanwar Durg RAIPUR 9425211784 [email protected]

22 Sh Praveen Kumar Khairagarh RAIPUR 9968168695 [email protected]

23 Sh U N Kurrey NO.1 Raipur RAIPUR 9406251455 [email protected]

24 Sh D P Chaubey Jhagrakhand RAIPUR 9826678187 [email protected]

25 Sh E T Babu Kashmunda RAIPUR 9926643399 [email protected]

26 Sh Vivekanand Pradhan NTPC Korba RAIPUR 9424149309 [email protected]

kvbilaspurlibrary.files.wordpress.com · 2 श्री संतोष कु ा ल्ल,...

Documents