As per Revised Syllabus of

MSBTE - I Scheme

Basic Mathematics Vinayak K. Nirmale

Arjun D. Wandhekar

M.Sc. Mathematics Lecturer in Mathematics MIT Polytechnic, Pune

M.Sc. Mathematics Lecturer in Mathematics, (Sel. Grade) Govt. Polytechnic, Ahmednagar

Vitthal B. Shinde

Sourabh B. Joshi

M.Sc. (Maths), M.Phil. (Mathematics), M.Sc. (Mathematics), B.Ed., Lecturer in Mathematics, Sr. Lecturer in Mathematics, Govt. Polytechnic, Pune Gramin Polytechnic, Nanded

Sadashiv N. Nirmale M.Sc. (Applied Mathematics) LMISTE, Formerly Lecturer in Mathematics GH Raisoni College of Engg. Ahmednagar

®

TM

TECHNICAL

Price : ` 260 /ISBN 978-93-332-1475-9

PUBLICATIONS An Up-Thrust for Knowledge 9 789333 214759

(i)

Basic Mathematics (Common to All Branches) Semester - I

First Edition : July 2017 Second Revised Edition : June 2018 Third Revised Edition : July 2018 Fourth Revised Edition : June 2019

ã Copyright with Authors All publishing rights (printed and ebook version) reserved with Technical Publications. No part of this book should be reproduced in any form, Electronic, Mechanical, Photocopy or any information storage and retrieval system without prior permission in writing, from Technical Publications, Pune.

Published by : ® TECHNICAL

TM

PUBLICATIONS

Amit Residency, Office No.1, 412, Shaniwar Peth, Pune - 411030, M.S. INDIA. Ph.: +91-020-24495496/97, Telefax : +91-020-24495497 Email : [email protected] Website : www.technicalpublications.org

Price : ` 260 /ISBN 978-93-332-1475-9

MSBTE I 9 789333 214759

9789333214759 [4]

(ii)

Preface The importance of Basic Mathematics is well known in various engineering fields. Overwhelming response to our books on various subjects inspired us to write this book. The book is structured to cover the key aspects of the subject Basic Mathematics. The book uses plain, lucid language to explain fundamentals of this subject. The book provides logical method of explaining various complicated concepts and stepwise methods to explain the important topics. Each chapter is well supported with necessary illustrations, practical examples and solved problems. All chapters in this book are arranged in a proper sequence that permits each topic to build upon earlier studies. All care has been taken to make students comfortable in understanding the basic concepts of this subject. The book not only covers the entire scope of the subject but explains the philosophy of the subject. This makes the understanding of this subject more clear and makes it more interesting. The book will be very useful not only to the students but also to the subject teachers. The students have to omit nothing and possibly have to cover nothing more. We wish to express our profound thanks to all those who helped in making this book a reality. Much needed moral support and encouragement is provided on numerous occasions by our whole family. We wish to thank the Publisher and the entire team of Technical Publications who have taken immense pain to get this book in time with quality printing. Any suggestion for the improvement of the book will be acknowledged and well appreciated.

Authors V. K. Nirmale A.D. Wandhekar V.B. Shinde S.B. Joshi S. N. Nirmale

(iii)

Teaching

Total Credits

Scheme (In Hours)

(L+T+P)

Examination Scheme Theory Marks

Practical Marks

L

T

P

C

ESE

PA

ESE

PA

4

2

-

6

70

30*

-

-

Total Marks 100

Unit

Major learning outcomes (in cognitive domain)

Topics and Sub-topics

Unit - I

1a. Solve the given simple problem based on laws of logarithm. 1b. Calculate the area of the given triangle by determinant method. 1c. Solve given system of linear equations using matrix inversion method and by Cramer’s rule. 1d. Obtain the proper and improper partial fraction for the given simple rational function.

1.1 Logarithm: Concept and laws of logarithm 1.2 Determinant and matrices a. Value of determinant of order 3x3 b. Solutions of simultaneous equations in three unknowns by Cramer’s rule. c. Matrices, algebra of matrices, transpose adjoint and inverse of matrices. Solution of simultaneous equations by matrix inversion method. d. Types of partial fraction based on nature of factors and related problems.

Algebra

(iv)

Unit - II Trigonometry

Unit - III Coordinate Geometry

2a. Apply the concept of Compound angle, allied angle, and multiple angles to solve the given simple engineering problem(s). 2b. Apply the concept of Submultiple angle to solve the given simple engineering related problem(s). 2c. Employ concept of factorization and defactorization formulae to solve the given simple engineering problem(s). 2d. Investigate given simple problems utilizing inverse trigonometric ratios. 3a. Calculate angle between given two straight lines. 3b. Formulate equation of straight lines related to given engineering problems. 3c. Identify perpendicular distance from the given point to the line. 3d. Calculate perpendicular distance between the given two parallel lines. .

(v)

2.1 Trigonometric ratios of Compound, allied, multiple and sub-multiple angles (without proofs) 2.2 Factorization and defactorization formulae(without proofs) 2.3 Inverse trigonometric ratios and related problem. 2.4 Principle values and relation between trigonometric and inverse trigonometric ratio.

3.1 Straight line and slope of straight line a. Angle between two lines. b. Condition of parallel and perpendicular lines. 3.2 Various forms of straight lines. a. Slope point form, two point form. b. Two points intercept form. c. General form. d. Perpendicular distance from a point on the line. e. Perpendicular distance between two parallel lines.

Unit - IV Mensuration

Unit - V Statistics

4a. Calculate the area of given triangle and circle. 4b. Determine the area of the given square, parallelogram, rhombus and trapezium. 4c. Compute surface area of given cuboids, sphere, cone and cylinder. 4d. Determine volume of given cuboids, sphere, cone and cylinder.

4.1 Area of regular closed figures, Area of triangle, square, parallelogram, rhombus, trapezium and circle. 4.2 Volume of cuboids, cone, cylinders and sphere.

5a. Obtain the range and coefficient of range of the given grouped and ungrouped data. 5b. Calculate mean and standard deviation of discrete and grouped data related to the given simple engineering problem. 5c. Determine the variance and coefficient of variance of given grouped and ungrouped data. 5d. Justify the consistency of given simple sets of data.

5.1 Range, coefficient of range of discrete and grouped data. 5.2 Mean deviation and standard deviation from mean of grouped and ungrouped data, weighted means 5.3 Variance and coefficient of variance. 5.4 Comparison of two sets of observation.

(vi)

Table of Contents Chapter 1

Logarithms

(1 - 1) to (1 - 26)

1.1 Types of Logarithm .................................................................. 1 - 2 1.2 Laws of Logarithm (Without Proof) .......................................... 1 - 3

Chapter 2

Determinant

(2 - 1) to (2 - 46)

2.1 Introduction ............................................................................... 2 - 1 2.2 Definition of 2  2 Determinant ................................................ 2 - 1 2.2.1

Value of 2  2 Order Determinants .......................................... 2 - 2

2.2.2

2  2 Determinant Equation ..................................................... 2 - 3

2.2.3

Solution of Simultaneous Equations using Determinants (Cramer’s Rule) ....................................................................... 2 - 3

2.3 Definition of 3  3 Determinant ................................................ 2 - 5 2.3.1

3  3 Order Determinant Equation ........................................... 2 - 9

2.3.2

Solution of Simultaneous Equations using Determinants (Cramer’s Rule) ..................................................................... 2 - 10

Chapter 3

Matrices

(3 - 1) to (3 - 82)

3.1 Introduction ............................................................................... 3 - 1 3.2 Types of Matrices ...................................................................... 3 - 2 3.3 Algebra of Matrices ................................................................... 3 - 4 3.4 Transpose of Matrix................................................................. 3 - 30 3.5 Minor, Co-factor of an Element of a Matrix ............................ 3 - 44 (vii)

3.6 Adjoint of a Square Matrix ...................................................... 3 - 45 3.7 Inverse of Matrix ..................................................................... 3 - 56 3.8 Solution of Simultaneous Equation by Matrix Inversion Method .................................................... 3 - 66

Chapter 4

Partial Fraction

(4 - 1) to (4 - 74)

4.1 Introduction ............................................................................... 4 - 1 4.2 Case 1 : Non-Repeated Linear Factor in the Denominator ........ 4 - 2 4.2.1

Problem Reducible to Case I after Suitable Substitution........ 4 - 19

4.3 Case 2 : Repeated Line as Factors in the Denominator ........... 4 - 27 4.4 Case 3 : Non Repeated Irreducible Quadratic Factor in Denominator ............................................................................ 4 - 45 4.5 Convert Improper Fraction to Proper Fraction ........................ 4 - 63

Chapter 5

Trigonometric Ratio of Allied Compound Multiple & Submultiple Angles (5 - 1) to (5 - 72)

5.1 Compound Angles ..................................................................... 5 - 1 5.2 Allied Angles ........................................................................... 5 - 27 5.3 Multiple Angles and Submultiple Angles ............................... 5 - 41

Chapter 6

Factorization & Defactorization (6 - 1) to (6 - 60)

6.1 Defactorization .......................................................................... 6 - 1 6.2 Factorization ............................................................................ 6 - 22 (viii)

Chapter 7

Inverse Trigonometric Ratios (7 - 1) to (7 - 54)

7.1 Introduction ............................................................................... 7 - 1 7.2 Definition................................................................................... 7 - 2 7.3 Principle Values of Inverse Functions ....................................... 7 - 4 7.4 Properties of Inverse Function OR Relation between Inverse Trigonometric Functions ................................ 7 - 4

Chapter 8

Straight Line

(8 - 1) to (8 - 86)

8.1 Introduction to Straight Line ..................................................... 8 - 1 8.2 Inclination of a Line .................................................................. 8 - 1 8.3 Slope of a Line........................................................................... 8 - 2 8.3.1

Slope of a Line Passing through Two Point ............................. 8 - 2

8.3.2

Slope of General Form of a Line Ax + By + C = 0 .................. 8 - 3

8.4 Condition for Parallel and Perpendicular Lines ......................... 8 - 4 8.5 Intercept of a Line ..................................................................... 8 - 4 8.6 Equations of a Lines .................................................................. 8 - 5 8.6.1

Standard Forms of Equations of Lines ..................................... 8 - 5

8.6.2

Equations of Co-ordinates Axes ............................................. 8 - 11

8.6.3

Equations of Lines Parallel to Co-ordinate Axes ................... 8 - 11

8.7 Point of Intersection of Two Lines .......................................... 8 - 44 8.8 Angle between Two Straight Lines ......................................... 8 - 60 8.9 Perpendicular Distance between Point and Line ..................... 8 - 71 8.10 Perpendicular Distance between Two Parallel Lines............... 8 - 80 (ix)

Chapter 9

Mensuration

(9 - 1) to (9 - 72)

9.1 Introduction ............................................................................... 9 - 1 9.2 Volume and Surface Area........................................................ 9 - 24 9.2.1

Cuboid.................................................................................... 9 - 24

9.2.2

Cube ....................................................................................... 9 - 31

9.2.3

Cylinder ................................................................................. 9 - 35

9.2.4

Cone ....................................................................................... 9 - 45

9.2.5

Sphere .................................................................................... 9 - 57

Summary ............................................................................... 9 - 69

Chapter 10 Measure of Dispersion

(10 - 1) to (10 - 60)

10.1 Introduction ............................................................................ 10 - 1 10.2 Range ....................................................................................... 10 - 2 10.3 Mean Deviation ..................................................................... 10 - 10 10.4 Standard Deviation (S.D.) ..................................................... 10 - 21 10.5 Variance................................................................................. 10 - 39 10.6 Comparsion of Two Sets of Observations ............................. 10 - 52

Solved Sample Question Paper

Solved MSBTE Question Papers

(x)

(S - 1) to (S - 4)

(S - 5) to (S - 12)

Basic Mathematics

1-1

Logarithms

Introduction :

While solving crucial engineering problems logarithm is one of the best tool to simplify the given engineering problems. Definition :

Broadly logarithm is nothing but a equivalent from of an exponential expressions. x

i.e. if y = a , a > 0, a  1, a  R, then same expression can be expressed in equivalent logarithmic form as x = loga y, and read as x is called logarithm of y to the base a. For example, 5

then

1) If 32 = 2

1/2

2) If 10 = 100

5 = log2 32 1 2 = log100 10

then

x

Note : In the definition of logarithm a = y is called exponential form or Index form and x = loga y is called Logarithmic form.

If one is given we can write the other.

(1 - 1)

TM

Technical Publications

- An up thrust for knowledge

Basic Mathematics

1-2

Logarithmic form

Exponential form OR Index form 2

 9 = 3

2 = log3 9 1/2 = log16 4

4 = (16)

1/2

1 –2 25 = (5)

1  2 = log5   25  

0

0 = loga1

1 = a

2

 2 = log10 0.01

1.1

Logarithms

0.01 = 10

Types of Logarithm

i) N atural logarithm : If base of the logarithm is a nepier’s number e then it is called a natural logarithm where e = 2.718281 For example loge 7, loge 2 etc. Note : For natural logarithm it is not necessary to write base i.e. loge 7 can be written as log 7.

ii) Common logarithm : If base of the logarithm is 10 number then it is called common logarithm. For example log10 3, log10 7 etc. Example : Which of the following are common logarithm and natural logarithm 1) log10x 2) log25 3) loge5 4) log103 Ans. : 1)

2) 3)

4)

log10x is common logarithm. log5x is natural logarithm. loge5 is natural logarithm. log103 is common logarithm. TM

Technical Publications

- An up thrust for knowledge

Basic Mathematics

1-3

Logarithms

Deduction from the definition

1) We know that 0

a = 1



then

loga 1 = 0

Logarithm of number 1 to any base is zero.

e.g.

log3 1 = 0,

log1/7 1 = 0,

log1000 1 = 0

etc.

2) We know that 1

a = a

then

loga a = 1

 Logarithm of a number to the same base is equal to one.

e.g. log4 4 = 1, log29 29 = 1,

log73 73 = 1 etc.

3) We know that x

a = y

then

x = loga y

x



a = y loga y



a

log3 7

e.g. 3

= y = 7,

log5 3

5

= 3, 2 5

log 49 2 5 = 49 etc.

4) We know that

If

loga y = x,



loga y = x

x

then

y =a

x

 loga a = x 2

From (i)

e.g. log3 3 = 2,

1.2

…(i)

3

4

log4 4 = 3,

log29 29 = 4 etc.

Laws of Logarithm (Without Proof)

1) Logarithm of product If m, n, a are positive real number and a  1 then loga (m n) = loga m + loga n TM

Technical Publications

- An up thrust for knowledge

Basic Mathematics

1-4

Logarithms

“The logarithm of a product is equal to the sum of their logarithms”. e.g. log5 (2  3) = log5 2 + log5 3 log (mnp) = log m + log n + log p

Corollary :

2) Logarithm of quotient If m, n, a are positive real number and a  1 m loga   = loga m  loga n n

then

“The logarithm of the quotient of two numbers is equal to the difference between their logarithms”. 8 log2   = log2 8  log2 7 7 1 loga   = loga x x

e.g. Corollary : e.g.

1)

 loga 0.1 = loga 10

2)

loga sec  = loga cos 

3) Logarithm of power

If ‘m’ and ‘a’ are positive real numbers. a  1 and n  R then n

loga (m) = n loga m “The logarithm of a power of a number is equal to the product of the index by the logarithm of the number”. e.g.

1)

4

loga (17) = 4 loga 17 3

log2 8 = log2 (2) = 3 log2 2 = 3

3) Corollary 1 :

loga

e.g

log4

.

Corollary 2 :

–3

2) log x = – 3 log x

n

1 m = n loga m

5

1 71 = 5 log4 71

p

q

loga (m  n ) = p loga m + q loga n TM

Technical Publications

- An up thrust for knowledge

Basic Mathematics

1-5

Logarithms 4

2

log5 (144) = log5 (2  3 ) = 4 log5 2 + 2 log5 3

e.g.

4) Change of base rule

If m, n, a are positive real numbers. n  1, a  1

Deduction :

logn m =

loga m loga n

logn m =

loga m loga n

changing the base to m

logm m logn m = log n m

 logn m =

1 logm n

logn m  logm n = 1 e.g.

log2 5 where new base is 2 log2 7

1)

log7 5 =

2)

log5 3 log8 3 = log 8 5

Example 1.1 : Write the following term in Logarithmic form. 3

i) 5 = 125 1 2 iv) 5 = 25

4

0

ii) 3 = 81

iii) 7 = 1

v) 0.001 = 10

3

y

vi) x = z

Solution : x

i) We know if y = a

then

x = loga y

3

then

3 = log5 125

4

then

4 = log3 81

0

then

0 = log7 1

then

2 = log5 



5 = 125

ii)

3 = 81

iii) iv)

7 =1 1 2 5 = 25

TM

Technical Publications

1  25

- An up thrust for knowledge