Description
McGraw Hill Integral Calculus 3D Geometry & Vector Booster With Problems & Solutions For Jee Main & Advanced 2Nd Edition by Makshud, Rejaul
Integral Calculus, 3D Geometry & Vector Booster for JEE Main and Advanced has been conceptualized and produced for aspirants of various engineering entrance examinations. It renders complex concepts and problems of Integral Calculus, 3D Geometry & Vector easy for the students and provides them as many opportunities for guided practice as needed. It covers all topics, and is imbued with a strong conceptual approach, with relevant theory being followed by plenty of solved exercise questions, graded LODs and past years’ papers. The various levels of difficulties have numerous MCQs, integer type questions, comprehensive link passage questions and matrix match questions for ample practice and thorough understanding. Step-by-step guidance in problem solving and alternative methods of approaching a problem are included to boost aspirants’ analytical and mathematical skills and are intended to teach them independent problem-solving techniques based upon the solutions.
SALIENT FEATURES
Covers the latest JEE (Main & Advanced) syllabus as per the JEE Pattern
o Exercise questions graded into Level I, Level II, Level III and Level IV
o Level I questions based on Fundamentals
o Level II based on Mixed Problems
o Level III consist of Problems for JEE Advanced
o Level IV comprise of Tougher Problem
o Chapter-wise previous years’ questions with complete solution
o Solutions to questions have been provided in a step-by-step manner and alternate methods of solution have also been used wherever applicable
TABLE OF CONTENTS
Preface vii
1. Indefinite Integrals 1.1.1.161
1. Definition 1.1
2. Geometrical interpretation of Integration 1.1
3. Basic formulae on integration 1.1
4. Standard Methods of Integration 1.2
5. Integration by Parts 1.5
6. Choice of the first function and the second function 1.5
7. Partial Fractions 1.6
8. Integration of Irrational Functions 1.8
9. Eulerfs Substitution 1.9
10. Integration by Reduction Formula 1.9
11. Inexpressible integrals 1.12
Exercises 1.12
Answers 1.39
Hints and Solutions 1.41
2. Definite Integrals 2.1.2.131
1. What is definite Integrals? 2.1
2. Evaluation of Definite Integrals 2.1
3. Evaluation of definite integrals by substitution 2.1
4. Geometrical interpretation of the definite integral 2.1
5. Definite integral as the limit of sum 2.2
6. Evaluation of the limit of the sum using Newton-Leibnitz Formula 2.2
7. Properties of Definite Integrals 2.2
8. Mean Value of a Function over an Interval 2.7
9. Improper Integrals 2.7
10. Gamma Function 2.7
11. Beta Function 2.7
12. Wallifs Formula 2.7
Exercises 2.8
Answers 2.37
Hints and Solutions 2.38
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3. Area Bounded by the Curves 3.1.3.62
3.1 ** 3.1
3.2 Area of the Cartesian Curve 3.2
3.3 area of the region bounded by a single curve and the co-ordinate axes 3.2
3.4 Area between two curves 3.3
3.5 Area of the region bounded by the several curves 3.4
3.6 Least value of a variable area 3.4
Exercises 3.4
Answers 3.17
Hints and Solutions 3.17
4. Differential Equation 4.1.4.103
1. Introduction 4.1
2. Definition 4.1
3. Ordinary Differential Equation 4.1
4. Partial differential equation 4.2
5. Order of a differential equation 4.2
6. Degree of a differential equation 4.2
7. Linear and non-linear differential equation 4.3
8. Formation of a differential equation 4.3
9. Differential equation of first order and first degree 4.3
10. Orthogonal Trajectories 4.6
11. First Order Higher Degree Differential Equation 4.6
12. Higher Order Differential Equation 4.7
13. Applications of Differential Equation 4.8
Exercises 4.9
Answers 4.26
Hints and Solutions 4.28
5. The Vectors 5.1.5.80
1.1 Introduction 5.1
1.2 Physical quantities 5.1
1.3 Representation of vectors 5.1
1.4 types of vectors 5.2
1.5 Algebra of vectors 5.3
1.6 Left and right handed orientation 5.4
1.7 Position vector of a point in a space 5.4
1.8 Linear Combination 5.4
1.9 Linearly dependent vectors 5.4
1.10 Linearly independent vectors 5.4
1.11 Section formulae 5.4
1.12 Bisector of angle between vectors A and b 5.6
1.13 straight line 5.6
1.14 Plane 5.6
2.1 Definition 5.7
2.2 Geometrical interpretation of a . b 5.7
2.3 Properties of dot product of two vectors 5.7
2.4 Component of a vector __ . B along and perpendicular to vector __ . A 5.8
Contents xi
2.5 Physical significance of the dot product of two vectors 5.8
3.1 Introduction 5.9
3.2 Geometrical Interpretation of _ . a ~ _ . b 5.9
3.3 Properties of vector product of two vectors 5.9
4.1 Introduction 5.10
4.2 Properties of scalar triple product of vectors 5.10
5.1 Introduction 5.11
5.2 Geometrical significance of a ~ (b ~ c) 5.11
5.3 ** 5.11
5.4 Properties of vector triple product 5.11
5.5 Scalar product of four vectors 5.12
5.6 Vector product of four vectors 5.13
5.7 Geometrical interpretation of (a ~ b) ~ (c ~ d) 5.13
5.8 Reciprocal system of vectors 5.13
Exercises 5.15
Answers 5.35
Hints and Solutions 5.36
6. 3-D Co-ordinate Geometry 6.1.6.69
1.1 Introduction 6.1
1.2 Rectangular co-ordinate system 6.1
1.3 Position vector of a point in a space 6.1
1.4 Distance between two points in space 6.2
1.5 Section formulae 6.2
1.6 Direction cosines and direction ratios of a vector or a line 6.3
2.1 Definition 6.4
2.2 Equation of a line passing through a point and parallel to a vector 6.4
2.3 Equation of a Line Passing Through Two Points A (r1) and B(r2) is r = r1 + l(r2 . r1) 6.5
2.4 Angle between two straight lines 6.5
2.5 Skew lines 6.5
3.1 Definition 6.5
3.2 General Form 6.5
3.3 Equation of a plane possing through a point (x, y, z,) 6.6
3.4 Equation of a plane passing through three non-collinear points 6.6
3.5 Co-planarity of four points 6.6
3.6 Intercept form of a plane 6.6
3.7 Normal to a plane 6.6
3.8 Vector form 6.6
3.9 Equation of a plane in normal form 6.6
3.10 Normal form of a Plane 6.7
3.11 Theorem 6.7
3.12 Angle between two planes 6.7
3.13 Angle between a line and a plane 6.7
3.14 Equation of a plane parallel to a given plane 6.7
3.15 Equation of a plane parallel to the axes 6.7
3.16 Equation of a plane passing through (x1, y1, z1), (x2, y2, z2) and parallel to the line having
direction ratios a, b, c 6.8
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3.17 Equation of the plane passing through a point (x1, y1, z1) and parallel to two lines
having direction ratios (a1, b1, c1) and (a2, b2, c2) is 6.8
3.18 Equation of a plane passing through the line of intersection of planes 6.8
3.19 Distance of a point from a plane 6.9
3.20 Distance between two parallel planes 6.9
3.21 Sides of a plane 6.9
3.22 Intersection of a line and a plane 6.9
3.23 Condition for a line to lie in a plane 6.9
3.24 Condition of co-planarity of two lines 6.9
3.25 Equation of the planes bisecting the angle between the planes 6.9
3.26 Bisector of the acute and obtuse angles between two planes 6.9
3.27 Foot of perpendicular of a point w.r.t. a plane 6.10
3.28 Image of a point w.r.t. a plane 6.10
3.29 Equation of the plane containing a given line and parallel to a given line 6.10
3.30 Equation of the plane containing two given lines 6.10
Exercises 6.10
Answers 6.24
Hints and Solutions 6.25