MATHEMATICS. for JEE MAIN. Objective. with BOARDS SCORE BOOSTER. Fully Solved

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2 Objective MATHEMATICS for JEE MAIN with BOARDS SCORE BOOSTER Fully Solved F JEE Main Solved Papers F Key Concepts with Illustrations F 3 levels of Exercises F 100+Questions based on NCERT F 1500+Past Competitive Exam MCQ's F Practice MCQ's for JEE Main F Aligned as per Class 11 & 1 NCERT

3 Head Office : B-3, Shivalik Main Road, Malviya Nagar, New Delhi Sales Office : B-48, Shivalik Main Road, Malviya Nagar, New Delhi Tel. : / Typeset by Disha DTP Team DISHA PUBLICATION ALL RIGHTS RESERVED Copyright Author No part of this publication may be reproduced in any form without prior permission of the author and the publisher. The author and the publisher do not take any legal responsibility for any errors or misrepresentations that might have crept in. We have tried and made our best efforts to provide accurate up-to-date information in this book. For further information about the books from DISHA Log on to or to info@dishapublication.com

4 PREFACE With the Boards having a weightage (40%) for admission through JEE Main the need of a book that can help students both in JEE Main and Boards becomes more prominent. The All New Objective Mathematics for JEE Main with Boards Score Booster encapsulates the formula for cracking Boards & JEE Main simultaneously. Salient features of Disha's Objective Mathematics for JEE Main with Boards Score Booster books are: Fully Solved 015 JEE Main Question paper has been added. Exhaustive theory, with solved examples, explaining all fundamentals/ concepts to build a strong base. Illustrations to master applications of concepts & sharpen problem-solving skills. 3 levels of graded exercises to ensure sufficient practice NCERT based Questions (Boards Score Booster) for Board exams covered in a separate exercise past Competitive Exam MCQ s of JEE Main/ AIEEE and other entrance exams to provide a better exposure covered in the exercise Window to Competitive Exams Practice MCQ s for the JEE Main exam. The book contains In-chapter MCQ s after every significant topic covered in the theory portion. Finally a Practice exercise at the end of each chapter containing both Conceptual and Applied questions. The book covers all variety of questions as per the format of the previous year AIEEE/ JEE Main Papers. Covers entire syllabus as per the latest NCERT books and JEE Main syllabus. The complete book has been aligned as per the chapter flow of NCERT class 11 & 1 books. At places certain chapters have been divided into sub-chapters. For example - Binomial Theorem is divided into parts Part A Binomial Theorem; Part B Infinite Series. More than 7 out of the 30 JEE Main questions were directly or indirectly covered in our book and the readers must have really benefitted from it. In the end, Chapter-wise Exemplar Problems have been provided. These problems have been designed in such a manner that they cover all the important concepts of each and every chapter. So these problems will help the students in revising the complete Mathematics. The students are adviced to approach these problems only once he has completed his preparation. This uphill task would not have been accomplished without the valuable support, encouragement and suggestions from my students, my family, my friends and the publishers. So, I take this opportunity to express my sincerest gratitude to all of them. I must mention my wife REENA and my sons OM and DEV for their continuous assistance day and night. All efforts have been made to keep the book free from errors. However, the revised edition has been enlarged, incorporated with a lot of changes and included many new topics therefore some errors might have crept inadvertently. I shall be thankful to all those readers who point-out these errors. Any suggestions for improvement of book will always be entertained keenly. Er. Anoop K Srivastava NEW DELHI

5 INDEX JEE MAIN 015 Solved Paper (with solutions) JEE MAIN 014 Solved Paper (with solutions) Sets, Relations and Functions 1-3 Description of a set Different kinds of sets, subsets, power set and universal set Venn Diagram and operations on set : union, intersection, difference, symmetric difference, complement Infinite and Finite sets and cardinal number Ordered pairs and Cartesian product of sets, and operations Relations, domain and range of a relation & Inverse relation Type of relations : void relation, universal relation, reflexive, symmetric, transitive and antisymmetric relations Equivalence relations and Partial order relations Relation of congruence modulo 'm' Composition of relations Partition of a set One-one correspondence, function (mapping) Kinds of function : one-one, many-one, onto and into functions & Bijective functions A. Trigonometric Ratios and Identities Angles and measurement of angles, Sexagesimal system, Centesimal system and Circular system Trigonometric functions, their domain and range Basic formulae and sign of trigonometric functions and periods of functions Trigonometric ratios of compound angles Trigonometric ratios of sum and difference of angles Trigonometric ratios of multiple and submultiple angles Conditional trigonometric identities Graphs of Trigonometric functions Table for the values of Trigonometric ratios of some important angles Expression of sin A/ and cos A/ in terms of sin A Periodic form of a sinq + b cosq and range of its values. B. Trigonometric Equations & Inequations Trigonometric Equations, general solutions of elementary equations Different methods of solving trigonometric equations of various kinds Solving Trigonometric Inequations. C. Properties of Triangles and Height & Distances Representations of angles and sides of a triangle Sine rule, Cosine rule, Projection formula Trigonometrical ratios of half of the angles of a triangle Napier's Analogy and formulae for area of triangle Circles connected to a triangle Formulae for circum-radius, in-radius and ex-radii Solution of a right angled triangle Solution of triangles in general Heights and distances, problems of plane figures Bearing of a line Problems of heights and distances of figures in three dimension. 3. Principle of Mathematical Induction, Permutation and Combination Peano's Axioms and Principles of Mathematical Induction Fundamental Principles of Counting, principle of addition and principle of multiplication Notations, factorials, combination and permutation notations Permutations and formulae for permutations in different situations Combinations and formulae for combinations in different situations Some additional formulae involving conditional permutations and combinations Use of multinomial theorem for counting. 4. Complex Numbers Definition of complex numbers and Fundamental Operations of addition, multiplication, subtraction, division and equality Algebra of complex number. Conjugate of complex numbers and properties Graphical representation and properties of modulus Polar form of complex numbers, Argument, Evaluation of principal argument and its properties Euler's Notation, logarithm of a complex number DeMoivre theorem, nth roots of a complex number and properties, cube roots of unity, nth roots of unity Geometry of complex numbers, distance formula, section formula, angle between two lines and propositions about triangle Equation of straight lines in argand plane, complex slope and condition of parallel & perpendicular lines Equation of circle and some loci in complex plane Some imporant inequalities. 5. Quadratic Equations 1-60 Quadratic equation and its roots Quadratic equation with real coefficients Nature of roots, real roots, non-real roots, equal roots, rational and irrational roots Symmetric function of roots Formation of equation with given roots Sign of roots, positive roots, negative roots, zero roots, infinite roots, reciprocal roots Common roots Equations reducible to quadratic forms Sign of quadratic expressions and application in solving inequations and locating the roots Finding extreme values of quadratic expressions and rational expressions.

6 6. Linear Inequalities Interval notations in real numbers Modulus of a real number and properties General theory of binomial equations Wavy Curve Method of solving inequalities Ration inequalities and their solutions Logarithmic Equations and Inequations Equations and Inequations containing greatest integer function[x]. 7. Sequences and Series Definition of sequences and representations Series and Progression Arithmetic Progression, formula for general term and sum of n terms and characteristics Arithmetic Mean of two terms and inserting n arithmetic means between two terms Geometric Progression, formula for general term, sum of n terms and sum of infinite terms and characteristics Geometric Mean of two terms and inserting n geometric means between two terms Harmonic Progression, formula for general term Harmonic Mean of two terms and inserting n harmonic means between two terms Relation between A.M., G.M. and H.M. Arithmetic-Geometric Series Sequences of natural numbers, sum of n consecutive natural numbers, sum of their squares and cubes Sum of sequences using sigma notation nd method of difference. 8A. Binomial Theorem Introduction Binomial theorem for positive integral index and characteristics of expansion Middle term, greatest coefficient and numerically greatest term in the expansion Binomial coefficients and properties. 8B. Infinite Series Binomial theorem for any index and characteristics of expansion Exponential Series and properties of the series Logarithms and Laws Logarithmic Series and its properties Calculation of Naperian log and common log. 9A. Points & Straight Lines Introduction to Cartesian Coordinate system Formula for distance between points, section formula Results of a triangle, formula for area, coordinates of centroid, incentre, excentres & circumcentre Conditions of collinearity of three points Conditions for a given quadrilateral to be parallelogram, rectangle, rhombus and square Locus, definition and equation of a locus Slope of a straight line, slope of a line joining two points Intercepts of a line on coordinate axes Equations of straight line in various forms Normal form and parametric form of a straight line General form of equation of a straight line Relative positions of two points with respect to a straight line Point of intersection of two straight lines, angle between two straight lines Angular bisectors of two lines Condition of concurrency of three straight lines Family of straight lines. 9B. Pair of Straight Lines Pair of straight lines passing through origin, homogeneous equation of second degree Angle between the pair of lines through origin, perpendicular lines and coincident lines, pair of bisectors of angle between the lines General form of pair of straight lines, general equation of second degree, point of intersection of lines Separation of equation of lines, parallel and perpendicular lines Pair of straight lines joining the origin to the points of intersection of a curve and a straight line Translation and rotation of axes. 10A. Circles Equations of circles in various forms, general equation of a circle, real circle, imaginary circle and point circle, parametric equation of circle Intercepts of circle on the coordinate axes, position of a point with respect to circle Intersection of a line and a circle, tangent and normal to circle at a point, tangents from an external point, length of tangent Equations of pair of tangents and chord of contact System of two circles, intersecting circles, angle of intersection of two circles, case of orthogonal circles Radical axis and system of coaxal circles Family of circles

7 10B. Parabola Sections of a cone, degenerate and non-degenerate conics, pair of straight lines, circle, ellipse, parabola and hyperbola Analytical definition of conic section, focus, directrix and eccentricity, general equation of second degree Conditions that general equation of second degree represents pair of striaght lines, circle, parabola, ellipse and hyerbola, method of finding centre of conic section Parabola, standard equation and properties, Parabolas with different directions of concavity and parametric equations Parabola with vertex not at origin Intersection of a straight line and parabola, tangent at a point, chord joining two points, focal chord and other results Tangents to a parabola from an external point, equation of pair of tangents and chord of contact Equation of normal in different forms Three concurrent normals to parabola and conormal points. 10C. Ellipse and Hyperbola Ellipse, standard equation and properties, parametric equation, equation of chord joining two points Equation of ellipse in other standard forms and position of a point with respect to an ellipse Intersection of a line and ellipse, tangent at a point and conditions of tangency Pair of tangents from a point and chord of contact Equation of normals in different forms Hyperbola, standard equation and properties Parametric equation of hyperbola, equation of chord joining two points Proposition of tangents and normals to hyperbola Pair of tangents from a point and chord of contact Asymptotes Rectangular hyperbola xy = c and its propositions. 11. Mathematical Reasoning Introduction Sentence, Assertive Sentence, Imperative Sentence, Exclamatory Sentence, Interrogative Sentence, Optative Sentence Statement, Open Statement, Truth Value of a Statement Logical Variables Simple Statement, Compound Statement, Sub Statements Basic Logical Connectives or Logical Operators, Connectives, Conjunction, Disjunction or Alternation Negation Negation of Compound Statements, Negation of conjunction, Negation of Disjunction, Negation of a Negation Implication or conditional Statements Converse, Inverse and Contrapositive of an implication Biconditional or Equivalence Statement Negation of Conditional Statement, Negation of Implication, Negation of Biconditional Statement Joint Denial Proposition Tautologies Contradictions (or Fallacy) Logical Equivalence Duality Algebra of statements. 1. Statistics Measures of central tendencies, Mathematical averages and averages of position Arithmatic mean for unclassified data, frequency distribution and classified data Deviation, short-cut method in computing arithmetic mean, step deviation method in a classified frequency Algebraic properties of arithmetic mean, combined mean, weighted arithmetic mean, geometric mean and harmonic mean Median, determination of median for simple distribution, unclassified frequency distribution and classified data Quartiles, determination for raw data and continuous distribution Mode, mode for a raw data, unclassified frequency distribution and classified data Symmetric and skew distribution, relation between mean, median and mode Dispersion, measure of dispersion, mean deviation and standard deviation Variance, root mean square deviation, short cut method and step deviation for standard deviation. 13. Probability Basic Trial and events (cases) exhaustive number of cases, mutually exclusive events, equally likely events, favourable events and classical definition of probability Axiomatic approach, random experiment, sample space, events, mutually exclusive events, definition of probability Algebra of events, addition theorem of probability Finite probability spaces, infinite sample space and uncountable probability spaces.

8 14. Functions Functions, introduction, domain, codomain, range, single valued functions, dependent and independent variables, real functions Way of defining real functions, uniform and piecewise definitions, explicit and implicit functions Classification of real functions, algebraic and transcendental functions Graph of some important functions, constant function, identity function, linear function, quadratic function, square root function, modulus function, signum function, greatest integer function, exponential and logarithmic functions, power function, their properties Algebraic operations on functions, composition of functions, one-one, many-one, into and onto functions, inverse of function Even and odd functions Periodic functions and their properties. 15. Inverse Trigonometric Functions Inverse Trigonometric Functions, their domain and range Properties of Inverse trigonometric functions Solving equations containing inverse Trigonometric function. 16. Matrices Matrices and types of matrices Equality of two matrices Addition of matrices and properties of addition Multiplication of a matrix by a scalar Multiplication of two matrices and properties of multiplication Positive integral powers of matrix Transpose of a matrix, symmetric and skew-symmetric matrices Determinant of a matrix, singular and non-singular matrices, minors and cofactors Adjoint of a matrix, inverse, trace and rank of a matrix Solution of simultaneous linear equations using matrix method. 17. Determinants Determinants and their expansion Properties of Determinants Multiplication of two determinants Differentiation of a determinant Solution of system of linear equations using determinants : Cramer's Rule Consistency and Inconsistency of simultaneous linear equations System of homogeneous linear equations, trivial and non-trivial solutions. 18. Limits, Continuity and Differentiability Limits of a function, fundamental theorems on limits, methods of evaluating limits Existence of limit, left hand and right hand limits and their evaluation Continuity, Cauchy's definition, continuity at a point and continuity in an interval Fundamental theorems on continuity Differentiability of a function at a point and in an interval Geometrical Interpretation. 19. Differentiation & Application of Derivatives Differentiation as operator, some standard differentiations, fundamental rules of differentiation Differentiation of composite functions (chain rule), differentiation of inverse trigonometric functions Differentiation of implicit functions, differentiation of parametric functions Logarithmic differentiation, differentiation of a function with respect to another function Successive differentiation Application of derivatives, tangents and normals, angle of intersection of two curves, orthogonal curves Monotonicity of functions, Maxima and minima Necessary condition for existence of local extremum, first derivative test and second derivative tests for local minima and maxima Absolute maximum and minimum values in an interval and application Mean value Theorems : Rolle's theorem and Lagrange's theorem Application of dy/dx as rate measurer, differentials, errors and approximations. 0. Indefinite Integration Primitive or antiderivative, indefinite integral, elementary results Different methods of integration, integration by substitution, integration of rational algebraic functions using partial fractions Integration by parts and application Integration using trigonometrical identities Integration of Irrational Algebraic functions Reduction Formula

9 1. Definite Integration & Applications Definite Integration, second fundamental theorem of calculus Evaluation of definite integrals by substitution Fundamental properties of definite integrals Geometrical interpretation of definite integrals, area function, and first fundamental theorem of calculus Definite integral as limit of sum Areas of region bounded by curves Some Important definite integrals.. Differential Equations Introduction of Differential equations, order and degree, linear and non-linear differential equations Solution of differential equation, general and particular solution Formation of differential equation Solution of differential equations of first order and first degree, variable separable form, homogenous and linear differential equations Differential equations reducible to variable separable form, homogeneous and linear forms, Bernoulli's equations. 3. Vector Algebra Introduction, scalars and vectors, representation of vectors, types of vectors Addition of vectors, difference of vectors, multiplication of a vector by a scalar, and properties, collinear vectors and points Linear combination of vectors, linearly independent and dependent system of vectors Resolution or components of a vector in a plane and in space Section formula, centroid of a triangle, collinearity of three points, coplanarity of four points Scalar product of two vectors, geometrical interpretation, and properties Application of scalar product, angle between two vectors, projection and resolved parts, work done by a force Vector (cross) product of two vectors, geometrical interpretation and properties Application of vector product, angle between two vectors, area of a triangle and parallelogram, vector moment of a force about a point, moment of a force about a line, moment of a couple Scalar triple product, geometrical interpretation and properties Volume of a parallelopiped, triangular prism and tetrahedron Vector tirple product, scalar product of four vectors, vector product of four vectors Reciprocal system of vectors. 4. Three Dimensional Geometry Introduction to three dimensional Cartesian coordinate system, coordinates of a point in three dimensional space and octants Distance between two points and section formulae Direction cosines and direction ratios of a line, direction ratios of a line joining two points Equation of a straight line in space, Angle between two intersecting lines, conditions of perpendicularity and parallelism of two lines Projection of join of two points on a straight line Shortest distance between two lines and coplanar lines Plane, general equation of the plane, equation of a plane passing through a fixed point Normal form and intercept form of equation of plane Trace, angle between two planes, perpendicularity and parallelism of two planes Angle between a line and a plane, relative position of two points with respect to a plane, distance of a point from a plane, distance between two parallel planes Bisectors of angles between two planes Equations of some particular planes under given conditions. 5. Probability Advanced Random variables, probability distribution and expectation of a finite random variable, variance and standard deviation Binomial distribution and multinomial distribution Conditional probability, multiplication theorem Addition theorem for independent events and independence Repeated trials, finite stochastic process and tree diagram Theorem of total probability and Baye's Theorem. CHAPTERWISE EXEMPLER PROBLEMS

10 JEE MAIN 015 MATHEMATICS (Held on 4th April-015) 1. Let a, b and c be three non-zero vectors such that no two of 1 them are collinear and (a b) c = b c a. If q is the angle 3 between vectors bandc, then a value of sin q is : (a) (c) 3 3 (b) (d) Let O be the vertex and Q be any point on the parabola, x = 8y. If the point P divides the line segment OQ internally in the ratio 1 : 3, then locus of P is : (a) y = x (b) x = y (c) x = y (d) y = x 3. If the angles of elevation of the top of a tower from three collinear points A, B and C, on a line leading to the foot of the tower, are 30, 45 and 60 respectively, then the ratio, AB : BC, is : (a) 1: 3 (b) : 3 (c) 3 :1 (d) 3: 4. The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices (0, 0), (0, 41) and (41, 0) is : (a) 80 (b) 780 (c) 901 (d) The equation of the plane containing the line x 5y + z = 3; x + y + 4z = 5, and parallel to the plane, x + 3y + 6z = 1, is : (a) x + 3y + 6z = 7 (b) x + 6y + 1z = 13 (c) x + 6y + 1z = 13 (d) x + 3y + 6z = 7 6. Let A and B be two sets containing four and two elements respectively. Then the number of subsets of the set A B, each having at least three elements is : (a) 75 (b) 510 (c) 19 (d) Locus of the image of the point (, 3) in the line (x 3y + 4) + k (x y + 3) = 0, k Î R, is a : 8. (a) circle of radius. (b) circle of radius 3. (c) straight line parallel to x-axis (d) straight line parallel to y-axis (1 - cos x)(3+ cosx) lim x 0 x tan4x is equal to : (a) (b) 1 (c) 4 (d) 3 9. The distance of the point (1, 0, ) from the point of intersection of the line x - y + 1 z - = = and the plane x y + z = 16, is (a) 3 1 (b) 13 (c) 14 (d) The sum of coefficients of integral power of x in the binomial expansion ( 1- x) 50 is : (c) ( ) (d) ( 3 50 ) 11. The sum of first 9 terms of the series (a) ( - ) (b) ( + ) (a) 14 (b) 19 (c) 71 (d) The area (in sq. units) of the region described by {(x, y) : y x and y ³ 4x 1} is (a) (c) (b) (d) The set of all values of l for which the system of linear equations : x 1 x + x 3 = lx 1 x 1 3x + x 3 = lx x 1 + x = lx 3 has a non-trivial solution, (a) contains two elements. (b) contains more than two elements (c) is an empty set. (d) is a singleton 14. A complex number z is said to be unimodular if z = 1. Suppose z1- z z 1 and z are complex numbers such that - zz is 1 unimodular and z is not unimodular. Then the point z 1 lies on a: (a) circle of radius. (b) circle of radius. (c) straight line parallel to x-axis (d) straight line parallel to y-axis. 15. The number of common tangents to the circles x + y 4x 6x 1 = 0 and x + y + 6x + 18y + 6 = 0, is : (a) 3 (b) 4 (c) 1 (d)

11 Objective Mathematics for JEE Main with Boards Score Booster 13th Edition 30% OFF Publisher : Disha Publication ISBN : Author : Disha Publication Type the URL : Get this ebook