This syllabus can be also called as Universal Maths Syllabus, where we are covering almost all topics, which will be very helpful to you for crack any Competitive exams of the world.
MATHS SYLLABUS  
CONTENT INDEX Trigonometry 
Trigonometric Functions and Identities  – Trigonometric Ratio & Identities Angle and its Measurement – Measurement of an Angle – Sexagesimal System (degree measure) – Centesimal System (grade measure) – Circular System (radian measure) – Number of radians in an Angle subtended by an Arc of a Circle at the Centre (Arc length radian fomula) – Definitions of Circular Functions or Trigonometric Ratio – Coplanar vectors – Signs of Trigonometric Ratios – Trigonometrical Identities – Trigonometric Ratio of Standard Angles, Allied Angles – Trigonometric Ratios of Angle(90^{0} – θ) in Terms of θ – Trigonometric Ratios of (90^{0} + θ) in Terms of θ – Trigonometric Ratios of (180^{0} – θ) in Terms of θ – Trigonometric Ratios of (180^{0} + θ) in Terms of θ – Trigonometric Ratios of (360^{0} – θ) in Terms of θ – Trigonometric Ratios of Compound Angles – Addition Formulae – Subtraction Formulae – Transformation Formulae – Trigonometric Ratios of Multiple Angles – Trigonometric Ratios of Sub multiple Angles Identities – Identities of Trigonometry &Trigonometric equations – General solution of trigonometric equations – Periodicity & Graphs of Trigonometric Functions – Trigonometric Functions & Identities – Maximum & Minimum Values of Trigonometric Functions Trigonometry Right triangles – Radian measure – Law of cosines – Law of sines – Equations – Double angle formulas Relations between sides and angles of a triangle – Halfangle formula and the area of a triangle – Inverse trigonometric functions (principal value only)– Problems on Heights & Distances 
Trigonometric Equation  – Trigonometric Equations & Inequations – Properties and Solutions of Triangles – Napier’s analogy Straight Line – Cartesian equation of straight line – Cartesian form – Perpendicular distance of a point from a line (Vector form, Cartesian form) – Reflection or image of a point in a straight line (Vector form, Cartesian form) 

Logarithms and Their Properties  – Logarithmic Function – Graph of Logarithmic Function – Properties of Logarithmic Function – Changing of Base 

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Algebra 
Complex Number  – Complex Numbers – Explain Complex numbers as ordered pairs of real numbers – Sum of two complex numbers – Product of two complex numbers – Difference of two complex numbers – Division of two complex numbers – Modulus & argument of a complex number – Addition – Multiplication – Conjugation – Representation of complex numbers, Argand diagram – Polar form (Representation) of a Complex Number – Geometrical representation of complex numbers – Vector representation of complex numbers – Properties of modulus & principal argument – Modulus & argument (amplitude) of a complex number, – Conjugate to complex numbers – Algebra of complex numbers Method for square root of a complex numbers – Cube roots of unity 1,ω,ω^{2}– De Moivre’s Theorem – nth roots of a complex number – Representation of roots of unity – Common properties of roots of unity – Square root of a complex number – Triangle inequality – Cube roots of unity – Geometric interpretations Properties of Moduli (Modulus) – Properties of Arguments – Some useful formulae for Modulus & Amplitude– Concept on Anticlockwise rotation – General formula for rotation – Passage type questions. 
Theory of Quadratic Equations  – Quadratic equations in real & complex number system and their solutions – The relation between roots & coefficients – Roots of the equation – Sum & Product of the roots – To find the equation whose roots are α and β – Nature of the roots – The formation of quadratic equations with given roots – Symmetric functions of roots – Sign of the expression – Modulus Function – Greatest Integer function – Common roots – Inequalities – Certain important definitions 

Permutations and Combinations  – Definitions of Permutation – Definitions of Combination – The Fundamental principle (Theorem) of counting – Permutation as an arrangement & Combination as selection – The meaning of P (n,r) & C (n,r) – Simple applications – Important Results Other Important Concepts on Permutations & Combinations 

Mathematical Induction  – The principle of Mathematical Induction and its simple applications  
Sets, Relations, and Functions  – Sets and their representation – Union, intersection, & complement of sets and their algebraic properties – Powerset – Relation – Types of relations – Equivalence relations 

Statistics  Measures of Dispersion – Mean absolute difference – Mode – Range – Interquartile range – Median absolute deviation – Variance – Standard Deviation – Mean deviation – Graphs & plots – Least squares regression (linear, quadratic, exponential) 

Probability  – Some Definitions – Sample Space – Probability of Events – Mutually Exclusive Events – Equally Likely Events – Exhaustive Events – Classical Definition of Probability – Odds in favour & odds against an event – Axiomatic approach to probability Theory – Axioms of Probability – Conditional Probability – Multiplication Theorem of probability – Addition theorems of probability – Independent events – Important Remarks – Partition of a sample Space – Theorem of Total Probability for Compound Events – Computation of probability of events using permutations & combinations – Bernoulli trials – Binomial Distributions & Probability distribution – Theorem – Use of Multinomial Theorem – Alternative Concept of Inverse Probability (Bayes’ Theorem) 

Progressions & Series  – Arithmetical Progression – Geometrical Progression – Insertion of arithmetic – Arithmetic, Geometric, & Harmonic means – The relation between Arithmetic Mean & Geometric Mean – Definition, nth term of a Geometrical Progression – Sum of n terms of a Geometrical Progression – Sums of finite arithmetic & geometric progressions – Geometric Mean Between two given numbers (a & b) – Product of n geometric means = Gn, An Important Note – Infinite geometric series – Sums of squares & cubes of the first n natural numbers – Sum up to n terms of special series (Sn, Sn2, Sn3) – Arithmetico Geometric progression. Arithmeticgeometric series, – Definition, – Natural Numbers, – Method of difference, – Harmonical Progression (H.P.), – nth term of Harmonic Progression, – Harmonic mean (single), – n Harmonic Means Between a & b – Relations between A, G, & H 

Binomial Theorem  – Statement of binomial theorem for positive integral index – General term and middle term – Properties of Binomial coefficients and their simple applications – Tr + 1, Binomial coefficients of terms equidistant from the beginning and the are equal – Number of terms & middle term – Values of Binomial Coefficients – Term containing xr will occur in Tr + 1 for (1 + x)n and it will be 〖(_ ^n)C〗_r x^r – Term independent of x in the expansion of (x + a)n – Terms equidistant from the beginning & end of the binomial expansion (x+a)n 

Matrix (plural: Matrices)  – Definition – Algebra of matrices – Various Types of Matrix – Matrices of order two & three – Properties of Matrix Multiplication – Matrices as a rectangular array of real numbers – Equality of matrices, addition, multiplication by a scalar & product of matrices Transpose of Matrix – Properties of Transpose – Properties of these matrix operations – Symmetric & skew symmetric matrices and their properties – Definition of Adjoint of a Matrix – Working rule for finding the adjoint of A – Rule to write the cofactor of an element aij – The Inverse of Matrix – Properties – Inverse of a matrix is unique – Rank of a Matrix – Sub –matrix of order r – The rank of a given matrix A is said to be r if – Working Rule – Solution of Equations – Representation of equations in matrix form – Nature of Solution 

Determinants  – Definition – Aid to memory – Rule – Expansion with respect to first column – Properties of determinants – Evaluation of determinants – The area of triangles using determinants – Determinant of a square matrix of order up to three – Adjoint & evaluation of inverse of a square matrix using determinants & elementary transformations – Test of consistency and solution of simultaneous linear equations in two or three variables using determinants & matrices Cofactor & Minors – Differentiation of a determinant – Integration of a Determinant – Factor of certain standard determinants Remember the results – System of linear Equations – Non – homogeneous Equations in two unknowns – Homogeneous linear equation in two unknowns – Non – homogeneous linear equations in three unknowns – Homogeneneous linear equation – Three equations in two unknowns – Gist of discussion in simple language 

Mean and Variance  – Introduction – Discrete Random Variable – Probability Distribution – Mean of a Discrete Random Variable – Variance of a Discrete Random Variable 

Linear Programming  – Some Definitions & Results – Feasible Solution – Infeasible solution – Feasible Region – Optimal Feasible Solution – Convex Set – Theorem – Fundamental Extreme point Theorem – Graphical Methods of Solving Linear Programming Problems – CornerPoint Method – IsoProfit or IsoCost Method – Different Types of linear Programming Problems – Diet Problems – Optimal Product Line Problems – Transportation Problems 

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Differential Calculus 
Fundamentals and Functions  – Fundamentals – Basic Definitions – Closed & Open intervals – Modulus or Absolute Value Function, – Generalized Results, – Wavy Curve Method/Number Line Rule/ Sign Scheme For Rational Function – Fundamentals Of Quadratic Equations – The sum of several nonnegative terms Definition – Independent & dependent variable – Graphical representation of function – Real Function – Content function – Identity function – Modulus function – Properties of modulus function – The greatest integer function – Properties of greatest integral function – Fractional part function – Properties of fractional part of x – Least integer function – Properties of least integer function – Real valued functions – Algebra of functions – Linear functions – Trigonometric functions – Piecewise function – Recursive function – Parametric function – Signum Function – Reciprocal function – Logarithmic function – Exponential function – Square root function – Polynomial function – Rational function – Inverse of trigonometric function – Operations on real function – Composition of functions – Rule for Finding Domain – Graphs of simple functions – Logarithms and their properties Monotonic Function – Nature of derivative of function – Nature of derivative – Range – Method to find the range of a function y = f(x) – Odd & even function – Properties of odd & even function – Periodic function – Properties of periodic function – While taking LCM we should always remember Mapping of Function – Kinds of function – Oneone Mapping or injective or monomorphic – Method to check oneone mapping – Number of one – one Mapping – Method to check ManyOne – Onto Function or (Subjective) – Method of show onto or subjective – Number of onto Functions – Number of one – one onto mapping or bijection – Equal & identical function – Inverse of function, – Graph of the inverse of an invertible function – Properties of inverse of a function – Composite functions – The adjacent figure shows the steps to be taken – Properties of Composition of function 
Limit  – Basic concept – Fundamental algebraic operation on limits of function – Standard limits – Indeterminate forms – Sandwich theorem – Some important expansions – Factorization method – Rationalization Method – Based on standard formula – Algebraic function of ∞ type ∞/∞ form – How to solve problems – Trigonometrical Limits – Logarithmic Limits – Exponential Limits – Based on definition of ‘e’, Evaluation of exponential limits of the form 1∞ – Particular cases – Miscellaneous Forms – Use of Sandwich theorem (Squeeze theorem) – Use of NewtonLeibnitz’s formula in evaluating the limits – L’ Hospital’s Theorem 

Continuity and Differentiability  – Continuity of a function – Graphical View – Continuity at end points – Jump discontinuity – Properties of continuity function – Differentiability Differentiability in a set – Some standard results on differentiability 

Differentiation  – Introduction – Derivative of standards functions – Rule (i) – Rule (ii) ( Product Rule) – Generalization of the product rule – Rule (iii) (Quotient rule) – Differentiation of a function of a function – Differentiation by using trigonometrical transformations – Differentiation of the sum, difference, product, & quotient of two functions Differentiation of implicit functions – Logarithmic Differentiation – Differentiation of parametric functions – Differentiation of a function with respect to another function – Differentiation of determinants – Differentiation of trigonometric, inverse trigonometric, logarithmic, exponential, composite and implicit functions 

Application and Derivatives  – Derivatives of order up to two – Derivative as a rate measurer – Derivative as the time rate of change – Rate of change of quantities – Differentials – Errors and Approximations Rolle’s Theorem – Geometrical proof of Rolle’s Theorem – Lagrange’s mean value Theorem – Geometrical interpretation of Lagrange’s Theorem 

Tangents and Normals  – Basic definition – Slope (Gradient) of a line – Slope of a line in terms of coordinates of any two points on it – Slope of a line when its equation is given – Angle between two lines – Equation of a straight line 

Monotonicity  – Monotonic function – monotonic increasing & decreasing functions – Properties of Monotonic Functions – Advance level Include Subjective type questions 

Maxima Minima  – Definition – Maxima & Minima at end point – Maxima & minima of functions of one variable – Method of finding extrema of continuous functions – Method of 2nd derivative – Concept of Global Maximum / Minimum – Global Maximum / Minimum in [a, b] – Global Maximum / Minimum in (a, b) Applied problems in maxima & minima – Maxima & Minima in Discontinuous Function – Minimum of discontinuous functions – Maximum of discontinuous functions – Method of finding the greatest & least values of a continuous function 

Graphical Transformation  – Some Standard Graphs – Straight Line – Ellipse – Hyperbola – Rectangular Hyperbola – Transformation of Graphs – How to draw graph of Polynomial – Draw the graph – Plotting graph of f(x – [x]) 

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Integral calculus 
Indefinite integral  – What is Integration – Integral as an antiderivative (inverse process of differentiation) – What is Geometrical Significance – What is the Importance of constant ‘C’ – Fundamental Integration Formulas – Fundamental Theorem of Integral Calculus involving algebraic, trigonometric, exponential & logarithmic functions. – Integration using trigonometric identities – Integral as limit of a sum. – Indefinite integrals of standard functions What are the various Operations on Integration. Integration by methods of Substitution, by Parts, & by Partial fraction – Integral of the form, – Integral of the form. – Some special Integrals, – Some Important Substitutions, – Integration of The Type, – Integration reducible to type. – Integral of the type, – Integral reducible of Type. – Integration of the type. – Integration of type, – Integral of the form, – Integrals of the form, – Integral of the form. What is integration by parts? – Integrals of the type – Integrals of the form – What is rational function? – What are the standard rules for integrating rational functions? – Formula, Integral of the form – Integral of the form 
Definite Integral  – What is Definite Integral? – Geometrical Interpretation of Definite Integral – Evaluation of Definite integrals by substitution – Properties of Definite Integrals (Properties I to III) – Generalization Properties of Definite Integrals (Properties IV to VII) – Properties of Definite Integrals (Properties VIII to XIV) – Application of Integrals to determining areas of the regions bounded by simple curves in standard form (Area under simple curves, Area between Two Curves). 

Area  – Enquiry – How it is possible to find the area enclosed by a curve and xaxis? – Enquiry – What about change in sign of area according to the position of curve. (Above or below x axis), Shaded portion (P1P2Q2Q1) 

Differential Equation  – Ordinary differential equations – Order & degree of a differential equation – Linear & NonLinear Differential equation – Separation of variables method (Fourier method) – Solution of a differential equation (by the method of separation of variables) – Formation of differential equation – Algorithm for formation of differential equation – Method of solving a first order of degree differential equation – Reducible to variables separable – Homogeneous Equations – The solution of homogeneous differential equations – Solution of linear differential of first order – Linear differential equations of the form – Equations reducible to linear form 

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Vector and 3d Geometry 
Vectors  – Scalars & Vectors – Definitions – Null vector – Unit vector – Like/Unlike vectors – Parallel vectors – Position vector – Collinear vectors – Coplanar vectors – Coinitial vectors – Free vectors – Localised vectors – Equal vectors – Polygon law of addition – Vector Addition – Vector subtraction – Multiplication Vector – Vector division – Position Vector of a Point – Section Formula – Internal division – External division – Coplanar Vectors – Components of a Vector in Two Dimension – Components of a Vector (AB) ⃗ in terms of Coordinates of A and B – Addition, Subtraction, Multiplication of a Vector by Equality in Terms – Components of a Vector in 2D & 3D space – Co linearity of Vectors – Co linearity of Points – Linearly Dependent & Independent Vectors Product of Two Vectors – Scalar product (Dot product) of two vectors – Properties of Scalar product – Application of Dot Product on Plane Trigonometry Vector Product (cross product) of two Vectors – Properties – Scalar Triple Product – Properties – Vector Triple Product – Tetrahedron – Geometrical interpretation 
Three Dimensional Coordinate System  – Three Dimensional Coordinate System Position vector of a point on space – Solids, surface area & volume (cylinders, pyramids, cones, prisms, spheres) – G Signs of Coordinates of a Point in Various Octants – Distance Formula (the distance between two points) – Section Formula – Direction Cosines and Direction Ratio’s of a Vector – Coordinates of P are (r cos〖α ,r cosβ,r cosγ 〗 ) – Direction Ratios – Directions cosines of parallel vector – Angle between direction cosines, direction ratios, & two intersecting lines – Straight Line – Cartesian equation of straight line – Cartesian form – Perpendicular distance of a point from a line (Vector form, Cartesian form) – Reflection or image of a point in a straight line (Vector form, Cartesian form) – Skew lines, the shortest distance between them & its equation. – Equations of a line & a plane in different forms, the intersection of a line & a plane, coplanar lines 

Plane  – Plane – Equation of a plane passing through a given point – Intercept form of a plane – Vector equation of a plane passing through a given point and normal to a given vector – Cartesian form. – Equation of plane in normal form, vector form, Cartesian form, – Angle between the two planes, – Angle between a line and a plane, – Equation of Plane forming through three given points – Cartesian equivalence – Equation of plane that passes through a point A with position vector a ⃗ and is parallel to given vector (b ) ⃗ & (c ) ⃗ – Cartesian form – Equation of any Plane Passing through the Line of Intersection of Plane, +++= and +++=is (+++)+(+++–=) – Vector form – Two Sides of a Plane Distance of a Point from a Plane – Vector form – Cartesian form – Equation of the Planes Bisecting the Angle between two Planes – Vector form – Bisector of the angle between the two planes containing the origin – Bisector of the acute & obtuse angles between two planes – Intersection of a line and a plane – Condition for a line to lie parallel to a plane 

Sphere  – Definition, – Equation of a sphere, – Equation of a sphere passing through four noncoplanar points. 

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Coordinate Geometry 
– Euclidean plane  
Coordinate Systems and Coordinates  – Co ordinate Axes – Cartesian coordinates system (Rectangular coordinate system) – Distance formula (Distance Between Two Points) Section Formulae – Formula For Internal Division – Formula For External Division nbsp; – Centroid of a Triangle – Incenter of a triangle – Orthocenter of a triangle – Circumcentre of a Triangle – Area of a Triangle – Shift of origin Locus & Its Equation – Change of Axes or the Transformations of Axes – Removal of the term xy from f (x, y) = ax2 + 2hxy + by2 without changing the origin Symmetry– Transformations 

The Straight Lines  – Angle of inclination of line – Slope or gradient of a line – Angle between two lines – Parallel & perpendicular lines – Line parallel to coordination axes Intercepts of a line on the coordinate axis – Various forms of the equation of a straight line – Intersection of lines – Reduction of general equation to standard form – The distance form or symmetric form or parametric form of a line Theorem – Position of a point relative to a line – Position of two points relative to a given line – Distance of a point from a line – Working Rule – Distance between two parallel lines Area of parallelogram – Point of intersection of two lines – Concurrent lines (Concurrency of lines) – Family of lines – How to find Circumcentre & orthocenter by slopes – How to find Orthocenter – Line equally inclined with two lines Equation of the bisectors – Bisector of the angle containing the origin – Working Rule – Equation of internal & external bisectors of the angle between two lines – How to distinguish the acute (internal) & obtuse (external) angle bisectors – Short cut method for finding acute (internal) & obtuse (external) angle bisectors – Reflection Image or reflection of a point (x1, y1) about a line mirror – Optimization (Minimization or Maximization) 

The Pairs of Straight Lines  – Pair of Straight Line – Homogeneous Equation in Two Variables – Two Very Useful Identities – Angle between the pair of straight line y = m2x and ax2 + 2hxy + by2 = 0 – Bisectors of the Angle Between the Lines Given by a Homogeneous Equation General Equation of Second Degree – Important Facts – To find the two lines – Angle – Removal of First Degree Terms, Removal of XY term: Rotate the axis with angle θ, Equation of the lines joining the origin to the points of intersection of a given line and a given curve, Curve 

Circles  – Circle – Equation of a Circle – Equation of Center, Chord, & Diameter of circle – Equation of Circles in Different Forms – General Form – Equation of a circle when the endpoints of a diameter are given – Rule For Finding the Centre & Radius of a Circle – Parametric Form – General Form – Conditions For a Circle – Nature of the Circle – Concentric Circle Diameter Form of a Circle – Equation of Circle Passing Through Three NonCollinear Points – Cyclic Quadrilateral – Intercepts Made on the Axes by a Circle – Different Form of the Equation of a Circle – Position of a Point With Respect to a Circle – Maximum and Minimum Distance of a Point From the Circle – Intersection of a Line & a Circle – Product of Algebraic Distances PA & PB is the constant when from P – A secant be drawn to cut the circle in the points A, B – Length of intercept cut off from a line by a circle – Tangent to a Circle at a Given Point – Wrong Process – Find its tangent only by calculus – Parametric Forms – Normal to a Circle at a Given Point – Different Forms – Tangents From a Point to a Circle – Points of intersection of a line & a circle with the centre at the origin and condition for a line to be tangent to a circle, equation of the tangent Power of a Point With Respect to a Circle – Chord of Contact, Definition – Chord Bisected At Given Point – Pair of Tangents – Director circle – Diameter of A circle – Pole & Polar – Coordination of pole of a line – Properties Two Circle Touching Each Other – Common tangents to two circles – How to find the direct common tangents – How to find transverse common tangents – Transverse imaginary – Common chord of two circles – Length of common chord – Family of Circles, Angle of intersection of two circles, Radical Axis, Some properties of radical axis, Radical Centre, Coaxial System of Circles, Limiting Point 

Parabola  – Conic Section – Recognisation of Conics – Standard Equation of Parabola General Equation of Parabola – The Generalized Form – Parabolic Curve – Position of a Point (x1, y1) with respect to a Parabola y2 = 4ax – Parametric Relation Between the Coordinates of the Ends of a Focal Chord of a Parabola Intersection of a Line and a Parabola – Condition for Tangent – The Point of Contact – Equation of Tangent in Different Forms – Point of Intersection of Tangents at Any Two Points of the Parabola Equations of Normals in Different Forms – Point of intersection of normals at any two points on the parabola – Relation between ‘t1’ and ‘t2’ If Normal at ‘t1’ Meets the Parabola at ‘t2’ Again – Co – Normal Points – Pair of Tangents – Chord of Contact: T = 0 – Chord Bisected at a Given Point (Middle Point Chord) T=S1 – Circle Through Co – Normal Points – Diameter – Lengths of Tangent, Subtangent, Normal and Subnormal – Reflection Property of a Parabola 

Ellipse  – Ellipse Definition – Standard Equation of Ellipse – Length of the Latus Rectum – Focal Distance of a Point – Another Definition of Ellipse – The shape of the ellipse x^2/a^2 +y^2/b^2 =1,when b > a, Position of a point with respect to an ellipse – Intersection of a line and an ellipse – Condition of tangency – Equation of Tangent in Different Forms – Equation of Normal in Different Forms – Properties of Eccentric Angles of the Co – Normal Points – Pair of Tangents Chord of Contact – Chord Bisected at a Given Point, Director Circle, Sub – Tangent and Sub – Normal – Reflection Property of an Ellipse – Equation of an ellipse referred to two perpendicular lines – Centre 

Hyperbola  – Hyperbola – Standard Equation of Hyperbola – General Equation – The Foci and Directrix of a Hyperbola – Length of Latus Rectum Focal Distance – Conjugate Hyperbola – Position of a Point With Respect To a Hyperbola – Inter section of a line y = mx + c and a Hyperbola x^{2}/a^{2} – y^{2}/b^{2} =1 – Condition for Tangency – Point of contact Equations of Tangent in Different Forms – Equation of Normal in Different Form – Pair of Tangent – Chord of Contact – Chord Bisected at a Given Point (Middle Point Chord) – Diameter – Conjugate Diameters.Director Circle – Sub Tangent and Sub Normal – Co – Normal Points – Asymptotes – Method to Find – The Rectangular Hyperbola xy = c2 – Properties of Rectangular Hyperbola xy = c2 – Reflection Property of a Hyperbola 
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