## Units and Topics

# | Unit/Topic | Marks |
---|---|---|

100 | Relations and Functions (Section A) | - |

200 | Algebra (Section A) | - |

300 | Calculus (Section A) | - |

400 | Probability (Section A) | - |

500 | Vectors (Section B) | - |

600 | Three - Dimensional Geometry (Section B) | - |

700 | Application of Integrals (Section B) | - |

800 | Application of Calculus (Section C) | - |

900 | Linear Regression (Section C) | - |

1000 | Linear Programming (Section C) | - |

Total | - |

## Syllabus

- Introduction of Relations and Functions
- Types of Relations
- One-One Relation(Injective)
- Many-one relation
- Into relation
- Onto relation (Surjective)

Reflexive, symmetric, transitive, not reflexive, not symmetric and not transitive.

- Concept of Relations
- Types of Functions
- one-one (or injective)
- many-one
- onto (or surjective)
- one-one and onto (or bijective)

- Composition of Functions and Invertible Function
- Inverse of a Function
- Concept of Binary Operations
Definition, Commutative Binary Operations, Associative Binary Operations , Identity Binary Operation, Invertible Binary Operation

- All Axioms and Properties
- Conditions of Invertibility
- Basic Concepts of Trigonometric Functions
sine, cosine, tangent, cotangent, secant, cosecant function

- Inverse Trigonometric Functions
- Properties of Inverse Trigonometric Functions
Inverse of Sin, Inverse of cosin, Inverse of tan, Inverse of cot, Inverse of Sec, Inverse of Cosec

(i) Types of relations:- reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, inverse of a function. Binary operations.

Relations as:-

- Relation on a set A
- Identity relation, empty relation, universal relation.
- Types of Relations:- reflexive, symmetric, transitive and

equivalence relation.

Binary Operation: all axioms and properties

Functions:-

- As special relations, concept of writing “y is a function of x” as y =

f(x). - Types: one to one, many to one, into, onto.
- Real Valued function.
- Domain and range of a function.
- Conditions of invertibility.
- Composite functions and invertible functions (algebraic functions only).

(ii) Inverse Trigonometric Functions

Definition, domain, range, principal value branch. Graphs of inverse trigonometric functions. Elementary properties of inverse

trigonometricfunctions.

- Principal values
- sin
^{-1}x, cos^{-1}x, tan^{-1}x etc. and their graphs. `sin^(-1)x=cos^(-1)sqrt(1-x^2)=tan^(-1)(x/sqrt(1-x^2))`

`sin^(-1)x=cosec^(-1)1/x;sin^(-1)x+cos^(-1)x=pi/2`

`sin^(-1)x+-sin^(-1)y=sin^(-1)(xsqrt(1-y^2)+-ysqrt(1-x^2))`

`cos^(-1)x+-cos^(-1)y=cos^(-1)(xy+-sqrt(1-y^2)sqrt(1-x^2))`

`tan^(-1)x-tan^(-1)y=tan^(-1)((x-y)/(1+xy))`

- Formulae for 2sin-1x, 2cos-1x, 2tan-1x, 3tan-1x etc. and application of these formulae.

- Introduction of Matrices
- Matrices
- Matrices Notation
Matrices Notation

- Matrices Notation
- Order of a Matrix
- Equality of Matrices
- Types of Matrices
- Symmetric and Skew Symmetric Matrices
- Concept of Transpose of a Matrix
- Operations on Matrices
- Multiplication of Matrices
Non-commutativity of multiplication of matrices, Zero matrix as the product of two non zero matrices

- Multiplication of Matrices
- Multiplication of Two Matrices
- Elementary Operation (Transformation) of a Matrix
- Invertible Matrices
- Introduction of Determinant
- Determinants of Matrix of Order One and Two
- Determinant of a Square Matrix
up to 3 x 3 matrices

- Determinant of a Matrix of Order 3 × 3
- 1st, 2nd and 3rd Row
- 1st, 2nd and 3rd Columns
- Expansion along first Row (R1), Expansion along second row (R2),Expansion along first Column (C1)

- Properties of Determinants
- Property 1 - The value of the determinant remains unchanged if its rows are turned into columns and columns are turned into rows.
- Property 2 - If any two rows (or columns) of a determinant are interchanged then the value of the determinant changes only in sign.
- Property 3 - If any two rows ( or columns) of a determinant are identical then the value of the determinant is zero.
- Property 4 - If each element of a row (or column) of a determinant is multiplied by a constant k then the value of the new determinant is k times the value of the original determinant.
- Property 5 - If each element of a row (or column) is expressed as the sum of two numbers then the determinant can be expressed as the sum of two determinants
- Property 6 - If a constant multiple of all elements of any row (or column) is added to the corresponding elements of any other row (or column ) then the value of the new determinant so obtained is the same as that of the original determinant.
- Property 7 - (Triangle property) - If all the elements of a determinant above or below the diagonal are zero then the value of the determinant is equal to the product of its diagonal elements.

- Minors and Co-factors
- Area of a Triangle
- Inverse of a Matrix
- Applications of Determinants and Matrices
- Martin’S Rule
Martin’s Rule (i.e. using matrices)

**(i) Matrices**

Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operation on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order upto 3). Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it exists (here all matrices will have real entries).

**(ii) Determinants**

Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.

Types of matrices (m × n; m, n`<=` 3), order; Identity matrix, Diagonal matrix.

Symmetric, Skew symmetric.

Operation :– addition, subtraction, multiplication of a matrix with scalar, multiplication of two matrices (the compatibility).

E.g.`[(1,1),(0,2),(1,1)][(1,2),(2,2)]` =AB(say) but BA is not possible.

singular and non-singular matrices.

Existence of two non-zero matrices whose product is a zero matrix.

Inverse (2x2, 3x3) `A^(-1)=(AdjA)/|A|`

**Martin’s Rule (i.e. using matrices):-**

a_{1}x + b_{1}y + c_{1}z = d_{1}

a_{2}x + b_{2}y + c_{2}z = d_{2}

a_{3}x + b_{3}y + c_{3}z = d_{3}

`A=[(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)]B=[(d_1),(d_2),(d_3)]X=[(x),(y),(z)]`

`AX=BrArrX=A^(-1)B`

Problems based on above.

NOTE 1:- The conditions for consistency of equations in two and three variables, using

matrices, are to be covered.

NOTE 2:- Inverse of a matrix by elementary operations to be covered.

**Determinants:-**

- Order.
- Minors.
- Cofactors.
- Expansion.
- Applications of determinants in finding the area of triangle and collinearity.
- Properties of determinants. Problems based on properties of determinants.

- Concept of Continuity
- Continuous Function of Point
Continuous left hand limit

Continuous right hand limit

- Algebra of Continuous Functions
- Exponential and Logarithmic Functions
- Derivatives of Composite Functions - Chain Rule
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Implicit Functions
- Logarithmic Differentiation
- Derivatives of Functions in Parametric Forms
- Mean Value Theorem
- Second Order Derivative
- L' Hospital'S Theorem

Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions.

Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives. Rolle's and Lagrange's Mean Value Theorems (without proof) and their geometric interpretation.

**Continuity**

Continuity of a function at a point x = a.

Continuity of a function in an interval.

Algebra of continues function.

Removable discontinuity.

**Differentiation**

Concept of continuity and differentiability of |x| , [x], etc.

Derivatives of trigonometric functions.

Derivatives of exponential functions.

Derivatives of logarithmic functions.

Derivatives of inverse trigonometric functions - differentiation by means of substitution.

Derivatives of implicit functions and chain rule for composite functions.

Derivatives of Parametric functions.

Differentiation of a function with respect to another function e.g. differentiation of sinx^{3} with respect to x^{3}.

Logarithmic Differentiation - Finding dy/dx when `y = x^(x^x)`

Successive differentiation up to 2^{nd} order.

**NOTE 1:-** Derivatives of composite functions using chain rule.

**NOTE 2:-** Derivatives of determinants to be covered.

L' Hospital's theorem.

`0/0`

Rolle's Mean Value Theorem - its geometrical interpretation.

Lagrange's Mean Value Theorem - its geometrical interpretation

- Introduction to Applications of Derivatives
- Rate of Change of Bodies Or Quantities
- Increasing and Decreasing Functions
- Tangents and Normals
- Approximations
- Maxima and Minima
- Maximum and Minimum Values of a Function in a Closed Interval
- Simple Problems on Applications of Derivatives
Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations)

Applications of derivatives:- rate of change of bodies, increasing/decreasing functions, tangents and normals, use of derivatives in approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-lifesituations).

Equation of Tangent and Normal

Approximation.

Rate measure.

Increasing and decreasing functions.

Maxima and minima.

- Stationary/turning points.
- Absolute maxima/minima
- local maxima/minima
- First derivatives test and second derivatives test
- Point of inflexion.
- Application problems based on maxima and minima.

- Introduction of Integrals
- Integration as an Inverse Process of Differentiation
- Methods of Integration - Integration by Substitution
- Methods of Integration - Integration Using Partial Fractions
- Methods of Integration - Integration by Parts
- Properties of Indefinite Integral
- Anti-derivatives of Polynomials and Functions
Anti-derivatives of polynomials and functions (ax +b)

^{n}, sinx, cosx, sec^{2}x, cosec^{2}x etc . - Evaluation of Simple Integrals of the Following Types and Problems
- Definite Integral as the Limit of a Sum
- Fundamental Theorem of Calculus
Area function, First fundamental theorem of integral calculus and Second fundamental theorem of integral calculus

- Properties of Definite Integrals
- Evaluation of Definite Integrals by Substitution

Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.

Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

**Indefinite integral**:-

Integration as the inverse of differentiation.

Anti-derivatives of polynomials and functions (ax +b)^{n} , sinx, cosx, sec^{2}x, cosec^{2}x etc .

Integrals of the type sin^{2}x, sin^{3}x, sin^{4}x, cos^{2}x, cos^{3}x, cos^{4}x.

Integration of 1/x, e^{x}.

Integration by substitution.

Integrals of the type f'(x)[f(x)]^n,`(f'(x))/(f(x))`

Integration of tanx, cotx, secx, cosecx.

Integration by parts.

Integration using partial fractions.

Expression of the form`f(x)/g(x)` when degree of f(x) < degree of g(x)

`(x+2)/((x-2)(x-1)^2)=A/(x-1)+B/(x-1)^2 +C/(x-2)`

`(x+1)/((x^2+3)(x-1))=(Ax+B)/(x^2+3)+c/(x-1)`

When degree of `f (x) >=` degree of g(x),

**Integrals of the type:-**

`int(dx)/(x^2+-a^2)`

and

`intsqrt(a^2+-x^2)dx`

`int(dx)/(acosx+bsinx),`

`int(dx)/(a+bcosx),int(dx)/(a+bsinx)int(dx)/(acosx+bsinx+c),`

`int((acosx+bsinx)dx)/(`

`int(dx)/(acos^2x+bsin^2x+c)`

`int(1+-x^2)/(1+x^4)dx,`

`int(dx)/(1+x^4),intsqrt(tanxdx),intsqrt(cotxdx)`

**Definite Integral:-**

Definite integral as a limit of the sum.

Fundamental theorem of calculus (without proof)

Properties of definite integrals.

Problems based on the following properties of definite integrals are to be covered.

`int_a^bf(x)dx=int_a^bf(t)dt`

`int_a^bf(x)dx=-int_b^af(x)dx`

`int_a^bf(x)dx=int_a^cf(x)dx+int_c^bf(x)dx`

where a < c < b

`int_a^bf(x)dx=int_a^bf(a+b-x)dx`

`int_0^af(x)dx=int_0^af(a-x)dx`

`int_0^(2a)f(x)dx={(2int_0^af(x),`

`int_(-a)^af(x)dx={(2int_0^af(x)dx,`

- Basic Concepts of Differential Equation
- Order and Degree of a Differential Equation
- General and Particular Solutions of a Differential Equation
- Formation of a Differential Equation Whose General Solution is Given
- Formation of Differential Equation by Eliminating Arbitary Constant
- Methods of Solving First Order, First Degree Differential Equations
- Solutions of Linear Differential Equation
Solutions of linear differential equation of the type:-

- dy/dx + py= q, where p and q are functions of x or constants.
- dx/dy + px = q, where p and q are functions of y or constants.

- Application on Growth and Decay
- Solve Problems on Velocity, Acceleration, Distance and Time
- Solve Population Based Problems on Application of Differential Equations
- Application on Coordinate Geometry

Definition, order and degree, general and particular solutions of a differential equation. Formation of differential equation whose general solution is given. Solution of differential equations by method of separation of variables solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:

`(dy)/(dx)+py=q,`

where p and q are functions of x or constants.

`(dx)/(dy)+px=q,`

where p and q are functions of y or constants.

- Differential equations, order and degree.
- Formation of differential equation by eliminating arbitrary constant(s).
- Solution of differential equations.
- Variable separable.
- Homogeneous equations.
- Linear form
`(dy)/(dx)+Py=Q`

where P and Q are functions of x only. Similarly for dx/dy. - Solve problems of application on growth and decay.
- Solve problems on velocity, acceleration, distance and time.
- Solve population based problems on application of differential equations.
- Solve problems of application on coordinate geometry.

**NOTE 1:-** Equations reducible to variable separable type are included.

**NOTE 2:-** The second order differential equations are excluded.

- Introduction of Probability
- Random experiment
- Outcome
- Equally likely outcomes
- Sample space
- Event

- Dependent Events
- Conditional Event
- Conditional Probability
- Multiplication Theorem on Probability
- Independent Events
- Baye'S Theorem
- Partition of a sample space
- Theorem of total probability

- Addition Theorem of Probability
- Random Variables and Its Probability Distributions
- Mean of a Random Variable
- Bernoulli Trials and Binomial Distribution
- Laws of Probability
- Probability Distribution Function
- Mean of Binomial Distribution (P.M.F.)
- Variance of Binomial Distribution (P.M.F.)

Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean and variance of random variable. Repeated independent (Bernoulli) trials and Binomial distribution.

- Independent and dependent events conditional events.
- Laws of Probability, addition theorem, multiplication theorem, conditional probability.
- Theorem of Total Probability.
- Baye’s theorem.
- Theoretical probability distribution, probability distribution function; mean and variance of random variable, Repeated independent (Bernoulli trials), binomial distribution – its mean and variance.

- Magnitude and Direction of a Vector
- Types of Vectors
Zero Vector, Unit Vector, Coinitial Vectors, Collinear Vectors, Equal Vectors, Negative of a Vector (Free Vector)

- Basic Concepts of Vector Algebra
- Position Vector
- Direction Cosines and Direction Ratios of a Vector

- Components of a Vector
- Addition of Vectors
- Operations - Sum and Difference of Vectors
- Multiplication of a Vector by a Scalar
- Position Vector of a Point Dividing a Line Segment in a Given Ratio
- Geometrical Interpretation of Scalar
- Product of Two Vectors
- Scalar Triple Product of Vectors
volume of a parallelepiped, co-planarity

- Section formula
for internal and external division

Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors, scalar triple product of vectors.

- As directed line segments.
- Magnitude and direction of a vector.
- Types:- equal vectors, unit vectors, zero vector.
- Position vector.
- Components of a vector.
- Vectors in two and three dimensions.
- iˆ, ˆj, kˆ as unit vectors along the x, y and the z axes; expressing a vector in terms of the unit vectors.
- Operations:- Sum and Difference of vectors; scalar multiplication of a vector.
- Section formula.
- Triangle inequalities.
- Scalar (dot) product of vectors and its geometrical significance.
- Cross product - its properties - area of a triangle, area of parallelogram, collinear vectors.
- Scalar triple product - volume of a parallelepiped, co-planarity.

**NOTE:- Proofs of geometrical theorems by using Vector algebra are excluded.**

- Direction Cosines and Direction Ratios of a Line
- Equation of a Line in Space
- Shortest Distance Between Two Lines
- Distance between two skew lines
- Distance between parallel lines

- Vector and Cartesian Equation of a Plane
- Angle Between Two Lines
- Angle Between Two Planes
- Angle Between Line and a Plane
- Plane
- Distance of a Point from a Plane
- Direction Ratios of the Normal to the Plane.
- Intersection of the Line and Plane
- Concept of Line

Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane.

Angle between (i) two lines, (ii) two planes, (iii) a line and a plane.

Distance of a point from a plane.

Equation of x-axis, y-axis, z axis and lines parallel to them.

Equation of xy - plane, yz – plane, zx – plane.

Direction cosines, direction ratios.

Angle between two lines in terms of direction cosines /direction ratios.

Condition for lines to be perpendicular / parallel.

**Lines:-**

- Cartesian and vector equations of a line through one and two points.
- Coplanar and skew lines.
- Conditions for intersection of two lines.
- Distance of a point from a line.
- Shortest distance between two lines.

**NOTE:- Symmetric and non-symmetric forms of lines are required to be covered.**

**Planes:-**

- Cartesian and vector equation of a plane.
- Direction ratios of the normal to the plane.
- One point form.
- Normal form.
- Intercept form.
- Distance of a point from a plane.
- Intersection of the line and plane.
- Angle between two planes, a line and a plane.
- Equation of a plane through the intersection of two planes i.e. P
_{1}+ kP_{2}= 0.

- Area Under Simple Curves
- Area of the Region Bounded by a Curve and a Line
circle-line, elipse-ine, parabola-line

- Area Between Two Curves
- Applications of the Integrations

Application in finding the area bounded by simple curves and coordinate axes. Area enclosed between two curves.

- Application of definite integrals - area bounded by curves, lines and coordinate axes is required to be covered.
- Simple curves:- lines, circles / parabolas / ellipses, polynomial functions, modulus function, trigonometric function, exponential functions, logarithmic functions

- Application of Calculus in Commerce and Economics in the Cost Function
- Application of Calculus in Commerce and Economics in the Average Cost
- Application of Calculus in Commerce and Economics in the Marginal Cost and Its Interpretation
- Application of Calculus in Commerce and Economics in the Demand Function
- Application of Calculus in Commerce and Economics in the Revenue Function
- Application of Calculus in Commerce and Economics in the Marginal Revenue Function and Its Interpretation
- Application of Calculus in Commerce and Economics in the Profit Function and Breakeven Point
- Rough Sketching
- Rough sketching of the following curves:- AR, MR, R, C, AC, MC and their mathematical interpretation using the concept of maxima & minima and increasing- decreasing functions.

Application of Calculus in Commerce and Economics in the following:-

- Cost function,
- average cost,
- marginal cost and its interpretation
- demand function,
- revenue function,
- marginal revenue function and its interpretation,
- Profit function and breakeven point.
- Rough sketching of the following curves:- AR, MR, R, C, AC, MC and their mathematical interpretation using the concept of maxima & minima and increasing- decreasing functions.

Self-explanatory

**NOTE:- Application involving differentiation, integration, increasing and decreasing function and maxima and minima to be covered.**

- Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
- Statistics
- Scatter Diagram
(I) a) Perfect positive correlation, b) Positive correlation with high degree, c) Positive correlation with low degree

(II) a) Perfect negative correlation, b) Negative correlation with high degree, c) Negative correlation with low degree

(III) No correlation (Zero correlation)

- Scatter Diagram
- The Method of Least Squares
- Lines of Best Fit
- Regression Coefficient of X on Y and Y on X
- Identification of Regression Equations
- Angle Between Regression Line and Properties of Regression Lines
- Estimation of the Value of One Variable Using the Value of Other Variable from Appropriate Line of Regression

- Lines of regression of x on y and y on x.
- Scatter diagrams
- The method of least squares.
- Lines of best fit.
- Regression coefficient of x on y and y on x.
`b_(xy)xxb_(yx)=r^2,0<+b_(xy)xxb_(yx)<+1`

- Identification of regression equations
- Angle between regression line and properties of regression lines.
- Estimation of the value of one variable using the value of other variable from appropriate line of regression.
- Self-explanatory

- Introduction of Linear Programming
- Mathematical Formulation of Linear Programming Problem
- Different Types of Linear Programming Problems
Different types of linear programming (L.P.) problems:-

- Manufacturing problem
- Diet Problem
- Transportation problem

- Graphical Method of Solving Linear Programming Problems
- Advantages and Limitations of Linear Programming

Introduction, related terminology such as constraints, objective function, optimization, advantages of linear programming, limitations of linear programming

application areas of linear programming:-

- different types of linear programming (L.P.) problems,
- mathematical formulation of L.P. problems,
- graphical method of solution for problems in two variables,
- feasible and infeasible regions(bounded and unbounded),
- feasible and infeasible solutions,
- optimal feasible solutions (up to three non-trivialconstraints).