## Topics with syllabus and resources

(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.

- Concepts :
- Introduction of Relations and Functions videos (3)
- Types of Relations videos (18) question (90)
- Types of Relations - Identity Relation
- Types of Functions videos (11) question (30)
- Composition of Functions and Invertible Function videos (11) question (22)
- Inverse of a Function question (3)
- Concept of Binary Operations videos (8) question (35)
- All Axioms and Properties
- Conditions of Invertibility
- Basic Concepts of Trigonometric Functions videos (4) question (37)
- Inverse Trigonometric Functions - Principal Value Branch videos (3) question (40)
- Graphs of Inverse Trigonometric Functions
- Properties of Inverse Trigonometric Functions videos (25) question (48)

**(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.

- Concepts :
- Introduction of Matrices videos (9) question (1)
- Matrices Notation question (1)
- Order of a Matrix question (18)
- Equality of Matrices videos (1) question (15)
- Types of Matrices videos (3) question (23)
- Symmetric and Skew Symmetric Matrices videos (4) question (28)
- Concept of Transpose of a Matrix videos (3) question (3)
- Addition of Matrices videos (4) question (23)
- Multiplication of Two Matrices videos (9) question (3)
- Elementary Operation (Transformation) of a Matrix videos (3) question (11)
- Multiplication of Matrices videos (2) question (10)
- Invertible Matrices videos (1) question (25)
- Proof of the Uniqueness of Inverse question (1)
- Introduction of Determinant videos (2) question (2)
- Determinants of Matrix of Order One and Two videos (2) question (16)
- Determinant of a Square Matrix videos (3) question (1)
- Determinant of a Matrix of Order 3 × 3 videos (3) question (7)
- Properties of Determinants videos (14) question (36)
- Minors and Co-factors videos (2) question (9)
- Area of a Triangle videos (3) question (9)
- Adjoint and Inverse of a Matrix videos (6) question (23)
- Applications of Determinants and Matrices videos (7) question (21)
- Martin’S Rule

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

- Concepts :
- Concept of Continuity videos (10) question (25)
- Continuous Function of Point videos (11) question (11)
- Algebra of Continuous Functions videos (4) question (16)
- Exponential and Logarithmic Functions videos (6) question (16)
- Derivatives of Composite Functions - Chain Rule videos (4) question (2)
- Derivatives of Inverse Trigonometric Functions videos (7) question (23)
- Derivatives of Implicit Functions videos (5) question (13)
- Logarithmic Differentiation videos (8) question (27)
- Derivatives of Functions in Parametric Forms videos (8) question (28)
- Mean Value Theorem videos (5) question (14)
- Second Order Derivative videos (3) question (22)
- L' Hospital'S Theorem question (1)

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.

- Concepts :
- Introduction to Applications of Derivatives videos (1)
- Rate of Change of Bodies Or Quantities videos (8) question (28)
- Increasing and Decreasing Functions videos (8) question (40)
- Tangents and Normals videos (9) question (45)
- Approximations videos (3) question (33)
- Maxima and Minima videos (9) question (68)
- Maximum and Minimum Values of a Function in a Closed Interval videos (12) question (5)
- Simple Problems on Applications of Derivatives videos (4) question (5)

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,`

- Concepts :
- Introduction of Integrals videos (15)
- Integration as an Inverse Process of Differentiation videos (1) question (52)
- Methods of Integration - Integration by Substitution videos (13) question (63)
- Methods of Integration - Integration Using Partial Fractions videos (7) question (32)
- Methods of Integration - Integration by Parts videos (6) question (33)
- Properties of Indefinite Integral videos (1) question (3)
- Anti-derivatives of Polynomials and Functions question (1)
- Evaluation of Simple Integrals of the Following Types and Problems question (2)
- Definite Integral as the Limit of a Sum videos (9) question (31)
- Fundamental Theorem of Calculus videos (3) question (26)
- Properties of Definite Integrals videos (11) question (36)
- Evaluation of Definite Integrals by Substitution videos (2) question (27)

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.

- Concepts :
- Basic Concepts of Differential Equation videos (3) question (2)
- Order and Degree of a Differential Equation videos (4) question (21)
- General and Particular Solutions of a Differential Equation videos (7) question (50)
- Formation of a Differential Equation Whose General Solution is Given videos (5) question (15)
- Formation of Differential Equation by Eliminating Arbitary Constant question (3)
- Differential Equations with Variables Separable videos (4) question (28)
- Homogeneous Differential Equations videos (6) question (25)
- Solutions of Linear Differential Equation videos (2) question (6)
- Application on Growth and Decay
- Solve Problems on Velocity, Acceleration, Distance and Time
- Solve Population Based Problems on Application of Differential Equations question (2)
- Application on Coordinate Geometry

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.

- Concepts :
- Introduction of Probability videos (1)
- Dependent Events
- Conditional Event
- Conditional Probability videos (12) question (49)
- Multiplication Theorem on Probability videos (1) question (2)
- Independent Events videos (5) question (27)
- Baye'S Theorem videos (8) question (18)
- Addition Theorem
- Random Variables and Its Probability Distributions videos (10) question (30)
- Mean of a Random Variable videos (4) question (7)
- Bernoulli Trials and Binomial Distribution videos (3) question (25)
- Laws of Probability
- Probability Distribution Function question (1)
- Mean of Binomial Distribution (P.M.F.)
- Variance of Binomial Distribution (P.M.F.) question (1)

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.**

- Concepts :
- Magnitude and Direction of a Vector question (7)
- Types of Vectors videos (4) question (2)
- Basic Concepts of Vector Algebra videos (4) question (26)
- Components of a Vector videos (5) question (5)
- Addition of Vectors videos (2) question (10)
- Operations - Sum and Difference of Vectors
- Multiplication of a Vector by a Scalar videos (1) question (5)
- Position Vector of a Point Dividing a Line Segment in a Given Ratio question (1)
- Geometrical Interpretation of Scalar question (1)
- Scalar (Or Dot) Product of Two Vectors videos (9) question (14)
- Vector (Or Cross) Product of Two Vectors videos (8) question (15)
- Scalar Triple Product of Vectors videos (1) question (13)
- Section formula videos (3) question (7)

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.

- Concepts :
- Direction Cosines and Direction Ratios of a Line videos (5) question (16)
- Equation of a Line in Space videos (7) question (27)
- Shortest Distance Between Two Lines videos (6) question (15)
- Vector and Cartesian Equation of a Plane videos (6) question (26)
- Angle Between Two Lines videos (8) question (9)
- Angle Between Two Planes videos (3) question (6)
- Angle Between Line and a Plane videos (1) question (7)
- Intercept Form of the Equation of a Plane videos (3) question (5)
- Distance of a Point from a Plane videos (3) question (13)
- Direction Ratios of the Normal to the Plane.
- Intersection of the Line and Plane question (1)
- Equation of Plane Passing Through the Intersection of Two Given Planes videos (1) question (3)
- Equation of Line Passing Through Given Point and Parallel to Given Vector videos (1) question (3)

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

- Concepts :
- Area Under Simple Curves videos (9) question (34)
- Area of the Region Bounded by a Curve and a Line videos (5) question (23)
- Area Between Two Curves videos (4) question (15)
- Application of Integrals - Polynomial Functions
- Application of Integrals - Modulus Function
- Application of Integrals - Trigonometric Function
- Application of Integrals - Exponential Functions
- Application of Integrals - Logarithmic 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.**

- Concepts :
- Application of Calculus in Commerce and Economics in the Cost Function
- Application of Calculus in Commerce and Economics in the Average Cost question (1)
- 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 question (4)
- Rough Sketching

- 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

- Concepts :
- Lines of Regression of X on Y and Y on X Or Equation of Line of Regression videos (1) question (2)
- Scatter Diagram
- The Method of Least Squares
- Lines of Best Fit
- Regression Coefficient of X on Y and Y on X question (2)
- 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

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).

- Concepts :
- Introduction of Linear Programming videos (1)
- Mathematical Formulation of Linear Programming Problem videos (2) question (1)
- Different Types of Linear Programming Problems videos (2) question (19)
- Graphical Method of Solving Linear Programming Problems videos (6) question (23)
- Advantages and Limitations of Linear Programming

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Mathematics - I.S.C. - CISCE (Council for the Indian School Certificate Examinations) question paper - 2013 - 2014 March

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