# Mathematics Syllabus XII

Teaching hours 150

### Unit 1: Permutation and Combination (10 Hours)

• Basic Principle of counting
• Permutation of (a) set of objects all different
• (b) Set of objects not all different
• (c) circular arrangement
• (d) repeated use of the same object
• Combination of things all different
• Properties of combination.

### Unit 2: Binomial Theorem (10 Hours)

• Binomial theorem for a positive integral index
• General term
• Binomial coefficients
• Binomial theorem for any index (Without proof)
• Application to approximation
• Euler's number
• Expansion of ex, ax and log(1+x) (without proof)

### Unit 3: Elementary Group Theory (8 Hours)

• Binary operation
• Binary operation on sets of integers and their properties
• Definition of a Group
• Groups whose element are not numbers
• Finite and infinite groups
• Uniquences of identity
• Uniquences of inverse
• Cancellation law
• Abelian Group

### Unit 4: Conic Sections (12 Hours)

• Standard equation of parabola, Ellipse and Hyperbola
• Equations of tangent and normal to a parabola at a given point

### Unit 5: Co - Ordinates in Space (12 Hours)

• Co - ordinate axes
• Co - ordinate planes
• The octants
• Distance between two points
• External and internal point of division
• Direction cosines and ratios
• fundamental relation between direction cosines
• Projections
• Angle between two lines
• General equation of a plane
• Equation of a plane in intercept and normal form
• Plane through three given points
• Plane through the intersection of two given planes
• Parallel and perpendicular planes
• Angle between two planes
• Distance of a point from a plane.

### Unit 6: Vectors and Its Applications (14 Hours)

• Cartesian representation of vectors
• Collinear and non - collinear vectors
• Coplanar and non-Coplanar vectors
• Linear combination of vectors
• Scalar product of two vectors
• Angle between two vectors
• Geometric interpretation of scalar product
• Properties of Scalar Product
• Condition of perpendicularity
• Vector product of two vectors
• Geometric interpretation of vector product
• Properties of vector product
• Application of product of vectors in plane trigonometry.

### Unit 7: Derivative and Its Application (14 Hours)

• Derivative of inverse trigonometric, exponential and logarithmic functions by definition
• Relationship between continuity and differentiability
• Rules for differentiating hyperbolic function and inverse hyperbolic function
• Composite function and function of the type f(x)^g(x)
• L'Hospital's rule (for 0/0, ∞/∞)
• Differentials
• Tangent and Normal
• Geometric interpretation and application of Rolle's theorem and Mean value theorem.

### Unit 8: Antiderivatives (7 Hours)

• Antiderivatives
• Standard Integrals
• Integrals reducible to standard forms
• Integrals of rational functions.

### Unit 9: Differential Equations and their Applications (7 Hours)

• Differential equation and its order and degree
• Differential equations of first order and first degree
• Differential equations with separable variables
• Homogeneous and exact differential equations.

### Unit 10: Dispersion, Correlation and Regression (12 Hours)

• Dispersion
• Measures of dispersion (Range, Semi interquartile range, Mean deviation, Standard deviation)
• Variance
• Coefficient of variation
• Skewness
• Karl Pearson's and Bowley's Coefficient of Skewness
• Bivariate distribution
• Correlation
• Nature of correlation
• Correlation coefficient by Karl Pearson's method
• Interpretation of correlation coefficient
• Properties of correlation coefficient (Without proof)
• Regression equation
• Regression line of y on x and x on y.

### Unit 11: Probability (8 Hours)

• Random experiment
• Sample space
• Event
• Equally likely cases
• Mutually exclusive events
• Exhaustive cases
• Favourable cases
• Independent and dependent cases
• Mathematical and empirical definition of probability
• Two basic laws of probability
• Conditional probability (without proof)
• Binomial distribution
• Mean and Standard deviation of binomial distribution (without proof)

### Unit 12: Statics (9 Hours)

• Force and Resultant forces
• Parallelogram of forces
• Composition and resolution of forces
• Resultant of coplanar forces acting at a point
• Triangle of forces and Lami's theorem

### Unit 13: Statics (Continued) (9 Hours)

• Resultant of like and unlike parallel forces
• Moment of a force
• Varignon's theorem
• Couple and its properties (without proof).

### Unit 14: Dynamics (9 Hours)

• Motion of particle in a straight line
• Motion with uniform acceleration
• Motion under gravity
• Motion down a smooth inclined plane
• The concepts and theorems can be restated and formulated as application of calculus.

### Unit 15: Dynamics (Continued) (9 hours)

• Newton's laws of motion
• Impulse
• Work
• Energy and Power
• Projectiles.

### Unit 16: Linear Programming (11 Hours)

• Introduction of a linear programming problem (LPP)
• Graphical solution of LPP in two variables
• Solution of LPP by simplex method (two variables)

### Unit 17: Computational Method (9 Hours)

• Introduction to Numerical computing (Characteristics of Numerical computing, Accuracy, Rate of Convergence, Numerical Stability, Efficiency)
• Number systems (Decimal, Binary, Octal & Hexadecimal system conversion of one system into another)
• Approximation and error in computing Roots of nonlinear equation
• Algebric, Polynomial & Transcendental equations and their solution by bisection and Newton - Raphson Methods.

### Unit 18: Computational Method (Continued) (8 Hours)

• Solution of system of linear equations by Gauss elimination method
• Gauss - Seidel method
• Ill Conditioned systems
• Matrix inversion method

### Unit 19: Numerical Integration (8 Hours)

• Trapezoidal and Simpson's rules
• Estimation of errors

### Evaluation Scheme:

 No. of questions Marks Total Remarks 15 2 30 Covering all Units 10 4 40 With four OR - questions from the same. 5 6 30 With two OR - questions from the same.