Class 12 Maths
1 Relations and Functions – This Chapter is consists of 5 exercises based on the following concepts: Equivalence relations: reflexive, symmetric, transitive, Injective(one-to-one) and Surjective(onto), composite functions, concepts of inverse function and Binary operations.
2 Inverse Trigonometric Functions – This chapter has 2 exercises and 1 Miscellaneous based on the following concepts: Definition, Range, Domain, Principal value of the trigonometric function, Graphs of inverse trigonometric functions. Elementary properties and use of formulas of inverse trigonometric functions.
3 Matrices – This Chapter has 4 exercises and Miscellaneous exercise: Concept, notation, order, equality, types of matrices: Square, Rectangular, Diagonal, Scalar, zero and identity matrix, transpose of a matrix, concepts of symmetric and skew-symmetric matrices and how to express as a sum of both. Operation on matrices. properties related to addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of inverse matrices
Elementary row and column operations to find inverse of matrices. Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).
4 Determinants – This Chapter Includes 6 exercises and 1 Miscellaneous exercise which has the following concepts: Introduction, Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency, and number of solutions of a system of linear equations by examples, solving system of linear equations in two or three variables with the use of inverse matrix concepts.
5 Continuity and Differentiability – This Chapter comprising of 6 exercises and 1 Miscellaneous exercise with the following concepts: Definition of Continuity and differentiability and question based on this, Chain rule, derivative of the following function: composite functions, inverse trigonometric functions, implicit functions. exponential and logarithmic functions, derivatives of functions expressed in parametric forms. Second-order derivatives. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretation.
6 Applications of Derivatives – This Chapter has 5 exercises and 1 Miscellaneous exercise which includes following concepts: rate of change, Concept of increasing/decreasing functions, how to tangents and normal and equation of tangent and normal, use of derivatives in approximation, maxima, and minima using first derivative test and second derivative test. Simple problems (that illustrate basic principles and concepts of the chapters as well as real-life situations).
7 Integrals – This Chapter has 11 exercises with 1 Miscellaneous exercise and contains following concept: Methods of Integration; Simple Integration, Integration of a variety of functions by substitution, by partial fractions and by parts, Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.
8 Applications of the Integrals – This chapter has 2 exercises and 1 Miscellaneous exercise with the following concepts: finding the area under simple curves especially lines, circles/parabolas/ellipses (in standard form only), Area between any of the two curves, circle, parabola, and ellipse.
9 Differential Equations – This chapter is comprising of 6 exercises and 1 Miscellaneous exercise which includes the following concepts: Definition, order and degree, how to general and particular solutions of a differential equation. Formation of differential equations whose general solution is given. Solution of differential equations by the method of homogeneous differential equations of first order and first degree. Solutions of linear differential equation.
10 Vectors – This Chapter has 4 exercises and one Miscellaneous exercise. It includes the following concepts; Vectors or scalars, graphical displacement, 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 using triangle law and parallelogram law, 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, Projection of vector on a line, scalar triple product of vectors.
11 Three–Dimensional Geometry – This chapter has 3 exercises and one Miscellaneous exercise which contains questions based on Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, equation of plane in: Normal form, Perpendicular to vector and passing through point, Passing through three non collinear points, Intercept form and passing through intersection of planes, Coplanarity of two lines, equations of lines under planes conditions, coplanar and skew lines, shortest distance between two lines. Angle between
· two lines
· two planes
· a line and a plane.
· Distance of a point from a plane.
12 Linear Programming – This Chapter has 2 exercises and 1 Miscellaneous exercise which includes questions based on: Introduction, Linear Programming problem and its Mathematical Formulation, mathematical formulation of problem, definition of constraints, objective function, optimization, 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-trivial constraints).
13 Probability – This Chapter has 5 exercises and 1 Miscellaneous exercise which includes questions based on: Conditional probability and its properties, multiplication theorem, independent events, Bayes’ theorem (partition of a sample space), Theorem of total Probability, Random variable and its probability distribution of a random variable, mean and variance of a random variable. Bernoulli trials and Binomial distribution.
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