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Mathematics

Algebra

• Complex Number:
  • Algebra of complex numbers, addition and multiplication of complex number
  • conjugation of complex number
  • polar representation
  • properties of modulus and principal argument
  • triangle inequality
  • cube roots of unity
  • geometric interpretations
• Quadratic equations :
  • Quadratic equations with real coefficients
  • relations between roots and coefficients
  • formation of quadratic equations with given roots
  • symmetric functions of roots
• Progression:
  • Arithmetic, geometric and harmonic progressions (AP , GP , HP)
  • Arithmetic, geometric and harmonic means(AM , GM ,HM)
  • Sums of finite arithmetic and geometric progressions
  • Infinite geometric series
  • sums of squares and cubes of the first n natural numbers
• Logarithms and their properties
• Permutations and combinations
• Binomial theorem for a positive integral index, properties of binomial coefficients

Matrices

• Matrices as a rectangular array of real numbers
• Equality of matrices
• Addition, multiplication by a scalar and product of matrices
• Transpose of a matrix
• Determinant of a square matrix of order up to three
• Inverse of a square matrix of order up to three
• Properties of these matrix operations
• Diagonal, symmetric and skew-symmetric matrices and their properties
• Solutions of simultaneous linear equations in two or three variables

Probability

• Addition and multiplication rules of probability
• Conditional probability
• Bayes Theorem
• Independence of events
• Computation of probability of events using permutations and combinations

Trigonometry

• Trigonometric functions
• periodicity and graphs
• Addition and subtraction formulae
• General solution of trigonometric equations.
• Relations between sides and angles of a triangle,
• Sine rule, cosine rule , half angle formula
• The area of a triangle
• Inverse trigonometric functions

Analytical geometry

• Two dimensions:
  • Cartesian coordinates, distance between two points, section formulae, shift of origin.
  • Straight Line :
    • Equation of a straight line in various forms
    • angle between two lines, distance of a point from a line
    • Lines through the point of intersection of two given lines
    • equation of the bisector of the angle between two lines
    • concurrency of lines
    • Centroid, orthocentre, incentre and circumcentre of a triangle.
  • Circle :
    • Equation of a circle in various forms
    • equations of tangent, normal and chord.
    • Parametric equations of a circle
    • Intersection of a circle with a straight line or a circle
    • equation of a circle through the points of intersection of two circles and those of a circle and a straight line.
  • Equations of a parabola, ellipse and hyperbola in standard form, their foci, directrices and eccentricity, parametric equations, equations of tangent and normal. Locus problems.
• Three dimensions:
• Direction cosines and direction ratios
• equation of a straight line in space,
• equation of a plane
• distance of a point from a plane.

Differential calculus

• Function :
  • Real valued functions of a real variable, into, onto and one-to-one functions, sum, difference, product and quotient of two functions
  • composite functions, absolute value, polynomial, rational, trigonometric, exponential and logarithmic functions.
  • Even and odd functions, inverse of a function
• Limit and Continuity :
  • Limit and continuity of a function
  • limit and continuity of the sum, difference, product and quotient of two functions
  • L’Hospital rule of evaluation of limits of functions
  • continuity of composite functions
  • intermediate value property of continuous functions
• Differentiation :
  • Derivative of a function
  • derivative of the sum, difference, product and quotient of two functions
  • chain rule
  • derivatives of polynomial, rational, trigonometric, inverse trigonometric, exponential and logarithmic functions.
  • Derivatives of implicit functions
  • derivatives up to order two
• Application :
  • geometrical interpretation of the derivative
  • tangents and normals
  • increasing and decreasing functions
  • maximum and minimum values of a function
  • Rolle’s theorem and Lagrange’s mean value theorem

Integral calculus

• Integration as the inverse process of differentiation
• indefinite integrals of standard functions
• definite integrals and their properties
• fundamental theorem of integral calculus
• Integration by parts
• integration by the methods of substitution and partial fractions
• application of definite integrals to the determination of areas involving simple curves.
• Differential Equation :
  • Formation of ordinary differential equations
  • solution of homogeneous differential equations
  • separation of variables method
  • linear first order differential equations

Vectors

• Addition of vectors
• scalar multiplication
• dot and cross products
• scalar triple products and their geometrical interpretations.