**Mathematics**

**Algebra: **
Algebra of complex numbers, addition,
multiplication, conjugation, polar representation, properties of modulus and
principal argument, triangle inequality, cube roots of unity, geometric
interpretations.

Quadratic equations
with real coefficients, relations between roots and coefficients, formation of
quadratic equations with given roots, symmetric functions of roots.

Arithmetic, geometric
and harmonic progressions, arithmetic, geometric and harmonic means, 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 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.

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, their periodicity and
graphs, addition and subtraction formulae, formulae involving multiple and
submultiple angles, general solution of trigonometric equations.

Relations between
sides and angles of a triangle, sine rule, cosine rule, half-angle formula and
the area of a triangle, inverse trigonometric functions (principal value only).

**Analytical
geometry (2 dimensions): **Cartesian coordinates,
distance between two points, section formulae, shift of origin.

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.

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.

**Analytical
geometry (3 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: **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.

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.

Even and odd
functions, inverse of a function, continuity of composite functions,
intermediate value property of continuous functions.

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, 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.

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.

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