| Mathematics |
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| 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. |
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| Trigonometry: |
Trigonometric functions, their periodicity and graphs,
addition and subtraction formulae, formulae involving multiple
and sub-multiple 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). |
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| Analytical geometry: |
Two 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. |
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. |
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| 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. |
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| 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. |
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| Vectors: |
| Addition of vectors, scalar multiplication, dot and
cross products, scalar triple products and their geometrical
interpretations. |
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