IIST: ISAT 2012 Mathematics
PERMUTATIONS AND COMBINATIONS:
Fundamental principle of counting. Permutations and Combinations, derivation
of formulae and their connections and simple applications.
Principle of Mathematical Induction and its simple applications.
BINOMIAL THEOREM AND ITS SIMPLE APPLICATIONS:
Binomial theorem for positive integral indices, general term and middle term,
properties of Binomial coefficients and simple applications.
SEQUENCES AND SERIES:
Arithmetic and Geometric progressions, insertion of arithmetic, geometric
means between two given numbers. Relation between A.M. and G.M. Sum upto n terms
of special series n, n2, n3. Arithmetico - Geometric sequence.
Trigonometric functions. Trigonometrical identities and equations. Inverse
Trigonometric functions, their properties and applications.
COMPLEX NUMBERS AND QUADRATIC EQUATIONS:
Complex numbers as ordered pairs of reals. Representation of
complex numbers in a plane. Argand plane and polar representation of complex
numbers. Algebra of complex numbers, modulus and argument (or amplitude) of a
complex number, square root of a complex number, triangle inequality. Quadratic
equations in real and complex number system and their solutions. Relation
between roots and coefficients, nature of roots, formation of quadratic
equations with given roots.
SETS, RELATIONS AND FUNCTIONS:
Sets and their representations. Union, intersection and
complement of sets and their algebraic properties. Power Set. Relation, types of
relations and equivalence relation. One-one, into and onto functions and
composition of functions. Real - valued functions, algebra of functions,
polynomials, rational, trigonometric, logarithmic and exponential functions,
inverse functions. Graphs of simple functions. Even and odd functions.
LIMIT, CONTINUITY AND DIFFERENTIABILITY:
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. Differentiability of functions.
Differentiation of the sum, difference, product and quotient of two functions.
Differentiation of trigonometric, inverse trigonometric, logarithmic,
exponential, composite and implicit functions; derivatives of order up to two.
Rolle’s and Lagrange’s Mean Value Theorems. Applications of derivatives: rate of
change of quantities, monotonic - increasing and decreasing functions, maxima
and minima of functions of one variable, tangents and normals.
Integral as an anti-derivative. Fundamental integrals
involving algebraic, trigonometric, exponential and logarithmic functions.
Integration by substitution, by parts and by partial fractions. Integration
using trigonometric identities. Definite Integral as limit of a sum. Fundamental
Theorem of Calculus. Properties of definite integrals. Evaluation of definite
integrals. Applications of the integrals: determining areas of the regions
bounded by simple curves in standard form.
Ordinary differential equations, their order and degree.
Formation of differential equation whose general solution is given. Solution of
differential equations by the method of separation of variables. Solution of
homogeneous differential equations and linear first order differential
Cartesian coordinate system, distance formula, section formula, locus and its
equation, translation of axes, slope of a line, parallel and perpendicular
lines, intercepts of a line on the coordinate axes.
Straight lines : Various forms of equations of a line,
intersection of lines, angles between two lines, conditions for concurrence of
three lines, distance of a point from a line, equations of internal and external
bisectors of angles between two lines, coordinates of centroid, orthocentre and
circumcentre of a triangle, equation of family of lines passing through the
point of intersection of two lines. Circles, Conic sections : Standard equation
of a circle, general form of the equation of a circle, its radius and centre,
equation of a circle when the end points of a diameter are given, points of
intersection of a line and a circle with the centre at the origin and condition
for a line to be tangent to a circle, equation of the tangent. Sections of a
cone, standard equations and properties of conic sections (parabola, ellipse and
hyperbola), condition for y = mx + c to be a tangent and point (s) of tangency.
THREE DIMENSIONAL GEOMETRY:
Coordinates of a point in space, distance between two points,
section formula. Direction ratios and direction cosines of a line joining two
points, angle between two intersecting lines. Coplanar and Skew lines, the
shortest distance between two lines. Equations of a line and a plane in
different forms, intersection of a line and a plane.
Scalars and vectors, addition of vectors, components of a vector in two
dimensional and three dimensional spaces, scalar and vector products scalar and
vector triple product.
MATRICES AND DETERMINANTS:
Matrices, algebra of matrices, types of matrices, elementary
row and column operations. Determinant of matrices of order two and three.
Properties of determinants, area of triangles using determinants. Adjoint and
inverse of a square matrix. Test of consistency and solution of system of linear
equations in two or three variables using inverse of a matrix.
STATISTICS AND PROBABILITY:
Measures of Dispersion: Calculation of mean, median, mode of
grouped and ungrouped data. Calculation of standard deviation, variance and mean
deviation for grouped and ungrouped data. Probability: Probability of an event,
addition and multiplication theorems of probability, Baye’s theorem, probability
distribution of a random variable, Bernoulli trials and Binomial distribution.