ALGEBRA
THEORY OF INDICES AND LOGARITHMS
- Recapitulation of theory of Indices
- problems
- Laws of logarithms (with proof) -
problems
|
|
|
|
PROGRESSIONS |
- Recapitulation of sequences of real
numbers, finite and infinite sequences as mappings.
- Definition of infinite series, A.P.,
G.P., H.P,. nth term of an AP, GP, HP, sum to n terms of an AP, GP (with proof) - problems
- Sum to infinity of a G.P. when the
common ratio r is such that -1 < r < 1. Recurring decimal numbers - problems.
- A.M., G.M., H.M. of two numbers a
and b. Proofs of G2 = AH and A P G P H , where A, G H are the A.M., G.M., and
H.M. respectively of any two numbers a and b. To insert n arithmetic means, n geometric
means and n harmonic means between any two given numbers - problems
|
|
MATHEMATICAL INDUCTION |
- Principle of mathematical induction.
Problems on induction including Sn, Sn2, Sn3
|
|
THEORY OF EQUATIONS |
- Recapitulation of quadratic
equations and the formula for the roots of a quadratic equation.
- The equation x2 + 1 = 0
and introducing complex numbers, square roots, cube roots and fourth roots of unity.
- The relations between the roots and
coefficients of a quadratic equation, a cubic equation and a biquadratic - equation.
Solutions of quadratic, cubic and biquadratic equations given certain conditions and given
that the roots are in A.P., G.P., H.P. - problems.
- Symmetric functions of the roots of
quadratic, cubic and biquadratic equations - problems.
- Proofs of (i) irrational roots of a
polynomial equation occur in conjugate pairs, (ii) complex roots of a polynomial equation
occur in conjugate pairs - Problems of solving equations given an irrational root and
given a complex root - problems.
- Solution of a standard cubic
equation X3 + 3HX + G = 0 by Cardan's method only - problems.
|
|
PERMUTATIONS AND COMBINATIONS
|
- Definition of linear permutation,
derivation of the formula for nPr from first principles. Formula for
the number of permutations when some things are alike of one kind, etc. - problems
- Circular permutation - formula -
problems.
- Definition of combination,
derivation of the formula for nCr, from first principles. Proofs of nCr
= nCn-r and
n Cr-1 + nCr = n+1Cr
- problems
|
|
BINOMIAL THEOREM |
- Statement and proof of Binomial
theorem for a positive integral index by induction. To find the middle terms, terms
independent of x and term containing a definite power of x - problems.
- Binomial coefficients - problems.
|
|
PARTIAL FRACTIONS |
- Rational fractions, proper and
improper fractions, reduction of an improper fraction into a sum of a polynomial and a
proper fraction - problems
- Rules for resolving a proper
fraction into partial fractions. - problems
|
|
ELEMENTS OF NUMBER THEORY AND
CONGRUENCES |
- Divisibility - Definition and
properties of divisibility, statement of Division Algorithm.
- Greatest Common Divisor (G.C.D.) of
any two integers, using Euclid,s Algorithm., to find the G.C.D. of any two integers. To
express the G.C.D. of two integers a and b as ax + by for integers x and y - problems
- Relatively prime numbers, prime
numbers and composite numbers, the number of positive divisors of a number and sum of all
positive divisors of a number - statements of the formulae without proof - problems.
- Proofs of the following properties
(1) The smallest divisor > 1 of an integer > 1 is a prime number.
(2) There are infinity of primes.
(3) If c and a are relatively prime and c|ab then c|b
(4) If p is prime and p|ab then p/a or p|b
(5) If there exist integers x and y such that ax + by = 1 then (a,b) = 1
(6) If (a,b) = 1, (a,c) = 1 then (a, bc) = 1
(7) If p is prime and a is any integer then either (p,a) = 1 or p | a
(8) The smallest positive divisor of a composite number "a" does not axceed a
- Congruence modulo m - Definition,
Proofs of the following properties
(1) "Lmode m" is an equivalence relation
(2) a L b (mod m) => a ExLbEx (mod m) and ax L bx (mod m)
(3) If c is relatively prime to m and ca Lcb (mod m) then a L b (mod m) - cancellation law
(4) If a L b (mod m) and n is a positive divisor of m then a L b (mod n)
(5) a L b (mod m) => a and b leave the same remainder when divided by m
- Conditions for the existence of the
solution of linear congruence ax L b (mod m) (statements only). to find the solution of ax
L b (mod m) - problems
|
|
- Set theory: Recapitulation of
sets. Relations and functions. Pmblems.
|
|
- Mathematical Logic:
Proposition and truth values, connectives, their truth tables, inverse, converse,
contrapositive of a proposition. Tautology and contradiction, Logical Equivalence-
standard theorems. Examples hom switching cirruits. Truth tables. Problems.
|
|
- Matrices and determinants:
Racapitulation of types of matrices and problems. Determinant of a square matrix defined
as mappings
D : M (2,R) Y R and D: M(3,R) Y R
Properties of determinants including D (AB) = D (A) D (B). Problerns.
Minor and cofactor of an element of a square matrix, adjoint, singular and non-singular
matrices. Inverse of a matrix, proof of a A (adjA) = (adjA) A = | A | I and hence formula
for A-1 . Problems.
Solution of a system of linear equations in two and three variables - (i) Matrix method,
(ii) Cramer's rule. Problems.
Characteristic equation of a square matrix.
Charactedstic roots of a square matrix, Cayley Hamilton theorem (statement
only), Verification of Cayley Hamilton theorem for square matrices of order 2 and 3
only. Finding A-1 by Cayley Hamilton theorem. Problems.
|
|
- Vector: Recapitulation of a
vector as directed line segment, magnitude and direction of a vector, equal vectors, unit
vector, position vector of a point. Problems.
Two and three dimensional vectors as ordered pair and ordered triplet respectively of real
numbers-components of a vector, Addition, subtraction, scalar multiplication of a vector.
Problems.
Position vector of the point dividing a given line segment in the given ratio.
Sclar (dot) product, Vector (Cross) product of two vectors, Scalar triple (Box) product of
three vectors, vector triple product of three vectors and their properties. Application of
these to the area of a parallelogram, area of a triangle, Volume of a parallelopiped,
orthogonal vectors and coplanarity of three vectors, projection of one vector on another
vector. Problems.
Work done, moment of a force about a point, moment of a couple about a point, the
diagonals of a parallelogram bisect each other, angle in a semi circle is a right angle,
medians of a triangle are concurrent. Derivations and problems.
|
|
- Groups: Binary operation,
Algebraic structures. Definition of semigroup, group, Albelian group - Examples from real
and complex numbers. Finite and infinite groups, order of a group, composition tables,
modular systems, modular groups, group of matrices - Problems.
Permutations, symmetric group of order 3.
Proof of "The set of all permutations of the set S={1,2,3}from a non-abelian group
w.r.t. product of permutations"
Square roots, cube roots and fourth
roots of unity from abelian groups w.r.t. multiplication
Proofs of the following properties
- Identity of a group is unique.
- The inverse of an element of a group
is unique.
- (a-1)-1 = a"
aX G where G is a group.
- (a * b)-1 = b-1
* b-1 in a group.
- Left and right cancellation laws.
- Solution of a * x = b and y * a = b
exist and are unque in a group.
Subgroups, proofs of necessary and sufficient conditions for a subgroup.
- A non-empty subset H of a group G is
a subgroup of G iff
i) " a, b X H, a * b X H and
ii) for each a X H, a-1X H
- A non-empty subset H of a group G is
a subgroup of G iff a,b X H, a * b-1X H - Problems
Problems of the type
i) If (ab)-1 = a-1
b-1, then G is abelian
ii) If every element of a group is its own inverse, then G is abelian.
iii) In a group of even order there exists an element a K e such that a-1
= a
|
|
CALCULUS
- Functions of a real variable, types
of functions, periodic functions, functional value - problems.
- Limit of a function - definition,
statements of the algebra of limits - problems
- Standard limits (with proofs)
Statements of the limits
(i) lim nYh (1+ 1/n)n = e
(ii) lim xY 0 (1 + x) 1/x = e
(iii) lim xY 0 loge (1 + x) /x = 1
(iv) lim xY 0 ex - 1 / x = 1
(v) lim xY 0 ax - 1 / x = loge a
Problems on these limits
Evaluation of limits where degree of f(n) O degree g (n) problems
- Continuity and differentiation:
Continuity of a fnction, sum of two functions, polynomial, trigonometric function,
exponential function, inverse trigonometric function. Problems.
- Differentiation -
Differenctiability. Derivative of a function by first principles. Differentialbility
implies continuity by the converse is not true (proof and example respectively).
Derivatives of sum, difference, product of a constant and a function, constant, product of
two functions, quotient of two functions by first principles.
- Derivatives of xn, ex,
ax, sinx, cos x, tan x, cosec x, sec x, cot x, log x by first
principles. Problems.
- Derivatives of inverse trigonometric
functions by first principles hyperbolic and inverse hyperbolic functions and their
derivatives w.r.t. x. Problems.
- Composite functions - Chain rule.
Problems.
- Differenctiation of inverse
trigonometric functions by substitution. Problems.
- Differenctiation of implicit
functions, parametric functions, a function w.r.t. another function, logarithmic
differenctiation. Problems.
- Successive differentiation -
Problems of finding second derivatives, deriving second order differential equations.
- Applications of Derivatives: Geometrical
meaning of dy / dx, Equations of tangent and normal, angle between two curves.
Problems.
- Subtangent and subnormal. Problems.
- Derivative as the rate measure.
Problems.
- Maxima and minima of a function of a
single variable - Problems.
Also problems involving two dimensional figures only.
- Interation: Statement of
fundamental theorem of integral calculus.
Intereation as the reverse process of differentiation. Standard for mulae, methods of
integration (i) substitution (ii) partial fractions (iii) integration by parts. Problems.
- Definite Intregrals: Evaluation
of definite integrals, properties of definite integrals. Problems.
- Application of Definite
Integrals: Area under a curve, area enclosed between two curves usig definite
integrals, standard areas likearea of circle, ellipse, parabola etc. Problems.
- Differential Equations:
Definition of order and degree of a differential equation. Formation of a first order
differential equation. Problems. solution of first order differential equations by the
method of separation of variables. Probles.
|
|
TRIGONOMETRY
MEASUREMENT OF ANGLES AND
TRIGONOMETRIC FUNCTIONS
- Radian measure - Definition. Proofs
of (i) p radians = 1800, (ii) 1 radian is constant, (iii ) s = rq
where q is in radians, (iv) Area of the sector of a circle given by A=1/2 r2q
where q is in radians - problems
- Trigonometric functions -
definitions.
- Trigonometric ratios of an acute
angle.
- Trigonometric identities (with
proofs), problems
- Trigonometric functions of
standard angles, problems.
- Trigonometric functions of allied
angles, compound angles, multiple angles, submultiple angles and Transformation formulae
(with proofs) - problems.
- Heights and distances - Angle of
elevation, angle of depression, problems.
- Graphs of Trigonometric functions
RELATIONS BETWEEN SIDES AND ANGLES OF A TRIANGLE
- Sine rule, Cosine rule, Tangent
rule, Half-angle formulae, area of a triangle, projection rule (with proofs) - problems.
- Solution of triangles given
(i) three sides, (ii) two sides and the included angle, (iii) two angles and a side, (iv)
two sides and the angle opposite to one of these sides .
- Problems.
- Inverse Trigonometric Functions:
Definition of inverse trigonometric functions, domain and range. Derivations of standard
formulae. Problems.
Solutions of inverse trigonometric equations. Problems.
- General Solutions of
trigonometric equations:
General Solutions of sin x = k, cos x = k, (-1 O k O 1), tan x = k, a cos x + b sin x = c,
derivations. Problems. Complex
Numbers: Definition of a complex number as an ordered pair, real and imaginary parts,
modulus and amplitude of a complex number, equality of complex numbers. Algebra of complex
numbers, polar form of a complex number, Argand Diagram. Exponential form of a complex
number. Problems.
De Moivre's theorem - statement and
proof, method of finding square roots, cube roots and fourth roots of a complex number and
their representation in the Argand diagram. Problems.
|
|
ANALYTICAL GEOMETRY
COORDINATE GEOMETRY
- Coordinate system in a plane
(cartesian)
- Distance formula, section formula,
mid-point formula, centroid of a triangle, area of a triangle - Derivations, problems
- Locus of a point, problems
- Straight lines, slope of a line m =
tanq where q is the angle made by the line with the positive x-axis, slope of the
line joining any two points, general equation of a line. Derivation and problems
- Conditions for parallelism and
perpendicularity of two lines - problems
- Various forms of the equation of a
straight line : slope - point form, slope - intercept form , two point form, intercept
form, Normal form - Derivations - problems
- Angle between two lines, point of
intersection of two lines, condition for concurrency of three lines, Length of the
perpendicular from the origin and from a point to a line, Equation of the inernal and
external bisector of the angle between two lines - Derivations, problems
- Pair of lines - Homogeneous equation
of second degree, general equation of second degree, derivations of (1) condition for pair
of lines, (2) condition for a pair of parallel lines, perpendicular lines and distance
between the pair of parallel lines, (3) condition for a pair of coincident lines (4) angle
and point of intersections of a pair of lines - problems.
- Circles: Definition, equation
of a circle, with centre (0,0) and radius r, with centre (h,k) and radius r. Equation of a
circle with (x1, y1) and (x2, y2) as the ends
of a diameter, general equation of a circle, centre and radius. Derivations of all these.
Problems.
Equation of the tangent to a circle - Derivation. Problems
Condition for y = mx + c to be the tangent to the circle x2 + y2
= r2 Derivation. Problems.
Length of the tangent from an external point to a circle - Derivation, Problems.
Power of a point, radical axis of two circles, radical centre of a system of three circles
- Derivation, Problems.
Condition for a point to be inside or outside or on a circle. Proof and problems. Proof of
"The radical axis of two circles is perpendicular to the line joining their
centres." Problems.
Orthogonal circles - Derivation of the condition. problems.
Co-axal system, limiting points, conjugate system. Problems.
- Conic Sections: Definition by
focus - directrix property, eccentricity, definition of Parabola, Ellipse, Hyperbola,
Rectangular hyperbola.
Derivation of standard equation of ellipse. Equation of other forms of ellipse (statements
only). Standard properties of parabola. Problems.
Derivation of standard equation of ellipse. Equation of other forms of ellipse (statements
only). Standard properties of ellipse. Problems.
Derivation of standard equation of hyperbola. Equations of other forms of hyperbola
(statemnets only) Standard properties of hyperbola. Problems.
Equations of tangent, Derivations and problems.
|
|