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Mathematics - Syllabus...

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
  •              nCr-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.

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