**MATHEMATICS – I**

**ALGEBRA**

PARTIAL FRACTIONS

Rational functions, proper and improper fractions, reduction of improper fractions as a sum of a polynomial and a proper fraction.

Rules of resolving a rational function into partial fractions in which denominator contains

(i) Linear distinct factors, (ii) Linear repeated factors, (iii) Non repeated non factorizable quadratic factors [problems limited to evaluation of three constants].

LOGARITHMS

(i) Definition Of logarithm

(ii) Indices leading to logarithms and vice versa

(iii) Laws with proofs:

(a) logam+logan = loga(mn)

(b) logam – logan = loga(m/n)

(c) logamn = nlogam

(d) log b m = logam/logab (change of base rule)

(iv) Common Logarithm: Characteristic and mantissa; use of logarithmic tables,problems theorem

MATHEMATICAL INDUCTION

(i) Recapitulation of the nth terms of an AP and a GP which are required to find the general term of the series

(ii) Principle of mathematical Induction proofs of

a. ∑n =n(n+1)/2

b.∑n2 =n(n+1)(2n+1)/6

c. ∑n3 = n2 (n+1)2/4

By mathematical induction

Sample problems on mathematical induction

SUMMATION OF FINITE SERIES

(i) Summation of series using ∑n, ∑n2, ∑n3

(ii) Arithmetico-Geometric series

(iii) Method of differences (when differences of successive terms are in AP)

(iv) By partial fractions

THEORY OF EQUATIONS

(i) FUNDAMENTAL THEOREM OF ALGEBRA: An nth degree equation has n roots(without proof)

(ii) Solution of the equation x2 +1=0.Introducing square roots, cube roots and fourth roots of unity

(iii) Cubic and biquadratic equations, relations between the roots and the co-efficients. Solutions of cubic and biquadratic equations given certain conditions

(iv) Concept of synthetic division (without proof) and problems. Solution of equations by finding an integral root between – 3 and +3 by inspection and then using synthetic division.

Irrational and complex roots occur in conjugate pairs (without proof). Problems based on this result in solving cubic and biquadratic equations.

BINOMIAL THEOREM

Permutation and Combinations:

Recapitulation of nPr and nCr and proofs of

(i) general formulae for nPr and nCr

(ii) nCr = nCn-r

(iii) nCr-1 + n C r = n+1 C r

(1) Statement and proof of the Binomial theorem for a positive integral index by induction. Problems to find the middle term(s), terms independent of x and term containing a definite power of x.

(2) Binomial co-efficient – Proofs of

(a) C 0 + C 1 + C 2 + …………………..+ C n = 2 n

(b) C 0 + C 2 + C 4 + …………………..= C 1+ C 3 + C 5 + ………2 n – 1

MATHEMATICAL LOGIC

Proposition and truth values, connectives, their truth tables, inverse, converse, contrapositive of a proposition, Tautology and contradiction, Logical Equivalence – standard theorems, Examples from switching circuits, Truth tables, problems.

GRAPH THEORY

Recapitulation of polyhedra and networks

(i) Definition of a graph and related terms like vertices, degree of a vertex, odd vertex, even vertex, edges, loop, multiple edges, u-v walk, trivial walk, closed walk, trail, path, closed path, cycle, even and odd cycles, cut vertex and bridges.

(ii) Types of graphs: Finite graph, multiple graph, simple graph, (p,q) graph, null graph, complete graph, bipartite graph, complete graph, regular graph, complete graph, self complementary graph, subgraph, supergraph, connected graph, Eulerian graph and trees.

(iii) The following theorems: p

p

(1) In a graph with p vertices and q edges ∑deg n i = 2 q

i=1

(2) In any graph the number of vertices of odd degree is even.

(iv) Definition of connected graph, Eulerian graphs and trees – simple probles.

ANALYTICAL GEOMETRY

1. Co-ordinate system

(i) Rectangular co-ordinate system in a plane (Cartesian)

(ii) Distance formula, section formula and mid-point formula, centroild of a triangle, area of a triangle – derivations and problems.

(iii) Locus of a point. Problems.

2 .Straight line

(i)Straight line: Slope m = (tanθ) of a line, where θ 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.

(ii) Conditions for two lines to be (i) parallel, (ii) perpendicular. Problems.

(iii) Different forms of the equation of a straight line: (a) slope – point form (b) slope intercept form (c) two points form(d) intercept form and (e) normal form – derivation; Problems.

(iv) 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 any point to a line. Equations of the internal and external bisectors of the angle between two lines- Derivations and Problems.

3. Pair of straight lines

(i) Pair of lines, homogenous equations of second degree. General equation of second degree. Derivation of (1) condition for pair of lines (2) conditions for pair of parallel lines, perpendicular lines and distance between the pair of parallel lines.(3) Condition for pair of co-incidence lines and (4) Angle and point of intersection of a pair of lines.

LIMITS AND CONTINUTY

(1) Limit of a function – definition and algebra of limits.

(2) Standarad limits (with proofs)

(i) Lim x n – a n/x – a= na n-1 (n rational) x→a

(ii) Lim sin θ / θ = 1 (θ in radian) and Lim tan θ / θ = 1 (θ in radian)

θ→0 θ →0

(3) Statement of limits (without proofs):

(i) Lim (1 + 1/n) n = e (ii) Lim (1 + x/n) n = ex

n→ ∞ n→∞

(iii) Lim (1 + x)1/x = e (iv) Lim log(1+x)/x = 1

x→0 x→0

v) Lim (e x – 1)/x= 1 vi) Lim (a x – 1)/x = logea

x→0 x→0

Problems on limits

(4) Evaluation of limits which tale the form Lim f(x)/g(x)[0/0 form]’ Lim f(n)/g(n)

x→0 x→∞ [∞ /∞ form] where degree of f(n) ≤ degree of g(n). Problems.

(5) Continuity: Definitions of left- hand and right-hand limits and continuity. Problems.

TRIGONOMETRY

Measurement of Angles and Trigonometric Functions

Radian measure – definition, Proofs of:

(i) radian is constant

(ii) p radians = 1800

(iii) s = rθ where θ is in radians

(iv) Area of the sector of a circle is given by A = ½ r2θ where θ is in radians. Problems

Trigonometric functions – definition, trigonometric ratios of an acute angle, Trigonometric identities (with proofs) – Problems.Trigonometric functions of standard angles. Problems. Heights and distances – angle of elevation, angle of depression, Problems. Trigonometric functions of allied angles, compound angles, multiple angles, submultiple angles and Transformation formulae (with proofs). Problems. Graphs of sinx, cosx and tanx.

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.

**MATHEMATICS – II**

**ALGEBRA**

ELEMENTS OF NUMBER THEORY

(i) Divisibility – Definition and properties of divisibility; statement of division algorithm.

(ii) Greatest common divisor (GCD) of any two integers using Eucli’s algorithm to find the GCD of any two integers. To express the GCD of two integers a and b as ax + by for integers x and y. Problems.

(iii) Relatively prime numbers, prime numbers and composite numbers, the number of positive divisors of a number and sum of all positive division of a number – statements of the formulae without proofs. Problems.

(iv) Proofs of the following properties:

(1) the smallest divisor (>1) of an integer (>1) is a prime number

(2) there are infinitely many 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 ineger then either (p,a)=1 or p|a

(8) the smallest positive divisor of a composite number a does not exceed √a

Congruence modulo m – definition, proofs of the following properties:

(1) ≡mod m” is an equivalence relation

(2) a ≡ b (mod m) => a ± x ≡ b ± x (mod m) and ax ≡ bx (mod m)

(3) If c is relatively prime to m and ca ≡ cb (mod m) then a ≡ b (mod m) – cancellation law

(4) If a ≡ b (mod m) – and n is a positive divisor of m then a ≡ b (mod n)

(5) a ≡ 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 ≡ b (mod m) (statement only), Problems on finding the solution of ax ≡ b (mod m)

GROUP THEORY

Groups – (i) Binary operation, Algebraic structures. Definition of semigroup, group, Abelian 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.

(ii) Square roots, cube roots and fourth roots of unity from abelian groups w.r.t. multiplication (with proof).

(iii) Proofs of the following properties:

(i) Identity of a group is unique

(ii)The inverse of an element of a group is unique

(iii) (a-1)-1 = a, ” a Є G where G is a group

(iv)(a*b)-1 = b-1*a-1 in a group

(v)Left and right cancellation laws

(vi)Solutions of a* x = b and y* a = b exist and are unique in a group

(vii)Subgroups, proofs of necessary and sufficient conditions for a subgroup.

(a) A non-empty subset H of a group G is a subgroup of G iff (i) ” a, b Є H, a*b Є H and (ii) For each a Є H,a-1Є H (b) A non-empty subset H of a group G is a subgroup of G iff a, b Є H, a * b-1 Є H. Problems.

VECTORS

(i) Definition of vector as a directed line segment, magnitude and direction of a vector, equal vectors, unit vector, position vector of point, problems.

(ii) Two-and three-dimensional vectors as ordered pairs and ordered triplets respectively of real numbers, components of a vector, addition, substraction, multiplication of a vector by a scalar, problems.

(iii) Position vector of the point dividing a given line segment in a given ratio.

(iv) Scalar (dot) product and vector (cross) product of two vectors.

(v) Section formula, Mid-point formula and centroid.

(vi) Direction cosines, direction ratios, proof of cos2 α + cos2β +cos2γ = 1 and problems.

(vii) Application of dot and cross products to the area of a parallelogram, area of a triangle, orthogonal vectors and projection of one vector on another vector, problems.

(viii) Scalar triple product, vector triple product, volume of a parallelepiped; conditions for the coplanarity of 3 vectors and coplanarity of 4 points.

(ix) Proofs of the following results by the vector method:

(a) diagonals of parallelogram bisect each other

(b) angle in a semicircle is a right angle

(c) medians of a triangle are concurrent; problems

(d) sine, cosine and projection rules

(e) proofs of 1. sin(A±B) = sinAcosB±cosAsinB

2. cos(A±B) = cosAcosB μ sinAsinB

MATRICES AND DETERMINANTS

(i) Recapitulation of types of matrices; problems

(ii) Determinant of square matrix, defined as mappings ∆: M (2,R) → R and ∆ :M(3,R)→ R. Properties of determinants including ∆(AB)=∆(A) ∆(B), Problems.

(iii) Minor and cofactor of an element of a square matrix, adjoint, singular and non-singular matrices, inverse of a matrix,. Proof of A(Adj A) = (Adj A)A = |A| I and hence the formula for A-1. Problems.

(iv) Solution of a system of linear equations in two and three variables by (1) Matrix method, (2) Cramer’s rule. Problelms.

(v) Characteristic equation and characteristic roots of a square matrix. Cayley-Hamilton therorem |statement only|. Verification of Cayley-Hamilton theorem for square matrices of order 2 only. Finding A-1 by Cayley-Hamilton theorem. Problems.

**ANALYTICAL GEOMETRY**

CIRCLES

(i) Definition, equation of a circle with centre (0,0) and radius r and 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, its centre and radius – derivations of all these, problems.

(ii) Equation of the tangent to a circle – derivation; problems. Condition for a line y=mx+c to be the tangent to the circle x2+y2 = r2 – derivation, point of contact and problems.

(iii) Length of the tangent from an external point to a circle – derivation, problems

(iv) Power of a point, radical axis of two circles, Condition for a point to be inside or outside or on a circle – derivation and problems. Poof of the result “the radical axis of two circles is straight line perpendicular to the line joining their centres”. Problems.

(v) Radical centre of a system of three circles – derivation, Problems.

(vi) Orthogonal circles – derivation of the condition. Problems

CONIC SECTIONS (ANANLYTICAL GEOMETRY)

Definition of a conic

1. Parabola

Equation of parabola using the focus directrix property (standard equation of parabola) in the form y2 = 4 ax ; other forms of parabola (without derivation), equation of parabola in the parametric form; the latus rectum, ends and length of latus rectum. Equation of the tangent and normal to the parabola y2 = 4 ax at a point (both in the Cartesian form and the parametric form) (1) derivation of the condition for the line y=mx+c to be a tangent to the parabola, y2 = 4 ax and the point of contact. (2) The tangents drawn at the ends of a focal chord of a parabola intersect at right angles on the directix – derivation, problems.

2. Ellipse

Equation of ellipse using focus, directrix and eccentricity – standard equation of ellipse in the form x2/a2 +y2/b2 = 1(a>b) and other forms of ellipse (without derivations). Equation of ellips in the parametric form and auxillary circle. Latus rectum: ends and the length of latus rectum. Equation of the tangent and the normal to the ellipse at a point (both in the cartesian form and the parametric form)

Derivations of the following:

(1) Condition for the line y=mx+c to be a tangrent to the ellipsex2/a2 +y2/b2 = 1 at (x1,y1) and finding the point of contact

(2) Sum of the focal distances of any point on the ellipse is equal to the major axis

(3) The locus of the point of intersection of perpendicular tangents to an ellipse is a circle (director circle)

3 Hyperbola

Equation of hyperbola using focus, directrix and eccentricity – standard equation hyperbola in the form x2/a2 -y2/b2 = 1 Conjugate hyperbola x2/a2 -y2/b2 = -1 and other forms of hyperbola (without derivations). Equation of hyperbola in the parametric form and auxiliary circle. The latus rectum; ends and the length of latus rectum. Equations of the tangent and the normal to the hyperbola x2/a2 -y2/b2 = 1 at a point (both in the Cartesian from and the parametric form). Derivations of the following results:

(1) Condition for the line y=mx+c to be tangent to the hyperbola x2/a2 -y2/b2 = 1 and the point of contact.

(2) Differnce of the focal distances of any point on a hyperbola is equal to its transverse axis.

(3) The locus of the point of intersection of perpendicular tangents to a hyperbola is a circle (director circle)

(4) Asymptotes of the hyperbola x2/a2 -y2/b2 = 1

(5) Rectangular hyperbola

(6) If e1 and e2 are eccentricities of a hyperbola and its conjugate then 1/e12+1/e22=1

**TRIGONOMETRY**

COMPLEX NUMBERS

(i) 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.

(ii) 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.

DIFFERENTIATION

(i) Differentiability, derivative of function from first principles, Derivatives of sum and difference of functions, product of a constant and a function, constant, product of two functions, quotient of two functions from first principles. Derivatives of Xn , e x, a x, sinx, cosx, tanx, cosecx, secx, cotx, logx from first principles, problems.

(ii) Derivatives of inverse trigonometric functions, hyperbolic and inverse hyperbolic functions.

(iii) Differentiation of composite functions – chain rule, problems.

(iv) Differentiation of inverse trigonometric functions by substitution, problems.

(v) Differentiation of implicit functions, parametric functions, a function w.r.t another function, logarithmic differentiation, problems.

(vi) Successive differentiation – problems upto second derivatives.

APPLICATIONS OF DERIVATIVES

(i) Geometrical meaning of dy\dx, equations of tangent and normal, angle between two curves. Problems.

(ii) Subtangent and subnormal. Problems.

(iii) Derivative as the rate measurer. Problems.

(iv) Maxima and minima of a function of a single variable – second derivative test. Problems.

INVERSE TRIGONOMETRIC FUNCTIONS

(i) Definition of inverse trigonometric functions, their domain and range. Derivations of standard formulae. Problems.

(ii) Solutions of inverse trigonometric equations. Problems.

GENERAL SOLUTIONS OF TRIGONOMETRIC EQUATIONS

General solutions of sinx = k, cosx=k, (-1≤ k ≤1), tanx = k, acosx+bsinx= c – derivations. Problems.

INTEGRATION

(i) Statement of the fundamental theorem of integral calculus (without proof). Integration as the reverse process of differentiation. Standarad formulae. Methods of integration, (1) substitution, (2) partial fractions, (3) integration by parts. Problems.

(4) Problems on integrals of:

1/(a+bcosx); 1/(a+bsinx); 1/(acosx+bsinx+c); 1/asin2x+bcos2x+c; [f(x)]n f ‘ (x);

f'(x)/ f(x); 1/√(a2 – x2 ) ; 1/√( x2 – a2); 1/√( a2 + x2); 1/x √( x2± a2 ) ; 1/ (x2 – a2);

√( a2 ± x2); √( x2- a2 ); px+q/(ax2+bx+c; px+q/√(ax2+bx+c); pcosx+qsinx/(acosx+bsinx); ex[f(x) +f1 (x)]

DEFINITE INTEGRALS

(i) Evaluation of definite integrals, properties of definite integrals, problems.

(ii) Application of definite integrals – Area under a curve, area enclosed between two curves using definite integrals, standard areas like those of circle, ellipse. Problems.

DIFFERENTIAL EQUATIONS

Definitions 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, equations reducible to the variable separable form. General solution and particular solution. Problems.