V SEMESTER

Course Outcomes :

CO1:Understand finite and infinite sets, Countable and Uncountable sets, Cantor’s theorem.

CO 2:  Understand Roots of equations, Relations connecting the roots and coefficients of an equation,Transformation of equations, The cubic equation, Character and position of roots of an equation.

CO 3: Understand Descarte’s rule of signs, De Gua’s Rule, Limits to the roots of an equation, Rational roots of equations, Newton’s method of divisors, Symmetric functions of roots of an equation, Symmetric functions involving only the difference of the roots of f(x)=0, Equations whose roots are symmetric functions of α,β, γ. 

CO4: Understand Reciprocal equations. 

CO5 :Understand Cubic equation, Equation whose roots are the squares of the difference of the roots, Character of the Roots, Cardan’s Solution 

CO6 :Understand Roots of complex numbers, General form of De Moivre’s theorem, the nth roots of unity, the nth roots of -1, Factors of xn-1 and xn+1, the imaginary cube roots of unity. 

CO7: Understand polar form of complex numbers, powers and roots.  

Unit I - Finite and Infinite Sets 

Finite and infinite sets, Countable sets, Uncountable sets, Cantor’s theorem (Section 1.3 of Text 1).

Unit II - Theory of equations I  

Roots of equations, Relations connecting the roots and coefficients of an equation, Transformation of equations, Special cases, The cubic equation, Character and position of roots of an equation, Some general theorems, Descarte’s rule of signs, Corollaries, De Gua’s Rule, Limits to the roots of an equation, To find the rational roots of an equation, Newton’s method of divisors, Symmetric functions of roots of an equation, Symmetric functions involving only the difference of the roots of f(x) =0, Equations whose roots are symmetric functions of α, β, γ (Sections 1 to 17 in chapter VI of Text 2). 

Unit III - Theory of equations II 

Reciprocal equation (Proof of theorems excluded) (Section 1 in chapter XI of Text 2) The Cubic equation, Equation whose roots are the squares of the difference of the roots, Character of the Roots, Cardan’s Solution (Section 5 of chapter VI and sections 1 to 4 of chapter XI I in Text 2). 

Unit IV – Complex numbers  

Quick review of a complex number, equality of complex numbers, fundamental operations, zero product, geometrical representation of complex numbers, addition and subtraction, product and quotients, conjugate numbers (Sections 1 to14 in chapter V of Text 2) [Questions should not be included in the End Semester Examination from these topics for Quick review],Roots of complex numbers, General form of De Moivre’s theorem, the nthroots of unity, the nth roots of -1, Factors of xn -1 and xn +1, the imaginary cube roots of unity (Sections 15 to 20 of chapter V of Text 2).  Polar form of complex numbers, powers and roots (Section 13.2 of Text 3). 

Texts :

• 1. R.G. Bartle and D. R. Sherbert, Introduction to Real Analysis (4thedition), Wiley
• 2. Bernard and Child, Higher Algebra, A.I.T.B.S. Publishers
• 3. E. Kreyszig, Advanced Engineering Mathematics (10th edition), Wiley.