## I SEMESTER MATHEMATICS – PAPER – I

### (ALGEBRA I AND DIFFERENTIAL CALCULUS I)

THEORY
Matrices (14hrs) Rank of a matrix – Elementary row/column operations – Invariance of rank under elementary operations – inverse of a non-singular matrix by elementary operations. System of m linear equations in n unknowns – matrices associated with linear equations –trivial and nontrivial solutions – criterion for existence of non-trivial solution of homogeneous and non-homogeneous systems – Criterion for uniqueness of solutions – Problems. Eigen values and eigenvectors of a square matrix – Properties – Diagonalization of a real symmetric matrix – Cayley -Hamilton theorem – Applications to determine the powers of square matrices and inverses of non-singular matrices.

Theory of Equations (14 hrs) Theory of equations – Euclid’s algorithm – Polynomials with integral coefficients – Remainder theorem – Factor theorem – Fundamental theorem of algebra (statement only) – Irrational and complex roots occurring in conjugate pairs – Relation between roots and coefficients of a polynomial equation – symmetric functions – transformation – Reciprocal equations – Descartes’ rule of signs – multiple roots – solving cubic equations by Cardon’s method – solving quartic equations by Descarte’s Method.

Differential Calculus (28 hrs) Recapitulation of limits, Continuity and differentiability - Derivatives of higher order –nth derivatives of the functions: eax , (ax + b)n , log(ax + b) , sin(ax + b) , cos(ax + b) , eaxsin(bx+ c) , eaxcos(bx + c) – Problems, Leibnitz theorem (with proof) – Monotonic functions – Maxima and Minima – Concavity Convexity and points of inflection. Polar coordinates –angle between the radius vector and the tangent at a point on a curve – angle of intersection between two curves – Pedal equations – Derivative of arc length in Cartesian, parametric and polar form, Coordinates of center of curvature – radius of curvature – circle of curvature – evolutes.

Books for References:

1. Natarajan, Manicavasagam Pillay and Ganapathy – Algebra

2. Serge Lang – First Course in Calculus

3. LipmanBers – Calculus, Volumes 1 and 2

4. N. Piskunov – Differential and Integral Calculus

5. B S Vatssa, Theory of Matrices, New Delhi: New Age International Publishers, 2005.

6. A R Vashista, Matrices, Krishna PrakashanaMandir, 2003.

7. G B Thomasand R L Finney, Calculus and analytical geometry,Addison Wesley, 1995.

8. J Edwards, An elementary treatise on the differential calculus: withApplications and numerous example, Reprint. Charleston, USABiblioBazaar, 2010.

9. N P Bali, Differential Calculus, India: Laxmi Publications (P) Ltd.., 2010.

10. S Narayanan & T. K. Manicavachogam Pillay, Calculus.:S. Viswanathan Pvt. Ltd., vol. I & II 1996.

11. Frank Ayres and Elliott Mendelson, Schaum's Outline of Calculus, 5th ed.USA: Mc. Graw Hill., 2008.

12. Shanti Narayan and P K Mittal, Text book of Matrices, 5th edition, New Delhi, S Chand and Co. Pvt. Ltd.,2013.

13. Shanthi Narayan and P K Mittal, Differential Calculus, Reprint. New Delhi: S Chand and Co. Pvt. Ltd., 2014.

#### PRACTICALS – I Mathematics Practical with FOSS tools for computer programs (3 hours/ week per batch of not more than 15 students) LIST OF PROBLEMS

1. Introduction to Scilab and commands connected with matrices.

2. Computations with matrices.

3. Row reduced echelon form and normal form.

4. Establishing consistency or otherwise and solving system of linear equations.

5. Introduction to Maxima and commands for derivatives and nth derivatives.

6. nth derivative without Leibnitz rule.

7. nth derivative with Leibnitz rule.

8. Scilab and Maxima commands for plotting functions.

9. Plotting of standard Cartesian curves using Scilab/Maxima.

10. Plotting of standard Cartesian curves using Scilab/Maxima.

11. Plotting of standard Polar curves using Scilab/Maxima.

12. Plotting of standard parametric curves using Scilab/Maxima.

Note: The above list may be changed annually with the approval of the BOS in UG (Mathematics). Geogebra/Octave may also be used in place of scilab/maxima