Syllabus (MATHEMATICS)

Course Type: MAJ-1

Semester: 1

Course Code: BMTMMAJ01T

Course Title: Classical Algebra, Analytical Geometry (2D) & Calculus

(L-P-Tu): 5-0-1

Credit: 6

Practical/Theory: Theory

Course Objective: TO BE DONE

Learning Outcome: The whole course will have the following outcomes: Upon successful completion of this course, students will be able to CO1: solve tangent and area problems using concepts of limit, derivatives and integrals. CO2: calculate higher order derivatives and fin

Syllabus:

Unit -1: Classical Algebra [Credit: 2, 30L]

• Complex Numbers: De-Moivre’s Theorem and its applications, Direct and inverse circular and hyperbolic functions, Exponential, Sine, Cosine and Logarithm of a complex number.
• Polynomial equation, Fundamental theorem of Algebra (Statement only), Multiple roots, Statement of Rolle’s theorem only and its applications, Equation with real coefficients, Complex roots, Descarte’s rule of sign, relation between roots and coefficients, transformation of equation, reciprocal equation, binomial equation– special roots of unity, solution of cubic equations–Cardan’s method, solution of biquadratic equation– Ferrari’s method.
• Inequalities involving arithmetic, geometric and harmonic means. Schwarz and Weierstrass’s inequalities.

Unit -2: Analytical Geometry (2D) [Credit: 2, 30L]

• Transformation of Rectangular axes: Translation, Rotation and Rigid body motion, Invariants.
• Pair of straight lines: Condition that the general equation of second degree in two variables should represent a pair of straight lines, Angle between pair of straight lines, Bisectors of angle between the pair of straight lines, Equation of two lines joining the origin to the points in which a line meets a conic.
• General Equation of second degree in two variables: Reduction into canonical form.
• Polar Equations: Polar Co-ordinates, Polar equation of straight lines, Circles, conics referred to a focus as pole. Equations of tangents, normal.

Unit -3: Calculus [Credit: 2, 30L]

• Differential Calculus: Higher order derivatives, Leibnitz rule of successive differentiation and its applications; Indeterminate forms, L’Hospital’s rule; Basic ideas of Partial derivatives, Chain Rules, Jacobian, Euler’s theorem and its converse; Tangents and Normals, Sub-tangent and sub-normals, Derivatives of arc lengths, Pedal equation of a curve; Curvature and radius of curvature; Asymptotes; Envelopes; Concavity, convexity and point of inflexion.
• Integral Calculus: Reduction formulae; Rectification & quadrature of plane curves, length of a curve; Arc length as a parameter; Volume and surface area of revolution.

Reading References:

1. S.K. Mapa, Higher Algebra (Classical), Levant.
2. Titu Andreescu and Dorin Andrica, Complex Numbers from A to Z, Birkhauser, 2006.
3. W.S. Burnstine and A.W. Panton, Theory of equations, Wentworth Press.
4. J.G. Chakraborty & P.R. Ghosh, Advanced Analytical Geometry, U.N. Dhur & Sons Pvt. Ltd.
5. S.L. Loney, The Elements of Coordinate Geometry (Part 1), Arihant.
6. R. M. Khan, Analytical Geometry of two and three dimensions and vector analysis, New Central Book Agency (P) Ltd.
7. Arup Mukherjee and Naba Kumar Bej, Analytical Geometry of Two and Three Dimensions (Advanced level), Books & Allied Pvt. Ltd.
8. G.B. Thomas and R.L. Finney, Calculus, 9th Ed., Pearson Education, Delhi, 2005
9. M.J. Strauss, G.L. Bradley and K. J. Smith, Calculus, 3rd Ed., Dorling Kindersley (India) P. Ltd. (Pearson Education), Delhi, 2007.
10. R. Courant and F. John, Introduction to Calculus and Analysis (Volumes I & II), Springer-Verlag, New York, Inc., 1989.
11. T.M. Apostol, Calculus, Volumes I and II, Wiley Edition.
12. Shanti Narayan, P. K. Mittal, Differential Calculus, S. Chand.
13. Maity & Ghosh, Integral Calculus, New central book agency (P) Ltd.