### ENGINEERING MATHEMATICS – I | Diploma 2020

 Course Title: ENGINEERING MATHEMATICS – I Course Code : 15SC01M Semester : I Core / Elective : Core Teaching Scheme in Hrs (L:T:P) : 4:0:0 Credits : 4 Credits Type of course : Lecture + Assignments Total Contact Hours : 52 CIE : 25 Marks SEE : 100 Marks Programmes: Common to all Engineering Diploma Programmes

Pre-requisites:

Basics in Algebra, Trigonometry and Coordinate Geometry in Secondary Education.

### Course Objectives:

1.                        Apply the concept of matrices and determinants and their applications to solve the linear equation in engineering field.

2.                        Apply the vector algebra in solving the problems of statics and mechanics.

3.                        Analyse the civil engineering problems using concepts of probability.

4.                        Evaluate the advanced engineering mathematical problems using logarithms.

5.                        Apply and evaluate trigonometric concept in vector engineering field.

6.                        Create the basic concept of calculus.

### Course Content:

 Topic and Contents Hours Marks LINEAR ALGEBRA UNIT-1: MATRICES AND DETERMINANTS 10 31 (a) Matrices: Basic concepts of matrices: Definition, types of matrices and mathematical operations on matrices (addition, subtraction and multiplication of matrices). 02 (b) Determinant: Definition, problems on finding the determinant value of 2nd and 3rd order. Problems on finding unknown quantity in a 2nd and 3rd order determinants using expansion. Solving simultaneous linear equations using determinant method (Cramer’s rule up to 3rd order). 04

 (c) Inverse and applications of matrices: Minors and Cofactors of elements of matrix. Adjoint and Inverse of matrices of order 2nd and 3rd order. Elementary row and column operations on matrices. Characteristic equation and characteristic roots (eigen values) of 2x2 matrix. Statement of Cayley-Hamilton theorem and its verification for 2x2 matrix. Solution of system of linear equations using Gauss Elimination method (for 3 unknowns only). 04 ALGEBRA UNITS-2: VECTORS 08 27 Definition of vector. Representation of vector as a directed line segment. Magnitude of a vector. Types of vectors. Position vector. Expression of vector by means of position vectors. Addition and subtraction of vectors in terms of line segment. Vector in plane and vector in a space in terms of unit vector i, j and k respectively. Product of vectors. Scalar product and vector product of two vectors. Geometrical meaning of scalar and vector product. Applications of dot (scalar) and cross (vector) products. Projection of a vector on another vector. Area of parallelogram and area of triangle. Work done by force and moment of force. UNITS-3: PROBABILITY AND LOGARITHMS 08 14 (a) Probability: Introduction. Random experiments: outcomes and sample space. Event: Definition, occurrence of an event, types of events. Algebra of events- complementary event, the events A or B, A and B, A but not B, mutually exclusive events, exhaustive events, defining probability of an event. Addition rule of probability. Conditional probability: definition, properties of conditional probability, simple problems. 06 (b) Logarithms: Definition of common and natural logarithms. Laws of logarithms (no proof). Simple problems on laws of logarithms. 02

 TRIGONOMETRY UNIT-4: ALLIED ANGLES AND COMPOUND ANGLES. 16 47 (a)  Recapitulation of angle measurement, trigonometric ratios and standard angles. Allied angles: Meaning of allied angle. Signs of trigonometric ratios. Trigonometric ratios of allied angles in terms of 8. Problems on allied angles. (b)     Compound angles: Geometrical proof of sin(A+B) and cos(A+B) and hence deduce tan(A+B). Write the formulae for sin(A-B), cos(A-B) and tan(A-B), problems. Multiple and sub multiple angle formulae for 2A and 3A. Simple problems. Transformation formulae. Expression for sum or difference of sine and cosine of angles into product form. Expression for product of sine and cosine of angles into sum or differences form. 02 06 08 UNIT-5:COMPLEX NUMBERS 04 09 Meaning of imaginary number i and its value. Definition  of  complex  number  in  the  form  of    a + ib. Argand diagram of complex numbera + ib (Cartesian system). Equality of complex numbers. Conjugate of complex number. Algebra of complex numbers, modulus of complex number, principal value of argument of complex number, polar form: Z = r(cos8 + i sin8) and exponential form Z = rei8of complex number, where r is modulus and 8 is principal value of argument of complex number. UNIT-6: INTRODUCTION TO CALCULUS 06 17 Limits: Constants and variables. Definition of function. Types of functions: Explicit and implicit function, odd and even functions(definition with example). Concept of x → a.Definition of limit of a function. Indeterminate forms. Evaluation of limit of functions by factorization, rationalization. Algebraic limits. Statement of sn–an lim                = nan–1 where n is any rational number. s→a s–a Proof of lim sin 8 = 1 where 8 is in radian. Related 8→0 8 problems.   Standard limit (statement only)   x                                                                                               x 1. lim a –1 = log a,                         2. Lim e –1 = 1 s→0        s                     e                                   s→0        s n                                                                     1 3. lim (1 + 1)   = e,                           4.lim(1 + n)n  = e n→∞                n                                              n→0 Simple problems on standard limits. TOTAL 52 145