### DIPLOMA 1st and 2nd Semester Civil engineering Common to all Engineering Diploma Programmes

DIPLOMA 1st and 2nd Semester Civil engineering Common to all Engineering Diploma Programmes

Board of Technical Examinations

 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

On successful completion of the course, the student will be able to:

1.     Find the product of matrices, value of determinants, and inverse of matrix and solve the simultaneous linear equation.

2.     Find the product of vectors and their geometrical applications in finding moment of force, work done.

3.     Determine probability of various types of events.

4.     Solve the problems related to logarithms.

5.     Solve the problems on trigonometric functions with angle of any magnitude.

6.     Evaluate the limiting value of algebraic and trigonometric functions.

 CO Course Outcome PO Mapped Cognitive Level Theory Sessions Allotted marks on cognitive levels TOTAL R U A CO1 Find the product of matrices, value of determinants, and inverse of matrix and solve the simultaneous linear equation 1,2,3 R/U/A 10 9 10 12 31 CO2 Find the product of vectors and their geometrical applications in finding moment of force, work done 1,2,3 R/U/A 8 6 15 6 27 CO3 Determine probability of various types of events 1,2, R/U/A 8 3 5 6 14 CO4 Evaluate    the                integrations               of algebraic,                  trigonometric            and exponential function 1,2,3,10 R/U/A 16 15 20 12 47 CO5 Solve the problems related to logarithms. 1,2 R/A 4 3 0 6 09 CO6 Evaluate the limiting value of algebraic     and                  trigonometric functions 1,2,10 R/U/A 6 6 5 6 17 Total Hours of instruction 52 Total marks 145
##### R-Remember; U-Understanding; A-Application Course outcomes –Program outcomes mapping strength
 Course Programme Outcomes 1 2 3 4 5 6 7 8 9 10 Engineering Mathematics-I 3 3 3 - - - - - - 3

Method is to relate the level of PO with the number of hours devoted to the COs which address the given PO. If >40% of classroom sessions addressing a particular PO, it is considered that PO is addressed at Level 3

If 25 to 40% of classroom sessions addressing a particular PO, it is considered that PO is addressed at Level 2 If 5 to 25% of classroom sessions addressing a particular PO, it is considered that PO is addressed at Level 1

If < 5% of classroom sessions addressing a particular PO, it is considered that PO is considered not-addressed.

### Reference:

1.     NCERT Mathematics Text books of class XI and XII.

2.     Karnataka State PUC mathematics Text Books of I & II PUC by H.K. Dass and Dr.Ramaverma published by S.Chand & Co.Pvt.Ltd.

3.     CBSE Class Xi & XII by Khattar&Khattar published PHI Learning Pvt. ltd.,

4.       First and Second PUC mathematics Text Books of different authors.

### Course Assessment and Evaluation:

The Course will be delivered through lectures, class room interaction, exercises and self- study cases.

 Method What To whom When/where (Frequency in the course) Max Marks Evidence collected Contributing to course outcomes *CIE Internal Assessment Tests Student Three tests (Average of Three tests will be computed). 20 Blue books 1 to 6 Assignments Two Assignments based on CO's (Average marks of Two Assignments shall be  rounded off to         the  next higher digit.) 5 Log of record 1 to 6 Total 25 *SEE Semester End Examination End of the course 100 Answer scripts at BTE 1 to 6 Student feedback Students Middle of the course -NA- Feedback forms 1 to 3, delivery of the course End of Course survey End of course Questionnaire 1 to 6, Effectiveness of delivery of instructions and   assessment methods

*CIE – Continuous Internal Evaluation        *SEE – Semester End Examination

Note: I.A. test shall be conducted for 20 marks. Average marks of three tests shall be rounded off to the next higher digit.

Composition of Educational Components:                                                                              Questions for CIE and SEE will be designed to evaluate the various educational components (Bloom's taxonomy) such as:

 Sl. No. Educational Component Weightage (%) 1 Remembering 25 2 Understanding 40 3 Applying the knowledge acquired from the course 30 Analysis and Evaluation 5