DIPLOMA IN ENGINEERING I YEAR

Transcription

1 GOVERNMENT OF TAMILNADU DIRECTORATE OF TECHNICAL EDUCATION DIPLOMA IN ENGINEERING I YEAR SEMESTER SYSTEM L - SCHEME 0-0 I SEMESTER ENGINEERING MATHEMATICS - I CURRICULUM DEVELOPMENT CENTER

2 STATE BOARD OF TECHNICAL EDUCATION & TRAINING, TAMILNADU DIPLOMA IN ENGINEERING - SYLLABUS L-SCHEME (Implements from the Academic Year 0-0 on wards) Course Name : All Branches of Diploma in Engineering and Technology and Special Programmes except DMOP, HMCT and Film & TV Subject Code : 00 Semester : I Semester Subject Title : ENGINEERING MATHEMATICS - I TRAINING AND SCHEME OF EXAMINATION: TOPICS AND ALLOCATION OF HOURS: No. of Weeks per Semester: 6 Weeks Subject Instructions Examination Hours / Hours / Duration Week Semester Marks Engineering Mathematics - I Internal Board Total 5 Hrs. 80 Hrs. Assessment Examination Hrs Sl.No. Topic Time (Hrs.) Determinants and Matrices 4. Binomial Theorem and Partial Fraction 4 3. Straight lines 4 4. Trigonometry I 4 5. Trigonometry II 4 Tutorial 0 Total 80 Rationale: This subject being a branch of Logic is classified as one of the basic sciences and intends to teach students, basic facts, concepts and principles of mathematics as a tool to analyse Engineering problems. Mathematics lay down foundation for understanding core technology subjects. Objectives: This subject helps the students to develop logical thinking which is useful in Comprehending the principles of all other subjects. Analytical and systematic approach towards any problem is developed through learning of this subject. Mathematics being a versatile subject can be used at every stage of human life. CDC- DOTE ENGINEERING MATHEMATICS-I Page of

3 LEARNING STRUCTURE: Application To understand the techniques and methods for solving Engg. Problems such as simultaneous equation involved in vibrations, ckts. Laws of friction, projections, Lami s thorem, stress strain curves etc. Work done moment of force about a point and line. Procedure To explain use of properties of determinants to solve problems, use of cramer s rule, matrix inversion Binomial Theorem for positive integral index and for rational index partial fraction of proper and improper fractions To find distance of a point from a fixed line, distance between parallel lines, derivation of angle between two lines and pair of lines, parallel line and perpendicular line condition for two lines and pair of lines. Using various techniques proving standard results which will be used in solving engineering problems. Using various technique proving standard results which will be used in solving engineering problems. Concepts Cramer s rule, Applications of properties to solve problems, Cramer s rule, Algebra of matrices. Inverse of a matrix Binomial Expansion for positive and negative integral index. Cases of finding partial fractions. Angle between lines combined equation of two lines through origin and not through origin. Angle between pair of lines. Trigonometrical ratios of allied angles compound angles and multiple angles. Inverse Trigonometrical ratios. Facts Definition of determinants properties of determinants definition of matrix. Definition of permutation and combination. Definition of polynomial fractions. Combined equation of two lines. Definition of Trigonometrical ratios, allied angles, compound angles and multiple angles. Definition of inverse Trigonometric functions. CDC- DOTE ENGINEERING MATHEMATICS-I Page 3 of

4 DETAILED SYLLABUS CONTENTS UNIT NAME OF TOPICS Hours Mark I II III DETERMINANTS. Definition and expansion of determinants of order and 3. Properties of determinants. Cramer s rule to solve simultaneous equations in and unknowns- Simple Problems. Problems involving properties of Determinants 4 7 MATRICES.3 Definition of matrix. Types of matrices. Algebra of matrices such as equality, addition, subtraction, scalar multiplication and multiplication of matrices. Transpose a matrix, Adjoint matrix and Inverse matrix. Simple Problems 5 8 BINOMIAL THEOREM. Definition of factorial notation, definition of permutation and combinations with formula. Binomial theorem for positive integral index 5 8 (statement only), finding of general and middle terms. Simple Problems. Problems finding coefficient of x n, independent terms. Simple Problems. Binomial Theorem for rational index, expansions only upto for negative integers. Simple Expansions. PARTIAL FRACTIONS.3 Definition of Polynomial fraction, proper and improper fractions and definition of partial fractions. To resolve proper fraction into partial fraction with denominator containing 5 7 non repeated linear factors, repeated linear factors and irreducible non repeated quadratic factors. Simple Problems. STRAIGHT LINES 3. Length of perpendicular distance from a point to the line and perpendicular distance between parallel lines. Simple Problems 4 7 Angle between two straight lines and condition for parallel and perpendicular lines. Simple Problems PAIR OF STRAIGHT LINES THROUGH ORIGIN 3. Pair of lines passing through the origin ax +hxy+by =0 expressed in the form (y-m x)(y-m x)=0. Derivation of tan = ± parallel and perpendicular lines. Simple Problems h -ab, condition of a + b 5 8 CDC- DOTE ENGINEERING MATHEMATICS-I Page 4 of

5 UNIT NAME OF TOPICS Hours Mark III IV V PAIR OF STRAIGHT LINES NOT THROUGH ORIGIN 3.3 Condition for General equation of the second degree ax +hxy+by +gx+fy+c=0 to represent pair of lines. a h g h b f = 0 (statement only) g f c Angle between them, condition for parallel and perpendicular lines. Simple Problems 5 7 TRIGONOMENTRY 4. Trigonometrical ratios of allied angles. Expansion of sin (A ± B) and 5 8 cos (A ± B). Problems using above expansion. 4. Expansion of tan (A ± B) and problems using this expansion Trigonometrical ratios of multiple angles (A only) and sub multiple angles. Simple Problems. 5 7 TRIGNOMENTRY 5. Trigonometrical ratios of multiple angels (3A only). Simple Problems Sum and product formulae. Simple Problems Definition of inverse trigonometric ratios relation between inverse trigonometric ratios. Simple Problems 5 7 Text Book: Mathematics for Higher Secondary I year and II year (Tamil nadu Text Book Corporation) Reference Book: Engineering Mathematics - Dr.M.K.Venkatraman, National Publishing Co, Chennai Engineering Mathematics Dr.P.Kandasamy & Others, S.Chand & Co Ltd, New Delhi CDC- DOTE ENGINEERING MATHEMATICS-I Page 5 of

6 MODEL QUESTION PAPER ENGINEERING MATHEMATICS I Time : 3Hrs PART A Max Marks:75 I. Answer Any 5 Questions 5x=5. Solve : x 3 x x = 0. Find the value of a b c b c a c =0 a b If A =, B Find AB Find the adjoint matrix of 5. Find the value of 0C 7 6. Find the general term of 3x y 8 7. Expand 3 x upto three terms when x 8. Split x into partial fraction without finding the constant x x 9. Find the value of m if the lines x+my=4 and x+5y-6=0 are perpendicular. 0. Find the combined the equation of the lines x+5y=0 and x+3y=0.. Show that the pair of lines x 8y 6y 0 are parallel. Write down the condition for the equation ax hxy by gx fy c 0 to represent a pair of straight lines. CDC- DOTE ENGINEERING MATHEMATICS-I Page 6 of

7 3. Show that sin (-330) x sin 40 = Find the value of cos50 cos40 sin 50 sin If tana= tan B = 3 Find tan (A+B) 6. Find the value of sin75 cos75 7 If sin=/3,find the value of sin3 8. Find the value of 4cos 3 0-3cos0 9. Show that SinA-sinB = -Cot(A+B) CosA-cosB 0. Show that tan - x = tan - x -x PART B (Answer Any two subdivision in each question ) 5x=60 All Questions carry Equal Marks a. Solve by using Cramer s Rule : x+y+z=3, x-y+z = and 3x+y-z=3 b. Short that x x =x (x+3) x c. Find the inverse of a 3 Find the middle terms in the expansion of x 3 x b. Find the term independent of x in the expansion of (x +/x) c. Resolve x 3 in to partial fraction x( x 5)( x 6) CDC- DOTE ENGINEERING MATHEMATICS-I Page 7 of

8 3. a Find the angle between the lines 7x+y= and x-5y=0 b. Find the separate equation of the pair of straight lines 9x +xy+4y =0. Also prove that the lines are parallel. c. Show that the equations represented by x 7xy 3y 5x 5y 0 is a pair of straight lines 4. a If sin A = 0 and sin B prove that A+B= 5 4 b. If A+B =45 prove that (+tana) (+tanb)= and hence deduce the value of tan c. sin A sin A Prove that tan A cos A cos A 5. a Prove that sin0 sin 40 sin 60 sin80 = 3 6 b. Prove that (cosα-cosβ) + (sinα-sinβ) =4 sin c. Show that tan - x-y +xy =tan - x+tan - y ******** α-β CDC- DOTE ENGINEERING MATHEMATICS-I Page 8 of

9 MODEL QUESTION PAPER ENGINEERING MATHEMATICS I Time : 3Hrs PART A Max Marks:75 I. Answer Any 5 Questions 5x=5. Find x if x 0 0 x - =0 x x 3x. Prove that 4x 5x 6x 0 7x 8x 9x If A = Find A 4. Find the inverse of 0 5. Find the th term of 3x x 0 6. How many middle terms are in the expansion of (5x-y) 9 7. Write the first three terms in the expansion of (-x) - 8. Without finding the constants split 3 x 3 in to partial fraction. ( x )( x ) 9. Find the perpendicular distances from the point (,) to the straight line 3x+y+=0. 0. Write down the condition for the pair of lines given by ax +hxy+by =0 to be parallel.. Find a if the lines represented by 3x +4xy+ay =0 are perpendicular. State the expression for angle between pair of line given by ax +hxy+by +gx+fy+c=0 3 Find the value sin5 o without using tables or calculator. 4. Find the value of cos40 o Cos 0 o + sin 40 o sin0 o CDC- DOTE ENGINEERING MATHEMATICS-I Page 9 of

10 5. Simplify tan tan 3 tan tan 3 6. Prove that sin A = tana cos A 7. Find the value of 3sin0 o 4 sin 3 0 o 8. If Cos A = 5 3 find the value of cos3a. 9. Show that Cos0 +Cos00+Cos40=0. x 0. Show that sin - = tan - x. x PART B (Answer any two subdivision in each Question) 5x=60 All Questions carry Equal Marks.a) Solve the equations 4x+y+z=6, x-y-z=-6 and x+y+z=3, using Cramer s rule x a b c b) Prove that a x b c =x (x+a+b+c) a b x c c) If A = show that A 5A 4I 0. a Find the middle terms in the expansion of (x+/x) 3 b. Find the term independent of x in the expansion of 7x 4 c. Resolve in to a partial function. ( x )( x ) x x 3. a Derive the expression for angle between two lines y=m x+c and y=m x+c b. If the slope of one of the straight lines ax hxy by 0 is twice that of the other. Show that 8h =9ab. c. Find the value of ' ' so that the equation 3x +4xy+8y -8x-y+λ=0 represents a pair of straight lines. CDC- DOTE ENGINEERING MATHEMATICS-I Page 0 of

11 3 4. a If sin A =, cos B =, find the values of sin (A-B) and cos (A-B) 5 3 b. If A+B=45 o Prove that (cota-) (cotb-)=. Also find the value of cot c. Show that +sin -cos = tan/ +sin +cos 3 3 cos A cos3a sin A sin 3A 5. a Prove that 3 cos A sin A b. If a=sina+sinb, B=cosA+cosB, Show that tan A-B = c. Show that tan - 3x-x 3 = 3tan - x -3x a +b ******** 4-(a +b ) a +b CDC- DOTE ENGINEERING MATHEMATICS-I Page of