Engineering Mathematics 2018 : MA6151

Transcription

1 Engineering Mathematics 08 NAME OF THE SUBJECT : Mathematics I SUBJECT CODE : MA65 NAME OF THE METERIAL : Part A questions REGULATION : R 03 WEBSITE : wwwhariganeshcom UPDATED ON : November 07 TEXT BOOK FOR REFERENCE : Sri Hariganesh Publications (Author: C Ganesan) To bu the book visit wwwhariganeshcom/tetbook (Scan the above QR code for the direct download of this material) Unit I (Matrices) Find the sum and product of the eigenvalues of a 3 3 matri A whose characteristic 3 equation is Find the sum and product of all the eigenvalues of Tet Book Page No: 8 3 Given : A 3 0 Find the eigenvalues of A (Tet Book Page No: 7) 4 4 Find the eigenvalues of A where 3 4 A 0 6 (Tet Book Page No: 8) 5 5 Find the eigenvalues of the inverse of the matri 0 A Sri Hariganesh Publications (Ph: , ) Page

2 Engineering Mathematics If 3 and 6 are two eigenvalues of A 5, write down all the eigenvalues of 3 A (Tet Book Page No: 9) Two eigenvalues of the matri A are 3 and 0 What is the third 4 3 eigenvalue? What is the product of the eigenvalues of A? 8 If and are the eigenvalues of a X matri A, what are the eigenvalues of A and A -? 6 9 The product of two eigenvalues of the matri A 3 is 6 Find the third 3 eigenvalue of A (Tet Book Page No: 0) 0 0 If, 3 are the eigenvalues of 0 0, then find the value of b b 0 If,, 3 are the eigenvalues of the matri A, then find the eigenvalues of the matri A I If the sum of two eigenvalues and trace of a 3 X 3 matri A are equal, find the value of A (Tet Book Page No: 0) 3 If the eigenvalues of the matri A of order 3 X 3 are, 3 and, then find the eigenvalues of adjoint of A 4 If is the eigenvalue of the matri A, then prove that 5 If 0 is an eigenvalue of a square matri A, then show that A is the eigenvalue of A is an eigenvalue of 6 State Cale- Hamilton theorem (Tet Book Page No: 4) 7 Find the constants a and b such that the matri eigenvalues a 4 b has 3 and - as its Sri Hariganesh Publications (Ph: , ) Page

3 Engineering Mathematics 08 when A Use Cale- Hamilton theorem to find A 4 4A 3 5A A I Tet Book Page No: Write down the quadratic form corresponding to the matri A Tet Book Page No: 93 0 Check whether the matri B is orthogonal? Justif cos sin 0 B sin cos 0 Tet Book Page No: 57 Can 0 A 0 be diagonalized? Wh? Find the nature of the Quadratic Form 3 3 Tet Book Page No: 94 3 Write down the matri of the quadratic form 8z 4 0z z Tet Book Page No: 94 4 Give the nature of a quadratic form whose matri is Find the smmetric matri A, whose eigenvalues are and 3 with corresponding eigenvectors and (Tet Book Page No: 88) Unit II (Sequences and Series) Define limit of a sequence (Tet Book Page No: ) Distinguish between a sequence and series Sri Hariganesh Publications (Ph: , ) Page 3

4 Engineering Mathematics 08 3 Define convergent sequence (Tet Book Page No: 3) 4 Define convergence series with eample 5 Give an eample for conditionall convergent series 6 Find the nature of the series 3 n (Tet Book Page No: 9) 7 Eamine the convergence of the sequence u n n 8 Test the convergence of sequence n n (Tet Book Page No: 6) 9 Test the convergence of the series Test the convergence of the series Tet Book Page No: 90 Define bounded sequence (Tet Book Page No: ) Define oscillating sequence with eample (Tet Book Page No: 4) 3 Define monotone sequence 4 Given an eample for decreasing and increasing sequence 5 State the necessar condition for convergence 6 Define comparison test (Tet Book Page No: 6) 7 Define Cauch s integral test (Tet Book Page No: 37) 8 Test the convergence of the series (Tet Book Page No: 39) n n n 9 Determine the convergence or divergence of the series n n, using integral test 0 Eamine the convergence of the series (Tet Book Page No: 4) n nlog n Define D Alembert sratio test (Tet Book Page No: 49) Sri Hariganesh Publications (Ph: , ) Page 4

5 Engineering Mathematics 08 Discuss the convergence of the series n 3 n n 3 Using comparison test, prove that the series (Tet Book Page No: 53) 3 is divergent Tet Book Page No: 9 4 Using integral test, determine the convergence of 3 5 n 5 Find the coefficient of 6 in the epansion of e 6 Define alternating series (Tet Book Page No: 8) 7 State the Leibnitz s rule (Tet Book Page No: 8) 8 Define absolute and conditional convergent (Tet Book Page No: 98) 9 Show that an absolute convergent series is convergent Unit III (Applications of Differential Calculus) For the catenar ccosh, find the curvature (Tet Book Page No: 35) c Find the radius of curvature for e at the point where it cuts the ais Tet Book Page No: 3 3 Define circle of curvature (Tet Book Page No: 3) 4 Define geometricall curvature of the curve and centre of curvature at a point 5 Find the center of curvature of at the origin 6 Find the curvature of the curve 5 0 Tet Book Page No: 3 7 Find the radius of curvature of the curve Tet Book Page No: Find the radius of curvature of the curve c at, cc Sri Hariganesh Publications (Ph: , ) Page 5

6 Engineering Mathematics 08 9 What is the curvature of the circle 6 at an point on it? 0 Write down the formula for Radius of curvature in terms of Parametric Coordinates Sstem (Tet Book Page No: 35) Define Involutes and Evolutes (Tet Book Page No: 347) Write the properties of Evolutes (Tet Book Page No: 347) 3 Find the envelope of the lines Tet Book Page No: 379 m a m b where m is the parameter a 4 Find the envelope of famil of straight lines m, m being the parameter m Tet Book Page No: Find the envelope of the famil of straight lines m, where m is a parameter m Tet Book Page No: Find the envelope of the lines t c t, ' t ' being a parameter Tet Book Page No: Find the envelope of the famil of lines cos sin, being the parameter a b 8 Find the envelope of the famil of straight lines cos sin where is the parameter (Tet Book Page No: 380) 9 Find the envelope of the lines cos ec cot a, being the parameter Tet Book Page No: 38 0 Find the envelope of the famil of circles Tet Book Page No: 399 r, being the parameter Sri Hariganesh Publications (Ph: , ) Page 6

7 Engineering Mathematics 08 Unit IV (Differential Calculus of Several Variables) z u u u If u, find z (Tet Book Page No: 44) z z Using Euler s theorem, given u(, ) is a homogeneous function of degree n, prove that u u u n( n ) u (Tet Book Page No: 4) 3 Find u and u if u 4 If u, show that u u (Tet Book Page No: 40) 5 Given u(, ) tan, find the value of u u u, find u u u (Tet Book Page No: 4) z 6 If u f z, z, 7 Using the definition of total derivative, find the value of du dt given u 4a ; at, at (Tet Book Page No: 49) 8 Find du dt when u, t and t e 9 If 0 If u and 3 3 u where Tet Book Page No: 438 at, at, find du dt at and at then find du dt? Find du dt, if u, where t e, log t Find du dt if u sin( / ), where e t t, Sri Hariganesh Publications (Ph: , ) Page 7

8 Engineering Mathematics 08 3 If, then find d (Tet Book Page No: 4) d 4 If, then find d d 5 If, u v, find ( uv, ) (Tet Book Page No: 443) (, ) 6 If u v and uv, find the Jacobian of and with respect to u and v Tet Book Page No: If u, v, r cos, r sin then compute ( uv, )? ( r, ) Tet Book Page No: 44 8 If rcos, r sin find 9 If rcos, r sin find (, ) (Tet Book Page No: 44) ( r, ) ( r, ) (Tet Book Page No: 44) (, ) 0 If u v and v u, find (, ) ( uv, ) Write the sufficient condition for f (, ) to have a maimum value at (a,b) Tet Book Page No: 469 State the conditions for maima and minima of f (, ) Tet Book Page No: 469 Unit V (Multiple Integrals) a b Evaluate ( ) dd (Tet Book Page No: 5) 5 Evaluate dd Sri Hariganesh Publications (Ph: , ) Page 8

9 Engineering Mathematics 08 3 Evaluate 4 Evaluate b a dd (Tet Book Page No: 56) 3 dd (Tet Book Page No: 58) 5 Evaluate 0 dd (Tet Book Page No: 59) dd (Tet Book Page No: 58) 6 Evaluate 7 Sketch the region of integration in dd 8 Write down the double integral, to find the area between the circles r sin and r 4sin (Tet Book Page No: 594) a 9 Evaluate r drd (Tet Book Page No: 56) 0 Evaluate r sin ddr (Tet Book Page No: 57) sin Evaluate r drd (Tet Book Page No: 54) Evaluate / sin r ddr (Tet Book Page No: 59) 3 Evaluate d d where C is the path C, from 0,0 to 4 Find the area bounded b the lines 0,, and 0 5 Find the area bounded b the lines 0, and, using double integration Tet Book Page No: Find the area bounded b the line and parabola Sri Hariganesh Publications (Ph: , ) Page 9

10 Engineering Mathematics 08 7 Evaluate R dd, where R is the shaded region in the figure (Tet Book Page No: 56) 8 Change the order of integration in I f (, ) dd (Tet Book Page No: 560) 0 9 Change the order of integration for the double integral f (, ) dd a a 0 Change the order of integration in f (, ) dd (Tet Book Page No: 560) 0 Change the order of integration in a f (, ) dd (Tet Book Page No: 560) Change the order of integration dd (Tet Book Page No: 53) 3 Epress f dd 0 (, ) in polar co-ordinates 4 Plot the region of integration to evaluate the integral f (, ) dd where D is the region bounded b the line and the parabola 6 D 5 Evaluate 6 Evaluate a b c dddz 0 3 z dddz Evaluate z dddz 3 Sri Hariganesh Publications (Ph: , ) Page 0

11 Engineering Mathematics 08 8 Evaluate the triple integral 33 z dddz 9 Evaluate 0 dddz (Tet Book Page No: 53) Tetbook for Reference: ENGINEERING MATHEMATICS - I Publication: Sri Hariganesh Publications Author: C Ganesan Mobile: , To bu the book visit wwwhariganeshcom/tetbook ---- All the Best---- Sri Hariganesh Publications (Ph: , ) Page