MA6151 MATHEMATICS I PART B UNIVERSITY QUESTIONS. (iv) ( i = 1, 2, 3,., n) are the non zero eigen values of A, then prove that (1) k i.

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1 UNIT MATRICES METHOD EIGEN VALUES AND EIGEN VECTORS Find he Eigen values and he Eigenvecors of he following marices (i) ** (ii) 6 (iii) 3 (iv) 0 3 Prove ha he Eigen values of a real smmeric mari are real 6 0 (v) If he Eigen values of A are 0, 3, 5, find he eigenvecors of A and diagonalize he mari A If for i ( i =,, 3,, n) are he non zero eigen values of A, hen prove ha () k i are he eigen values of ka, where k being a non zero scalar; () are he eigen values of A i METHOD II CAYLEY HAMILTON THEOREM Using Cale Hamilon heorem find A and 4 A, if A Verif Cale Hamilon heorem for he mari 0 3, hence find is A 3 3 Verif Cale Hamilon heorem for A 4, hence use i o find A 3 4 Show ha he mari 0 0 saisfies is own characerisic equaion Find also is inverse\ Verif Cale Hamilon heorem for he mari 0 0 0, hence find 0 6 Show ha he mari 7 Find A and 4 A saisfies is own characerisic equaion Find also is inverse\ n A using Cale Hamilon heorem, aking hence find METHOD III QUADRATIC FORM INTO CANONICAL FORM Reduce he quadraic form6 3 3z 4 z 4z ino a canonical form b an orhogonal reducion Hence, find is rank and naure Reduce he quadraic form 5 6 ino a canonical form b an orhogonal reducion 3 A Hence, find is rank and naure 3 **Reduce he quadraic form3 5 3z z z ino a canonical form b an orhogonal reducion 4 Reduce he quadraic form 3 3 ino a canonical form 5 Reduce he quadraic form z z z ino a canonical form hrough an orhogonal ransformaion Wrie down he ransformaion

2 6 Reduce he quadraic form 5 3z 4 ino a canonical form b an orhogonal reducion and sae is naure UNIT IV DIFFERENTIAL CALCULUS OF SEVERAL VARIALBES METHOD I TAYLER S SEREIES **Epand e log Epand e cos in powers of and up o he hird degree erms using Talor s heorem a 0, up o hird erm using Talors series 3 Epand e sin in powers of and as far as he erms of hird degree using Talor s epansion 4 Epand + 3 in powers of ( ) and ( + ) up o hird degree erms 5 Find he Talor s series epansion of in powers of ( + ) and ( - ) up o hird degree erms METHOD II MAXIMA AND MINIMA Discuss he maima and minima of f, 3 Find he ereme value of z subjec o he condiion + + z = 3a 3 Discuss he maima and minima of f, 3 4 Tes for he maima and minima of he funcion f, If z r, show ha he maimum value of z + z + is r and he minimum value is 6 Tes for he maima and minima of he funcion Find he ereme values of he funcion 3 3 f, 3 0 f, 3 0 r 8 Find he dimensions of he recangular bo, open a he op, of maimum capaci whose surface area is 43 square meer 9 Find he maimum and minimum values of z A recangular bo open a he op, is o have a volume of 3cc find he dimension of he bo, ha requires he leas maerial for is consrucion METHOD III JACOBIAN METHOD **If z z u, v, w, hen find u, v, w z,, z If = u( - v), = uv, compue J and J ' and prove ha 3 If + +z = u, + z = uv, z = uvw, prove ha METHOD IV PARTIAL DERIVATIVES,, z ' JJ u, v, w uv W W W z **IfW f z, z,, hen show ha 0 u f, hen find If 3 If u e,show ha u u u z u u u u u u u 4 If F is a funcion of and and if e sin v, e cos v, prove ha 5 If u = + z + z where, e and z e find du d F F u F F e u v

3 6 Transform he equaion Z Z Z 0 b changing he independen variables using u = and v = + 7 If z = f(, ), where u v z z z z u v, = uv, prove ha 4u v

4 ROEVER COLLEGE OF ENGINEERING & TECHNOLOGY ELAMBALUR, PERAMBALUR-6 DEPARTMENT OF MATHEMATICS MATHEMATICS I SUB CODE : MA65 DATE : CLASS : B &C Secion MARKS : 50 PART A (0 = 0 ) ANSWER ALL THE QUESTIONS: Find he characerisics equaion of he mari A= Find he Sum and Produc of he Eigen values of he mari 3 A 3 The produc of wo eigen values of he mari A= is 6 Find he hird eigen value 4 If and are he Eigen values of a mari A Wha are he Eigen values of A? 5 If he Sum of he Eigen values and race of 33 mari A are equal hen find he value of A 6 Verif cale Hamilon heorem A= 7 Sae Cale Hamilon heorem 8 Wrie he mari of he Quadraic form z-5z 9 Deermine he naure of he Quadraic form +3 +z + 0 Define Inde, Signaure and Rank of he mari PART B( 3 0=30 ) a) Verif Cale Hamilon heorem for A= (or) b) Use Cale Hamilon heorem and hence find A - and A 4 A= a) Find he eigen values and eigen vecors of he mari A= (or) b) Diagonalise he mari A = o Diagonal form 3) a) Reduce he quadraic form z -4-z+4z ino canonical form b an orhogonal ransformaion (or) b) Reduce he quadraic form z ino canonical form b an orhogonal ransformaion Discuss is naure

5 METHOD I RADIUS OF CONVERGENCE MA65 MATHEMATICS I PART B UNIVERSITY QUESTIONS UNIT III - DIFFERENTIAL CALCULUS Find he ROC a an poin on e cos, e sin Find he ROC of he ccloid aa sin, a cos 3 Find he ROC a an poin of ccosh c 4 Find he ROC of he curve acos sin, asin cos 5 Find he ROC and cenre of curvaure of he parabola 6 If a a, prove ha a 3 a 4a a he poin also find he equaion of he evolue where is he ROC METHOD II ENVELOPE n n n Find he envelope of, subjec o c, where c is consan 3 Find he envelope of he famil of sraigh lines m am am, where m is he parameers 3 Find he, where he parameers are relaed b he equaion c 4 Find he envelope of he circle drawn upon he radius vecors of he ellipse as diameer l m 5 Find he envelope of, where l and m are conneced b he relaion (a and m are consans) l m 6 Find he envelope of he ssem of ellipse 4 METHOD III CIRCLE OF CURVATUER a, where he parameers a and b are conneced b he relaion ab = b Find he equaion of he COC of a (, 3) 4 9 a a ****Find he equaion of he COC a, 4 4 on a METHOD IV CENTRE OF CURVATURE Find he cenre of he curvaure of he parabola Find he cenre of curvaure of he curve 3 4a a an poin using parameric equaions 6 3 a he poin (, -) METHOD V EVOLUTE Find he Evolue of a ***Find he equaion of he evolues of he parabola 4a 3 Find he evolue of he parabola 4a

6 METHOD I VOLUME INTEGRAL Find he volume of he region bounded b he parabolaid Evaluae V log 3 Evaluae 4 Evaluae 5 Evaluae z 3 UNIT V MULTIPLE INTEGRAL z and he plane z = 4 dzdd where V is he region bounded b = 0, = 0, z = 0, and + + z = log z e dzdd dddz z dddz z 6 Find he volume of he porion of he clinder 4 z for all posiive values of,, z for which he inegral is real METHOD II CHANGE THE ORDER OF INTEGRATION **Change he order of inegraion ** Change he order of inegraion aa a a 0 a 0 inerceped beween he plane = 0 and he parabolaid dd and hence evaluae i dd and hence evaluae i 3 Change he order of inegraion dd and hence evaluae i METHOD III AREA 0 Find he area of he cardioids ra cos Find he surface area of he secion of he clinder a made b he plane + + z = a 3 Using double inegral find he area of he ellipse 4 Find he smaller of he areas bounded b he ellipse and he sraigh line + 3 = 6 4 Using double inegral find he areounded b he parabolas 4a and 4a 5 Find, b double inegraion, he areeween he wo parabolas 3 5 and Find he area which is inside he circle r 3acos METHOD IV DOUBLE INTEGRAL Evaluae dd over he posiive quadran of he circle and ouside he cardioids r a cos a

7 B changing o polar coordinaes, evaluae 3 B ransforming ino polar coordinaes, evaluae a and b 00 e dd dd over he annular region beween he circles 4 Transform he double inegral a a 0 a dd a ino polar coordinaes and hen evaluae i