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Engineering Mathematics II www.BrainKart.com Click Here !!! for Engineering Mathematics II full study material. Click Here !!! for other subjects (Anna University) Click Here !!! for Anna University Notes Android App. Click Here !!! for BrainKart Android App. Click Here for Engineering Mathematics II full study material. VALLIAMMAI ENGINEERING COLLEGE S.R.M. Nagar, Kattankulathur - 603203 DEPARTMENT OF MATHEMATICS IMPORTANT QUESTIONS II SEMESTER II SEMESTER (COMMON TO ALL BRANCHES) MA 8251- MATHEMATICS –II Regulation – 2017 Academic Year – 2016 - 17 Prepared by Dr.T.Isaiyarasi Mr.V.Dhanasekaran Ms.C.V.Dhanya Ms.N.Prabhavathy www.BrainKart.com Click Here for Engineering Mathematics II full study material. VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur – 603 203 DEPARTMENT OF MATHEMATICS SUBJECT : MA8251- ENGINEERING MATHEMATICS-II SEM / YEAR: II SEMESTER / I YEAR (COMMON TO ALL BRANCHES) UNIT I MATRICES Eigen values and Eigenvectors of a real matrix – Characteristic equation – Properties of Eigen values and Eigenvectors – Cayley-Hamilton theorem – Diagonalization of matrices – Reduction of a quadratic form to canonical form by orthogonal transformation – Nature of quadratic forms. Q.No. Question BT Level Competence BTL-1 Remembering BTL-1 Remembering 1 2 −2 Find the sum and product of the eigen values of A= ( 1 0 3) −2 −1 −3 3 1 4 Find the sum and squares of the eigen values of A= (0 2 6) 0 0 5 6 −2 2 The product of the 2 eigen values of A= (−3 3 −1) is 14. Find the 2 −1 3 3rdeigen value. If the sum of 2 eigen values and the trace of a 3×3 matrix are equal , find the value of |𝑨| State Cayley-Hamilton theorem. 2 1 Use Cayley Hamilton theorem to find 𝑨−1 if A=( ) 1 −5 1 0 If A=( ) find A3 using Cayley Hamilton theorem 4 5 Write any 2 applications of Cayley Hamilton theorem 1 1 .................. Prove that the eigen values of 𝑨−1 are , 1 , 1 BTL-1 Remembering BTL-1 Remembering BTL-1 Remembering BTL-1 Remembering BTL-2 BTL-2 Understanding Understanding BTL-2 Understanding BTL-2 BTL-3 Understanding Applying Prove that sum of eigen values of a matrix is equal to its trace. 8 −6 2 Find the sum of the eigen values of 2A, if A= (−6 7 −4) 2 −4 3 a 4 Find the constants a and b such that the matrix ( ) has 3,-2 be the 1 b eigen values of A 1 2 For what values of c, the eigen values of the matrix ( )are real and c 4 unequal , real and equal, complex conjugates. BTL-3 BTL-3 Applying Applying BTL-4 Analyzing BTL-4 Analyzing PART-A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 3 2 Obtain the eigen values of A3 where A=( ) 1 2 4 1 Find the eigen values of 2A2 if A=( ) 3 2 𝝀1 𝝀 2 𝝀 3 www.BrainKart.com 𝝀n Click Here for Engineering Mathematics II full study material. 16 Find the matrix corresponding to the quadratic form 2xy+2yz+2zx. BTL-4 Analyzing 17 Find the quadratic form corresponding to the matrix 10 −2 −5 A= (−2 2 3) −5 3 6 What is the nature of the quadratic form x2+y2+z2 in 4 variables? BTL-5 Evaluating BTL-5 Evaluating If 2,-1,-3 are the eigen values of the matrix A, then find the eigen values of A2-2I Find the symmetric matrix A, whose eigen values are 1 and 3 with 1 1 corresponding eigen vectors ( −1) and ( )1 BTL- 6 Creating BTL- 6 Creating 18 19 20 www.BrainKart.com Click Here for Engineering Mathematics II full study material. UNIT II- VECTORCALCULUS Gradient and directional derivative – Divergence and curl – Vector identities – Irrotational and Solenoidal vector fields – Line integral over a plane curve – Surface integral – Area of a curved surface – Volume integral – Green’s, Gauss divergence and Stoke’s theorems – Verification and application in evaluating line, surface and volume integrals. PART-A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 2 2 2 Find 𝛁𝝋, if 𝝋 = x + y + z at (1, -1, 1). Find the Directional derivative of † = xyz at (1,1,1)in the direction t⃗ + y⃗ + ⃗⃗ k. Find the Directional derivative of 𝝋 = 4xz2 + x2yz at (1,-2,1) in the ⃗⃗. direction 2t⃗ + 3y⃗ + 4k State Gauss Divergence theorem BTL-1 BTL-1 Remembering Remembering BTL-1 Remembering BTL-1 Remembering State Stokes theorem. State Greens theorem Give the unit normal vector to the surface xyz = 2 at (2, 1, 1). Give the unit normal vector to the surface x2 + y2 + z2 = 1 at (1,1,1). If 𝝋 = 3xy − yz ., Give grad 𝝋 at (1,1,1). If ⃗r⃗ is the position vector, Give div ⃗r⃗. Show that(rn ) = nrn−2r ⃗⃗. Show that the vector ⃗⃗ 𝑭 = (y2 − z2 + 3yz − 2x)t⃗ + (3xz + 2xy)y⃗ + (3xy − 2xz + 2z)⃗k⃗is solenoidal Show that curl(grad 𝝋) = 0. If ⃗𝑭⃗ = 3xyt⃗ − y2 y⃗⃗, Evaluate ƒ ⃗𝑭⃗. d⃗r⃗, where C is the arc of the parabola y = 2x2 from the point (0,0) to the point (1,2). If ⃗𝑭⃗= (x2)t⃗ + (xy2)y⃗ , evaluate ƒ ⃗𝑭⃗.d⃗r⃗from (0,0) to (1,1) along the path y = x. Using Green’s theorem evaluate ƒ𝑪 [(2x2 − y2)dx + (x2 + y2) dy where C is the boundary of the square enclosed by the lines x = 0, y = 0, x = 2, y = 3. BTL-1 BTL-1 BTL-2 BTL-2 BTL-2 BTL-2 BTL-3 BTL-3 Remembering Remembering Understanding Understanding Understanding Understanding Applying Applying BTL-3 BTL-4 Applying Analyzing BTL-3 Applying BTL-6 Creating BTL-5 Evaluating BTL-5 Evaluating BTL-6 Creating BTL-6 Creating 17 Is the position vector r x i y j z k irrotational? Justify. 18 Evaluate using Gauss Divergence theorem for ⃗𝑭⃗ = 4xzt⃗ − y2y⃗ + +yz⃗k⃗ taken over the cube x = 0, y = 0, z = 0, x = 1, y = 1, z = 1. 19 What is the value of m if the vector ⃗⃗ is solenoidal ⃗𝑭⃗ = (x + 3y)t⃗ + (y − 2z)y⃗ + (x + mz)k 20 What is the value of a, b, c if the vector ⃗𝑭⃗= (x + y + az)t⃗ + (by + 2y − z)y⃗ + (−x + cy + 2z)⃗k⃗may be irrotational. www.BrainKart.com Click Here for Engineering Mathematics II full study material. UNIT III ANALYTIC FUNCTIONS Analytic functions – Necessary and sufficient conditions for analyticity in Cartesian and polar coordinates - Properties – Harmonic conjugates – Construction of analytic function - Conformal mapping – Mapping by functions w = z + 𝑪, 𝑪z, 1/z, z2 - Bilinear transformation. PART-A 1 2 Examine if †(z) = z3 analytic ? Identify the constants a, b, c if analytic. Define conformal mapping. BTL-1 BTL-1 Remembering Remembering BTL-1 Remembering BTL-1 Remembering BTL-1 BTL-1 Remembering Remembering z 1 BTL-2 Understanding z1 Estimate the invariant point of the bilinear transformation z−1−i w = z+2 BTL-2 Understanding BTL-2 Understanding BTL-2 Understanding BTL-3 Applying Show that an analytic function in a region R with constant imaginary part is constant. Show that is harmonic. BTL-3 Applying BTL-3 Applying BTL-4 Analyzing BTL-4 Analyzing BTL-4 Analyzing 17 If f(z) is an analytic function whose real part is constant, Point out f(z) is a constant function. Explain that a bilinear transformation has at most 2 fixed points. Examine whether the function xy2 can be real part of analytic function. Test the analyticity of the function BTL-5 Evaluating 18 Evaluate the image of hyperbola BTL-5 Evaluating 3 4 5 6 7 8 9 is Can u = 3x2y − y3 be the real part of an analytic function? Justify your answer State necessary and sufficient condition for f(z) to be analytic. Identify the invariant point of the bilinear transformation Estimate the invariant points of the transformation w 10 Give the image of the circle w = 5z. Under the transformation 11 in the complex plane. Show that z is not analytic at any point. 12 13 14 15 16 under the transformation give the image of the circle 2 under the transformation 19 Formulate the critical points of the transformation BTL-6 Creating 20 Formulate the bilinear transformation which maps z = 0, −i, −1 into w = i, 1, 0 respectively. BTL-6 Creating www.BrainKart.com Click Here for Engineering Mathematics II full study material. UNIT IV COMPLEX INTEGRATION Line integral - Cauchy’s integral theorem – Cauchy’s integral formula – Taylor’s and Laurent’s series – Singularities – Residues – Residue theorem – Application of residue theorem for evaluation of real integrals – Use of circular contour and semicircular contour. PART –A 1 2 State Cauchy’s integral theorem Identify the type of singularity of function Sin BTL-1 BTL-1 Remembering Remembering 3 4 State Cauchy’s residue theorem and Cauchy’s integral formula Identify the value of e z dz , where C is |z| = 1? BTL-1 BTL-1 Remembering Remembering BTL-2 Understanding BTL-2 Understanding BTL-2 Understanding BTL-2 Understanding BTL-3 Applying BTL-3 Applying BTL-3 Applying BTL-4 Analyzing BTL-4 Analyzing BTL-4 Analyzing BTL-5 Evaluating BTL-5 Evaluating BTL-6 Creating 1 . 1 z C 5 Estimate the residue of the function 6 Give the Laurent’s series of f (z) 4 at z 3 (z 2) f (z) 1 a simple pole valid inthe region z 1 1 . z(1 z) 7 8 Give the Laurent’s series expansion of f (z) Give the Taylor’s series for f (z) Sinz about z e z (z 1) 2 about z = 1. . 4 9 Calculate the residue at z = 0 of f (z) 1 ez z 3 z−3 10 Calculate the residue of the function(z+1)(z+2) at poles. 11 Determine the residues at poles of the function f (z) z 4 (z 1)(z 2) 12 1 Expand as Laurent’s series about z = 0 in the annulus 13 z(z 1) 0 < |z| < 1. Obtain the expansion of log(1 z) when z 1. 14 Evaluate ƒ z 15 Evaluate z2 𝑪 (z−2) z C 16 Evaluate 1 2i C dz where C is a)|z| = 1 b) |z| = 3 dz whereC isthecircle z 2 z 2 5 z 3 in the z –plane. dz whereC is z 4 using Cauchy’s integral . formula. 17 Integrate dz where C is the circle Z =2. C z4 z dz i† 𝑪 is |z| = 2. 18 Integrateƒ 19 Expand †(z) = 1 2as Taylor’s series about the point z = 2 BTL-4 Analyzing 20 Find the residues of †(z) = BTL-5 Evaluating e 𝑪 z−1 BTL- 6 Creating z www.BrainKart.com z+2 (z−2)(z+1)2 about each singularity. Click Here for Engineering Mathematics II full study material. UNIT V LAPLACE TRANSFORM Existence conditions – Transforms of elementary functions – Transform of unit step function and unit impulse function – Basic properties – Shifting theorems -Transforms of derivatives and integrals – Initial and final value theorems – Inverse transforms – Convolution theorem – Transform of periodic functions – Application to solution of linear second order ordinary differential equations with constant coefficients. PART-A 1 State the sufficient conditions for the existence of Laplace BTL-1 Remembering transform. 2 State first and second shifting theorem. BTL-1 Remembering 3 State and prove change of scale property BTL-1 Remembering 4 State Initial value and final value theorems. BTL-1 Remembering 5 State Convolution theorem BTL-1 Remembering 6 BTL-1 Remembering cost Tell whether L exist? Justify. t 7 Find the inverse Laplace transform of F(s) = 1 BTL-2 Understanding s(s−2) 8 Estimate L[t cost] BTL-2 Understanding sin at Estimate L t 10 Find L1 cot 1 s BTL-2 Understanding BTL-2 Understanding 11 Apply and verify the initial value theorem for the function f(t) = 3 e−2t 12 Apply and verify the final value theorem of the function †(t) = t2 e−3t 13 Give example of two functions for which Laplace Transform do not exist? 14 Verify initial value theorem for the function 1+e−2t. BTL-3 Applying BTL-3 Applying BTL-3 Applying BTL-4 Analyzing BTL-4 Analyzing BTL-4 Analyzing BTL-5 Evaluating BTL-5 Evaluating BTL- 6 Creating BTL- 6 Creating 9 15 16 17 18 19 20 e at ebt Find L1 t s 1 Find L1 log s 1 1 Evaluate L1 4 (s 2) 3s Evaluate L1 2s 9 3s 2 Formulate L1 s 2 4 1 1 Formulate L s(s 4) www.BrainKart.com