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268 - MA8251 Engineering Mathematics II - 2 marks Important Questions

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Engineering Mathematics II
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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
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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
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𝝀n
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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
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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 ƒð‘ª [(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.
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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
z1
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
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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
z2
𝑪 (z−2)
z
C
16
Evaluate
1
2i 
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
z4
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
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z+2
(z−2)(z+1)2
about each singularity.
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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 L1 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  ebt 
Find L1 

t


s  1

Find L1 log

s 1  

1

Evaluate L1

4
(s  2) 


3s


Evaluate L1 

 2s  9 
 3s  2
Formulate L1 

 s 2  4 
1 
1 
Formulate L 

s(s  4) 
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