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Reg. No._____________ Name:_____________________

APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY

THIRD SEMESTER B.TECH DEGREE EXAMINATION, MARCH 2017

MA 201: LINEAR ALGEBRA AND COMPLEX ANALYSIS

Max. Marks: 100 Duration: 3 Hours

PART A

Answer any 2 questions

1. a. Check whether the following functions are analytic or not. Justify your answer.

i) zzf z (4)

ii) 2

zzf

(4)

b. Show that zzf sin is analytic for all z. Find zf (7)

2. a. Show that 323 yyxv is harmonic and find the corresponding analytic function

yxivyxuzf ,, (8)

b. Find the image of 10 x , 12

1 y under the mapping zew (7)

3. a. Find the linear fractional transformation that carries �� = −2, �� = 0 and �� = 2

on to the points �� = ∞, �� = 14� and �� = 3

8� . Hence find the image of x-axis.(7)

b. Find the image of the rectangular region x , bya under the mapping

zw sin (8)

PART B

Answer any 2 questions

4. a. Evaluate ∫ |�|���

where

i) C is the line segment joining -i and i (3)

ii) C is the unit circle in the left of half plane (4)

b. Verify Cauchy’s integral theorem for �� taken over the boundary of the rectangle

with vertices -1, 1, 1+i, -1+i in the counter clockwise sense. (8)

5. a. Find the Laurent’s series expansion of 21

1

zzf

which is convergent in

i) |� − 1| < 2 (4)

ii) |� − 1| > 2 (4)

b. Determine the nature and type of singularities of

i) 2

2

z

e z

(3)

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ii) � sin (�

�)

(4)

6. a. Use residue theorem to evaluate

dzzz

zz

C

1312

523302

2

where C is 1z (7)

b. Evaluate

dxx

0

221

1 using residue theorem. (8)

PART C

Answer any 2 questions

7. a. Solve the following by Gauss elimination

y + z – 2w = 0, 2x – 3y – 3z + 6w = 2, 4x + y + z – 2w = 4 (6)

b. Reduce to Echelon form and hence find the rank of the matrix

1502121

5424426

2203

(6)

c. Find a basis for the null space of

402

840

022

(8)

8. a. i) Are the vectors (3 -1 4), (6 7 5) and (9 6 9) linearly dependent or

independent? Justify your answer. (5)

ii) Is all vectors zyx ,, in ℝ� with 04 zxy form a vector space over the field

of real numbers? Give reasons for your answer. (5)

b. i) Find a matrix C such that xCxTQ where

2331

2221

21 5243 xxxxxxxQ

(4)

ii) Obtain the matrix of transformation

y1 = cos θ x1 – sin θ x2, y2 = sin θ x1 + cos θ x2

Prove that it is orthogonal. Obtain the inverse transformation. (6)

9. a. Find the eigenvalues, eigenvectors and bases and dimensions for each Eigen space

of

021

612

322

A

(10)

b. Find out what type of conic section, the quadratic form 128173017 2221

21 xxxx

and transform it to principal axes. (10)

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Reg. No.______________ Name:_______________________

APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY

THIRD SEMESTER B.TECH DEGREE EXAMINATION, JULY 2017

Course Code: MA 201

Course Name: LINEAR ALGEBRA AND COMPLEX ANALYSIS.

Max. Marks :100 Duration: 3 hours

PART A

Answer any two questions.

1. (a) Does the limit Limz→0 �

� exit? If yes find the value. If no, explain why? (8)

(b) If f(z) = u + iv is analytic, prove that � = constant and � = constant are families of

curves cutting orthogonally (7)

2. (a) Find the image of the semi-circle � = +√4 − �� under the transformation � = ��

(7)

(b) Find the image of the half-plane Re(z) ≥ 2 under the map � = �� (8)

3. (a) Find the points, if any, in complex plane where the function �(�) = 2�� + � +

�(�� − �) is

(i) differentiable (ii) analytic. (8)

(b) Prove that the function �(�, �) = �� − 3��� − 5� is harmonic everywhere. Also

find the harmonic conjugate of �. (7)

PART B

Answer any two questions.

4. (a) Evaluate ∫ � ����

where � is given by � = 3�, � = ��, −1 ≤ � ≤ 4. (8)

(b) Show that ∫ (2 + �)��� = −�

�� where � is any path connecting the points -2 and

-2 + i (7)

5. (a) Evaluate ∫����

���������

� where � is the circle |� − 2| = 2. (8)

(b) Find the Laurent’s series expansion of �

���� in 1 < |� + 1| < 2. (7)

6. (a) Use Cauchy’s integral formula to evaluate ∫���

���������

� where � is |�| = 1.

(8)

(b) Using Contour integration, evaluate ∫������

����� ������

�

�� (7)

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PART C

Answer any two questions.

7. (a) Using Gauss elimination method, find the solution of the system of equations

� + 2� − � = 3, 3� − � + 2� = 1, 2� − 2� + 3� = 2 and � − � + � = −1 (7)

(b) Find the values of � for which the system of equations � + � + � = 1, � + 2� +

3� = � and � + 5� + 9� = �� will be consistent. For each value of � obtained,

find the solution of the system. (7)

(c) Prove that the vectors (2,3,0). (1,2,0) and (8,13,0) are linearly dependent in ��.

(6)

8. (a) Find the rank of the matrix � =

⎣⎢⎢⎡2 31 −1

−1 −1−2 −1

3 16 3

3 −20 −7 ⎦

⎥⎥⎤

(7)

(b) Find the eigen values and eigen vectors of the matrix �1 0 −11 2 12 2 3

� (7)

(c) Write the canonical form of the quadratic form �(�, �, �) = 3�� + 5�� + 3�� −

2�� + 2�� − 2�� and hence show that �(�, �, �) > 0 for all non-zero values of

�, �, �. (6)

9. (a) Diagonalize the matrix � = �2 0 10 2 01 0 2

� and hence find ��. (7)

(b) If 2 is an eigen value of �−3 −1 11 5 −11 −1 3

�, without using its characteristic equation,

find the other eigen values. Also find the eigen values of ��, ��, ���, 5�, � − 3� and

��� �. (7)

(c) Show that 17x2 – 30xy + 17y2 = 128 represents an ellipse. Also find the equations

of the major and minor axes of the ellipse in terms of � and �. (6)

***

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Total Pages: 2 Reg No.:_______________ Name:__________________________

APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY

THIRD SEMESTER B.TECH DEGREE EXAMINATION, DECEMBER 2017

Course Code: MA201

Course Name: LINEAR ALGEBRA AND COMPLEX ANALYSIS

Max. Marks: 100 Duration: 3 Hours

PART A Answer any two full questions, each carries 15 marks. Marks

1 a) Find the points where Cauchy-Riemann equations are satisfied for the function

f(z) = xy2 + i x2 y. Where does f |(z) exist? Is the function f(z) analytic at those

points?

(7)

b) If v = ex (x sin y + y cos y), find an analytic function f(z)=u+iv. (8)

2 a) Show that u = x2-y2-y is harmonic. Also find the corresponding conjugate harmonic

function.

(7)

b) (i) Find a bilinear transformation which maps (−𝑖, 0, i) onto (0, -1, ∞).

(ii) Test the continuity at z = 0, if f(z) = 𝐼𝑚 𝑧

|𝑧|, 𝑧 ≠ 0

= 0, z = 0

(8)

3 a) Find the image of the lines x=1, y=2 and x>0, y<0 under the mapping W= z2 (8)

b) Find the image of the semi-infinite strip x > 0, 0< 𝑦 < 2 under the transformation

w=iz+1. Draw the regions.

(7)

PART B

Answer any two full questions, each carries 15 marks.

4 a) Evaluate ∮ 𝑅𝑒 z2dz over the boundary C of the square with vertices 0, i, 1+ i,1

clockwise

(8)

b) Evaluate ∫4−3𝑧

𝑧(𝑧−1) dz over the circle |z|=

3

2 (4)

c) Evaluate ∫3𝑧2+7𝑧+1

𝑧+1 dz over the circle |z+ i |=1 (3)

5 a) Expand 𝑧

(𝑧−1)(𝑧−2) in (1) 0<|z-2|<1, (2) |z-1|>1 (8)

b) Evaluate ∫1

2+cos 𝜃

2𝜋

0 d𝜃 (7)

6 a) Using Residue theorem evaluate ∫𝑧2

(𝑧−1)2(𝑧+2) dz over the circle |z|=3 (7)

b) Find the Taylor series of sin 𝑧

𝑧−𝜋 about the point z= 𝜋 (4)

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c) Evaluate ∫sin 𝑧

𝑧6 dz over the circle |z|=2 using Cauchy’s Residue theorem. (4)

PART C

Answer any two full questions, each carries 20 marks.

7 a) Solve by Gauss-Elimination method x + y + z = 6, x+ 2y- 3z = -4, -x-4y+9z =18. (7)

b) Find the values of ‘a’ and ‘b’ for which the system of equations x + y + 2z =2,

2x-y+3z=10,5x-y+az=b has:

(i) no solution (ii) unique solution (iii) infinite number of solutions.

(7)

c) Verify whether the vectors (1,2,1,2), (3,1,-2,1),(4,-3,-1,3) and (2,4,2,4) are linearly

independent in R4 .

(6)

8 a) Write down the matrix associated with the quadratic form 8x12+7x2

2+3x32-12x1x2

-8x2x3+4x3x1. By finding eigen values, determine nature of the quadratic form.

(7)

b) Diagonalise the matrix A = [

1 −2 0−2 0 20 2 −1

]

(7)

c) If A is a symmetric matrix, verify whether AAT and ATA are symmetric? (6)

9 a) Find the eigen vectors of A = [

3 0 05 4 03 6 1

] (8)

b)

Find the null space of AX=0 if A=[

1 1 0 2−2 −2 1 −51 1 −1 34 4 −1 9

]

(6)

c) Verify whether 𝐴 = [

1 0 00 cos 𝜃 −sin 𝜃0 sin 𝜃 cos 𝜃

] is orthogonal.

What can you say about determinant of an orthogonal matrix? Prove or disprove the

result.

(6)

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A B1A003 Total No. of pages:2

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Reg. No._______________ Name:__________________________

APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY

THIRD SEMESTER B.TECH DEGREE EXAMINATION, DEC 2016

Course Code: MA201

Course Name: LINEAR ALGEBRA AND COMPLEX ANALYSIS

Max. Marks: 100 Duration:3. Hours

PART A

(Answer any two questions)

1.a Show that � = �� − 3���is harmonic and hence find its harmonic conjugate. (8)

b Find the image of �� −�

�� ≤

�

�under the transformation =

�

� . Also find the fixed points

of the transformation � =�

� (7)

2.a Define an analytic function and prove that an analytic function of constant modulus is

constant. (8)

b Find the linear fractional transformation that maps �� = 0, �� = 1, �� = ∞onto

�� = −1, �� = −�, �� = 1 respectively. (7)

3.a Show that �(�) = ������� − �������� is differentiable everywhere. Find

its derivative. (8)

b Find the image of the lines � = � and � = �, where �&�are constants, under the

transformation � = ����. (7)

PART B

(Answer any two questions)

4.a Evaluate ∫ �� (�) ���

where � is a straight line from 0 to 1 + 2�. (7)

b Show that ∫��

���� = �

�√�

�

� (8)

5.a Integrate ��

���� counterclockwise around the circle |� − 1 − �| =

�

� by Cauchy’s

Integral Formula. (7)

b Evaluate ∫����

���������

� where � is |� − 2 − �| = 3.5 by Cauchy’s Residue Theorem

(8)

6.a If �(�) =�

�� find the Taylor series that converges in |� − �| < �and the Laurent’s

series that converges in |� − �| > �. (8)

b Define three types of isolated singularities with an example for each. (7)

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PART C

(Answer any two questions)

7.a Solve by Gauss Elimination:

�� − �� + �� = 0,

−�� + �� − �� = 0,

10 �� + 25 �� = 90,

20 �� + 10 �� = 80. (5)

b Find the rank. Also find a basis for the row space and column space for

� 0 1 0−1 0 −4 0 4 0

� (5)

c Find out what type of conic section the quadratic form

� = 17 �� − 30 �� + 17 �� = 128 represents and transform it to the principal

axes. (10)

8.a Find whether the vectors [1 2−1 3], [2 −13 2]��� [−1 8−9 5] are

linearly dependent. (5)

b Show that the matrix � = �1 22 −2

� is symmetric. Find the spectrum. (5)

c Diagonalise � = � 8 −6 2−6 7 −4 2 −4 3

� (10)

9. a. Determine whether the matrix

⎣⎢⎢⎡1 0 0

0 1√2

� −1√2

�

0 1√2

� 1√2

� ⎦⎥⎥⎤ is orthogonal? (5)

b. Find the Eigen values and Eigen vectors of � 1 1 2−1 2 1 0 1 3

� (5)

c. Define a Vector Space with an example. (10)