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A B3A005 Pages:2
Page 1 of 2
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) 2zzf
(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 �� = ∞, �� =1
4� 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)
A B3A005 Pages:2
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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
0221
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 222121 xxxx
and transform it to principal axes. (10)
A A3801 Pages: 2
Page 1 of 2
Reg No.:_______________ Name:__________________________
APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY
THIRD SEMESTER B.TECH DEGREE EXAMINATION, APRIL 2018
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) Let �(�) = �(�, �) + ��(�, �) be defined and continuous in some neighbourhood
of a point � = � + �� and differentiable at � itself. Then prove that the first
order partial derivatives of � and � exist and satisfy the Cauchy – Riemann
equations.
(7)
b) Prove that � = sin � cosh � is harmonic. Hence find its harmonic conjugate. (8)
2 a) Find the image of the region �� −�
�� ≤
�
� under the transformation � =
�
� (8)
b) Find a linear fractional transformation which maps −1, 0, 1 onto 1, 1 + �, 1 + 2�. (7)
3 a) Check whether the function �(�) = �
�� (��)
|�|� �� � ≠ 0
0 �� � = 0
� is continuous at � = 0. (7)
b) Find the image of the x-axis under the linear fractional transformation � =�� �
��� � (8)
PART B
Answer any two full questions, each carries 15 marks
4 a) Evaluate ∫ ��(��)���
where � is the triangle with vertices 0, 1, � counter-
clockwise.
(7)
b) Using Cauchy’s Integral Formula, evaluate ∫��
�������� � ��
�where � is taken
counter-clockwise around the circle:
i) |� + 1|=�
� ii) |� − 1 − �|=
�
�
(8)
5 a) Determine and classify the singular points for the following functions:
i) �(�) =��� �
(���)� ii) �(�) = (� + �)��
��
����
(7)
b) Evaluate ∫�
(�� ��)� ��
�
��. (8)
6 a) Evaluate ∫��� �
���� ��
� counter clockwise around �: |�|=
�
� using Cauchy’s Residue
Theorem.
(7)
b) Find all Taylor series and Laurent series of �(�) =���� �
������ � with centre 0 in
i) |�|< 1 ii) 1 < |�|< 2.
(8)
A A3801 Pages: 2
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PART C
Answer any two full questions, each carries 20 marks
7 a) Solve the system of equations by Gauss Elimination Method:
3� + 3� + 2� = 1, � + 2� = 4, 10� + 3� = −2, 2� − 3� − � = 5.
(8)
b) Prove that the vectors (1, 1, 2), (1, 2, 5), (5, 3, 4) are linearly dependent. (6)
c) Prove that the set of vectors � = {(��, ��, ��) ∈ ℝ� : − �� + �� + 4�� = 0} a
vector space over the field ℝ. Also find the dimension and the basis.
(6)
8 a) Find the Eigen values and the corresponding Eigen vectors of
� = � 1 1 −2−1 2 1 0 1 −1
�
(8)
b) What kind of conic section is given by the quadratic form 7��� + 6���� + 7��
� =
200. Also find its equation.
(6)
c) Determine whether the matrix � = �
1 0 00 ���� −����0 ���� ����
� symmetric, skew-
symmetric or orthogonal.
(6)
9 a)
Reduce the matrix � = �
21
3−1
−1 −1−2 −4
3 1 3 −26 3 0 −7
� to Row Echelon Form and hence
find its rank.
(8)
b) Diagonalize � = �
3 −1 1−1 3 −1 1 −1 3
� (12)
****
A B3A001S Pages: 2
Page 1 of 2
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)
A B3A001S Pages: 2
Page 2 of 2
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)
***
A A7046
Page 1 of 2
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, 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
A A7046
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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)
****
A B1A003 Total No. of pages:2
Page 1 of 2
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)
A B1A003 Total No. of pages:2
Page 2 of 2
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)