Science and Humanities Department

Vishwakarma Govt. Engineering College Chandkheda, Ahmedabad

Semester II (All Branches)
Academic year: Even 2024-25
Subject Name (Subject Code): Mathematics II (BE02000011)
Term Dates: 21/01/2025 to 10/05/2025

Tutorial – 1 Matrices

1 Find row echelon and reduced row echelon form of the given matrix
1 5 4  1 3 2 4 −3 1 3 5 8
 
(i )  0 3 2  (ii ) 
1 2 3 4 2 6 0 −1 −2 −2 7 3 10
 iii) [ ] iv) [ ]
 2 13 10   −2 0 5 7  0 0 6 2 −1 5 4 14 18
1 3 −1 4 2 2 1 5 6

2 Find the rank of the given matrices by reducing it into row echelon form
 1 5 3 −2   0 6 7 1 3 4 5  3 −2 4 1
(i)  2 0 4 1  (ii)  −5 4 2  (iii) 1 2 6 7  iv) [
2 −1 −3 5
]
    5 −3 1 6
 4 8 9 −1  1 −2 0  1 5 0 10 −1 0 10 −9
3 Find inverse of the following matrices by Gauss Jordan method
 1 2 3 2 6 6 5 −1 5 0 2 13
   
(i )  2 5 3 (ii)  2 7 6  iii) [ 0 2 0 ] iv) [ 4 5 −3]
 1 0 8  2 7 7  −5 3 −15 −1 −1 −1

4 Discuss the consistency of system of linear equations using concept of rank of matrix

5 Examine consistency of following equations using Gauss elimination method and find the
solution of following if exists
x + y + 2z = 8 2x + 2 y + 2z = 0 x − 2y + z = 4
(1) − x − 2 y + 3z = 1 (2) − 2 x + 5 y + 2 z = 1 (3) 3 x + 5 y + z = 6
3x − 7 y + 4 z = 10. 8 x + y + 4 z = −1 6x − y + 4z = 2
 Science and Humanities Department
Vishwakarma Govt. Engineering College Chandkheda, Ahmedabad
Semester II (All Branches)
Academic year: Even 2024-25
Subject Name (Subject Code): Mathematics II (BE02000011)
Term Dates: 21/01/2025 to 10/05/2025

Tutorial – 2 Matrices

Solve the given system of linear equation by using elementary row operation
1. 𝑥 − 𝑦 + 𝑧 = 3, 2𝑥 − 3𝑦 + 5𝑧 = 10, 𝑥 + 𝑦 + 4𝑧 = 4
2. 𝑥 + 2𝑦 + 𝑧 = 8, 2𝑥 + 3𝑦 + 𝑧 = 13, 𝑥 + 𝑦 = 5
3. 𝑥 − 2𝑦 + 𝑧 = 4, 3𝑥 + 5𝑦 + 𝑧 = 6, 6𝑥 − 𝑦 + 4𝑧 = 2
1 4. 5𝑥 + 𝑦 + 𝑧 + 𝑤 = 4, 𝑥 + 7𝑦 + 𝑧 + 6𝑤 = 2,
𝑥 + 𝑦 + 6𝑧 + 𝑤 = −5, 𝑥 + 𝑦 + 𝑧 + 𝑤 = 0
5. 𝑣 + 3𝑤 − 2𝑥 = 0, 2𝑢 + 𝑣 − 4𝑤 + 3𝑥 = 0, 2𝑢 + 3𝑣 + 2𝑤 − 𝑥 = 0,
−4𝑢 − 3𝑣 + 5𝑤 − 4𝑥 = 0

2 Solve the following linear system using Gauss-Jordan method

1. 2𝑥 + 𝑦 + 𝑧 = 12, 3𝑥 + 2𝑦 + 3𝑧 = 24, 𝑥 + 4𝑦 + 9𝑧 = 34
2. 𝑥 + 2𝑦 − 𝑧 = −1, 3𝑥 + 8𝑦 + 2𝑧 = 28, 4𝑥 + 9𝑦 − 𝑧 = 14
3. 𝑥 + 𝑦 + 𝑧 = 6, 𝑥 + 2𝑦 + 3𝑧 = 14, 2𝑥 + 4𝑦 + 7𝑧 = 30
3 Investigate for what values of a and b the following system of linear equations have
(1) No solution (2) Infinite solutions and (3) Unique solution
𝑥 + 𝑦 + 𝑧 = 6, 𝑥 + 2𝑦 + 3𝑧 = 10, 𝑥 + 2𝑦 + 𝑎𝑧 = 𝑏
4 Define symmetric, skew symmetric and orthogonal matrix.
5 Answer the following question
1. A square matrix equal to its transpose is called?
2. For a given matrix 𝐴, 𝐴𝑇 = −𝐴 then 𝐴 is called?
3. The addition of two symmetric matrices result in
4. Determinant of skew symmetric matrix having an odd order is?
6 3 2 6
1
Prove that 𝐴 = 7 [−6 3 2 ] is an orthogonal matrix of order 3 and also find its inverse.
2 6 −3
 Science and Humanities Department
Vishwakarma Govt. Engineering College Chandkheda, Ahmedabad
Semester II (All Branches)
Academic year: Even 2024-25
Subject Name (Subject Code): Mathematics II (BE02000011)
Term Dates: 21/01/2025 to 10/05/2025
Tutorial – 3 Matrices
1 2
Find the eigen value of 𝐴 A and 𝐴−1 , 𝐴 = [ ]
1 0 4
2 −4 2 −2
Find the eigen value of 𝐴 = [ 1 −3 −1] also find the eigen value for the matrix
−1 1 −5
𝐴2 , 𝐴3 , 𝐴 − 2𝐼, 𝐴 + 4𝐼, 𝐴𝑇

3 Find the eigenvalues and corresponding eigen vectors of the following matrices
 4 0 1  1 2 2 2 1 0 2 1 0
   
(i) A =  −2 1 0  (ii) A =  0 2 1  (iii) [−1 0 1] (iv) [0 2 1]
 −2 0 1   −1 2 2  0 0 1 0 0 2

4 2 1 0
If 1 is an eigen value of the matrix  −1 0 1  then find its corresponding eigen vectors.
 0 0 1 
5 State Cayley -Hamilton theorem. Find A−1 using
Caley-Hamilton theorem;
2 1 1
1 −1  
(i) A =   (ii) A =  0 1 0 
 2 3  1 1 2 

6 Find A3 using Cayley -Hamilton theorem if

1 0 1 
A = 1 −1 1 
0 1 0 

7 0 0 −2 
Diagonalize the matrix A= 1 2 1  . Hence find A25 .
1 0 3 
8 1 −1 2
Verify Caley Hamilton theorem for 𝐴 = [0 2 1 ] using it find 𝐴−1 and also compute
0 1 −1
𝐴7 − 2𝐴6 + 4𝐴5 − 6𝐴4 + 2𝐴3 + 𝐴2 − 5𝐴 + 𝐼

9 1 −1
Find the characteristic equation of the matrix 𝐴 = [ ] and hence find the matrix
−1 2
respected by 𝐴5 − 3𝐴4 + 𝐴3 + 4𝐴2

10 0 −2 2
Is the matrix 𝐴 = [1 2 0] diagonalizable?
3 2 0
 Science and Humanities Department
Vishwakarma Govt. Engineering College Chandkheda, Ahmedabad
Semester II (All Branches)
Academic year: Even 2024-25
Subject Name (Subject Code): Mathematics II (BE02000011)
Term Dates: 21/01/2025 to 10/05/2025

Tutorial – 4 Complex Variable (Basic)

1 Represent following complex numbers in complex plane.

2 Represent following complex numbers in complex plane.

3 Represent following in 𝑎 + 𝑖𝑏 or 𝑢 + 𝑖𝑣 form.

𝑧 2
𝑧 2 , 𝑧 3 , 𝑖𝑧 2 , 𝑧+1 , 𝑒 𝑧 sin 𝑧 , 𝑒 𝑧

4 Convert following complex numbers in polar and exponential form.

5 Represent following sets in complex plane.

6 Define
(1) Open set (2) Connected set (3) Domain (4) Bounded set
(5) Compact set

7 Discuss Analyticity of following functions using C-R equation.

1. 𝑓(𝑧) = 𝑧̅ 2. 𝑓(𝑧) = 𝑅𝑒(𝑧) 3. 𝑓(𝑧) = 𝑥 + 5𝑦𝑖
𝑧
4. 𝑓(𝑧) = cos 𝑧 5. 𝑓(𝑧) = 𝑒 6. 𝑓(𝑧) = 2𝑧 + 5𝑖
 Science and Humanities Department
Vishwakarma Govt. Engineering College Chandkheda, Ahmedabad
Semester II (All Branches)
Academic year: Even 2024-25
Subject Name (Subject Code): Mathematics II (BE02000011)
Term Dates: 21/01/2025 to 10/05/2025

Tutorial – 5 Complex Variable

1Fin Find derivative of following functions.

2 Define analytic function and state sufficient condition for analytic function.

3 Determine C-R equations in polar form.

4 If f(z) is analytic and 𝑓(𝑧) = 𝑐, then show that f(z) is constant.

5 Discuss analyticity of ln(z).

6 Define a harmonic function. Find harmonic conjugate for following functions

1. 𝑢 = 𝑥 2 − 𝑦 2 − 𝑦
2. 𝑢 = 𝑒 𝑥 cos 𝑦
𝑥
3. 𝑣 = 𝑥 2 − 𝑦 2 + 𝑥 2+𝑦 2
4. 𝑢 = sinh 𝑥 cos 𝑦
5. 𝑢 = 𝑥 3 − 3𝑥𝑦 2