F. Y. B. Tech (2023 Course)
Subject: Mathematics-I (Linear Algebra and Calculus)
A.Y. 2023-24 (Semester-I)
Practice Question Bank on Unit -I
Unit I: Matrices-System of Linear Equations
Q.
No
CO
BL
CO1
BL3
I: Rank of Matrix using Echelon Form
Q.1
Finding the rank by reducing the matrix A to Echelon Form.
1 1  1 1 


1) A  1  1 2  1
3 1
0 1 
1 1 2 3 5 


3) A  3 2 7 5 12 
3 3 6 9 15
 2  3 4 4


5) A  1 1 1 2
3  2 3 6
 4 2  1 2


7) A  1  1 2 1
2 2  2 0
 1 2 1 3 


9) A   3 4 0  1
 1 0  2 7 
1  3  1
0 1
1 
1 0
2

1 2 0 
1 1 1  1


13) A  1 2 3 4 
3 4 5 2 
0
1
11) A  
3

1
2)
 2 1  3  6
A  3  3 1
2 
1 1
1
2 
4)
 1 2 1 0
A   2 4 3 0
 1 0 2 8
6)
2 3  1  1 
1  1  2  4 

A
3 1
3  2


0  7
6 3
8)
10)
1 2 3 2 
A  2 3 5 1
1 3 4 5
2  3 5 1 
A  3 1  1 2
1 4  6 1 
1 1
1  1
12) A  
3 1

2  2
14)
1 6
2 5
1 8

3 7
 2  1  1 2
A  1 2
1 2
4  7  5 2
II: Non-Homogeneous system of Equation:
Q.2
Examine for consistency and if consistent then solve it.
i)
2x  y  z  2 ,
x  2y  z  2 ,
ii)
2x  2 y  2z  0 ,
iii)
2 x1  3x2  5 x3  1,
iv)
2 x1  x3  4 ,
v)
4x  2 y  6z  8 ,
vi)
2 x  y  3z  1 ,
BL3
CO1
BL3
4 x  7 y  5z  2
 2x  5 y  2z  1 ,
8 x  y  4 z  1
3x1  x2  x3  2 ,
x1  4 x2  6 x3  1
x1  2 x 2  2 x3  7 ,
CO1
3x1  2 x2  1
x  y  3 z  1 , 15 x  3 y  9 z  21
3x  2 y  z  3 ,
x  4 y  5 z  1
vii) 3 x  y  2 z  3 , 2 x  3 y  z  3 , x  2 y  z  4
viii) x  y  2 z  8 ,  x  2 y  3 z  1 , 3x  7 y  4 z  10
ix)
2x  3y  z  3 ,
x)
3 x  6 y  3 z  2 ,
xi) x  y  z  3,
x  2y  z  4 ,
6 x  6 y  3z  5 ,
x  2 y  3 z  4,
xii) x  2 y  2 z  1 ,
 2 y  3z  1
x  4 y  9z  6
2 x  2 y  3z  3 ,
xiii) 3 x  6 y  3 z  2 ,
5 x  4 y  3 z  2
x  y  3z  5
6 x  6 y  3 z  5 , 2 y  3 z  1
III: Non-Homogeneous equation with unknown:
Q.3
Investigate For what values of  ,  the system of equations has
(i ) No Solutions
(ii) An Infinite number of Solution
2 x  y  3z  2
a)
(iii) A Unique Solution
x yz 6
x  y  2z  2
b) x  2 y  3 z  10
x  2 y  z  
5 x  y  z  
2x  3 y  5z  9
x  2y  z  8
c) 7 x  3 y  2 z  8
2 x  3 y  z  
d) 2 x  2 y  2 z  13
3 x  4 y  z  
3x  4 y  5 z  
Q.4
Show that the system of equation 4 x  5 y  6 z   is consistent only when
5x  6 y  7 z  
 ,  ,  are in Arithmetic Progression.
IV: Homogeneous system of Equation
Q.5
. Solve the following system of equation
2 x  2 y  3z  0
5 x  2 y  3z  0
3x  2 y  z  0
i)
3x  y  z  0
ii)
x  2 y  5z  0
2x  y  6z  0
V: Homogeneous equation with unknown:
Q.6
Investigate for what values of k the system of simultaneous equations
CO1
BL3
CO1
BL3
has (i) trivial solution (ii) non-trivial solution. Find the non-trivial solution.
–
a)
b)
–
–
–
–
c)
d)
VI : Dependence and Independence Vectors:
Q.7
Test the following vectors for linear dependence or independence. If dependent
find the relation between them
i)
X 1  2,  1, 3, 2,
X 2  1, 3, 4, 2,
ii)
X 1  3, 1,  4,
iii)
X 1  1,  1, 1,
X 2  2, 1, 1,
iv)
X 1  3, 1, 1,
X 2  2, 0,  1,
v)
X 1  1, 1, 1,
X 2  1, 2, 3,
vi)
X 1  1,  1, 2,
vii)
1 
 3 
1 




X 1  2 , X 2   2 , X 3   6
3
 1 
  5
viii)
2
1
1 




X 1  2 , X 2  3 , X 3  2
1
1 
2
X 2  2, 2,  3,
X 3  3,  5, 2, 2
X 3  0,  4, 1
X 3  3, 0, 2
X 3  4, 2, 1
X 3  2, 3, 8
X 2  2, 3, 5,
X 3  3, 2, 1