Page 1
2024
Q1  2024 (01 Feb Shift 1)
If the system of equations
2 ?? + 3 ??  ?? = 5
?? + ???? + 3 ?? =  4
3 ??  ?? + ???? = 7
has infinitely many solutions, then 13 ???? is equal to
(1) 1110
(2) 1120
(3) 1210
(4) 1220
Q2  2024 (01 Feb Shift 2)
Let the system of equations ?? + 2 ?? + 3 ?? = 5 , 2 ?? + 3 ?? + ?? = 9 , 4 ?? + 3 ?? + ???? = ?? have
infinite number of solutions. Then ?? + 2 ?? is equal to :
(1) 28
(2) 17
(3) 22
(4) 15
Q3  2024 (27 Jan Shift 1)
Let ?? = [
2 0 1
1 1 0
1 0 1
] , ?? = [ ?? 1
, ?? 2
, ?? 3
], where ?? 1
, ?? 2
, ?? 3
are column matrices, and ?? 1
= [
1
0
0
],
AB
2
= [
2
3
0
] , AB
3
= [
3
2
1
]
Page 2
2024
Q1  2024 (01 Feb Shift 1)
If the system of equations
2 ?? + 3 ??  ?? = 5
?? + ???? + 3 ?? =  4
3 ??  ?? + ???? = 7
has infinitely many solutions, then 13 ???? is equal to
(1) 1110
(2) 1120
(3) 1210
(4) 1220
Q2  2024 (01 Feb Shift 2)
Let the system of equations ?? + 2 ?? + 3 ?? = 5 , 2 ?? + 3 ?? + ?? = 9 , 4 ?? + 3 ?? + ???? = ?? have
infinite number of solutions. Then ?? + 2 ?? is equal to :
(1) 28
(2) 17
(3) 22
(4) 15
Q3  2024 (27 Jan Shift 1)
Let ?? = [
2 0 1
1 1 0
1 0 1
] , ?? = [ ?? 1
, ?? 2
, ?? 3
], where ?? 1
, ?? 2
, ?? 3
are column matrices, and ?? 1
= [
1
0
0
],
AB
2
= [
2
3
0
] , AB
3
= [
3
2
1
]
If ?? =  ??  and ?? is the sum of all the diagonal elements of B, then ?? 3
+ ?? 3
is equal to
Q4  2024 (27 Jan Shift 2)
The values of ?? , for which

1
3
2
?? +
3
2
1
1
3
?? +
1
3
2 ?? + 3 3 ?? + 1 0
 = 0,
lie in the interval
(1) (  2 , 1 )
(2) (  3 , 0 )
(3) ( 
3
2
,
3
2
)
(4) ( 0 , 3 )
Q5  2024 (29 Jan Shift 2)
Let for any three distinct consecutive terms a , b , c of an A.P, the lines ???? + ???? + ?? = 0 be
concurrent at the point P and Q ( ?? , ?? ) be a point such that the system of equations
?? + ?? + ?? = 6,
2 ?? + 5 ?? + ???? = ?? and
?? + 2 ?? + 3 ?? = 4, has infinitely many solutions. Then ( PQ )
2
is equal to
Q6  2024 (30 Jan Shift 1)
Consider the system of linear equation x + y + z = 4 ?? , ?? + 2 ?? + 2 ???? = 10 ?? , ?? + 3 ?? +
4 ?? 2
?? = ?? 2
+ 15, where ?? , ?? ? R. Which one of the following statements is NOT correct?
(1) The system has unique solution if ?? ?
1
2
and ?? ? 1 , 15
(2) The system is inconsistent if ?? =
1
2
and ?? ? 1
(3) The system has infinite number of solutions if ?? =
1
2
and ?? = 15
(4) The system is consistent if ?? ?
1
2
Q7  2024 (30 Jan Shift 1)
Page 3
2024
Q1  2024 (01 Feb Shift 1)
If the system of equations
2 ?? + 3 ??  ?? = 5
?? + ???? + 3 ?? =  4
3 ??  ?? + ???? = 7
has infinitely many solutions, then 13 ???? is equal to
(1) 1110
(2) 1120
(3) 1210
(4) 1220
Q2  2024 (01 Feb Shift 2)
Let the system of equations ?? + 2 ?? + 3 ?? = 5 , 2 ?? + 3 ?? + ?? = 9 , 4 ?? + 3 ?? + ???? = ?? have
infinite number of solutions. Then ?? + 2 ?? is equal to :
(1) 28
(2) 17
(3) 22
(4) 15
Q3  2024 (27 Jan Shift 1)
Let ?? = [
2 0 1
1 1 0
1 0 1
] , ?? = [ ?? 1
, ?? 2
, ?? 3
], where ?? 1
, ?? 2
, ?? 3
are column matrices, and ?? 1
= [
1
0
0
],
AB
2
= [
2
3
0
] , AB
3
= [
3
2
1
]
If ?? =  ??  and ?? is the sum of all the diagonal elements of B, then ?? 3
+ ?? 3
is equal to
Q4  2024 (27 Jan Shift 2)
The values of ?? , for which

1
3
2
?? +
3
2
1
1
3
?? +
1
3
2 ?? + 3 3 ?? + 1 0
 = 0,
lie in the interval
(1) (  2 , 1 )
(2) (  3 , 0 )
(3) ( 
3
2
,
3
2
)
(4) ( 0 , 3 )
Q5  2024 (29 Jan Shift 2)
Let for any three distinct consecutive terms a , b , c of an A.P, the lines ???? + ???? + ?? = 0 be
concurrent at the point P and Q ( ?? , ?? ) be a point such that the system of equations
?? + ?? + ?? = 6,
2 ?? + 5 ?? + ???? = ?? and
?? + 2 ?? + 3 ?? = 4, has infinitely many solutions. Then ( PQ )
2
is equal to
Q6  2024 (30 Jan Shift 1)
Consider the system of linear equation x + y + z = 4 ?? , ?? + 2 ?? + 2 ???? = 10 ?? , ?? + 3 ?? +
4 ?? 2
?? = ?? 2
+ 15, where ?? , ?? ? R. Which one of the following statements is NOT correct?
(1) The system has unique solution if ?? ?
1
2
and ?? ? 1 , 15
(2) The system is inconsistent if ?? =
1
2
and ?? ? 1
(3) The system has infinite number of solutions if ?? =
1
2
and ?? = 15
(4) The system is consistent if ?? ?
1
2
Q7  2024 (30 Jan Shift 1)
If ?? ( ?? ) = 
2 c o s
4
? ?? 2 sin
4
? ?? 3 + sin
2
? 2 ?? 3 + 2 c o s
4
? ?? 2 sin
4
? ?? sin
2
? 2 ?? 2 c o s
4
? ?? 3 + 2 sin
4
? ?? sin
2
? 2 ??  then
1
5
?? '
( 0 ) is equal to
(1) 0
(2) 1
(3) 2
(4) 6
Q8  2024 (30 Jan Shift 2)
Consider the system of linear equations
?? + ?? + ?? = 5 , ?? + 2 ?? + ?? 2
?? = 9
?? + 3 ?? + ???? = ?? , where ?? , ?? ? ?? . Then, which of the following statement is NOT correct?
(1) System has infinite number of solution if ?? = 1 and ?? = 13
(2) System is inconsistent if ?? = 1 and ?? ? 13
(3) System is consistent if ?? ? 1 and ?? = 13
(4) System has unique solution if ?? ? 1 and ?? ? 13
Q9  2024 (31 Jan Shift 1)
If the system of linear equations
??  2 ?? + ?? =  4
2 ?? + ???? + 3 ?? = 5
3 ??  ?? + ???? = 3
has infinitely many solutions, then 12 ?? + 13 ?? is equal to
(1) 60
(2) 64
(3) 54
(4) 58
Q10  2024 (31 Jan Shift 1)
Page 4
2024
Q1  2024 (01 Feb Shift 1)
If the system of equations
2 ?? + 3 ??  ?? = 5
?? + ???? + 3 ?? =  4
3 ??  ?? + ???? = 7
has infinitely many solutions, then 13 ???? is equal to
(1) 1110
(2) 1120
(3) 1210
(4) 1220
Q2  2024 (01 Feb Shift 2)
Let the system of equations ?? + 2 ?? + 3 ?? = 5 , 2 ?? + 3 ?? + ?? = 9 , 4 ?? + 3 ?? + ???? = ?? have
infinite number of solutions. Then ?? + 2 ?? is equal to :
(1) 28
(2) 17
(3) 22
(4) 15
Q3  2024 (27 Jan Shift 1)
Let ?? = [
2 0 1
1 1 0
1 0 1
] , ?? = [ ?? 1
, ?? 2
, ?? 3
], where ?? 1
, ?? 2
, ?? 3
are column matrices, and ?? 1
= [
1
0
0
],
AB
2
= [
2
3
0
] , AB
3
= [
3
2
1
]
If ?? =  ??  and ?? is the sum of all the diagonal elements of B, then ?? 3
+ ?? 3
is equal to
Q4  2024 (27 Jan Shift 2)
The values of ?? , for which

1
3
2
?? +
3
2
1
1
3
?? +
1
3
2 ?? + 3 3 ?? + 1 0
 = 0,
lie in the interval
(1) (  2 , 1 )
(2) (  3 , 0 )
(3) ( 
3
2
,
3
2
)
(4) ( 0 , 3 )
Q5  2024 (29 Jan Shift 2)
Let for any three distinct consecutive terms a , b , c of an A.P, the lines ???? + ???? + ?? = 0 be
concurrent at the point P and Q ( ?? , ?? ) be a point such that the system of equations
?? + ?? + ?? = 6,
2 ?? + 5 ?? + ???? = ?? and
?? + 2 ?? + 3 ?? = 4, has infinitely many solutions. Then ( PQ )
2
is equal to
Q6  2024 (30 Jan Shift 1)
Consider the system of linear equation x + y + z = 4 ?? , ?? + 2 ?? + 2 ???? = 10 ?? , ?? + 3 ?? +
4 ?? 2
?? = ?? 2
+ 15, where ?? , ?? ? R. Which one of the following statements is NOT correct?
(1) The system has unique solution if ?? ?
1
2
and ?? ? 1 , 15
(2) The system is inconsistent if ?? =
1
2
and ?? ? 1
(3) The system has infinite number of solutions if ?? =
1
2
and ?? = 15
(4) The system is consistent if ?? ?
1
2
Q7  2024 (30 Jan Shift 1)
If ?? ( ?? ) = 
2 c o s
4
? ?? 2 sin
4
? ?? 3 + sin
2
? 2 ?? 3 + 2 c o s
4
? ?? 2 sin
4
? ?? sin
2
? 2 ?? 2 c o s
4
? ?? 3 + 2 sin
4
? ?? sin
2
? 2 ??  then
1
5
?? '
( 0 ) is equal to
(1) 0
(2) 1
(3) 2
(4) 6
Q8  2024 (30 Jan Shift 2)
Consider the system of linear equations
?? + ?? + ?? = 5 , ?? + 2 ?? + ?? 2
?? = 9
?? + 3 ?? + ???? = ?? , where ?? , ?? ? ?? . Then, which of the following statement is NOT correct?
(1) System has infinite number of solution if ?? = 1 and ?? = 13
(2) System is inconsistent if ?? = 1 and ?? ? 13
(3) System is consistent if ?? ? 1 and ?? = 13
(4) System has unique solution if ?? ? 1 and ?? ? 13
Q9  2024 (31 Jan Shift 1)
If the system of linear equations
??  2 ?? + ?? =  4
2 ?? + ???? + 3 ?? = 5
3 ??  ?? + ???? = 3
has infinitely many solutions, then 12 ?? + 13 ?? is equal to
(1) 60
(2) 64
(3) 54
(4) 58
Q10  2024 (31 Jan Shift 1)
If ?? ( ?? ) = 
?? 3
2 ?? 2
+ 1 1 + 3 ?? 3 ?? 2
+ 2 2 ?? ?? 3
+ 6
?? 3
 ?? 4 ?? 2
 2
 for all ?? ? R, then 2 ?? ( 0 ) + ?? '
( 0 ) is equal to
(1) 48
(2) 24
(3) 42
(4) 18
Q11  2024 (31 Jan Shift 2)
Let ?? be a 3 × 3 real matrix such that
A (
1
0
1
) = 2 (
1
0
1
) , A (
 1
0
1
) = 4 (
 1
0
1
) , A (
0
1
0
) = 2 (
0
1
0
).
Then, the system ( ??  3 ?? ) (
?? ?? ?? ) = (
1
2
3
) has
(1) unique solution
(2) exactly two solutions
(3) no solution
(4) infinitely many solutions
Answer Key
Q1 (2) ???? (2) Q3 (28) Q4 (2)
Q5 (113) Q6 (2) Q7 (1) Q8 (4)
Q9 (4) ?? ?? ?? ( ?? ) Q11 (1)
Solutions
Q1
Using family of planes
2x + 3y  z  5 = k
1
( x + ?? y + 3z + 4 ) + k
2
( 3x  y
Page 5
2024
Q1  2024 (01 Feb Shift 1)
If the system of equations
2 ?? + 3 ??  ?? = 5
?? + ???? + 3 ?? =  4
3 ??  ?? + ???? = 7
has infinitely many solutions, then 13 ???? is equal to
(1) 1110
(2) 1120
(3) 1210
(4) 1220
Q2  2024 (01 Feb Shift 2)
Let the system of equations ?? + 2 ?? + 3 ?? = 5 , 2 ?? + 3 ?? + ?? = 9 , 4 ?? + 3 ?? + ???? = ?? have
infinite number of solutions. Then ?? + 2 ?? is equal to :
(1) 28
(2) 17
(3) 22
(4) 15
Q3  2024 (27 Jan Shift 1)
Let ?? = [
2 0 1
1 1 0
1 0 1
] , ?? = [ ?? 1
, ?? 2
, ?? 3
], where ?? 1
, ?? 2
, ?? 3
are column matrices, and ?? 1
= [
1
0
0
],
AB
2
= [
2
3
0
] , AB
3
= [
3
2
1
]
If ?? =  ??  and ?? is the sum of all the diagonal elements of B, then ?? 3
+ ?? 3
is equal to
Q4  2024 (27 Jan Shift 2)
The values of ?? , for which

1
3
2
?? +
3
2
1
1
3
?? +
1
3
2 ?? + 3 3 ?? + 1 0
 = 0,
lie in the interval
(1) (  2 , 1 )
(2) (  3 , 0 )
(3) ( 
3
2
,
3
2
)
(4) ( 0 , 3 )
Q5  2024 (29 Jan Shift 2)
Let for any three distinct consecutive terms a , b , c of an A.P, the lines ???? + ???? + ?? = 0 be
concurrent at the point P and Q ( ?? , ?? ) be a point such that the system of equations
?? + ?? + ?? = 6,
2 ?? + 5 ?? + ???? = ?? and
?? + 2 ?? + 3 ?? = 4, has infinitely many solutions. Then ( PQ )
2
is equal to
Q6  2024 (30 Jan Shift 1)
Consider the system of linear equation x + y + z = 4 ?? , ?? + 2 ?? + 2 ???? = 10 ?? , ?? + 3 ?? +
4 ?? 2
?? = ?? 2
+ 15, where ?? , ?? ? R. Which one of the following statements is NOT correct?
(1) The system has unique solution if ?? ?
1
2
and ?? ? 1 , 15
(2) The system is inconsistent if ?? =
1
2
and ?? ? 1
(3) The system has infinite number of solutions if ?? =
1
2
and ?? = 15
(4) The system is consistent if ?? ?
1
2
Q7  2024 (30 Jan Shift 1)
If ?? ( ?? ) = 
2 c o s
4
? ?? 2 sin
4
? ?? 3 + sin
2
? 2 ?? 3 + 2 c o s
4
? ?? 2 sin
4
? ?? sin
2
? 2 ?? 2 c o s
4
? ?? 3 + 2 sin
4
? ?? sin
2
? 2 ??  then
1
5
?? '
( 0 ) is equal to
(1) 0
(2) 1
(3) 2
(4) 6
Q8  2024 (30 Jan Shift 2)
Consider the system of linear equations
?? + ?? + ?? = 5 , ?? + 2 ?? + ?? 2
?? = 9
?? + 3 ?? + ???? = ?? , where ?? , ?? ? ?? . Then, which of the following statement is NOT correct?
(1) System has infinite number of solution if ?? = 1 and ?? = 13
(2) System is inconsistent if ?? = 1 and ?? ? 13
(3) System is consistent if ?? ? 1 and ?? = 13
(4) System has unique solution if ?? ? 1 and ?? ? 13
Q9  2024 (31 Jan Shift 1)
If the system of linear equations
??  2 ?? + ?? =  4
2 ?? + ???? + 3 ?? = 5
3 ??  ?? + ???? = 3
has infinitely many solutions, then 12 ?? + 13 ?? is equal to
(1) 60
(2) 64
(3) 54
(4) 58
Q10  2024 (31 Jan Shift 1)
If ?? ( ?? ) = 
?? 3
2 ?? 2
+ 1 1 + 3 ?? 3 ?? 2
+ 2 2 ?? ?? 3
+ 6
?? 3
 ?? 4 ?? 2
 2
 for all ?? ? R, then 2 ?? ( 0 ) + ?? '
( 0 ) is equal to
(1) 48
(2) 24
(3) 42
(4) 18
Q11  2024 (31 Jan Shift 2)
Let ?? be a 3 × 3 real matrix such that
A (
1
0
1
) = 2 (
1
0
1
) , A (
 1
0
1
) = 4 (
 1
0
1
) , A (
0
1
0
) = 2 (
0
1
0
).
Then, the system ( ??  3 ?? ) (
?? ?? ?? ) = (
1
2
3
) has
(1) unique solution
(2) exactly two solutions
(3) no solution
(4) infinitely many solutions
Answer Key
Q1 (2) ???? (2) Q3 (28) Q4 (2)
Q5 (113) Q6 (2) Q7 (1) Q8 (4)
Q9 (4) ?? ?? ?? ( ?? ) Q11 (1)
Solutions
Q1
Using family of planes
2x + 3y  z  5 = k
1
( x + ?? y + 3z + 4 ) + k
2
( 3x  y
+ ?? ??  7 )
2 = k
1
+ 3 k
2
, 3 = k
1
??  k
2
,  1 = 3 k
1
+ ?? k
2
,  5 =
4 k
1
 7 k
2
On solving we get
?? 2
=
13
19
, ?? 1
=
 1
19
, ?? =  70 , ?? =
 16
13
13 ???? = 13 (  70 ) (
 16
13
)
= 1 1 2 0
Q2
?? + 2 ?? + 3 ?? = 5
2 ?? + 3 ?? + ?? = 9
4 ?? + 3 ?? + ???? = ??
for infinite following ? = ?
1
= ?
2
= ?
3
= 0
? = 
1 2 3
2 3 1
4 3 ??  = 0 ? ?? =  13
?
1
= 
5 2 3
9 3 1
?? 3 13
 = 0 ? ?? = 15
?
2
= 
1 5 3
2 9 1
4 15 13
 = 0
?
3
= 
1 2 5
2 3 9
4 3 15
 = 0
for ?? =  13 , ?? = 15 system of equation has infinite solution hence ?? + 2 ?? = 17
Q3
?? = [
2 0 1
1 1 0
1 0 1
] ? B = [ B
1
, B
2
, B
3
]
B
1
= [
x
1
y
1
z
1
] , ? B
2
= [
x
2
y
2
z
2
] , ? B
3
= [
x
3
y
3
z
3
]
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