Page 1
2024
Q1 - 2024 (01 Feb Shift 1)
If A=[
v2 1
-1 v2
],B=[
1 0
1 1
],C=ABA
T
and X=A
T
C
2
A, then det?X is equal to :
(1) 243
(2) 729
(3) 27
(4) 891
Q2 - 2024 (01 Feb Shift 2)
Let ?? =?? 2
-?? ?? ?? , where ?? is real matrix of order 2×1 such that the relation ?? ?? ?? =?? 1
holds. If ?? is a real number such that the relation AX=?? X holds for some non-zero real
matrix X of order 2×1, then the sum of squares of all possible values of ?? is equal to :
Q3 - 2024 (27 Jan Shift 1)
Consider the matrix ?? (?? )=[
cos??? -sin??? 0
sin??? cos??? 0
0 0 1
].
Given below are two statements :
Statement I: ?? (-?? ) is the inverse of the matrix ?? (?? ).
Statement II: ?? (?? )?? (?? )=?? (?? +?? ).
In the light of the above statements, choose the correct answer from the options given
below
(1) Statement I is false but Statement II is true
(2) Both Statement I and Statement II are false
(3) Statement I is true but Statement II is false
(4) Both Statement I and Statement II are true
Page 2
2024
Q1 - 2024 (01 Feb Shift 1)
If A=[
v2 1
-1 v2
],B=[
1 0
1 1
],C=ABA
T
and X=A
T
C
2
A, then det?X is equal to :
(1) 243
(2) 729
(3) 27
(4) 891
Q2 - 2024 (01 Feb Shift 2)
Let ?? =?? 2
-?? ?? ?? , where ?? is real matrix of order 2×1 such that the relation ?? ?? ?? =?? 1
holds. If ?? is a real number such that the relation AX=?? X holds for some non-zero real
matrix X of order 2×1, then the sum of squares of all possible values of ?? is equal to :
Q3 - 2024 (27 Jan Shift 1)
Consider the matrix ?? (?? )=[
cos??? -sin??? 0
sin??? cos??? 0
0 0 1
].
Given below are two statements :
Statement I: ?? (-?? ) is the inverse of the matrix ?? (?? ).
Statement II: ?? (?? )?? (?? )=?? (?? +?? ).
In the light of the above statements, choose the correct answer from the options given
below
(1) Statement I is false but Statement II is true
(2) Both Statement I and Statement II are false
(3) Statement I is true but Statement II is false
(4) Both Statement I and Statement II are true
Q4 - 2024 (27 Jan Shift 2)
Let A be a 2×2 real matrix and ?? be the identity matrix of order 2 . If the roots of the
equation |?? -???? |=0 be -1 and 3 , then the sum of the diagonal elements of the matrix
?? 2
is.
Q5 - 2024 (29 Jan Shift 1)
Let ?? =[
1 0 0
0 ?? ?? 0 ?? ?? ] and |2?? |
3
=2
21
where ?? ,?? ??? , Then a value of ?? is
(1) 3
(2) 5
(3) 17
(4) 9
Q6 - 2024 (29 Jan Shift 1)
Let ?? be a square matrix such that ?? ?? ?? =?? . Then
1
2
A[( A+A
T
)
2
+(A-A
T
)
2
] is equal
to
(1) A
2
+I
(2) ?? 3
+??
(3) ?? 2
+?? ??
(4) ?? 3
+?? ??
Q7 - 2024 (29 Jan Shift 2)
Let ?? =[
2 1 2
6 2 11
3 3 2
] and ?? =[
1 2 0
5 0 2
7 1 5
]. The sum of the prime factors of |P
-1
AP-2I| is
equal to
(1) 26
(2) 27
(3) 66
(4) 23
Page 3
2024
Q1 - 2024 (01 Feb Shift 1)
If A=[
v2 1
-1 v2
],B=[
1 0
1 1
],C=ABA
T
and X=A
T
C
2
A, then det?X is equal to :
(1) 243
(2) 729
(3) 27
(4) 891
Q2 - 2024 (01 Feb Shift 2)
Let ?? =?? 2
-?? ?? ?? , where ?? is real matrix of order 2×1 such that the relation ?? ?? ?? =?? 1
holds. If ?? is a real number such that the relation AX=?? X holds for some non-zero real
matrix X of order 2×1, then the sum of squares of all possible values of ?? is equal to :
Q3 - 2024 (27 Jan Shift 1)
Consider the matrix ?? (?? )=[
cos??? -sin??? 0
sin??? cos??? 0
0 0 1
].
Given below are two statements :
Statement I: ?? (-?? ) is the inverse of the matrix ?? (?? ).
Statement II: ?? (?? )?? (?? )=?? (?? +?? ).
In the light of the above statements, choose the correct answer from the options given
below
(1) Statement I is false but Statement II is true
(2) Both Statement I and Statement II are false
(3) Statement I is true but Statement II is false
(4) Both Statement I and Statement II are true
Q4 - 2024 (27 Jan Shift 2)
Let A be a 2×2 real matrix and ?? be the identity matrix of order 2 . If the roots of the
equation |?? -???? |=0 be -1 and 3 , then the sum of the diagonal elements of the matrix
?? 2
is.
Q5 - 2024 (29 Jan Shift 1)
Let ?? =[
1 0 0
0 ?? ?? 0 ?? ?? ] and |2?? |
3
=2
21
where ?? ,?? ??? , Then a value of ?? is
(1) 3
(2) 5
(3) 17
(4) 9
Q6 - 2024 (29 Jan Shift 1)
Let ?? be a square matrix such that ?? ?? ?? =?? . Then
1
2
A[( A+A
T
)
2
+(A-A
T
)
2
] is equal
to
(1) A
2
+I
(2) ?? 3
+??
(3) ?? 2
+?? ??
(4) ?? 3
+?? ??
Q7 - 2024 (29 Jan Shift 2)
Let ?? =[
2 1 2
6 2 11
3 3 2
] and ?? =[
1 2 0
5 0 2
7 1 5
]. The sum of the prime factors of |P
-1
AP-2I| is
equal to
(1) 26
(2) 27
(3) 66
(4) 23
Q8 - 2024 (30 Jan Shift 2)
Let ?? =(
?? 0 0
0 ?? 0
0 0 ?? ) be a non-zero 3×3 matrix, where sin??? =?? sin?(?? +
2?? 3
)=
?? sin?(?? +
4?? 3
) ?0,?? ?(0,2?? ). For a square matrix ?? , let trace (M) denote the sum of all
the diagonal entries of M. Then,
among the statements:
(I) Trace (R)=0
(II) If trace (adj?(adj?(?? ))=0, then ?? has exactly one non-zero entry.
(1) Both (I) and (II) are true
(2) Neither (I) nor (II) is true
(3) Only (II) is true
(4) Only (I) is true
Q9 - 2024 (31 Jan Shift 2)
Let ?? be a 3×3 matrix and det?(?? )=2. If
n=det?(adj?(adj?(……(adj??? )
?
2024- times
)))
Then the remainder when n is divided by 9 is equal to
Answer Key
Q1 (2) ???? (?? ) Q3 (4) Q4 (10)
Q9 (7)
Q5 (2) Q6 (4) Q7 (1) Q8 (2)
Page 4
2024
Q1 - 2024 (01 Feb Shift 1)
If A=[
v2 1
-1 v2
],B=[
1 0
1 1
],C=ABA
T
and X=A
T
C
2
A, then det?X is equal to :
(1) 243
(2) 729
(3) 27
(4) 891
Q2 - 2024 (01 Feb Shift 2)
Let ?? =?? 2
-?? ?? ?? , where ?? is real matrix of order 2×1 such that the relation ?? ?? ?? =?? 1
holds. If ?? is a real number such that the relation AX=?? X holds for some non-zero real
matrix X of order 2×1, then the sum of squares of all possible values of ?? is equal to :
Q3 - 2024 (27 Jan Shift 1)
Consider the matrix ?? (?? )=[
cos??? -sin??? 0
sin??? cos??? 0
0 0 1
].
Given below are two statements :
Statement I: ?? (-?? ) is the inverse of the matrix ?? (?? ).
Statement II: ?? (?? )?? (?? )=?? (?? +?? ).
In the light of the above statements, choose the correct answer from the options given
below
(1) Statement I is false but Statement II is true
(2) Both Statement I and Statement II are false
(3) Statement I is true but Statement II is false
(4) Both Statement I and Statement II are true
Q4 - 2024 (27 Jan Shift 2)
Let A be a 2×2 real matrix and ?? be the identity matrix of order 2 . If the roots of the
equation |?? -???? |=0 be -1 and 3 , then the sum of the diagonal elements of the matrix
?? 2
is.
Q5 - 2024 (29 Jan Shift 1)
Let ?? =[
1 0 0
0 ?? ?? 0 ?? ?? ] and |2?? |
3
=2
21
where ?? ,?? ??? , Then a value of ?? is
(1) 3
(2) 5
(3) 17
(4) 9
Q6 - 2024 (29 Jan Shift 1)
Let ?? be a square matrix such that ?? ?? ?? =?? . Then
1
2
A[( A+A
T
)
2
+(A-A
T
)
2
] is equal
to
(1) A
2
+I
(2) ?? 3
+??
(3) ?? 2
+?? ??
(4) ?? 3
+?? ??
Q7 - 2024 (29 Jan Shift 2)
Let ?? =[
2 1 2
6 2 11
3 3 2
] and ?? =[
1 2 0
5 0 2
7 1 5
]. The sum of the prime factors of |P
-1
AP-2I| is
equal to
(1) 26
(2) 27
(3) 66
(4) 23
Q8 - 2024 (30 Jan Shift 2)
Let ?? =(
?? 0 0
0 ?? 0
0 0 ?? ) be a non-zero 3×3 matrix, where sin??? =?? sin?(?? +
2?? 3
)=
?? sin?(?? +
4?? 3
) ?0,?? ?(0,2?? ). For a square matrix ?? , let trace (M) denote the sum of all
the diagonal entries of M. Then,
among the statements:
(I) Trace (R)=0
(II) If trace (adj?(adj?(?? ))=0, then ?? has exactly one non-zero entry.
(1) Both (I) and (II) are true
(2) Neither (I) nor (II) is true
(3) Only (II) is true
(4) Only (I) is true
Q9 - 2024 (31 Jan Shift 2)
Let ?? be a 3×3 matrix and det?(?? )=2. If
n=det?(adj?(adj?(……(adj??? )
?
2024- times
)))
Then the remainder when n is divided by 9 is equal to
Answer Key
Q1 (2) ???? (?? ) Q3 (4) Q4 (10)
Q9 (7)
Q5 (2) Q6 (4) Q7 (1) Q8 (2)
Solutions
Q1
?? =[
v2 1
-1 v2
]?det?(?? )=3
?? =[
1 0
1 1
]?det?(?? )=1
Now C=ABA
T
?det?(C)=(dct?(A))
2
xdet?(B)
|?? |=9
Now |X|=|A
T
C
2
A|
=|A
T
||C|
2
| A|
=|A|
2
|C|
2
=9×81
=729
Q2
A=I
2
-2MM
T
A
2
=(I
2
-2MM
T
)(I
2
-2MM
T
)
=I
2
-2MM
T
-2MM
T
+4MM
T
MM
T
=I
2
-4MM
T
+4MM
T
=I
2
AX=?? X
A
2
X=?? AX
X=?? (?? X)
X=?? 2
X
X(?? 2
-1)=0
?? 2
=1
?? =±1
Sum of square of all possible values =2
Page 5
2024
Q1 - 2024 (01 Feb Shift 1)
If A=[
v2 1
-1 v2
],B=[
1 0
1 1
],C=ABA
T
and X=A
T
C
2
A, then det?X is equal to :
(1) 243
(2) 729
(3) 27
(4) 891
Q2 - 2024 (01 Feb Shift 2)
Let ?? =?? 2
-?? ?? ?? , where ?? is real matrix of order 2×1 such that the relation ?? ?? ?? =?? 1
holds. If ?? is a real number such that the relation AX=?? X holds for some non-zero real
matrix X of order 2×1, then the sum of squares of all possible values of ?? is equal to :
Q3 - 2024 (27 Jan Shift 1)
Consider the matrix ?? (?? )=[
cos??? -sin??? 0
sin??? cos??? 0
0 0 1
].
Given below are two statements :
Statement I: ?? (-?? ) is the inverse of the matrix ?? (?? ).
Statement II: ?? (?? )?? (?? )=?? (?? +?? ).
In the light of the above statements, choose the correct answer from the options given
below
(1) Statement I is false but Statement II is true
(2) Both Statement I and Statement II are false
(3) Statement I is true but Statement II is false
(4) Both Statement I and Statement II are true
Q4 - 2024 (27 Jan Shift 2)
Let A be a 2×2 real matrix and ?? be the identity matrix of order 2 . If the roots of the
equation |?? -???? |=0 be -1 and 3 , then the sum of the diagonal elements of the matrix
?? 2
is.
Q5 - 2024 (29 Jan Shift 1)
Let ?? =[
1 0 0
0 ?? ?? 0 ?? ?? ] and |2?? |
3
=2
21
where ?? ,?? ??? , Then a value of ?? is
(1) 3
(2) 5
(3) 17
(4) 9
Q6 - 2024 (29 Jan Shift 1)
Let ?? be a square matrix such that ?? ?? ?? =?? . Then
1
2
A[( A+A
T
)
2
+(A-A
T
)
2
] is equal
to
(1) A
2
+I
(2) ?? 3
+??
(3) ?? 2
+?? ??
(4) ?? 3
+?? ??
Q7 - 2024 (29 Jan Shift 2)
Let ?? =[
2 1 2
6 2 11
3 3 2
] and ?? =[
1 2 0
5 0 2
7 1 5
]. The sum of the prime factors of |P
-1
AP-2I| is
equal to
(1) 26
(2) 27
(3) 66
(4) 23
Q8 - 2024 (30 Jan Shift 2)
Let ?? =(
?? 0 0
0 ?? 0
0 0 ?? ) be a non-zero 3×3 matrix, where sin??? =?? sin?(?? +
2?? 3
)=
?? sin?(?? +
4?? 3
) ?0,?? ?(0,2?? ). For a square matrix ?? , let trace (M) denote the sum of all
the diagonal entries of M. Then,
among the statements:
(I) Trace (R)=0
(II) If trace (adj?(adj?(?? ))=0, then ?? has exactly one non-zero entry.
(1) Both (I) and (II) are true
(2) Neither (I) nor (II) is true
(3) Only (II) is true
(4) Only (I) is true
Q9 - 2024 (31 Jan Shift 2)
Let ?? be a 3×3 matrix and det?(?? )=2. If
n=det?(adj?(adj?(……(adj??? )
?
2024- times
)))
Then the remainder when n is divided by 9 is equal to
Answer Key
Q1 (2) ???? (?? ) Q3 (4) Q4 (10)
Q9 (7)
Q5 (2) Q6 (4) Q7 (1) Q8 (2)
Solutions
Q1
?? =[
v2 1
-1 v2
]?det?(?? )=3
?? =[
1 0
1 1
]?det?(?? )=1
Now C=ABA
T
?det?(C)=(dct?(A))
2
xdet?(B)
|?? |=9
Now |X|=|A
T
C
2
A|
=|A
T
||C|
2
| A|
=|A|
2
|C|
2
=9×81
=729
Q2
A=I
2
-2MM
T
A
2
=(I
2
-2MM
T
)(I
2
-2MM
T
)
=I
2
-2MM
T
-2MM
T
+4MM
T
MM
T
=I
2
-4MM
T
+4MM
T
=I
2
AX=?? X
A
2
X=?? AX
X=?? (?? X)
X=?? 2
X
X(?? 2
-1)=0
?? 2
=1
?? =±1
Sum of square of all possible values =2
Q3
?? (-?? )=[
cos??? sin??? 0
-sin??? cos??? 0
0 0 1
]
?? (?? )·?? (-?? )=[
1 0 0
0 1 0
0 0 1
]=??
Hence statement- I is correct
Now, checking statement II
?? (?? )=[
cos??? -sin??? 0
sin??? cos??? 0
0 0 1
]
?? (?? )·?? (?? )=[
cos?(?? +?? ) -sin?(?? +?? ) 0
sin?(?? +?? ) cos?(?? +?? ) 0
0 0 1
]
?f(x)·f(y)=f(x+y)
Hence statement-II is also correct.
Q4
|?? -???? |=0
Roots are -1 and 3
Sum of roots =tr?(A)=2
Product of roots =|A|=-3
Let ?? =[
?? ?? ?? ?? ]
We have a+d=2
ad-bc=-3
?? 2
=[
?? ?? ?? ?? ]×[
?? ?? ?? ?? ]=[
?? 2
+???? ???? +????
???? +???? ???? +?? 2
]
We need a
2
+bc+bc+d
2
=?? 2
+2???? +?? 2
=(?? +?? )
2
-2???? +2????
=4-2(???? -???? )
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