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Week 6 Mathematics 2 Questions | Prasnya
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62Q
01
Let
T
:
R
3
→
R
3
T:\mathbb{R}^3\to\mathbb{R}^3
T
:
R
3
→
R
3
be a linear transformation such that
T
2
=
T
T^2=T
T
2
=
T
, and
rank
(
T
)
=
2
\operatorname{rank}(T)=2
rank
(
T
)
=
2
. Which of the following statements is correct?
Single correct
MEDIUM
4 marks
06 April 2026
02
Let
T
:
R
3
→
R
3
T:\mathbb{R}^3\to\mathbb{R}^3
T
:
R
3
→
R
3
be a linear transformation such that
T
(
1
,
0
,
1
)
=
(
1
,
2
,
1
)
T(1,0,1)=(1,2,1)
T
(
1
,
0
,
1
)
=
(
1
,
2
,
1
)
,
T
(
0
,
1
,
1
)
=
(
0
,
1
,
1
)
T(0,1,1)=(0,1,1)
T
(
0
,
1
,
1
)
=
(
0
,
1
,
1
)
, and
T
(
1
,
1
,
0
)
=
(
1
,
1
,
0
)
T(1,1,0)=(1,1,0)
T
(
1
,
1
,
0
)
=
(
1
,
1
,
0
)
. Let
U
=
span
{
(
1
,
0
,
1
)
,
(
0
,
1
,
1
)
}
U=\operatorname{span}\{(1,0,1),(0,1,1)\}
U
=
span
{(
1
,
0
,
1
)
,
(
0
,
1
,
1
)}
. Choose all correct options.
Multiple correct
MEDIUM
4 marks
06 April 2026
03
Let
T
:
R
3
→
R
3
T:\mathbb{R}^3\to\mathbb{R}^3
T
:
R
3
→
R
3
be defined by
T
(
x
,
y
,
z
)
=
(
x
+
y
,
y
+
z
,
z
+
x
+
k
y
)
T(x,y,z)=(x+y,\ y+z,\ z+x+ky)
T
(
x
,
y
,
z
)
=
(
x
+
y
,
y
+
z
,
z
+
x
+
k
y
)
, where
k
∈
R
k\in\mathbb{R}
k
∈
R
. Which statements are true?
Multiple correct
MEDIUM
4 marks
06 April 2026
04
Let
v
=
(
a
,
b
,
c
)
∈
R
3
v=(a,b,c)\in\mathbb{R}^3
v
=
(
a
,
b
,
c
)
∈
R
3
be a vector in the kernel of
T
T
T
such that
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Find the value of
a
+
2
b
+
3
c
a+2b+3c
a
+
2
b
+
3
c
up to two decimal places.
Comprehension
MEDIUM
4 marks
06 April 2026
05
Find the dimension of the image of
S
S
S
.
Comprehension
MEDIUM
4 marks
06 April 2026
06
Consider a vector
v
∈
U
v\in U
v
∈
U
such that
S
(
v
)
∈
span
{
(
1
,
1
,
1
)
}
S(v)\in\operatorname{span}\{(1,1,1)\}
S
(
v
)
∈
span
{(
1
,
1
,
1
)}
, and
∥
v
∥
2
=
6
\|v\|^2=6
∥
v
∥
2
=
6
, where
∥
⋅
∥
\|\cdot\|
∥
⋅
∥
is the norm with respect to the standard inner product on
R
3
\mathbb{R}^3
R
3
. Which options are the possible choices for
v
v
v
?
Comprehension
MEDIUM
3 marks
06 April 2026
07
Let
T
:
R
2
→
R
2
T:\mathbb{R}^2\to\mathbb{R}^2
T
:
R
2
→
R
2
be the linear transformation determined by
T
(
1
,
0
)
=
(
α
−
6
,
−
4
)
T(1,0)=(\alpha-6,-4)
T
(
1
,
0
)
=
(
α
−
6
,
−
4
)
and
T
(
0
,
1
)
=
(
3
,
α
+
1
)
T(0,1)=(3,\alpha+1)
T
(
0
,
1
)
=
(
3
,
α
+
1
)
, where
α
∈
R
\alpha\in\mathbb{R}
α
∈
R
. Let
α
1
\alpha_1
α
1
and
α
2
\alpha_2
α
2
be the minimum and maximum values, respectively, of
α
\alpha
α
for which
T
T
T
fails to be injective. Find
α
2
2
−
α
1
2
\alpha_2^2-\alpha_1^2
α
2
2
−
α
1
2
.
Numerical
MEDIUM
5 marks
03 August 2025
08
Find the number of columns in the matrix, with respect to any bases, of the linear transformation
T
T
T
.
Comprehension
EASY
2 marks
03 August 2025
09
Find the value of
k
k
k
for which the set
{
(
1
,
1
,
k
2
,
k
−
1
)
,
(
1
,
1
,
0
,
−
1
)
}
\{(1,1,k^2,k-1),(1,1,0,-1)\}
{(
1
,
1
,
k
2
,
k
−
1
)
,
(
1
,
1
,
0
,
−
1
)}
is a basis for the kernel of
T
T
T
.
Comprehension
MEDIUM
3 marks
03 August 2025
10
Which of the following is the matrix of the linear transformation
T
T
T
with respect to standard bases for both the domain and the codomain?
Comprehension
EASY
2 marks
03 August 2025
11
Which of the following spaces are isomorphic to the kernel of
T
T
T
?
Comprehension
MEDIUM
5 marks
03 August 2025
12
Which of the following statements is true?
Comprehension
MEDIUM
4 marks
16 March 2025
13
For the value of
k
k
k
obtained in the previous question, which of the following vectors belong to the range of
T
T
T
?
Comprehension
MEDIUM
4 marks
16 March 2025
14
Find
a
a
a
.
Comprehension
EASY
1 marks
01 December 2024
15
Find
b
b
b
.
Comprehension
EASY
1 marks
01 December 2024
16
Is
T
T
T
injective or surjective? If
T
T
T
is injective, write
1
1
1
and if
T
T
T
is surjective, write
−
1
-1
−
1
.
Comprehension
MEDIUM
1 marks
01 December 2024
17
Let
T
:
R
n
→
R
m
T:\mathbb{R}^n\to\mathbb{R}^m
T
:
R
n
→
R
m
be a linear transformation. Choose the correct option that guarantees that
T
T
T
is bijective.
Single correct
MEDIUM
3 marks
01 December 2024
18
Let
S
S
S
be any straight line in
R
2
\mathbb{R}^2
R
2
passing through the origin. If
T
(
S
)
T(S)
T
(
S
)
is the image of
S
S
S
under
T
T
T
in
R
3
\mathbb{R}^3
R
3
, what is the dimension of
T
(
S
)
T(S)
T
(
S
)
?
Comprehension
MEDIUM
1 marks
01 December 2024
19
Let
B
1
=
{
(
1
,
−
1
)
,
(
0
,
2
)
}
B_1=\{(1,-1),(0,2)\}
B
1
=
{(
1
,
−
1
)
,
(
0
,
2
)}
and
B
2
=
{
(
1
,
0
,
0
)
,
(
0
,
1
,
0
)
,
(
0
,
0
,
1
)
}
B_2=\{(1,0,0),(0,1,0),(0,0,1)\}
B
2
=
{(
1
,
0
,
0
)
,
(
0
,
1
,
0
)
,
(
0
,
0
,
1
)}
. Which matrix is the representation of
T
T
T
with respect to
B
1
B_1
B
1
for the domain and
B
2
B_2
B
2
for the codomain?
Comprehension
EASY
1 marks
01 December 2024
20
Let
V
=
span
{
(
3
,
0
,
0
)
,
(
0
,
−
2
,
2
)
,
(
−
1
,
1
,
−
1
)
}
V=\operatorname{span}\{(3,0,0),(0,-2,2),(-1,1,-1)\}
V
=
span
{(
3
,
0
,
0
)
,
(
0
,
−
2
,
2
)
,
(
−
1
,
1
,
−
1
)}
. Let
T
:
V
→
R
2
T:V\to\mathbb{R}^2
T
:
V
→
R
2
be defined by
T
(
x
,
y
,
z
)
=
(
c
x
−
y
,
y
+
z
)
T(x,y,z)=(cx-y,y+z)
T
(
x
,
y
,
z
)
=
(
c
x
−
y
,
y
+
z
)
. Choose all correct options.
Multiple correct
HARD
3 marks
01 December 2024
21
Let
A
=
[
x
−
4
3
y
]
A=\begin{bmatrix}x&-4\\3&y\end{bmatrix}
A
=
[
x
3
−
4
y
]
and
T
:
R
2
→
R
T:\mathbb{R}^2\to\mathbb{R}
T
:
R
2
→
R
be the linear transformation given by
T
(
x
,
y
)
=
trace
(
A
)
T(x,y)=\operatorname{trace}(A)
T
(
x
,
y
)
=
trace
(
A
)
. Which statement is true?
Single correct
EASY
2 marks
04 August 2024
22
Which matrix represents
T
T
T
with respect to the standard ordered basis for both the domain and codomain?
Comprehension
MEDIUM
2 marks
04 August 2024
23
Which option defines the transformation
T
T
T
?
Comprehension
EASY
1 marks
04 August 2024
24
Find the value of
a
a
a
.
Comprehension
EASY
1 marks
24 March 2024
25
Find the value of
b
b
b
.
Comprehension
EASY
1 marks
24 March 2024
26
Find the value of
c
c
c
.
Comprehension
EASY
1 marks
24 March 2024
27
Find the value of
d
d
d
.
Comprehension
EASY
1 marks
24 March 2024
28
Choose the correct option from the following.
Comprehension
MEDIUM
2 marks
24 March 2024
29
Choose the correct options from the following.
Comprehension
MEDIUM
2 marks
24 March 2024
30
Which of the following statements is true?
Comprehension
EASY
1 marks
24 March 2024
31
Is the statement true or false? If
{
v
1
,
v
2
,
v
3
}
\{v_1,v_2,v_3\}
{
v
1
,
v
2
,
v
3
}
are linearly independent vectors in
V
V
V
, then
{
T
(
v
1
)
,
T
(
v
2
)
,
T
(
v
3
)
}
\{T(v_1),T(v_2),T(v_3)\}
{
T
(
v
1
)
,
T
(
v
2
)
,
T
(
v
3
)}
are linearly independent vectors in
W
W
W
.
Comprehension
EASY
1 marks
24 March 2024
32
Is the statement true or false? Let
{
u
1
,
u
2
,
u
3
}
\{u_1,u_2,u_3\}
{
u
1
,
u
2
,
u
3
}
be a subset of
V
V
V
. If
{
T
(
u
1
)
,
T
(
u
2
)
,
T
(
u
3
)
}
\{T(u_1),T(u_2),T(u_3)\}
{
T
(
u
1
)
,
T
(
u
2
)
,
T
(
u
3
)}
is linearly independent in
W
W
W
, then
{
u
1
,
u
2
,
u
3
}
\{u_1,u_2,u_3\}
{
u
1
,
u
2
,
u
3
}
could also be linearly dependent in
V
V
V
.
Comprehension
MEDIUM
1 marks
24 March 2024
33
Find the nullity of
T
T
T
.
Comprehension
EASY
1 marks
24 March 2024
34
Which of the following is a basis for the kernel of
T
T
T
?
Comprehension
EASY
1 marks
24 March 2024
35
Which of the following is a basis for the image of
T
T
T
?
Comprehension
MEDIUM
1 marks
24 March 2024
36
Let
T
T
T
be a linear transformation from
R
2
\mathbb{R}^2
R
2
to
R
3
\mathbb{R}^3
R
3
. Let
S
⊂
R
2
S\subset\mathbb{R}^2
S
⊂
R
2
be a line passing through the origin. Which of the following are possible?
Multiple correct
EASY
2 marks
03 December 2023
37
Find the matrix corresponding to
T
T
T
if
γ
\gamma
γ
is used as the basis for both the domain and codomain.
Comprehension
MEDIUM
3 marks
03 December 2023
38
Is
T
T
T
an isomorphism?
Comprehension
EASY
1 marks
03 December 2023
39
Which of the following options are true?
Multiple correct
MEDIUM
2 marks
06 August 2023
40
Consider
V
=
{
[
a
b
&
¸
a
]
:
c
=
a
+
b
;
a
,
b
,
c
∈
R
}
V=\left\{\begin{bmatrix}a&b\c\&a\end{bmatrix}:c=a+b;\ a,b,c\in\mathbb{R}\right\}
V
=
{
[
a
b
&
¸
a
]
:
c
=
a
+
b
;
a
,
b
,
c
∈
R
}
and
T
:
V
→
R
4
T:V\to\mathbb{R}^4
T
:
V
→
R
4
defined by
T
(
A
)
=
(
a
,
b
,
c
,
a
+
b
−
c
)
T(A)=(a,b,c,a+b-c)
T
(
A
)
=
(
a
,
b
,
c
,
a
+
b
−
c
)
. Choose the correct options.
Multiple correct
MEDIUM
2 marks
06 August 2023
41
Define a linear transformation
T
:
V
2
→
R
2
T:V_2\to\mathbb{R}^2
T
:
V
2
→
R
2
by
T
(
x
,
y
,
z
)
=
(
x
,
x
+
y
+
z
)
T(x,y,z)=(x,x+y+z)
T
(
x
,
y
,
z
)
=
(
x
,
x
+
y
+
z
)
. What is the rank of
T
T
T
?
Numerical
MEDIUM
2 marks
06 August 2023
42
Let
S
:
V
1
→
V
2
S:V_1\to V_2
S
:
V
1
→
V
2
be a linear transformation. If
m
×
n
m\times n
m
×
n
is the order of the matrix
D
D
D
of
S
S
S
with respect to ordered bases for
V
1
V_1
V
1
and
V
2
V_2
V
2
, what is
2
m
−
3
n
2m-3n
2
m
−
3
n
?
Numerical
EASY
1 marks
06 August 2023
43
If
A
=
[
a
b
c
d
e
f
g
h
i
]
A=\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}
A
=
a
d
g
b
e
h
c
f
i
denotes the matrix of
T
T
T
with respect to
{
(
1
,
1
,
1
)
,
(
1
,
1
,
0
)
,
(
1
,
0
,
0
)
}
\{(1,1,1),(1,1,0),(1,0,0)\}
{(
1
,
1
,
1
)
,
(
1
,
1
,
0
)
,
(
1
,
0
,
0
)}
for domain and codomain, what is
2
b
+
2
e
+
2
h
2b+2e+2h
2
b
+
2
e
+
2
h
?
Comprehension
HARD
2 marks
06 August 2023
44
Consider the vector space
V
=
{
[
a
b
a
b
]
:
a
,
b
∈
R
}
V=\left\{\begin{bmatrix}a&b\\a&b\end{bmatrix}:a,b\in\mathbb{R}\right\}
V
=
{
[
a
a
b
b
]
:
a
,
b
∈
R
}
and
T
:
R
3
→
V
T:\mathbb{R}^3\to V
T
:
R
3
→
V
defined by
T
(
x
,
y
,
z
)
=
[
x
+
y
x
+
y
+
z
x
+
y
x
+
y
+
z
]
T(x,y,z)=\begin{bmatrix}x+y&x+y+z\\x+y&x+y+z\end{bmatrix}
T
(
x
,
y
,
z
)
=
[
x
+
y
x
+
y
x
+
y
+
z
x
+
y
+
z
]
. Choose the correct option.
Single correct
MEDIUM
2 marks
02 April 2023
45
Let
V
=
{
[
a
b
c
d
]
∈
M
2
×
2
(
R
)
:
a
+
b
=
c
+
d
}
V=\left\{\begin{bmatrix}a&b\\c&d\end{bmatrix}\in M_{2\times2}(\mathbb{R}):a+b=c+d\right\}
V
=
{
[
a
c
b
d
]
∈
M
2
×
2
(
R
)
:
a
+
b
=
c
+
d
}
and
T
:
V
→
R
2
T:V\to\mathbb{R}^2
T
:
V
→
R
2
be onto. What is the dimension of
ker
(
T
)
\ker(T)
ker
(
T
)
?
Numerical
EASY
3 marks
02 April 2023
46
If
A
=
[
a
b
c
d
e
f
]
A=\begin{bmatrix}a&b\\c&d\\e&f\end{bmatrix}
A
=
a
c
e
b
d
f
denotes the matrix of
T
T
T
with respect to
{
(
1
,
1
)
,
(
1
,
−
1
)
}
\{(1,1),(1,-1)\}
{(
1
,
1
)
,
(
1
,
−
1
)}
for
R
2
\mathbb{R}^2
R
2
and
{
(
1
,
1
,
1
)
,
(
1
,
1
,
0
)
,
(
−
1
,
0
,
0
)
}
\{(1,1,1),(1,1,0),(-1,0,0)\}
{(
1
,
1
,
1
)
,
(
1
,
1
,
0
)
,
(
−
1
,
0
,
0
)}
for
R
3
\mathbb{R}^3
R
3
, what is
a
+
d
+
e
a+d+e
a
+
d
+
e
?
Comprehension
MEDIUM
3 marks
02 April 2023
47
Define
T
:
W
→
R
2
T:W\to\mathbb{R}^2
T
:
W
→
R
2
by
T
(
x
,
y
,
z
)
=
(
0
,
x
−
z
)
T(x,y,z)=(0,x-z)
T
(
x
,
y
,
z
)
=
(
0
,
x
−
z
)
. What is the rank of
T
T
T
?
Comprehension
EASY
1 marks
02 April 2023
48
If the
m
×
n
m\times n
m
×
n
matrix
B
B
B
is the matrix of
T
T
T
with respect to some basis for
W
W
W
and the standard ordered basis for
R
2
\mathbb{R}^2
R
2
, what is
m
+
n
m+n
m
+
n
?
Comprehension
EASY
1 marks
02 April 2023
49
If
A
A
A
is the matrix representation of
S
∘
T
S\circ T
S
∘
T
with respect to
β
\beta
β
for both domain and codomain, and the order of
A
A
A
is
m
×
n
m\times n
m
×
n
, find
m
+
n
m+n
m
+
n
.
Comprehension
EASY
1 marks
20 November 2022
50
If
K
=
{
(
x
,
y
,
z
)
:
a
x
+
b
y
=
0
,
c
y
+
d
z
=
0
}
K=\{(x,y,z): ax+by=0,\ cy+dz=0\}
K
=
{(
x
,
y
,
z
)
:
a
x
+
b
y
=
0
,
cy
+
d
z
=
0
}
is the null space of
S
∘
T
S\circ T
S
∘
T
, find
a
b
−
2
c
d
\frac{a}{b}-2\frac{c}{d}
b
a
−
2
d
c
.
Comprehension
MEDIUM
2 marks
20 November 2022
51
Find the rank of
S
∘
T
S\circ T
S
∘
T
.
Comprehension
EASY
1 marks
20 November 2022
52
Find the nullity of the matrix
A
A
A
.
Comprehension
EASY
1 marks
20 November 2022
53
Which of the following options is true?
Comprehension
MEDIUM
2 marks
20 November 2022
54
If
T
(
x
,
y
,
z
)
=
(
m
x
+
n
y
+
a
z
,
p
x
+
q
y
+
r
z
)
T(x,y,z)=(mx+ny+az,px+qy+rz)
T
(
x
,
y
,
z
)
=
(
m
x
+
n
y
+
a
z
,
p
x
+
q
y
+
r
z
)
, find
(
m
+
n
+
a
)
−
3
(
p
+
q
+
r
)
(m+n+a)-3(p+q+r)
(
m
+
n
+
a
)
−
3
(
p
+
q
+
r
)
.
Comprehension
HARD
2 marks
20 November 2022
55
What is the value of
d
−
a
d-a
d
−
a
?
Comprehension
MEDIUM
1 marks
10 July 2022
56
What is the value of
e
−
b
e-b
e
−
b
?
Comprehension
MEDIUM
1 marks
10 July 2022
57
What is the value of
f
−
c
f-c
f
−
c
?
Comprehension
MEDIUM
1 marks
10 July 2022
58
What is the value of
m
m
m
?
Comprehension
EASY
1 marks
10 July 2022
59
What is the value of
n
n
n
?
Comprehension
EASY
1 marks
10 July 2022
60
If
A
A
A
is the matrix representation of
T
T
T
with respect to the standard bases of
R
3
\mathbb{R}^3
R
3
and
R
2
\mathbb{R}^2
R
2
, then
A
A
A
is:
Comprehension
EASY
1 marks
10 July 2022
61
If
{
(
l
,
m
,
n
)
}
\{(l,m,n)\}
{(
l
,
m
,
n
)}
is a basis of
ker
(
T
)
\ker(T)
ker
(
T
)
, find
l
l
l
when
n
=
1
n=1
n
=
1
.
Comprehension
MEDIUM
1 marks
10 July 2022
62
If
{
(
l
,
m
,
n
)
}
\{(l,m,n)\}
{(
l
,
m
,
n
)}
is a basis of
ker
(
T
)
\ker(T)
ker
(
T
)
, find
m
m
m
when
n
=
1
n=1
n
=
1
.
Comprehension
MEDIUM
1 marks
10 July 2022
Showing 62 questions.
Mathematics 2 > Week 6
All PYQs
Topic-Wise PYQs
Start Weekly Test
Mock tests
Week mock
Topic mock
Subject mock
Type:
All
Difficulty:
All
Year:
All
62Q
01
Let
T
:
R
3
→
R
3
T:\mathbb{R}^3\to\mathbb{R}^3
T
:
R
3
→
R
3
be a linear transformation such that
T
2
=
T
T^2=T
T
2
=
T
, and
rank
(
T
)
=
2
\operatorname{rank}(T)=2
rank
(
T
)
=
2
. Which of the following statements is correct?
Single correct
MEDIUM
4 marks
06 April 2026
02
Let
T
:
R
3
→
R
3
T:\mathbb{R}^3\to\mathbb{R}^3
T
:
R
3
→
R
3
be a linear transformation such that
T
(
1
,
0
,
1
)
=
(
1
,
2
,
1
)
T(1,0,1)=(1,2,1)
T
(
1
,
0
,
1
)
=
(
1
,
2
,
1
)
,
T
(
0
,
1
,
1
)
=
(
0
,
1
,
1
)
T(0,1,1)=(0,1,1)
T
(
0
,
1
,
1
)
=
(
0
,
1
,
1
)
, and
T
(
1
,
1
,
0
)
=
(
1
,
1
,
0
)
T(1,1,0)=(1,1,0)
T
(
1
,
1
,
0
)
=
(
1
,
1
,
0
)
. Let
U
=
span
{
(
1
,
0
,
1
)
,
(
0
,
1
,
1
)
}
U=\operatorname{span}\{(1,0,1),(0,1,1)\}
U
=
span
{(
1
,
0
,
1
)
,
(
0
,
1
,
1
)}
. Choose all correct options.
Multiple correct
MEDIUM
4 marks
06 April 2026
03
Let
T
:
R
3
→
R
3
T:\mathbb{R}^3\to\mathbb{R}^3
T
:
R
3
→
R
3
be defined by
T
(
x
,
y
,
z
)
=
(
x
+
y
,
y
+
z
,
z
+
x
+
k
y
)
T(x,y,z)=(x+y,\ y+z,\ z+x+ky)
T
(
x
,
y
,
z
)
=
(
x
+
y
,
y
+
z
,
z
+
x
+
k
y
)
, where
k
∈
R
k\in\mathbb{R}
k
∈
R
. Which statements are true?
Multiple correct
MEDIUM
4 marks
06 April 2026
04
Let
v
=
(
a
,
b
,
c
)
∈
R
3
v=(a,b,c)\in\mathbb{R}^3
v
=
(
a
,
b
,
c
)
∈
R
3
be a vector in the kernel of
T
T
T
such that
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Find the value of
a
+
2
b
+
3
c
a+2b+3c
a
+
2
b
+
3
c
up to two decimal places.
Comprehension
MEDIUM
4 marks
06 April 2026
05
Find the dimension of the image of
S
S
S
.
Comprehension
MEDIUM
4 marks
06 April 2026
06
Consider a vector
v
∈
U
v\in U
v
∈
U
such that
S
(
v
)
∈
span
{
(
1
,
1
,
1
)
}
S(v)\in\operatorname{span}\{(1,1,1)\}
S
(
v
)
∈
span
{(
1
,
1
,
1
)}
, and
∥
v
∥
2
=
6
\|v\|^2=6
∥
v
∥
2
=
6
, where
∥
⋅
∥
\|\cdot\|
∥
⋅
∥
is the norm with respect to the standard inner product on
R
3
\mathbb{R}^3
R
3
. Which options are the possible choices for
v
v
v
?
Comprehension
MEDIUM
3 marks
06 April 2026
07
Let
T
:
R
2
→
R
2
T:\mathbb{R}^2\to\mathbb{R}^2
T
:
R
2
→
R
2
be the linear transformation determined by
T
(
1
,
0
)
=
(
α
−
6
,
−
4
)
T(1,0)=(\alpha-6,-4)
T
(
1
,
0
)
=
(
α
−
6
,
−
4
)
and
T
(
0
,
1
)
=
(
3
,
α
+
1
)
T(0,1)=(3,\alpha+1)
T
(
0
,
1
)
=
(
3
,
α
+
1
)
, where
α
∈
R
\alpha\in\mathbb{R}
α
∈
R
. Let
α
1
\alpha_1
α
1
and
α
2
\alpha_2
α
2
be the minimum and maximum values, respectively, of
α
\alpha
α
for which
T
T
T
fails to be injective. Find
α
2
2
−
α
1
2
\alpha_2^2-\alpha_1^2
α
2
2
−
α
1
2
.
Numerical
MEDIUM
5 marks
03 August 2025
08
Find the number of columns in the matrix, with respect to any bases, of the linear transformation
T
T
T
.
Comprehension
EASY
2 marks
03 August 2025
09
Find the value of
k
k
k
for which the set
{
(
1
,
1
,
k
2
,
k
−
1
)
,
(
1
,
1
,
0
,
−
1
)
}
\{(1,1,k^2,k-1),(1,1,0,-1)\}
{(
1
,
1
,
k
2
,
k
−
1
)
,
(
1
,
1
,
0
,
−
1
)}
is a basis for the kernel of
T
T
T
.
Comprehension
MEDIUM
3 marks
03 August 2025
10
Which of the following is the matrix of the linear transformation
T
T
T
with respect to standard bases for both the domain and the codomain?
Comprehension
EASY
2 marks
03 August 2025
11
Which of the following spaces are isomorphic to the kernel of
T
T
T
?
Comprehension
MEDIUM
5 marks
03 August 2025
12
Which of the following statements is true?
Comprehension
MEDIUM
4 marks
16 March 2025
13
For the value of
k
k
k
obtained in the previous question, which of the following vectors belong to the range of
T
T
T
?
Comprehension
MEDIUM
4 marks
16 March 2025
14
Find
a
a
a
.
Comprehension
EASY
1 marks
01 December 2024
15
Find
b
b
b
.
Comprehension
EASY
1 marks
01 December 2024
16
Is
T
T
T
injective or surjective? If
T
T
T
is injective, write
1
1
1
and if
T
T
T
is surjective, write
−
1
-1
−
1
.
Comprehension
MEDIUM
1 marks
01 December 2024
17
Let
T
:
R
n
→
R
m
T:\mathbb{R}^n\to\mathbb{R}^m
T
:
R
n
→
R
m
be a linear transformation. Choose the correct option that guarantees that
T
T
T
is bijective.
Single correct
MEDIUM
3 marks
01 December 2024
18
Let
S
S
S
be any straight line in
R
2
\mathbb{R}^2
R
2
passing through the origin. If
T
(
S
)
T(S)
T
(
S
)
is the image of
S
S
S
under
T
T
T
in
R
3
\mathbb{R}^3
R
3
, what is the dimension of
T
(
S
)
T(S)
T
(
S
)
?
Comprehension
MEDIUM
1 marks
01 December 2024
19
Let
B
1
=
{
(
1
,
−
1
)
,
(
0
,
2
)
}
B_1=\{(1,-1),(0,2)\}
B
1
=
{(
1
,
−
1
)
,
(
0
,
2
)}
and
B
2
=
{
(
1
,
0
,
0
)
,
(
0
,
1
,
0
)
,
(
0
,
0
,
1
)
}
B_2=\{(1,0,0),(0,1,0),(0,0,1)\}
B
2
=
{(
1
,
0
,
0
)
,
(
0
,
1
,
0
)
,
(
0
,
0
,
1
)}
. Which matrix is the representation of
T
T
T
with respect to
B
1
B_1
B
1
for the domain and
B
2
B_2
B
2
for the codomain?
Comprehension
EASY
1 marks
01 December 2024
20
Let
V
=
span
{
(
3
,
0
,
0
)
,
(
0
,
−
2
,
2
)
,
(
−
1
,
1
,
−
1
)
}
V=\operatorname{span}\{(3,0,0),(0,-2,2),(-1,1,-1)\}
V
=
span
{(
3
,
0
,
0
)
,
(
0
,
−
2
,
2
)
,
(
−
1
,
1
,
−
1
)}
. Let
T
:
V
→
R
2
T:V\to\mathbb{R}^2
T
:
V
→
R
2
be defined by
T
(
x
,
y
,
z
)
=
(
c
x
−
y
,
y
+
z
)
T(x,y,z)=(cx-y,y+z)
T
(
x
,
y
,
z
)
=
(
c
x
−
y
,
y
+
z
)
. Choose all correct options.
Multiple correct
HARD
3 marks
01 December 2024
21
Let
A
=
[
x
−
4
3
y
]
A=\begin{bmatrix}x&-4\\3&y\end{bmatrix}
A
=
[
x
3
−
4
y
]
and
T
:
R
2
→
R
T:\mathbb{R}^2\to\mathbb{R}
T
:
R
2
→
R
be the linear transformation given by
T
(
x
,
y
)
=
trace
(
A
)
T(x,y)=\operatorname{trace}(A)
T
(
x
,
y
)
=
trace
(
A
)
. Which statement is true?
Single correct
EASY
2 marks
04 August 2024
22
Which matrix represents
T
T
T
with respect to the standard ordered basis for both the domain and codomain?
Comprehension
MEDIUM
2 marks
04 August 2024
23
Which option defines the transformation
T
T
T
?
Comprehension
EASY
1 marks
04 August 2024
24
Find the value of
a
a
a
.
Comprehension
EASY
1 marks
24 March 2024
25
Find the value of
b
b
b
.
Comprehension
EASY
1 marks
24 March 2024
26
Find the value of
c
c
c
.
Comprehension
EASY
1 marks
24 March 2024
27
Find the value of
d
d
d
.
Comprehension
EASY
1 marks
24 March 2024
28
Choose the correct option from the following.
Comprehension
MEDIUM
2 marks
24 March 2024
29
Choose the correct options from the following.
Comprehension
MEDIUM
2 marks
24 March 2024
30
Which of the following statements is true?
Comprehension
EASY
1 marks
24 March 2024
31
Is the statement true or false? If
{
v
1
,
v
2
,
v
3
}
\{v_1,v_2,v_3\}
{
v
1
,
v
2
,
v
3
}
are linearly independent vectors in
V
V
V
, then
{
T
(
v
1
)
,
T
(
v
2
)
,
T
(
v
3
)
}
\{T(v_1),T(v_2),T(v_3)\}
{
T
(
v
1
)
,
T
(
v
2
)
,
T
(
v
3
)}
are linearly independent vectors in
W
W
W
.
Comprehension
EASY
1 marks
24 March 2024
32
Is the statement true or false? Let
{
u
1
,
u
2
,
u
3
}
\{u_1,u_2,u_3\}
{
u
1
,
u
2
,
u
3
}
be a subset of
V
V
V
. If
{
T
(
u
1
)
,
T
(
u
2
)
,
T
(
u
3
)
}
\{T(u_1),T(u_2),T(u_3)\}
{
T
(
u
1
)
,
T
(
u
2
)
,
T
(
u
3
)}
is linearly independent in
W
W
W
, then
{
u
1
,
u
2
,
u
3
}
\{u_1,u_2,u_3\}
{
u
1
,
u
2
,
u
3
}
could also be linearly dependent in
V
V
V
.
Comprehension
MEDIUM
1 marks
24 March 2024
33
Find the nullity of
T
T
T
.
Comprehension
EASY
1 marks
24 March 2024
34
Which of the following is a basis for the kernel of
T
T
T
?
Comprehension
EASY
1 marks
24 March 2024
35
Which of the following is a basis for the image of
T
T
T
?
Comprehension
MEDIUM
1 marks
24 March 2024
36
Let
T
T
T
be a linear transformation from
R
2
\mathbb{R}^2
R
2
to
R
3
\mathbb{R}^3
R
3
. Let
S
⊂
R
2
S\subset\mathbb{R}^2
S
⊂
R
2
be a line passing through the origin. Which of the following are possible?
Multiple correct
EASY
2 marks
03 December 2023
37
Find the matrix corresponding to
T
T
T
if
γ
\gamma
γ
is used as the basis for both the domain and codomain.
Comprehension
MEDIUM
3 marks
03 December 2023
38
Is
T
T
T
an isomorphism?
Comprehension
EASY
1 marks
03 December 2023
39
Which of the following options are true?
Multiple correct
MEDIUM
2 marks
06 August 2023
40
Consider
V
=
{
[
a
b
&
¸
a
]
:
c
=
a
+
b
;
a
,
b
,
c
∈
R
}
V=\left\{\begin{bmatrix}a&b\c\&a\end{bmatrix}:c=a+b;\ a,b,c\in\mathbb{R}\right\}
V
=
{
[
a
b
&
¸
a
]
:
c
=
a
+
b
;
a
,
b
,
c
∈
R
}
and
T
:
V
→
R
4
T:V\to\mathbb{R}^4
T
:
V
→
R
4
defined by
T
(
A
)
=
(
a
,
b
,
c
,
a
+
b
−
c
)
T(A)=(a,b,c,a+b-c)
T
(
A
)
=
(
a
,
b
,
c
,
a
+
b
−
c
)
. Choose the correct options.
Multiple correct
MEDIUM
2 marks
06 August 2023
41
Define a linear transformation
T
:
V
2
→
R
2
T:V_2\to\mathbb{R}^2
T
:
V
2
→
R
2
by
T
(
x
,
y
,
z
)
=
(
x
,
x
+
y
+
z
)
T(x,y,z)=(x,x+y+z)
T
(
x
,
y
,
z
)
=
(
x
,
x
+
y
+
z
)
. What is the rank of
T
T
T
?
Numerical
MEDIUM
2 marks
06 August 2023
42
Let
S
:
V
1
→
V
2
S:V_1\to V_2
S
:
V
1
→
V
2
be a linear transformation. If
m
×
n
m\times n
m
×
n
is the order of the matrix
D
D
D
of
S
S
S
with respect to ordered bases for
V
1
V_1
V
1
and
V
2
V_2
V
2
, what is
2
m
−
3
n
2m-3n
2
m
−
3
n
?
Numerical
EASY
1 marks
06 August 2023
43
If
A
=
[
a
b
c
d
e
f
g
h
i
]
A=\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}
A
=
a
d
g
b
e
h
c
f
i
denotes the matrix of
T
T
T
with respect to
{
(
1
,
1
,
1
)
,
(
1
,
1
,
0
)
,
(
1
,
0
,
0
)
}
\{(1,1,1),(1,1,0),(1,0,0)\}
{(
1
,
1
,
1
)
,
(
1
,
1
,
0
)
,
(
1
,
0
,
0
)}
for domain and codomain, what is
2
b
+
2
e
+
2
h
2b+2e+2h
2
b
+
2
e
+
2
h
?
Comprehension
HARD
2 marks
06 August 2023
44
Consider the vector space
V
=
{
[
a
b
a
b
]
:
a
,
b
∈
R
}
V=\left\{\begin{bmatrix}a&b\\a&b\end{bmatrix}:a,b\in\mathbb{R}\right\}
V
=
{
[
a
a
b
b
]
:
a
,
b
∈
R
}
and
T
:
R
3
→
V
T:\mathbb{R}^3\to V
T
:
R
3
→
V
defined by
T
(
x
,
y
,
z
)
=
[
x
+
y
x
+
y
+
z
x
+
y
x
+
y
+
z
]
T(x,y,z)=\begin{bmatrix}x+y&x+y+z\\x+y&x+y+z\end{bmatrix}
T
(
x
,
y
,
z
)
=
[
x
+
y
x
+
y
x
+
y
+
z
x
+
y
+
z
]
. Choose the correct option.
Single correct
MEDIUM
2 marks
02 April 2023
45
Let
V
=
{
[
a
b
c
d
]
∈
M
2
×
2
(
R
)
:
a
+
b
=
c
+
d
}
V=\left\{\begin{bmatrix}a&b\\c&d\end{bmatrix}\in M_{2\times2}(\mathbb{R}):a+b=c+d\right\}
V
=
{
[
a
c
b
d
]
∈
M
2
×
2
(
R
)
:
a
+
b
=
c
+
d
}
and
T
:
V
→
R
2
T:V\to\mathbb{R}^2
T
:
V
→
R
2
be onto. What is the dimension of
ker
(
T
)
\ker(T)
ker
(
T
)
?
Numerical
EASY
3 marks
02 April 2023
46
If
A
=
[
a
b
c
d
e
f
]
A=\begin{bmatrix}a&b\\c&d\\e&f\end{bmatrix}
A
=
a
c
e
b
d
f
denotes the matrix of
T
T
T
with respect to
{
(
1
,
1
)
,
(
1
,
−
1
)
}
\{(1,1),(1,-1)\}
{(
1
,
1
)
,
(
1
,
−
1
)}
for
R
2
\mathbb{R}^2
R
2
and
{
(
1
,
1
,
1
)
,
(
1
,
1
,
0
)
,
(
−
1
,
0
,
0
)
}
\{(1,1,1),(1,1,0),(-1,0,0)\}
{(
1
,
1
,
1
)
,
(
1
,
1
,
0
)
,
(
−
1
,
0
,
0
)}
for
R
3
\mathbb{R}^3
R
3
, what is
a
+
d
+
e
a+d+e
a
+
d
+
e
?
Comprehension
MEDIUM
3 marks
02 April 2023
47
Define
T
:
W
→
R
2
T:W\to\mathbb{R}^2
T
:
W
→
R
2
by
T
(
x
,
y
,
z
)
=
(
0
,
x
−
z
)
T(x,y,z)=(0,x-z)
T
(
x
,
y
,
z
)
=
(
0
,
x
−
z
)
. What is the rank of
T
T
T
?
Comprehension
EASY
1 marks
02 April 2023
48
If the
m
×
n
m\times n
m
×
n
matrix
B
B
B
is the matrix of
T
T
T
with respect to some basis for
W
W
W
and the standard ordered basis for
R
2
\mathbb{R}^2
R
2
, what is
m
+
n
m+n
m
+
n
?
Comprehension
EASY
1 marks
02 April 2023
49
If
A
A
A
is the matrix representation of
S
∘
T
S\circ T
S
∘
T
with respect to
β
\beta
β
for both domain and codomain, and the order of
A
A
A
is
m
×
n
m\times n
m
×
n
, find
m
+
n
m+n
m
+
n
.
Comprehension
EASY
1 marks
20 November 2022
50
If
K
=
{
(
x
,
y
,
z
)
:
a
x
+
b
y
=
0
,
c
y
+
d
z
=
0
}
K=\{(x,y,z): ax+by=0,\ cy+dz=0\}
K
=
{(
x
,
y
,
z
)
:
a
x
+
b
y
=
0
,
cy
+
d
z
=
0
}
is the null space of
S
∘
T
S\circ T
S
∘
T
, find
a
b
−
2
c
d
\frac{a}{b}-2\frac{c}{d}
b
a
−
2
d
c
.
Comprehension
MEDIUM
2 marks
20 November 2022
51
Find the rank of
S
∘
T
S\circ T
S
∘
T
.
Comprehension
EASY
1 marks
20 November 2022
52
Find the nullity of the matrix
A
A
A
.
Comprehension
EASY
1 marks
20 November 2022
53
Which of the following options is true?
Comprehension
MEDIUM
2 marks
20 November 2022
54
If
T
(
x
,
y
,
z
)
=
(
m
x
+
n
y
+
a
z
,
p
x
+
q
y
+
r
z
)
T(x,y,z)=(mx+ny+az,px+qy+rz)
T
(
x
,
y
,
z
)
=
(
m
x
+
n
y
+
a
z
,
p
x
+
q
y
+
r
z
)
, find
(
m
+
n
+
a
)
−
3
(
p
+
q
+
r
)
(m+n+a)-3(p+q+r)
(
m
+
n
+
a
)
−
3
(
p
+
q
+
r
)
.
Comprehension
HARD
2 marks
20 November 2022
55
What is the value of
d
−
a
d-a
d
−
a
?
Comprehension
MEDIUM
1 marks
10 July 2022
56
What is the value of
e
−
b
e-b
e
−
b
?
Comprehension
MEDIUM
1 marks
10 July 2022
57
What is the value of
f
−
c
f-c
f
−
c
?
Comprehension
MEDIUM
1 marks
10 July 2022
58
What is the value of
m
m
m
?
Comprehension
EASY
1 marks
10 July 2022
59
What is the value of
n
n
n
?
Comprehension
EASY
1 marks
10 July 2022
60
If
A
A
A
is the matrix representation of
T
T
T
with respect to the standard bases of
R
3
\mathbb{R}^3
R
3
and
R
2
\mathbb{R}^2
R
2
, then
A
A
A
is:
Comprehension
EASY
1 marks
10 July 2022
61
If
{
(
l
,
m
,
n
)
}
\{(l,m,n)\}
{(
l
,
m
,
n
)}
is a basis of
ker
(
T
)
\ker(T)
ker
(
T
)
, find
l
l
l
when
n
=
1
n=1
n
=
1
.
Comprehension
MEDIUM
1 marks
10 July 2022
62
If
{
(
l
,
m
,
n
)
}
\{(l,m,n)\}
{(
l
,
m
,
n
)}
is a basis of
ker
(
T
)
\ker(T)
ker
(
T
)
, find
m
m
m
when
n
=
1
n=1
n
=
1
.
Comprehension
MEDIUM
1 marks
10 July 2022
Showing 62 questions.
