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43Q
01
Consider the system of linear equations
A
x
=
b
Ax=b
A
x
=
b
given by
−
x
+
y
−
z
=
1
-x+y-z=1
−
x
+
y
−
z
=
1
,
x
−
y
+
z
=
−
1
x-y+z=-1
x
−
y
+
z
=
−
1
, and
x
+
z
=
0
x+z=0
x
+
z
=
0
. Suppose
L
L
L
represents the affine space of solutions of
A
x
=
b
Ax=b
A
x
=
b
, and let
W
W
W
be the subspace in
R
3
\mathbb{R}^3
R
3
corresponding to the affine space
L
L
L
. Choose the correct option.
Single correct
MEDIUM
4 marks
06 April 2026
02
Let
⟨
⋅
,
⋅
⟩
\langle\cdot,\cdot\rangle
⟨
⋅
,
⋅
⟩
denote the standard inner product on
R
2
\mathbb{R}^2
R
2
, i.e.,
⟨
(
x
1
,
x
2
)
,
(
y
1
,
y
2
)
⟩
=
x
1
y
1
+
x
2
y
2
\langle(x_1,x_2),(y_1,y_2)\rangle=x_1y_1+x_2y_2
⟨(
x
1
,
x
2
)
,
(
y
1
,
y
2
)⟩
=
x
1
y
1
+
x
2
y
2
. For
v
∈
R
2
v\in\mathbb{R}^2
v
∈
R
2
, consider a linear transformation
T
v
:
R
2
→
R
T_v:\mathbb{R}^2\to\mathbb{R}
T
v
:
R
2
→
R
defined as
T
v
(
u
)
=
⟨
u
,
v
⟩
T_v(u)=\langle u,v\rangle
T
v
(
u
)
=
⟨
u
,
v
⟩
. Which options are true for
T
v
T_v
T
v
?
Multiple correct
MEDIUM
4 marks
06 April 2026
03
Let
A
A
A
and
B
B
B
be square matrices of the same order
n
n
n
. Which statements are sufficient to conclude that
A
A
A
is equivalent to
B
B
B
?
Multiple correct
MEDIUM
4 marks
06 April 2026
04
Suppose that
T
:
R
3
→
R
2
T:\mathbb{R}^3\to\mathbb{R}^2
T
:
R
3
→
R
2
is a linear transformation given by
T
(
x
,
y
,
z
)
=
(
x
−
2
y
,
y
+
z
)
T(x,y,z)=(x-2y,\ y+z)
T
(
x
,
y
,
z
)
=
(
x
−
2
y
,
y
+
z
)
. Let
A
A
A
be the matrix representation of
T
T
T
with respect to the ordered standard bases for both domain and codomain. If
B
B
B
is equivalent to
A
A
A
, find the rank of
B
B
B
.
Numerical
EASY
4 marks
06 April 2026
05
Consider the set
L
=
{
(
x
,
y
,
z
)
∈
R
3
∣
x
+
y
+
z
=
1
,
x
−
y
=
0
}
L=\{(x,y,z)\in\mathbb{R}^3\mid x+y+z=1,\ x-y=0\}
L
=
{(
x
,
y
,
z
)
∈
R
3
∣
x
+
y
+
z
=
1
,
x
−
y
=
0
}
. Find the
y
y
y
-coordinate of the point of intersection of the image of
L
L
L
under
T
T
T
, that is, the set
{
T
(
x
,
y
,
z
)
∣
(
x
,
y
,
z
)
∈
L
}
\{T(x,y,z)\mid (x,y,z)\in L\}
{
T
(
x
,
y
,
z
)
∣
(
x
,
y
,
z
)
∈
L
}
, and the
y
y
y
-axis in
R
2
\mathbb{R}^2
R
2
.
Comprehension
MEDIUM
2 marks
06 April 2026
06
Let
u
=
(
−
1
,
2
,
−
3
)
u=(-1,2,-3)
u
=
(
−
1
,
2
,
−
3
)
be a vector from the inner product space
R
3
\mathbb{R}^3
R
3
with the usual inner product. Which options are true?
Multiple correct
MEDIUM
6 marks
03 August 2025
07
Let
u
=
(
1
,
1
)
u=(1,1)
u
=
(
1
,
1
)
and
v
=
(
v
1
,
v
2
)
v=(v_1,v_2)
v
=
(
v
1
,
v
2
)
be vectors in
R
2
\mathbb{R}^2
R
2
with the usual inner product. Suppose
∥
v
∥
=
2
\|v\|=2
∥
v
∥
=
2
and the angle between
u
u
u
and
v
v
v
is
45
∘
45^\circ
4
5
∘
. Find
v
1
+
v
2
v_1+v_2
v
1
+
v
2
.
Numerical
EASY
4 marks
03 August 2025
08
Choose all the correct statements.
Comprehension
MEDIUM
4 marks
03 August 2025
09
If
α
=
−
2
\alpha=-2
α
=
−
2
, find
β
\beta
β
.
Comprehension
MEDIUM
3 marks
03 August 2025
10
Suppose
T
:
R
3
→
R
3
T:\mathbb{R}^3\to\mathbb{R}^3
T
:
R
3
→
R
3
is a linear transformation. Let
M
1
M_1
M
1
and
M
2
M_2
M
2
denote the matrix representations of
T
T
T
with respect to distinct bases for both domain and codomain
β
1
\beta_1
β
1
and
β
2
\beta_2
β
2
, respectively. Choose the correct statements.
Multiple correct
MEDIUM
4 marks
16 March 2025
11
Let
A
=
[
−
2
0
3
4
−
1
2
]
A=\begin{bmatrix}-2&0&3\\4&-1&2\end{bmatrix}
A
=
[
−
2
4
0
−
1
3
2
]
. Which of the following matrices are equivalent to
A
A
A
?
Multiple correct
MEDIUM
4 marks
16 March 2025
12
Let
A
=
[
2
−
3
4
1
]
A=\begin{bmatrix}2&-3\\4&1\end{bmatrix}
A
=
[
2
4
−
3
1
]
. Choose all the correct options.
Multiple correct
MEDIUM
4 marks
16 March 2025
13
Let
A
=
[
−
1
1
1
5
]
A=\begin{bmatrix}-1&1\\1&5\end{bmatrix}
A
=
[
−
1
1
1
5
]
and let
B
=
(
b
i
j
)
B=(b_{ij})
B
=
(
b
ij
)
be a matrix similar to
A
A
A
. If
b
11
=
7
b_{11}=7
b
11
=
7
, find
b
22
b_{22}
b
22
.
Numerical
EASY
2 marks
16 March 2025
14
Which of the affine spaces below correspond to the subspace
W
W
W
?
Comprehension
MEDIUM
2 marks
16 March 2025
15
If
A
A
A
is equivalent to
I
2
I_2
I
2
, then find the rank of
A
A
A
.
Comprehension
EASY
1 marks
01 December 2024
16
Find the value of
k
k
k
for which
A
A
A
is not equivalent to
I
2
I_2
I
2
.
Comprehension
MEDIUM
1 marks
01 December 2024
17
Find the number of values of
k
k
k
for which
A
A
A
is similar to
I
2
I_2
I
2
.
Comprehension
MEDIUM
1 marks
01 December 2024
18
For
k
=
0
k=0
k
=
0
, find the angle in degrees between
u
2
u_2
u
2
and
u
3
u_3
u
3
.
Comprehension
EASY
1 marks
01 December 2024
19
Consider the system
A
x
=
b
Ax=b
A
x
=
b
, where
A
A
A
is an
m
×
n
m\times n
m
×
n
matrix and
b
∈
R
m
b\in\mathbb{R}^m
b
∈
R
m
. Choose all options that guarantee that the solution set is an affine subspace of
R
n
\mathbb{R}^n
R
n
.
Multiple correct
MEDIUM
3 marks
01 December 2024
20
Consider
P
=
1
5
[
3
4
−
4
3
]
P=\frac15\begin{bmatrix}3&4\\-4&3\end{bmatrix}
P
=
5
1
[
3
−
4
4
3
]
and
Q
=
1
5
[
3
−
4
4
3
]
Q=\frac15\begin{bmatrix}3&-4\\4&3\end{bmatrix}
Q
=
5
1
[
3
4
−
4
3
]
. Let
A
A
A
be any
2
×
2
2\times2
2
×
2
matrix and
B
=
P
A
Q
B=PAQ
B
=
P
A
Q
. Which statement is true?
Single correct
MEDIUM
2 marks
04 August 2024
21
If
A
A
A
and
B
B
B
are similar matrices, then
A
T
A^T
A
T
and
B
T
B^T
B
T
are similar matrices.
Comprehension
MEDIUM
1 marks
04 August 2024
22
If
A
A
A
and
B
B
B
have the same rank, then they are similar.
Comprehension
EASY
1 marks
04 August 2024
23
Select all true statements.
Comprehension
EASY
2 marks
04 August 2024
24
Select all true statements.
Comprehension
MEDIUM
2 marks
04 August 2024
25
Let
A
A
A
and
B
B
B
be
n
×
n
n\times n
n
×
n
similar matrices. Suppose
A
A
A
has exactly
n
−
1
n-1
n
−
1
linearly independent columns. Then
det
(
B
)
\det(B)
det
(
B
)
is equal to _____.
Comprehension
EASY
1 marks
24 March 2024
26
Let
A
A
A
and
B
B
B
be
n
×
n
n\times n
n
×
n
matrices. Which statements are true?
Multiple correct
MEDIUM
3 marks
03 December 2023
27
Choose the correct options from the following.
Comprehension
MEDIUM
2 marks
03 December 2023
28
If the dimension of
L
L
L
is
m
m
m
and the dimension of
L
′
L'
L
′
is
n
n
n
, then find
m
+
n
m+n
m
+
n
.
Comprehension
EASY
1 marks
03 December 2023
29
An inner product on a vector space
V
V
V
satisfies positive-definiteness, additivity in the first input, symmetry, and scalar homogeneity. Let
V
=
R
2
V=\mathbb{R}^2
V
=
R
2
and define
⟨
(
x
1
,
x
2
)
,
(
y
1
,
y
2
)
⟩
=
x
1
y
1
−
x
2
y
1
−
x
2
y
2
\langle (x_1,x_2),(y_1,y_2)\rangle=x_1y_1-x_2y_1-x_2y_2
⟨(
x
1
,
x
2
)
,
(
y
1
,
y
2
)⟩
=
x
1
y
1
−
x
2
y
1
−
x
2
y
2
. Which conditions are satisfied?
Multiple correct
MEDIUM
3 marks
06 August 2023
30
Let
B
B
B
denote the matrix of
T
T
T
with respect to the standard ordered basis for both domain and codomain. Choose the correct options.
Comprehension
MEDIUM
2 marks
06 August 2023
31
An inner product on
V
V
V
satisfies positive-definiteness, additivity, symmetry, and homogeneity. Let
V
=
R
2
V=\mathbb{R}^2
V
=
R
2
and define
⟨
(
x
1
,
x
2
)
,
(
y
1
,
y
2
)
⟩
=
x
1
y
1
−
x
1
y
2
+
x
2
y
2
\langle(x_1,x_2),(y_1,y_2)\rangle=x_1y_1-x_1y_2+x_2y_2
⟨(
x
1
,
x
2
)
,
(
y
1
,
y
2
)⟩
=
x
1
y
1
−
x
1
y
2
+
x
2
y
2
. Which conditions are satisfied?
Multiple correct
MEDIUM
3 marks
02 April 2023
32
Let
B
B
B
denote the matrix of
T
T
T
with respect to the standard ordered bases for
R
2
\mathbb{R}^2
R
2
and
R
3
\mathbb{R}^3
R
3
. Choose the correct options.
Comprehension
MEDIUM
2 marks
02 April 2023
33
What is the dimension of
L
L
L
?
Comprehension
EASY
1 marks
02 April 2023
34
Which of the following options are true?
Multiple correct
MEDIUM
2 marks
20 November 2022
35
Find
∥
(
1
,
3
)
∥
2
\|(1,3)\|^2
∥
(
1
,
3
)
∥
2
.
Comprehension
EASY
1 marks
20 November 2022
36
Which of the following are unit vectors in
V
V
V
?
Comprehension
MEDIUM
2 marks
20 November 2022
37
Which affine subspace was considered by Soumya?
Comprehension
MEDIUM
2 marks
20 November 2022
38
Which affine subspace was considered by Sohini?
Comprehension
MEDIUM
2 marks
20 November 2022
39
Which function represents
f
f
f
correctly?
Comprehension
HARD
2 marks
20 November 2022
40
An inner product on
V
V
V
satisfies positive-definiteness, additivity, symmetry, and homogeneity. Let
V
=
R
2
V=\mathbb{R}^2
V
=
R
2
and define
⟨
(
x
1
,
x
2
)
,
(
y
1
,
y
2
)
⟩
=
x
1
y
1
−
x
1
y
2
−
x
2
y
1
+
x
2
y
2
\langle(x_1,x_2),(y_1,y_2)\rangle=x_1y_1-x_1y_2-x_2y_1+x_2y_2
⟨(
x
1
,
x
2
)
,
(
y
1
,
y
2
)⟩
=
x
1
y
1
−
x
1
y
2
−
x
2
y
1
+
x
2
y
2
. Which conditions are satisfied?
Multiple correct
MEDIUM
2 marks
10 July 2022
41
Let
U
U
U
be a subspace of
R
3
\mathbb{R}^3
R
3
with basis
{
(
1
,
0
,
1
)
,
(
0
,
1
,
2
)
}
\{(1,0,1),(0,1,2)\}
{(
1
,
0
,
1
)
,
(
0
,
1
,
2
)}
. Which subsets of
R
3
\mathbb{R}^3
R
3
are appropriate candidates for affine subspaces whose corresponding vector subspace is
U
U
U
?
Multiple correct
MEDIUM
2 marks
10 July 2022
42
Let
A
=
[
1
0
1
1
]
A=\begin{bmatrix}1&0\\1&1\end{bmatrix}
A
=
[
1
1
0
1
]
,
B
=
[
1
1
0
1
]
B=\begin{bmatrix}1&1\\0&1\end{bmatrix}
B
=
[
1
0
1
1
]
, and
C
=
[
1
0
0
1
]
C=\begin{bmatrix}1&0\\0&1\end{bmatrix}
C
=
[
1
0
0
1
]
. For Pair I:
A
,
B
A,B
A
,
B
; Pair II:
A
,
C
A,C
A
,
C
; Pair III:
B
,
C
B,C
B
,
C
, choose the correct option.
Single correct
MEDIUM
2 marks
10 July 2022
43
A norm on
V
V
V
satisfies triangle inequality, scalar homogeneity, and positive-definiteness. Consider
∥
(
x
1
,
x
2
,
x
3
)
∥
=
∣
x
1
+
x
2
+
x
3
∣
\| (x_1,x_2,x_3)\|=|x_1+x_2+x_3|
∥
(
x
1
,
x
2
,
x
3
)
∥
=
∣
x
1
+
x
2
+
x
3
∣
on
R
3
\mathbb{R}^3
R
3
. Which conditions are satisfied?
Multiple correct
MEDIUM
1 marks
10 July 2022
Showing 43 questions.
Mathematics 2 > Week 7
All PYQs
Topic-Wise PYQs
Start Weekly Test
Mock tests
Week mock
Topic mock
Subject mock
Type:
All
Difficulty:
All
Year:
All
43Q
01
Consider the system of linear equations
A
x
=
b
Ax=b
A
x
=
b
given by
−
x
+
y
−
z
=
1
-x+y-z=1
−
x
+
y
−
z
=
1
,
x
−
y
+
z
=
−
1
x-y+z=-1
x
−
y
+
z
=
−
1
, and
x
+
z
=
0
x+z=0
x
+
z
=
0
. Suppose
L
L
L
represents the affine space of solutions of
A
x
=
b
Ax=b
A
x
=
b
, and let
W
W
W
be the subspace in
R
3
\mathbb{R}^3
R
3
corresponding to the affine space
L
L
L
. Choose the correct option.
Single correct
MEDIUM
4 marks
06 April 2026
02
Let
⟨
⋅
,
⋅
⟩
\langle\cdot,\cdot\rangle
⟨
⋅
,
⋅
⟩
denote the standard inner product on
R
2
\mathbb{R}^2
R
2
, i.e.,
⟨
(
x
1
,
x
2
)
,
(
y
1
,
y
2
)
⟩
=
x
1
y
1
+
x
2
y
2
\langle(x_1,x_2),(y_1,y_2)\rangle=x_1y_1+x_2y_2
⟨(
x
1
,
x
2
)
,
(
y
1
,
y
2
)⟩
=
x
1
y
1
+
x
2
y
2
. For
v
∈
R
2
v\in\mathbb{R}^2
v
∈
R
2
, consider a linear transformation
T
v
:
R
2
→
R
T_v:\mathbb{R}^2\to\mathbb{R}
T
v
:
R
2
→
R
defined as
T
v
(
u
)
=
⟨
u
,
v
⟩
T_v(u)=\langle u,v\rangle
T
v
(
u
)
=
⟨
u
,
v
⟩
. Which options are true for
T
v
T_v
T
v
?
Multiple correct
MEDIUM
4 marks
06 April 2026
03
Let
A
A
A
and
B
B
B
be square matrices of the same order
n
n
n
. Which statements are sufficient to conclude that
A
A
A
is equivalent to
B
B
B
?
Multiple correct
MEDIUM
4 marks
06 April 2026
04
Suppose that
T
:
R
3
→
R
2
T:\mathbb{R}^3\to\mathbb{R}^2
T
:
R
3
→
R
2
is a linear transformation given by
T
(
x
,
y
,
z
)
=
(
x
−
2
y
,
y
+
z
)
T(x,y,z)=(x-2y,\ y+z)
T
(
x
,
y
,
z
)
=
(
x
−
2
y
,
y
+
z
)
. Let
A
A
A
be the matrix representation of
T
T
T
with respect to the ordered standard bases for both domain and codomain. If
B
B
B
is equivalent to
A
A
A
, find the rank of
B
B
B
.
Numerical
EASY
4 marks
06 April 2026
05
Consider the set
L
=
{
(
x
,
y
,
z
)
∈
R
3
∣
x
+
y
+
z
=
1
,
x
−
y
=
0
}
L=\{(x,y,z)\in\mathbb{R}^3\mid x+y+z=1,\ x-y=0\}
L
=
{(
x
,
y
,
z
)
∈
R
3
∣
x
+
y
+
z
=
1
,
x
−
y
=
0
}
. Find the
y
y
y
-coordinate of the point of intersection of the image of
L
L
L
under
T
T
T
, that is, the set
{
T
(
x
,
y
,
z
)
∣
(
x
,
y
,
z
)
∈
L
}
\{T(x,y,z)\mid (x,y,z)\in L\}
{
T
(
x
,
y
,
z
)
∣
(
x
,
y
,
z
)
∈
L
}
, and the
y
y
y
-axis in
R
2
\mathbb{R}^2
R
2
.
Comprehension
MEDIUM
2 marks
06 April 2026
06
Let
u
=
(
−
1
,
2
,
−
3
)
u=(-1,2,-3)
u
=
(
−
1
,
2
,
−
3
)
be a vector from the inner product space
R
3
\mathbb{R}^3
R
3
with the usual inner product. Which options are true?
Multiple correct
MEDIUM
6 marks
03 August 2025
07
Let
u
=
(
1
,
1
)
u=(1,1)
u
=
(
1
,
1
)
and
v
=
(
v
1
,
v
2
)
v=(v_1,v_2)
v
=
(
v
1
,
v
2
)
be vectors in
R
2
\mathbb{R}^2
R
2
with the usual inner product. Suppose
∥
v
∥
=
2
\|v\|=2
∥
v
∥
=
2
and the angle between
u
u
u
and
v
v
v
is
45
∘
45^\circ
4
5
∘
. Find
v
1
+
v
2
v_1+v_2
v
1
+
v
2
.
Numerical
EASY
4 marks
03 August 2025
08
Choose all the correct statements.
Comprehension
MEDIUM
4 marks
03 August 2025
09
If
α
=
−
2
\alpha=-2
α
=
−
2
, find
β
\beta
β
.
Comprehension
MEDIUM
3 marks
03 August 2025
10
Suppose
T
:
R
3
→
R
3
T:\mathbb{R}^3\to\mathbb{R}^3
T
:
R
3
→
R
3
is a linear transformation. Let
M
1
M_1
M
1
and
M
2
M_2
M
2
denote the matrix representations of
T
T
T
with respect to distinct bases for both domain and codomain
β
1
\beta_1
β
1
and
β
2
\beta_2
β
2
, respectively. Choose the correct statements.
Multiple correct
MEDIUM
4 marks
16 March 2025
11
Let
A
=
[
−
2
0
3
4
−
1
2
]
A=\begin{bmatrix}-2&0&3\\4&-1&2\end{bmatrix}
A
=
[
−
2
4
0
−
1
3
2
]
. Which of the following matrices are equivalent to
A
A
A
?
Multiple correct
MEDIUM
4 marks
16 March 2025
12
Let
A
=
[
2
−
3
4
1
]
A=\begin{bmatrix}2&-3\\4&1\end{bmatrix}
A
=
[
2
4
−
3
1
]
. Choose all the correct options.
Multiple correct
MEDIUM
4 marks
16 March 2025
13
Let
A
=
[
−
1
1
1
5
]
A=\begin{bmatrix}-1&1\\1&5\end{bmatrix}
A
=
[
−
1
1
1
5
]
and let
B
=
(
b
i
j
)
B=(b_{ij})
B
=
(
b
ij
)
be a matrix similar to
A
A
A
. If
b
11
=
7
b_{11}=7
b
11
=
7
, find
b
22
b_{22}
b
22
.
Numerical
EASY
2 marks
16 March 2025
14
Which of the affine spaces below correspond to the subspace
W
W
W
?
Comprehension
MEDIUM
2 marks
16 March 2025
15
If
A
A
A
is equivalent to
I
2
I_2
I
2
, then find the rank of
A
A
A
.
Comprehension
EASY
1 marks
01 December 2024
16
Find the value of
k
k
k
for which
A
A
A
is not equivalent to
I
2
I_2
I
2
.
Comprehension
MEDIUM
1 marks
01 December 2024
17
Find the number of values of
k
k
k
for which
A
A
A
is similar to
I
2
I_2
I
2
.
Comprehension
MEDIUM
1 marks
01 December 2024
18
For
k
=
0
k=0
k
=
0
, find the angle in degrees between
u
2
u_2
u
2
and
u
3
u_3
u
3
.
Comprehension
EASY
1 marks
01 December 2024
19
Consider the system
A
x
=
b
Ax=b
A
x
=
b
, where
A
A
A
is an
m
×
n
m\times n
m
×
n
matrix and
b
∈
R
m
b\in\mathbb{R}^m
b
∈
R
m
. Choose all options that guarantee that the solution set is an affine subspace of
R
n
\mathbb{R}^n
R
n
.
Multiple correct
MEDIUM
3 marks
01 December 2024
20
Consider
P
=
1
5
[
3
4
−
4
3
]
P=\frac15\begin{bmatrix}3&4\\-4&3\end{bmatrix}
P
=
5
1
[
3
−
4
4
3
]
and
Q
=
1
5
[
3
−
4
4
3
]
Q=\frac15\begin{bmatrix}3&-4\\4&3\end{bmatrix}
Q
=
5
1
[
3
4
−
4
3
]
. Let
A
A
A
be any
2
×
2
2\times2
2
×
2
matrix and
B
=
P
A
Q
B=PAQ
B
=
P
A
Q
. Which statement is true?
Single correct
MEDIUM
2 marks
04 August 2024
21
If
A
A
A
and
B
B
B
are similar matrices, then
A
T
A^T
A
T
and
B
T
B^T
B
T
are similar matrices.
Comprehension
MEDIUM
1 marks
04 August 2024
22
If
A
A
A
and
B
B
B
have the same rank, then they are similar.
Comprehension
EASY
1 marks
04 August 2024
23
Select all true statements.
Comprehension
EASY
2 marks
04 August 2024
24
Select all true statements.
Comprehension
MEDIUM
2 marks
04 August 2024
25
Let
A
A
A
and
B
B
B
be
n
×
n
n\times n
n
×
n
similar matrices. Suppose
A
A
A
has exactly
n
−
1
n-1
n
−
1
linearly independent columns. Then
det
(
B
)
\det(B)
det
(
B
)
is equal to _____.
Comprehension
EASY
1 marks
24 March 2024
26
Let
A
A
A
and
B
B
B
be
n
×
n
n\times n
n
×
n
matrices. Which statements are true?
Multiple correct
MEDIUM
3 marks
03 December 2023
27
Choose the correct options from the following.
Comprehension
MEDIUM
2 marks
03 December 2023
28
If the dimension of
L
L
L
is
m
m
m
and the dimension of
L
′
L'
L
′
is
n
n
n
, then find
m
+
n
m+n
m
+
n
.
Comprehension
EASY
1 marks
03 December 2023
29
An inner product on a vector space
V
V
V
satisfies positive-definiteness, additivity in the first input, symmetry, and scalar homogeneity. Let
V
=
R
2
V=\mathbb{R}^2
V
=
R
2
and define
⟨
(
x
1
,
x
2
)
,
(
y
1
,
y
2
)
⟩
=
x
1
y
1
−
x
2
y
1
−
x
2
y
2
\langle (x_1,x_2),(y_1,y_2)\rangle=x_1y_1-x_2y_1-x_2y_2
⟨(
x
1
,
x
2
)
,
(
y
1
,
y
2
)⟩
=
x
1
y
1
−
x
2
y
1
−
x
2
y
2
. Which conditions are satisfied?
Multiple correct
MEDIUM
3 marks
06 August 2023
30
Let
B
B
B
denote the matrix of
T
T
T
with respect to the standard ordered basis for both domain and codomain. Choose the correct options.
Comprehension
MEDIUM
2 marks
06 August 2023
31
An inner product on
V
V
V
satisfies positive-definiteness, additivity, symmetry, and homogeneity. Let
V
=
R
2
V=\mathbb{R}^2
V
=
R
2
and define
⟨
(
x
1
,
x
2
)
,
(
y
1
,
y
2
)
⟩
=
x
1
y
1
−
x
1
y
2
+
x
2
y
2
\langle(x_1,x_2),(y_1,y_2)\rangle=x_1y_1-x_1y_2+x_2y_2
⟨(
x
1
,
x
2
)
,
(
y
1
,
y
2
)⟩
=
x
1
y
1
−
x
1
y
2
+
x
2
y
2
. Which conditions are satisfied?
Multiple correct
MEDIUM
3 marks
02 April 2023
32
Let
B
B
B
denote the matrix of
T
T
T
with respect to the standard ordered bases for
R
2
\mathbb{R}^2
R
2
and
R
3
\mathbb{R}^3
R
3
. Choose the correct options.
Comprehension
MEDIUM
2 marks
02 April 2023
33
What is the dimension of
L
L
L
?
Comprehension
EASY
1 marks
02 April 2023
34
Which of the following options are true?
Multiple correct
MEDIUM
2 marks
20 November 2022
35
Find
∥
(
1
,
3
)
∥
2
\|(1,3)\|^2
∥
(
1
,
3
)
∥
2
.
Comprehension
EASY
1 marks
20 November 2022
36
Which of the following are unit vectors in
V
V
V
?
Comprehension
MEDIUM
2 marks
20 November 2022
37
Which affine subspace was considered by Soumya?
Comprehension
MEDIUM
2 marks
20 November 2022
38
Which affine subspace was considered by Sohini?
Comprehension
MEDIUM
2 marks
20 November 2022
39
Which function represents
f
f
f
correctly?
Comprehension
HARD
2 marks
20 November 2022
40
An inner product on
V
V
V
satisfies positive-definiteness, additivity, symmetry, and homogeneity. Let
V
=
R
2
V=\mathbb{R}^2
V
=
R
2
and define
⟨
(
x
1
,
x
2
)
,
(
y
1
,
y
2
)
⟩
=
x
1
y
1
−
x
1
y
2
−
x
2
y
1
+
x
2
y
2
\langle(x_1,x_2),(y_1,y_2)\rangle=x_1y_1-x_1y_2-x_2y_1+x_2y_2
⟨(
x
1
,
x
2
)
,
(
y
1
,
y
2
)⟩
=
x
1
y
1
−
x
1
y
2
−
x
2
y
1
+
x
2
y
2
. Which conditions are satisfied?
Multiple correct
MEDIUM
2 marks
10 July 2022
41
Let
U
U
U
be a subspace of
R
3
\mathbb{R}^3
R
3
with basis
{
(
1
,
0
,
1
)
,
(
0
,
1
,
2
)
}
\{(1,0,1),(0,1,2)\}
{(
1
,
0
,
1
)
,
(
0
,
1
,
2
)}
. Which subsets of
R
3
\mathbb{R}^3
R
3
are appropriate candidates for affine subspaces whose corresponding vector subspace is
U
U
U
?
Multiple correct
MEDIUM
2 marks
10 July 2022
42
Let
A
=
[
1
0
1
1
]
A=\begin{bmatrix}1&0\\1&1\end{bmatrix}
A
=
[
1
1
0
1
]
,
B
=
[
1
1
0
1
]
B=\begin{bmatrix}1&1\\0&1\end{bmatrix}
B
=
[
1
0
1
1
]
, and
C
=
[
1
0
0
1
]
C=\begin{bmatrix}1&0\\0&1\end{bmatrix}
C
=
[
1
0
0
1
]
. For Pair I:
A
,
B
A,B
A
,
B
; Pair II:
A
,
C
A,C
A
,
C
; Pair III:
B
,
C
B,C
B
,
C
, choose the correct option.
Single correct
MEDIUM
2 marks
10 July 2022
43
A norm on
V
V
V
satisfies triangle inequality, scalar homogeneity, and positive-definiteness. Consider
∥
(
x
1
,
x
2
,
x
3
)
∥
=
∣
x
1
+
x
2
+
x
3
∣
\| (x_1,x_2,x_3)\|=|x_1+x_2+x_3|
∥
(
x
1
,
x
2
,
x
3
)
∥
=
∣
x
1
+
x
2
+
x
3
∣
on
R
3
\mathbb{R}^3
R
3
. Which conditions are satisfied?
Multiple correct
MEDIUM
1 marks
10 July 2022
Showing 43 questions.
