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Topic Wise PYQs
Start Weekly Test
Type:
All
Difficulty:
All
Year:
All
30Q
01
Which of the following subsets are subspaces of
R
3
\mathbb{R}^3
R
3
under the usual addition and scalar multiplication?
Multiple correct
MEDIUM
4 marks
15 March 2026
02
Suppose the vectors
(
−
1
,
c
,
2
)
(-1,c,2)
(
−
1
,
c
,
2
)
,
(
0
,
1
,
3
)
(0,1,3)
(
0
,
1
,
3
)
, and
(
−
2
,
0
,
1
)
(-2,0,1)
(
−
2
,
0
,
1
)
in
R
3
\mathbb{R}^3
R
3
are linearly dependent. Find the value of
7
c
7c
7
c
.
Numerical
MEDIUM
4 marks
15 March 2026
03
Which of the following options is correct?
Comprehension
MEDIUM
4 marks
15 March 2026
04
Choose all the correct statements from the following.
Multiple correct
MEDIUM
6 marks
26 October 2025
05
The vectors
v
1
v_1
v
1
and
v
2
v_2
v
2
are linearly dependent vectors for some value of
a
a
a
.
Comprehension
EASY
3 marks
26 October 2025
06
Each of the options below contains a set
V
V
V
along with an addition operation
+
:
V
×
V
→
V
+:V\times V\to V
+
:
V
×
V
→
V
and a scalar multiplication operation
⋅
:
R
×
V
→
V
\cdot:\mathbb{R}\times V\to V
⋅
:
R
×
V
→
V
. Choose the options for which there is an additive identity in
V
V
V
with respect to
+
+
+
.
Multiple correct
HARD
5 marks
13 July 2025
07
Find the number of values of
k
k
k
for which the given three vectors are linearly dependent in
R
3
\mathbb{R}^3
R
3
.
Comprehension
MEDIUM
4 marks
13 July 2025
08
Find the largest value of
k
k
k
for which the given three vectors are linearly dependent in
R
3
\mathbb{R}^3
R
3
.
Comprehension
EASY
2 marks
13 July 2025
09
The vectors
v
1
v_1
v
1
and
v
2
v_2
v
2
are linearly dependent vectors for some value of
a
a
a
.
Comprehension
EASY
3 marks
23 February 2025
10
Which of the following options represents
W
1
∩
W
2
W_1\cap W_2
W
1
∩
W
2
? One or more options may be correct.
Comprehension
MEDIUM
4 marks
23 February 2025
11
Which of the following option is true?
Comprehension
EASY
2 marks
23 February 2025
12
B
=
{
v
1
,
v
2
,
v
3
}
B=\{v_1,v_2,v_3\}
B
=
{
v
1
,
v
2
,
v
3
}
is a linearly independent subset of
R
3
\mathbb{R}^3
R
3
. Select all linearly independent subsets of
R
3
\mathbb{R}^3
R
3
from the following.
Multiple correct
MEDIUM
3 marks
07 July 2024
13
Is the set
V
V
V
closed under addition?
Comprehension
EASY
1 marks
25 February 2024
14
Is the set
V
V
V
closed under scalar multiplication?
Comprehension
EASY
1 marks
25 February 2024
15
Which of the following is/are correct?
Comprehension
MEDIUM
2 marks
25 February 2024
16
W
1
=
{
[
a
0
0
0
b
0
0
0
c
]
:
a
,
b
,
c
∈
R
such that
a
+
b
+
c
=
1
}
W_1=\left\{\begin{bmatrix}a&0&0\\0&b&0\\0&0&c\end{bmatrix}:a,b,c\in\mathbb{R}\text{ such that }a+b+c=1\right\}
W
1
=
⎩
⎨
⎧
a
0
0
0
b
0
0
0
c
:
a
,
b
,
c
∈
R
such that
a
+
b
+
c
=
1
⎭
⎬
⎫
. If
W
1
W_1
W
1
is a subspace, find the dimension; else write the answer as
0
0
0
.
Comprehension
EASY
1 marks
25 February 2024
17
Unless otherwise stated, assume that we consider the usual vector addition and scalar multiplication. Choose the correct option(s) from the following statements.
Multiple correct
HARD
4 marks
29 October 2023
18
Let
v
1
,
v
2
v_1,v_2
v
1
,
v
2
and
v
3
v_3
v
3
be the column vectors of
A
A
A
and let
u
1
,
u
2
u_1,u_2
u
1
,
u
2
and
u
3
u_3
u
3
be the column vectors of
B
B
B
. Choose the correct option(s) from the following statements.
Comprehension
MEDIUM
3 marks
29 October 2023
19
Which of the following subsets of
R
2
\mathbb{R}^2
R
2
is/are vector spaces with respect to usual addition and usual scalar multiplication?
Multiple correct
MEDIUM
2 marks
16 July 2023
20
Select the true statement(s).
Multiple correct
EASY
2 marks
16 July 2023
21
Consider the set
W
=
{
A
∈
M
n
(
R
)
:
det
(
A
T
)
=
0
}
W=\{A\in M_n(\mathbb{R}):\det(A^T)=0\}
W
=
{
A
∈
M
n
(
R
)
:
det
(
A
T
)
=
0
}
with the usual addition and usual scalar multiplication of matrices. Which of the following is/are true?
Multiple correct
MEDIUM
2 marks
16 July 2023
22
If addition and scalar multiplication on
V
=
R
2
V=\mathbb{R}^2
V
=
R
2
are defined as follows: addition:
(
x
1
,
y
1
)
+
(
x
2
,
y
2
)
=
(
0
,
0
)
(x_1,y_1)+(x_2,y_2)=(0,0)
(
x
1
,
y
1
)
+
(
x
2
,
y
2
)
=
(
0
,
0
)
for
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
∈
V
(x_1,y_1),(x_2,y_2)\in V
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
∈
V
; scalar multiplication:
c
(
x
,
y
)
=
(
0
,
0
)
c(x,y)=(0,0)
c
(
x
,
y
)
=
(
0
,
0
)
for
(
x
,
y
)
∈
V
(x,y)\in V
(
x
,
y
)
∈
V
and
c
∈
R
c\in\mathbb{R}
c
∈
R
. Consider the statements: 1. There exists an element
0
0
0
in
V
V
V
such that
0
+
v
=
v
0+v=v
0
+
v
=
v
for all
v
∈
V
v\in V
v
∈
V
. 2. For each
v
∈
V
v\in V
v
∈
V
and
a
,
b
∈
R
a,b\in\mathbb{R}
a
,
b
∈
R
,
(
a
+
b
)
v
=
a
v
+
b
v
(a+b)v=av+bv
(
a
+
b
)
v
=
a
v
+
b
v
. 3. For each scalar
a
∈
R
a\in\mathbb{R}
a
∈
R
and
v
1
,
v
2
∈
V
v_1,v_2\in V
v
1
,
v
2
∈
V
,
a
(
v
1
+
v
2
)
=
a
v
1
+
a
v
2
a(v_1+v_2)=av_1+av_2
a
(
v
1
+
v
2
)
=
a
v
1
+
a
v
2
. 4. For each
v
∈
V
v\in V
v
∈
V
and
a
,
b
∈
R
a,b\in\mathbb{R}
a
,
b
∈
R
,
(
a
b
)
v
=
a
(
b
v
)
(ab)v=a(bv)
(
ab
)
v
=
a
(
b
v
)
. Which statement is not true? Enter the serial number.
Numerical
EASY
2 marks
16 October 2022
23
Consider the statements:
P
P
P
:
V
=
R
2
V=\mathbb{R}^2
V
=
R
2
with addition
(
x
1
,
y
1
)
+
(
x
2
,
y
2
)
=
(
x
1
x
2
,
y
1
y
2
)
(x_1,y_1)+(x_2,y_2)=(x_1x_2,y_1y_2)
(
x
1
,
y
1
)
+
(
x
2
,
y
2
)
=
(
x
1
x
2
,
y
1
y
2
)
and scalar multiplication
c
(
x
,
y
)
=
(
c
x
,
c
y
)
c(x,y)=(cx,cy)
c
(
x
,
y
)
=
(
c
x
,
cy
)
is a vector space.
Q
Q
Q
: Let
V
V
V
be a vector space. If
u
,
v
,
w
∈
V
u,v,w\in V
u
,
v
,
w
∈
V
are such that
a
u
+
b
v
+
c
w
=
0
au+bv+cw=0
a
u
+
b
v
+
c
w
=
0
for some scalars
a
,
b
,
c
∈
R
a,b,c\in\mathbb{R}
a
,
b
,
c
∈
R
and
a
c
≠
0
ac\ne0
a
c
=
0
, then
span
{
u
,
v
}
=
span
{
v
,
w
}
\operatorname{span}\{u,v\}=\operatorname{span}\{v,w\}
span
{
u
,
v
}
=
span
{
v
,
w
}
. Statement 1:
P
P
P
is true, but
Q
Q
Q
is false. Statement 2:
Q
Q
Q
is true, but
P
P
P
is false. Statement 3: Both
P
P
P
and
Q
Q
Q
are true. Statement 4: Both
P
P
P
and
Q
Q
Q
are false. Which statement is correct? Enter its serial number.
Numerical
MEDIUM
2 marks
16 October 2022
24
Which of the following options is true?
Comprehension
EASY
2 marks
16 October 2022
25
If addition and scalar multiplication on
V
=
R
2
V=\mathbb{R}^2
V
=
R
2
are defined by
(
x
1
,
y
1
)
+
(
x
2
,
y
2
)
=
(
x
1
+
x
2
,
y
1
+
y
2
)
(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2)
(
x
1
,
y
1
)
+
(
x
2
,
y
2
)
=
(
x
1
+
x
2
,
y
1
+
y
2
)
and
c
(
x
,
y
)
=
(
x
,
c
y
)
c(x,y)=(x,cy)
c
(
x
,
y
)
=
(
x
,
cy
)
, consider the statements: 1. There exists a zero vector
0
∈
V
0\in V
0
∈
V
such that
0
+
v
=
v
0+v=v
0
+
v
=
v
for all
v
∈
V
v\in V
v
∈
V
. 2. There exists a vector
v
′
∈
V
v'\in V
v
′
∈
V
such that
v
′
+
v
=
v
+
v
′
=
0
v'+v=v+v'=0
v
′
+
v
=
v
+
v
′
=
0
for all
v
∈
V
v\in V
v
∈
V
. 3. For each
v
∈
V
v\in V
v
∈
V
,
1
v
=
v
1v=v
1
v
=
v
. 4. For each
v
∈
V
v\in V
v
∈
V
and
a
,
b
∈
R
a,b\in\mathbb{R}
a
,
b
∈
R
,
(
a
+
b
)
v
=
a
v
+
b
v
(a+b)v=av+bv
(
a
+
b
)
v
=
a
v
+
b
v
. 5. For each
a
∈
R
a\in\mathbb{R}
a
∈
R
and
v
1
,
v
2
∈
V
v_1,v_2\in V
v
1
,
v
2
∈
V
,
a
(
v
1
+
v
2
)
=
a
v
1
+
a
v
2
a(v_1+v_2)=av_1+av_2
a
(
v
1
+
v
2
)
=
a
v
1
+
a
v
2
. 6. For each
v
∈
V
v\in V
v
∈
V
and
a
,
b
∈
R
a,b\in\mathbb{R}
a
,
b
∈
R
,
(
a
b
)
v
=
a
(
b
v
)
(ab)v=a(bv)
(
ab
)
v
=
a
(
b
v
)
. Which statement is not true? Enter the serial number.
Numerical
MEDIUM
4 marks
05 June 2022
26
Which of the following subsets are not vector subspaces of
R
3
\mathbb{R}^3
R
3
with respect to usual addition and scalar multiplication? 1)
W
=
{
(
x
,
y
,
z
)
:
x
,
y
,
z
∈
R
,
x
2
=
z
2
}
W=\{(x,y,z):x,y,z\in\mathbb{R},\ x^2=z^2\}
W
=
{(
x
,
y
,
z
)
:
x
,
y
,
z
∈
R
,
x
2
=
z
2
}
2)
W
=
{
(
x
,
y
,
z
)
:
x
,
y
,
z
∈
R
,
x
=
z
}
W=\{(x,y,z):x,y,z\in\mathbb{R},\ x=z\}
W
=
{(
x
,
y
,
z
)
:
x
,
y
,
z
∈
R
,
x
=
z
}
3)
W
=
{
(
x
,
y
,
z
)
:
x
,
y
,
z
∈
R
,
x
=
y
+
z
}
W=\{(x,y,z):x,y,z\in\mathbb{R},\ x=y+z\}
W
=
{(
x
,
y
,
z
)
:
x
,
y
,
z
∈
R
,
x
=
y
+
z
}
4)
W
=
{
(
x
,
y
,
z
)
:
x
,
y
,
z
∈
R
,
(
x
+
1
)
+
(
y
−
1
)
+
z
=
0
}
W=\{(x,y,z):x,y,z\in\mathbb{R},\ (x+1)+(y-1)+z=0\}
W
=
{(
x
,
y
,
z
)
:
x
,
y
,
z
∈
R
,
(
x
+
1
)
+
(
y
−
1
)
+
z
=
0
}
. Enter the serial number of the subset which is not a vector subspace.
Numerical
MEDIUM
4 marks
05 June 2022
27
Choose the set of correct options.
Multiple correct
MEDIUM
4 marks
05 June 2022
28
If
S
S
S
is a non-empty linearly independent subset of
R
3
\mathbb{R}^3
R
3
, what can be the maximum possible cardinality of
S
S
S
?
Comprehension
EASY
2 marks
05 June 2022
29
If
S
S
S
is a non-empty linearly dependent subset of
R
3
\mathbb{R}^3
R
3
, what can be the minimum possible cardinality of
S
S
S
?
Comprehension
EASY
2 marks
05 June 2022
30
The conditions that need to be checked to identify a subspace
W
W
W
of a vector space
V
V
V
are ______. Enter
2
2
2
best possible options as serial numbers in increasing order without commas or spaces.
Comprehension
EASY
2 marks
05 June 2022
Showing 30 questions.
Mathematics 2 > Week 3
All PYQs
Topic Wise PYQs
Start Weekly Test
Type:
All
Difficulty:
All
Year:
All
30Q
01
Which of the following subsets are subspaces of
R
3
\mathbb{R}^3
R
3
under the usual addition and scalar multiplication?
Multiple correct
MEDIUM
4 marks
15 March 2026
02
Suppose the vectors
(
−
1
,
c
,
2
)
(-1,c,2)
(
−
1
,
c
,
2
)
,
(
0
,
1
,
3
)
(0,1,3)
(
0
,
1
,
3
)
, and
(
−
2
,
0
,
1
)
(-2,0,1)
(
−
2
,
0
,
1
)
in
R
3
\mathbb{R}^3
R
3
are linearly dependent. Find the value of
7
c
7c
7
c
.
Numerical
MEDIUM
4 marks
15 March 2026
03
Which of the following options is correct?
Comprehension
MEDIUM
4 marks
15 March 2026
04
Choose all the correct statements from the following.
Multiple correct
MEDIUM
6 marks
26 October 2025
05
The vectors
v
1
v_1
v
1
and
v
2
v_2
v
2
are linearly dependent vectors for some value of
a
a
a
.
Comprehension
EASY
3 marks
26 October 2025
06
Each of the options below contains a set
V
V
V
along with an addition operation
+
:
V
×
V
→
V
+:V\times V\to V
+
:
V
×
V
→
V
and a scalar multiplication operation
⋅
:
R
×
V
→
V
\cdot:\mathbb{R}\times V\to V
⋅
:
R
×
V
→
V
. Choose the options for which there is an additive identity in
V
V
V
with respect to
+
+
+
.
Multiple correct
HARD
5 marks
13 July 2025
07
Find the number of values of
k
k
k
for which the given three vectors are linearly dependent in
R
3
\mathbb{R}^3
R
3
.
Comprehension
MEDIUM
4 marks
13 July 2025
08
Find the largest value of
k
k
k
for which the given three vectors are linearly dependent in
R
3
\mathbb{R}^3
R
3
.
Comprehension
EASY
2 marks
13 July 2025
09
The vectors
v
1
v_1
v
1
and
v
2
v_2
v
2
are linearly dependent vectors for some value of
a
a
a
.
Comprehension
EASY
3 marks
23 February 2025
10
Which of the following options represents
W
1
∩
W
2
W_1\cap W_2
W
1
∩
W
2
? One or more options may be correct.
Comprehension
MEDIUM
4 marks
23 February 2025
11
Which of the following option is true?
Comprehension
EASY
2 marks
23 February 2025
12
B
=
{
v
1
,
v
2
,
v
3
}
B=\{v_1,v_2,v_3\}
B
=
{
v
1
,
v
2
,
v
3
}
is a linearly independent subset of
R
3
\mathbb{R}^3
R
3
. Select all linearly independent subsets of
R
3
\mathbb{R}^3
R
3
from the following.
Multiple correct
MEDIUM
3 marks
07 July 2024
13
Is the set
V
V
V
closed under addition?
Comprehension
EASY
1 marks
25 February 2024
14
Is the set
V
V
V
closed under scalar multiplication?
Comprehension
EASY
1 marks
25 February 2024
15
Which of the following is/are correct?
Comprehension
MEDIUM
2 marks
25 February 2024
16
W
1
=
{
[
a
0
0
0
b
0
0
0
c
]
:
a
,
b
,
c
∈
R
such that
a
+
b
+
c
=
1
}
W_1=\left\{\begin{bmatrix}a&0&0\\0&b&0\\0&0&c\end{bmatrix}:a,b,c\in\mathbb{R}\text{ such that }a+b+c=1\right\}
W
1
=
⎩
⎨
⎧
a
0
0
0
b
0
0
0
c
:
a
,
b
,
c
∈
R
such that
a
+
b
+
c
=
1
⎭
⎬
⎫
. If
W
1
W_1
W
1
is a subspace, find the dimension; else write the answer as
0
0
0
.
Comprehension
EASY
1 marks
25 February 2024
17
Unless otherwise stated, assume that we consider the usual vector addition and scalar multiplication. Choose the correct option(s) from the following statements.
Multiple correct
HARD
4 marks
29 October 2023
18
Let
v
1
,
v
2
v_1,v_2
v
1
,
v
2
and
v
3
v_3
v
3
be the column vectors of
A
A
A
and let
u
1
,
u
2
u_1,u_2
u
1
,
u
2
and
u
3
u_3
u
3
be the column vectors of
B
B
B
. Choose the correct option(s) from the following statements.
Comprehension
MEDIUM
3 marks
29 October 2023
19
Which of the following subsets of
R
2
\mathbb{R}^2
R
2
is/are vector spaces with respect to usual addition and usual scalar multiplication?
Multiple correct
MEDIUM
2 marks
16 July 2023
20
Select the true statement(s).
Multiple correct
EASY
2 marks
16 July 2023
21
Consider the set
W
=
{
A
∈
M
n
(
R
)
:
det
(
A
T
)
=
0
}
W=\{A\in M_n(\mathbb{R}):\det(A^T)=0\}
W
=
{
A
∈
M
n
(
R
)
:
det
(
A
T
)
=
0
}
with the usual addition and usual scalar multiplication of matrices. Which of the following is/are true?
Multiple correct
MEDIUM
2 marks
16 July 2023
22
If addition and scalar multiplication on
V
=
R
2
V=\mathbb{R}^2
V
=
R
2
are defined as follows: addition:
(
x
1
,
y
1
)
+
(
x
2
,
y
2
)
=
(
0
,
0
)
(x_1,y_1)+(x_2,y_2)=(0,0)
(
x
1
,
y
1
)
+
(
x
2
,
y
2
)
=
(
0
,
0
)
for
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
∈
V
(x_1,y_1),(x_2,y_2)\in V
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
∈
V
; scalar multiplication:
c
(
x
,
y
)
=
(
0
,
0
)
c(x,y)=(0,0)
c
(
x
,
y
)
=
(
0
,
0
)
for
(
x
,
y
)
∈
V
(x,y)\in V
(
x
,
y
)
∈
V
and
c
∈
R
c\in\mathbb{R}
c
∈
R
. Consider the statements: 1. There exists an element
0
0
0
in
V
V
V
such that
0
+
v
=
v
0+v=v
0
+
v
=
v
for all
v
∈
V
v\in V
v
∈
V
. 2. For each
v
∈
V
v\in V
v
∈
V
and
a
,
b
∈
R
a,b\in\mathbb{R}
a
,
b
∈
R
,
(
a
+
b
)
v
=
a
v
+
b
v
(a+b)v=av+bv
(
a
+
b
)
v
=
a
v
+
b
v
. 3. For each scalar
a
∈
R
a\in\mathbb{R}
a
∈
R
and
v
1
,
v
2
∈
V
v_1,v_2\in V
v
1
,
v
2
∈
V
,
a
(
v
1
+
v
2
)
=
a
v
1
+
a
v
2
a(v_1+v_2)=av_1+av_2
a
(
v
1
+
v
2
)
=
a
v
1
+
a
v
2
. 4. For each
v
∈
V
v\in V
v
∈
V
and
a
,
b
∈
R
a,b\in\mathbb{R}
a
,
b
∈
R
,
(
a
b
)
v
=
a
(
b
v
)
(ab)v=a(bv)
(
ab
)
v
=
a
(
b
v
)
. Which statement is not true? Enter the serial number.
Numerical
EASY
2 marks
16 October 2022
23
Consider the statements:
P
P
P
:
V
=
R
2
V=\mathbb{R}^2
V
=
R
2
with addition
(
x
1
,
y
1
)
+
(
x
2
,
y
2
)
=
(
x
1
x
2
,
y
1
y
2
)
(x_1,y_1)+(x_2,y_2)=(x_1x_2,y_1y_2)
(
x
1
,
y
1
)
+
(
x
2
,
y
2
)
=
(
x
1
x
2
,
y
1
y
2
)
and scalar multiplication
c
(
x
,
y
)
=
(
c
x
,
c
y
)
c(x,y)=(cx,cy)
c
(
x
,
y
)
=
(
c
x
,
cy
)
is a vector space.
Q
Q
Q
: Let
V
V
V
be a vector space. If
u
,
v
,
w
∈
V
u,v,w\in V
u
,
v
,
w
∈
V
are such that
a
u
+
b
v
+
c
w
=
0
au+bv+cw=0
a
u
+
b
v
+
c
w
=
0
for some scalars
a
,
b
,
c
∈
R
a,b,c\in\mathbb{R}
a
,
b
,
c
∈
R
and
a
c
≠
0
ac\ne0
a
c
=
0
, then
span
{
u
,
v
}
=
span
{
v
,
w
}
\operatorname{span}\{u,v\}=\operatorname{span}\{v,w\}
span
{
u
,
v
}
=
span
{
v
,
w
}
. Statement 1:
P
P
P
is true, but
Q
Q
Q
is false. Statement 2:
Q
Q
Q
is true, but
P
P
P
is false. Statement 3: Both
P
P
P
and
Q
Q
Q
are true. Statement 4: Both
P
P
P
and
Q
Q
Q
are false. Which statement is correct? Enter its serial number.
Numerical
MEDIUM
2 marks
16 October 2022
24
Which of the following options is true?
Comprehension
EASY
2 marks
16 October 2022
25
If addition and scalar multiplication on
V
=
R
2
V=\mathbb{R}^2
V
=
R
2
are defined by
(
x
1
,
y
1
)
+
(
x
2
,
y
2
)
=
(
x
1
+
x
2
,
y
1
+
y
2
)
(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2)
(
x
1
,
y
1
)
+
(
x
2
,
y
2
)
=
(
x
1
+
x
2
,
y
1
+
y
2
)
and
c
(
x
,
y
)
=
(
x
,
c
y
)
c(x,y)=(x,cy)
c
(
x
,
y
)
=
(
x
,
cy
)
, consider the statements: 1. There exists a zero vector
0
∈
V
0\in V
0
∈
V
such that
0
+
v
=
v
0+v=v
0
+
v
=
v
for all
v
∈
V
v\in V
v
∈
V
. 2. There exists a vector
v
′
∈
V
v'\in V
v
′
∈
V
such that
v
′
+
v
=
v
+
v
′
=
0
v'+v=v+v'=0
v
′
+
v
=
v
+
v
′
=
0
for all
v
∈
V
v\in V
v
∈
V
. 3. For each
v
∈
V
v\in V
v
∈
V
,
1
v
=
v
1v=v
1
v
=
v
. 4. For each
v
∈
V
v\in V
v
∈
V
and
a
,
b
∈
R
a,b\in\mathbb{R}
a
,
b
∈
R
,
(
a
+
b
)
v
=
a
v
+
b
v
(a+b)v=av+bv
(
a
+
b
)
v
=
a
v
+
b
v
. 5. For each
a
∈
R
a\in\mathbb{R}
a
∈
R
and
v
1
,
v
2
∈
V
v_1,v_2\in V
v
1
,
v
2
∈
V
,
a
(
v
1
+
v
2
)
=
a
v
1
+
a
v
2
a(v_1+v_2)=av_1+av_2
a
(
v
1
+
v
2
)
=
a
v
1
+
a
v
2
. 6. For each
v
∈
V
v\in V
v
∈
V
and
a
,
b
∈
R
a,b\in\mathbb{R}
a
,
b
∈
R
,
(
a
b
)
v
=
a
(
b
v
)
(ab)v=a(bv)
(
ab
)
v
=
a
(
b
v
)
. Which statement is not true? Enter the serial number.
Numerical
MEDIUM
4 marks
05 June 2022
26
Which of the following subsets are not vector subspaces of
R
3
\mathbb{R}^3
R
3
with respect to usual addition and scalar multiplication? 1)
W
=
{
(
x
,
y
,
z
)
:
x
,
y
,
z
∈
R
,
x
2
=
z
2
}
W=\{(x,y,z):x,y,z\in\mathbb{R},\ x^2=z^2\}
W
=
{(
x
,
y
,
z
)
:
x
,
y
,
z
∈
R
,
x
2
=
z
2
}
2)
W
=
{
(
x
,
y
,
z
)
:
x
,
y
,
z
∈
R
,
x
=
z
}
W=\{(x,y,z):x,y,z\in\mathbb{R},\ x=z\}
W
=
{(
x
,
y
,
z
)
:
x
,
y
,
z
∈
R
,
x
=
z
}
3)
W
=
{
(
x
,
y
,
z
)
:
x
,
y
,
z
∈
R
,
x
=
y
+
z
}
W=\{(x,y,z):x,y,z\in\mathbb{R},\ x=y+z\}
W
=
{(
x
,
y
,
z
)
:
x
,
y
,
z
∈
R
,
x
=
y
+
z
}
4)
W
=
{
(
x
,
y
,
z
)
:
x
,
y
,
z
∈
R
,
(
x
+
1
)
+
(
y
−
1
)
+
z
=
0
}
W=\{(x,y,z):x,y,z\in\mathbb{R},\ (x+1)+(y-1)+z=0\}
W
=
{(
x
,
y
,
z
)
:
x
,
y
,
z
∈
R
,
(
x
+
1
)
+
(
y
−
1
)
+
z
=
0
}
. Enter the serial number of the subset which is not a vector subspace.
Numerical
MEDIUM
4 marks
05 June 2022
27
Choose the set of correct options.
Multiple correct
MEDIUM
4 marks
05 June 2022
28
If
S
S
S
is a non-empty linearly independent subset of
R
3
\mathbb{R}^3
R
3
, what can be the maximum possible cardinality of
S
S
S
?
Comprehension
EASY
2 marks
05 June 2022
29
If
S
S
S
is a non-empty linearly dependent subset of
R
3
\mathbb{R}^3
R
3
, what can be the minimum possible cardinality of
S
S
S
?
Comprehension
EASY
2 marks
05 June 2022
30
The conditions that need to be checked to identify a subspace
W
W
W
of a vector space
V
V
V
are ______. Enter
2
2
2
best possible options as serial numbers in increasing order without commas or spaces.
Comprehension
EASY
2 marks
05 June 2022
Showing 30 questions.
