Week 3 Mathematics 2 Questions

Q1

Which of the following subsets are subspaces of

\mathbb{R}^3

under the usual addition and scalar multiplication?

Multiple correctMARCH 15, 2026

MARCH 15, 2026

Q2

Suppose the vectors

(-1,c,2)

,

(0,1,3)

, and

(-2,0,1)

in

\mathbb{R}^3

are linearly dependent. Find the value of

7c

.

Integer answerMARCH 15, 2026

MARCH 15, 2026

Q3

Which of the following options is correct?

ComprehensionMARCH 15, 2026

MARCH 15, 2026

Q4

Choose all the correct statements from the following.

Multiple correctOCTOBER 26, 2025

OCTOBER 26, 2025

Q5

The vectors

v_1

and

v_2

are linearly dependent vectors for some value of

a

.

ComprehensionOCTOBER 26, 2025

OCTOBER 26, 2025

Q6

Each of the options below contains a set

V

along with an addition operation

+:V\times V\to V

and a scalar multiplication operation

\cdot:\mathbb{R}\times V\to V

. Choose the options for which there is an additive identity in

V

with respect to

+

.

Multiple correctJULY 13, 2025

JULY 13, 2025

Q7

Find the number of values of

k

for which the given three vectors are linearly dependent in

\mathbb{R}^3

.

ComprehensionJULY 13, 2025

JULY 13, 2025

Q8

Find the largest value of

k

for which the given three vectors are linearly dependent in

\mathbb{R}^3

.

ComprehensionJULY 13, 2025

JULY 13, 2025

Q9

The vectors

v_1

and

v_2

are linearly dependent vectors for some value of

a

.

ComprehensionFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q10

Which of the following options represents

W_1\cap W_2

? One or more options may be correct.

ComprehensionFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q11

Which of the following option is true?

ComprehensionFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q12

B=\{v_1,v_2,v_3\}

is a linearly independent subset of

\mathbb{R}^3

. Select all linearly independent subsets of

\mathbb{R}^3

from the following.

Multiple correctJULY 07, 2024

JULY 07, 2024

Q13

Is the set

V

closed under addition?

ComprehensionFEBRUARY 25, 2024

FEBRUARY 25, 2024

Q14

Is the set

V

closed under scalar multiplication?

ComprehensionFEBRUARY 25, 2024

FEBRUARY 25, 2024

Q15

Which of the following is/are correct?

ComprehensionFEBRUARY 25, 2024

FEBRUARY 25, 2024

Q16

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\}

. If

W_1

is a subspace, find the dimension; else write the answer as

0

.

ComprehensionFEBRUARY 25, 2024

FEBRUARY 25, 2024

Q17

Let

v_1,v_2

and

v_3

be the column vectors of

A

and let

u_1,u_2

and

u_3

be the column vectors of

B

. Choose the correct option(s) from the following statements.

ComprehensionOCTOBER 29, 2023

OCTOBER 29, 2023

Q18

Unless otherwise stated, assume that we consider the usual vector addition and scalar multiplication. Choose the correct option(s) from the following statements.

Multiple correctOCTOBER 29, 2023

OCTOBER 29, 2023

Q19

Which of the following subsets of

\mathbb{R}^2

is/are vector spaces with respect to usual addition and usual scalar multiplication?

Multiple correctJULY 16, 2023

JULY 16, 2023

Q20

Select the true statement(s).

Multiple correctJULY 16, 2023

JULY 16, 2023

Q21

Consider the set

W=\{A\in M_n(\mathbb{R}):\det(A^T)=0\}

with the usual addition and usual scalar multiplication of matrices. Which of the following is/are true?

Multiple correctJULY 16, 2023

JULY 16, 2023

Q22

If addition and scalar multiplication on

V=\mathbb{R}^2

are defined as follows: addition:

(x_1,y_1)+(x_2,y_2)=(0,0)

for

(x_1,y_1),(x_2,y_2)\in V

; scalar multiplication:

c(x,y)=(0,0)

for

(x,y)\in V

and

c\in\mathbb{R}

. Consider the statements: 1. There exists an element

0

in

V

such that

0+v=v

for all

v\in V

. 2. For each

v\in V

and

a,b\in\mathbb{R}

,

(a+b)v=av+bv

. 3. For each scalar

a\in\mathbb{R}

and

v_1,v_2\in V

,

a(v_1+v_2)=av_1+av_2

. 4. For each

v\in V

and

a,b\in\mathbb{R}

,

(ab)v=a(bv)

. Which statement is not true? Enter the serial number.

Integer answerOCTOBER 16, 2022

OCTOBER 16, 2022

Q23

Consider the statements:

P

:

V=\mathbb{R}^2

with addition

(x_1,y_1)+(x_2,y_2)=(x_1x_2,y_1y_2)

and scalar multiplication

c(x,y)=(cx,cy)

is a vector space.

Q

: Let

V

be a vector space. If

u,v,w\in V

are such that

au+bv+cw=0

for some scalars

a,b,c\in\mathbb{R}

and

ac\ne0

, then

\operatorname{span}\{u,v\}=\operatorname{span}\{v,w\}

. Statement 1:

P

is true, but

Q

is false. Statement 2:

Q

is true, but

P

is false. Statement 3: Both

P

and

Q

are true. Statement 4: Both

P

and

Q

are false. Which statement is correct? Enter its serial number.

Integer answerOCTOBER 16, 2022

OCTOBER 16, 2022

Q24

Which of the following options is true?

ComprehensionOCTOBER 16, 2022

OCTOBER 16, 2022

Q25

If addition and scalar multiplication on

V=\mathbb{R}^2

are defined by

(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2)

and

c(x,y)=(x,cy)

, consider the statements: 1. There exists a zero vector

0\in V

such that

0+v=v

for all

v\in V

. 2. There exists a vector

v'\in V

such that

v'+v=v+v'=0

for all

v\in V

. 3. For each

v\in V

,

1v=v

. 4. For each

v\in V

and

a,b\in\mathbb{R}

,

(a+b)v=av+bv

. 5. For each

a\in\mathbb{R}

and

v_1,v_2\in V

,

a(v_1+v_2)=av_1+av_2

. 6. For each

v\in V

and

a,b\in\mathbb{R}

,

(ab)v=a(bv)

. Which statement is not true? Enter the serial number.

Integer answerJUNE 05, 2022

JUNE 05, 2022

Q26

Which of the following subsets are not vector subspaces of

\mathbb{R}^3

with respect to usual addition and scalar multiplication? 1)

W=\{(x,y,z):x,y,z\in\mathbb{R},\ x^2=z^2\}

2)

W=\{(x,y,z):x,y,z\in\mathbb{R},\ x=z\}

3)

W=\{(x,y,z):x,y,z\in\mathbb{R},\ x=y+z\}

4)

W=\{(x,y,z):x,y,z\in\mathbb{R},\ (x+1)+(y-1)+z=0\}

. Enter the serial number of the subset which is not a vector subspace.

Integer answerJUNE 05, 2022

JUNE 05, 2022

Q27

If

S

is a non-empty linearly independent subset of

\mathbb{R}^3

, what can be the maximum possible cardinality of

S

?

ComprehensionJUNE 05, 2022

JUNE 05, 2022

Q28

If

S

is a non-empty linearly dependent subset of

\mathbb{R}^3

, what can be the minimum possible cardinality of

S

?

ComprehensionJUNE 05, 2022

JUNE 05, 2022

Q29

Choose the set of correct options.

Multiple correctJUNE 05, 2022

JUNE 05, 2022

Q30

The conditions that need to be checked to identify a subspace

W

of a vector space

V

are ______. Enter

2

best possible options as serial numbers in increasing order without commas or spaces.

ComprehensionJUNE 05, 2022

JUNE 05, 2022

All questions