Week 1 Mathematics 2 Questions

Q1

Let

A=\begin{bmatrix}1&a&a^2&a^3\\1&b&b^2&b^3\\1&c&c^2&c^3\\1&d&d^2&d^3\end{bmatrix}

where

a,b,c,d

are distinct real numbers. Which of the following statements are correct?

Multiple correctMARCH 15, 2026

MARCH 15, 2026

Q2

Suppose

a=(1,2)

and

b=(3,-1)

. Let

v=ka+lb

such that

v_1=v_2

and

k+l=5

. Find the value of

k

.

Integer answerMARCH 15, 2026

MARCH 15, 2026

Q3

Let

A

be a square matrix of order

n>1

. Select all the statements which imply that

\det(A)=0

.

Multiple correctOCTOBER 26, 2025

OCTOBER 26, 2025

Q4

If the sum of the two vectors

(1,2)

and

(2,x)

in

\mathbb{R}^2

is a scalar multiple of the vector

(2,1)

, then find the value of

x

.

Integer answerOCTOBER 26, 2025

OCTOBER 26, 2025

Q5

Find the minimum value of

\beta

such that the matrix

\begin{bmatrix}\beta-1&0&-1\\2&\beta+4&2\\\beta+1&\beta+1&\beta+1\end{bmatrix}

is not invertible.

Integer answerOCTOBER 26, 2025

OCTOBER 26, 2025

Q6

There are two kinds of coins, Type A and Type B. Let the unknown monetary values of coins of Type A and Type B be

x_1

and

x_2

, respectively. A collection of

2

coins of Type A and

3

coins of Type B is worth Rs

19

. A collection of

1

coin of Type A and

5

coins of Type B is worth Rs

27

. Use the above information to set up a system of linear equations and find the value of

x_1-x_2

.

Integer answerOCTOBER 26, 2025

OCTOBER 26, 2025

Q7

Choose the matrix which satisfies the equation

\det(A-cI)=c^2-5c+6

.

Single correctJULY 13, 2025

JULY 13, 2025

Q8

Let

A

be a

3\times3

matrix such that

\det(A)=-3

. For which of the options below is the matrix

B

, obtained from

A

by performing the operations in the option, such that

\det(B)=12

?

Multiple correctJULY 13, 2025

JULY 13, 2025

Q9

Let

x_1

be the price of

1

pencil and

x_2

be the price of

1

pen. Choose the correct option.

ComprehensionJULY 13, 2025

JULY 13, 2025

Q10

Suppose Shakti tries to find the price of

1

pencil and

1

pen and hopes to have unique values for the prices. Then

b

should not be equal to

ComprehensionJULY 13, 2025

JULY 13, 2025

Q11

Let

A

be an

m\times n

matrix. Choose the operations that are valid only when

m=n

.

Multiple correctFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q12

Find the number of

2\times2

scalar matrices with determinant equal to

4

.

Integer answerFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q13

If the sum of the two vectors

(1,2)

and

(2,x)

in

\mathbb{R}^2

is a scalar multiple of the vector

(2,1)

, then find the value of

x

.

Integer answerFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q14

Find the minimum value of

\beta

such that the matrix

\begin{bmatrix}\beta-1&0&-1\\2&\beta+4&2\\\beta+1&\beta+1&\beta+1\end{bmatrix}

is not invertible.

Integer answerFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q15

There are two kinds of coins, say of Type A and Type B. Let the unknown monetary values of coins of Type A and Type B be

x_1

and

x_2

, respectively. A collection of

2

coins of Type A and

3

coins of Type B is worth

₹19

. A collection of

1

coin of Type A and

5

coins of Type B is worth

₹27

. Use the above information to set up a system of linear equations and find the value of

x_1-x_2

.

Integer answerFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q16

Which of the following conditions are sufficient to deduce

\det(AB)=0

?

ComprehensionFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q17

For

\alpha=3

and

\beta=0

, find the value of

\det(-B^TA^2B^{-1})

.

ComprehensionFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q18

Choose the correct option(s) from the following:

Multiple correctOCTOBER 27, 2024

OCTOBER 27, 2024

Q19

Find the value of

\alpha

.

ComprehensionOCTOBER 27, 2024

OCTOBER 27, 2024

Q20

Find the value of

\beta

.

ComprehensionOCTOBER 27, 2024

OCTOBER 27, 2024

Q21

Let

A

be a

5\times5

matrix such that

A^T=-A

. Then

\det(A)=

ComprehensionOCTOBER 27, 2024

OCTOBER 27, 2024

Q22

Let

A

be a

2\times2

matrix such that

\det(A)=2

. If

B

is a matrix obtained from

A

by swapping the first and second row, and then multiplying the second row by

-2

, then

\det(B)=

ComprehensionOCTOBER 27, 2024

OCTOBER 27, 2024

Q23

Consider the matrix

A=\begin{bmatrix}1&0&5\\-2&3&-4\\1&4&a\end{bmatrix}

. Find the value of

a

if the system

Ax=0

has infinitely many solutions.

ComprehensionOCTOBER 27, 2024

OCTOBER 27, 2024

Q24

A=(a_{ij})\in M_{5\times5}(\mathbb{R})

is a matrix such that

a_{ij}+a_{ji}=0

for all

1\le i,j\le5

. Find the determinant of

A

.

Integer answerJULY 07, 2024

JULY 07, 2024

Q25

If determinant of

A

is

-2

, find the determinant of

A^T A

.

Integer answerJULY 07, 2024

JULY 07, 2024

Q26

Find the value of

\alpha

.

ComprehensionJULY 07, 2024

JULY 07, 2024

Q27

Find the value of

\beta

.

ComprehensionJULY 07, 2024

JULY 07, 2024

Q28

Find the determinant of the matrix

\begin{bmatrix}0&1&0&0&0\\0&0&0&1&0\\1&0&0&0&0\\0&0&1&0&0\\0&0&0&0&1\end{bmatrix}

.

ComprehensionFEBRUARY 25, 2024

FEBRUARY 25, 2024

Q29

Find the determinant of the matrix

\begin{bmatrix}a&b&0&c\\d&e&5&f\\g&h&4&i\\0&0&-5&0\end{bmatrix}

, given that

\det\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=2

.

ComprehensionFEBRUARY 25, 2024

FEBRUARY 25, 2024

Q30

Find the value of

c

.

ComprehensionFEBRUARY 25, 2024

FEBRUARY 25, 2024

Q31

Find the value of

d

.

ComprehensionFEBRUARY 25, 2024

FEBRUARY 25, 2024

Q32

Does there exist an

a

such that the system has no solution? If yes, find the value of

a

; else write the answer as

100

.

ComprehensionFEBRUARY 25, 2024

FEBRUARY 25, 2024

Q33

Consider the following images representing systems of linear equations. Choose the correct option(s) from the following statements.

Multiple correctOCTOBER 29, 2023

OCTOBER 29, 2023

Q34

Let

A

be a

3\times3

matrix such that

A^T=-A

. Then find

\det(A)

.

ComprehensionOCTOBER 29, 2023

OCTOBER 29, 2023

Q35

Let

A

be a

3\times3

matrix such that

\det(A)=2

. If

B

is a matrix obtained from

A

by swapping the second and third row, and then multiplying the first row by

-3

, then find

\det(B)

.

ComprehensionOCTOBER 29, 2023

OCTOBER 29, 2023

Q36

Let

A=\begin{bmatrix}1&-2\\-3&4\end{bmatrix}

and

\det(A-xI)=x^2-ax+b

. Find

a-b

.

ComprehensionOCTOBER 29, 2023

OCTOBER 29, 2023

Q37

What is the determinant of

B

?

ComprehensionOCTOBER 29, 2023

OCTOBER 29, 2023

Q38

Match the system of linear equations in Column A with their number of solutions in Column B and their geometric representation in Column C. Choose the correct option from the following:

Single correctJULY 16, 2023

JULY 16, 2023

Q39

Which of the following matrices satisfy

A^k=0

for some natural number

k

?

Multiple correctJULY 16, 2023

JULY 16, 2023

Q40

Let

A=\begin{bmatrix}2022&2023&2024\\2022&2021&2022\\2022&2022&2022\end{bmatrix}

. What is the determinant of

\frac{1}{2}A

?

Integer answerJULY 16, 2023

JULY 16, 2023

Q41

Choose the correct option(s).

ComprehensionJULY 16, 2023

JULY 16, 2023

Q42

Choose the set of correct options.

Multiple correctOCTOBER 16, 2022

OCTOBER 16, 2022

Q43

NOTE: Enter your answer to the nearest integer. If

A^4=\frac{\beta}{4}a^4I

, what is the value of

\beta

?

ComprehensionOCTOBER 16, 2022

OCTOBER 16, 2022

Q44

NOTE: Enter your answer to the nearest integer. Find the value of

a+\lambda

for which

\det(A-\lambda I)=0

, where

\lambda

is a real number and

a

is treated as a variable.

ComprehensionOCTOBER 16, 2022

OCTOBER 16, 2022

Q45

If

A+3I=0

, where

A

is a

2\times2

matrix and

I

is the identity matrix of order

2

, then find

\det(A)

.

Integer answerJUNE 05, 2022

JUNE 05, 2022

Q46

If

A=\begin{bmatrix}3&-3\\-3&3\end{bmatrix}

and

A^2=\lambda A

, then find the value of

\lambda

.

Integer answerJUNE 05, 2022

JUNE 05, 2022

Q47

If

\begin{bmatrix}a+4&3b\\8&-6\end{bmatrix}=\begin{bmatrix}2a+2&b+2\\8&a-8b\end{bmatrix}

, then find the value of

a+2b

.

Integer answerJUNE 05, 2022

JUNE 05, 2022

Q48

The matrix representation to find

x_1

,

x_2

, and

x_3

is:

ComprehensionJUNE 05, 2022

JUNE 05, 2022

All questions