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2024 V2 100102

B. Tech. Ist Semester Examination, 2024

Time 03 Hours
Full Marks 70
Instructions:
  • The marks are indicated in the right-hand margin.
  • There are NINE questions in this paper.
  • Attempt FIVE questions in all.
  • Question No. 1 is compulsory.

Q.1 Choose the correct option the following (Any seven question only):

Q1.1

Which operation is not an elementary row operation?

a)

Swapping two rows

b)

Adding a multiple of one row to another

c)

Multiplying a row by zero.

d)

Multiplying a row by a non-zero constant

Q1.2

The rank of a matrix is:

a)

Number of non-zero rows in echelon form

b)

Number of columns

c)

Number of rows

d)

None above

Q1.3

The inverse of a matrix using Gauss-Jordan:

a)

only determinant calculation

b)

Reducing matrix to diagonal form

c)

Augmenting with identity matrix and applying row operations

d)

None of the above

Q1.4

A matrix is diagonalizable:

a)

It has repeated eigenvalues only

b)

It is symmetric

c)

It has a full set of linearly independent eigenvectors

d)

It is square

Q1.5

A matrix is similar to a diagonal matrix if:

a)

It has complex entries

b)

It has n linearly independent eigenvectors

c)

It is invertible

d)

None of the above

Q1.6

A function is Riemann integrable if:

a)

It is continuous

b)

It has infinite discontinuities

c)

It is discontinuous everywhere

d)

None of the above

Q1.7

Jacobian is:

a)

A scalar

b)

A vector

c)

A matrix of partial derivatives

d)

A function

Q1.8

Taylor's expansion for multivariable function includes:

a)

Only first order terms

b)

Only second order

c)

All orders

d)

Only constant term

Q1.9

Change of variables in double integral involves:

a)

Matrix algebra

b)

Laplace transforms

c)

Jacobian

d)

Only constant term

Q1.10

Triple integral over a region gives:

a)

Area

b)

Length

c)

Volume

d)

Angle

Q.2 Solve both questions :

Q2.1

What is the difference between a vector space and a subspace? Illustrate with examples.

Q2.2

Define and explain rank, row space, and column space of a matrix with examples.

Q.3 Solve both questions :

Q3.1

Explain the Cayley-Hamilton Theorem and verify it for a 2x2 matrix.

Q3.2

Define and compute the Jacobian for transformation x=u+vx = u + v of u−vu - v.

Q.4 Solve both questions :

Q4.1

Explain the Gauss-Jordan method for finding the inverse of a matrix with an example.

Q4.2

Define Hermitian, Skew-Hermitian, and Unitary matrices. Provide examples for each.

Q.5 Solve both questions :

Q5.1

Explain and prove the Rank-Nullity theorem.

Q5.2

Describe the method of finding eigenvalues and eigenvectors of a matrix.

Q.6 Solve both questions :

Q6.1

State and prove Rolle's Theorem with a graphical example.

Q6.2

Define Beta and Gamma functions. Derive the relation between them.

Q.7 Solve both questions :

Q7.1

Derive the Taylor series for a function of two variables.

Q7.2

Solve a maxima-ratum problem using second derivative test.

Q.8 Solve both questions :

Q8.1

Evaluate a double integral to find area under a curve.

Q8.2

Convert a double integral into polar coordinates and evaluate.

Q.9 Write short notes on any two of the following:

2024 V3 100102

B. Tech. Ist Semester Examination, 2024

Time 03 Hours
Full Marks 70
Instructions:
  • The marks are indicated in the right-hand margin.
  • There are NINE questions in this paper.
  • Attempt FIVE questions in all.
  • Question No. 1 is compulsory.

Q.1 Choose the correct option the following (Any seven question only):

Q1.1

Which operation is not an elementary row operation?

a)

Swapping two rows

b)

Adding a multiple of one row to another

c)

Multiplying a row by zero.

d)

Multiplying a row by a non-zero constant

Q1.2

The rank of a matrix is:

a)

Number of non-zero rows in echelon form

b)

Number of columns

c)

Number of rows

d)

None above

Q1.3

The inverse of a matrix using Gauss-Jordan:

a)

only determinant calculation

b)

Reducing matrix to diagonal form

c)

Augmenting with identity matrix and applying row operations

d)

None of the above

Q1.4

A matrix is diagonalizable:

a)

It has repeated eigenvalues only

b)

It is symmetric

c)

It has a full set of linearly independent eigenvectors

d)

It is square

Q1.5

A matrix is similar to a diagonal matrix if:

a)

It has complex entries

b)

It has n linearly independent eigenvectors

c)

It is invertible

d)

None of the above

Q1.6

A function is Riemann integrable if:

a)

It is continuous

b)

It has infinite discontinuities

c)

It is discontinuous everywhere

d)

None of the above

Q1.7

Jacobian is:

a)

A scalar

b)

A vector

c)

A matrix of partial derivatives

d)

A function

Q1.8

Taylor's expansion for multivariable function includes:

a)

Only first order terms

b)

Only second order

c)

All orders

d)

Only constant term

Q1.9

Change of variables in double integral involves:

a)

Matrix algebra

b)

Laplace transforms

c)

Jacobian

d)

Only constant term

Q1.10

Triple integral over a region gives:

a)

Area

b)

Length

c)

Volume

d)

Angle

Q.2 Solve both questions :

Q2.1

What is the difference between a vector space and a subspace? Illustrate with examples.

Q2.2

Define and explain rank, row space, and column space of a matrix with examples.

Q.3 Solve both questions :

Q3.1

Explain the Cayley-Hamilton Theorem and verify it for a 2x2 matrix.

Q3.2

Define and compute the Jacobian for transformation $ x = u + v $ of $ u - v $.

Q.4 Solve both questions :

Q4.1

Explain the Gauss-Jordan method for finding the inverse of a matrix with an example.

Q4.2

Define Hermitian, Skew-Hermitian, and Unitary matrices. Provide examples for each.

Q.5 Solve both questions :

Q5.1

Explain and prove the Rank-Nullity theorem.

Q5.2

Describe the method of finding eigenvalues and eigenvectors of a matrix.

Q.6 Solve both questions :

Q6.1

State and prove Rolle's Theorem with a graphical example.

Q6.2

Define Beta and Gamma functions. Derive the relation between them.

Q.7 Solve both questions :

Q7.1

Derive the Taylor series for a function of two variables.

Q7.2

Solve a maxima-ratum problem using second derivative test.

Q.8 Solve both questions :

Q8.1

Evaluate a double integral to find area under a curve.

Q8.2

Convert a double integral into polar coordinates and evaluate.

Q.9 Write short notes on any two of the following:

Q9.1
a)

Properties of Eigen vectors

b)

Riemann Integration & Riemann Sum

c)

Application to area and volume using double and triple integral

d)

Scalar and vector fields

2024 V4 100102

B. Tech. Ist Semester Examination, 2024

Time 03 Hours
Full Marks 70
Instructions:
  • The marks are indicated in the right-hand margin.
  • There are NINE questions in this paper.
  • Attempt FIVE questions in all.
  • Question No. 1 is compulsory.

Q.1 Choose the correct option the following (Any seven question only):

Q1.1

Which operation is not an elementary row operation?

a)

Swapping two rows

b)

Adding a multiple of one row to another

c)

Multiplying a row by zero.

d)

Multiplying a row by a non-zero constant

Q1.2

The rank of a matrix is:

a)

Number of non-zero rows in echelon form

b)

Number of columns

c)

Number of rows

d)

None above

Q1.3

The inverse of a matrix using Gauss-Jordan:

a)

only determinant calculation

b)

Reducing matrix to diagonal form

c)

Augmenting with identity matrix and applying row operations

d)

None of the above

Q1.4

A matrix is diagonalizable:

a)

It has repeated eigenvalues only

b)

It is symmetric

c)

It has a full set of linearly independent eigenvectors

d)

It is square

Q1.5

A matrix is similar to a diagonal matrix if:

a)

It has complex entries

b)

It has n linearly independent eigenvectors

c)

It is invertible

d)

None of the above

Q1.6

A function is Riemann integrable if:

a)

It is continuous

b)

It has infinite discontinuities

c)

It is discontinuous everywhere

d)

None of the above

Q1.7

Jacobian is:

a)

A scalar

b)

A vector

c)

A matrix of partial derivatives

d)

A function

Q1.8

Taylor's expansion for multivariable function includes:

a)

Only first order terms

b)

Only second order

c)

All orders

d)

Only constant term

Q1.9

Change of variables in double integral involves:

a)

Matrix algebra

b)

Laplace transforms

c)

Jacobian

d)

Only constant term

Q1.10

Triple integral over a region gives:

a)

Area

b)

Length

c)

Volume

d)

Angle

Q.2 Solve both questions :

Q2.1

What is the difference between a vector space and a subspace? Illustrate with examples.

Q2.2

Define and explain rank, row space, and column space of a matrix with examples.

Q.3 Solve both questions :

Q3.1

Explain the Cayley-Hamilton Theorem and verify it for a 2x2 matrix.

Q3.2

Define and compute the Jacobian for transformation x=u+vx = u + v of u−vu - v.

Q.4 Solve both questions :

Q4.1

Explain the Gauss-Jordan method for finding the inverse of a matrix with an example.

Q4.2

Define Hermitian, Skew-Hermitian, and Unitary matrices. Provide examples for each.

Q.5 Solve both questions :

Q5.1

Explain and prove the Rank-Nullity theorem.

Q5.2

Describe the method of finding eigenvalues and eigenvectors of a matrix.

Q.6 Solve both questions :

Q6.1

State and prove Rolle's Theorem with a graphical example.

Q6.2

Define Beta and Gamma functions. Derive the relation between them.

Q.7 Solve both questions :

Q7.1

Derive the Taylor series for a function of two variables.

Q7.2

Solve a maxima-ratum problem using second derivative test.

Q.8 Solve both questions :

Q8.1

Evaluate a double integral to find area under a curve.

Q8.2

Convert a double integral into polar coordinates and evaluate.

Q.9 Write short notes on any two of the following:

Q9.1
  • Properties of Eigen vectors
  • Riemann Integration & Riemann Sum
  • Application to area and volume using double and triple integral
  • Scalar and vector fields
a)

Properties of Eigen vectors

b)

Riemann Integration & Riemann Sum

c)

Application to area and volume using double and triple integral

d)

Scalar and vector fields

2024 V6 100102

B. Tech. Ist Semester Examination, 2024

Time 03 Hours
Full Marks 70
Instructions:
  • The marks are indicated in the right-hand margin.
  • There are NINE questions in this paper.
  • Attempt FIVE questions in all.
  • Question No. 1 is compulsory.

Q.1 Choose the correct option the following (Any seven question only):

Q1.1

Which operation is not an elementary row operation?

a)

Swapping two rows

b)

Adding a multiple of one row to another

c)

Multiplying a row by zero.

d)

Multiplying a row by a non-zero constant

Q1.2

The rank of a matrix is:

a)

Number of non-zero rows in echelon form

b)

Number of columns

c)

Number of rows

d)

None above

Q1.3

The inverse of a matrix using Gauss-Jordan:

a)

only determinant calculation

b)

Reducing matrix to diagonal form

c)

Augmenting with identity matrix and applying row operations

d)

None of the above

Q1.4

A matrix is diagonalizable:

a)

It has repeated eigenvalues only

b)

It is symmetric

c)

It has a full set of linearly independent eigenvectors

d)

It is square

Q1.5

A matrix is similar to a diagonal matrix if:

a)

It has complex entries

b)

It has n linearly independent eigenvectors

c)

It is invertible

d)

None of the above

Q1.6

A function is Riemann integrable if:

a)

It is continuous

b)

It has infinite discontinuities

c)

It is discontinuous everywhere

d)

None of the above

Q1.7

Jacobian is:

a)

A scalar

b)

A vector

c)

A matrix of partial derivatives

d)

A function

Q1.8

Taylor's expansion for multivariable function includes:

a)

Only first order terms

b)

Only second order

c)

All orders

d)

Only constant term

Q1.9

Change of variables in double integral involves:

a)

Matrix algebra

b)

Laplace transforms

c)

Jacobian

d)

Only constant term

Q1.10

Triple integral over a region gives:

a)

Area

b)

Length

c)

Volume

d)

Angle

Q.2 Solve both questions :

Q2.1

What is the difference between a vector space and a subspace? Illustrate with examples.

Q2.2

Define and explain rank, row space, and column space of a matrix with examples.

Q.3 Solve both questions :

Q3.1

Explain the Cayley-Hamilton Theorem and verify it for a 2x2 matrix.

Q3.2

Define and compute the Jacobian for transformation x=u+vx = u + v of u−vu - v.

Q.4 Solve both questions :

Q4.1

Explain the Gauss-Jordan method for finding the inverse of a matrix with an example.

Q4.2

Define Hermitian, Skew-Hermitian, and Unitary matrices. Provide examples for each.

Q.5 Solve both questions :

Q5.1

Explain and prove the Rank-Nullity theorem.

Q5.2

Describe the method of finding eigenvalues and eigenvectors of a matrix.

Q.6 Solve both questions :

Q6.1

State and prove Rolle's Theorem with a graphical example.

Q6.2

Define Beta and Gamma functions. Derive the relation between them.

Q.7 Solve both questions :

Q7.1

Derive the Taylor series for a function of two variables.

Q7.2

Solve a maxima-ratum problem using second derivative test.

Q.8 Solve both questions :

Q8.1

Evaluate a double integral to find area under a curve.

Q8.2

Convert a double integral into polar coordinates and evaluate.

Q.9 Write short notes on any two of the following:

Q9.1
  • Properties of Eigen vectors
  • Riemann Integration & Riemann Sum
  • Application to area and volume using double and triple integral
  • Scalar and vector fields
a)

Properties of Eigen vectors

b)

Riemann Integration & Riemann Sum

c)

Application to area and volume using double and triple integral

d)

Scalar and vector fields

2024 V8 100102

B. Tech. Ist Semester Examination, 2024

Time 03 Hours
Full Marks 70
Instructions:
  • The marks are indicated in the right-hand margin.
  • There are NINE questions in this paper.
  • Attempt FIVE questions in all.
  • Question No. 1 is compulsory.

Q.1 Choose the correct option the following (Any seven question only):

Q1.1

Which operation is not an elementary row operation?

a)

Swapping two rows

b)

Adding a multiple of one row to another

c)

Multiplying a row by zero.

d)

Multiplying a row by a non-zero constant

Q1.2

The rank of a matrix is:

a)

Number of non-zero rows in echelon form

b)

Number of columns

c)

Number of rows

d)

None above

Q1.3

The inverse of a matrix using Gauss-Jordan:

a)

only determinant calculation

b)

Reducing matrix to diagonal form

c)

Augmenting with identity matrix and applying row operations

d)

None of the above

Q1.4

A matrix is diagonalizable:

a)

It has repeated eigenvalues only

b)

It is symmetric

c)

It has a full set of linearly independent eigenvectors

d)

It is square

Q1.5

A matrix is similar to a diagonal matrix if:

a)

It has complex entries

b)

It has n linearly independent eigenvectors

c)

It is invertible

d)

None of the above

Q1.6

A function is Riemann integrable if:

a)

It is continuous

b)

It has infinite discontinuities

c)

It is discontinuous everywhere

d)

None of the above

Q1.7

Jacobian is:

a)

A scalar

b)

A vector

c)

A matrix of partial derivatives

d)

A function

Q1.8

Taylor's expansion for multivariable function includes:

a)

Only first order terms

b)

Only second order

c)

All orders

d)

Only constant term

Q1.9

Change of variables in double integral involves:

a)

Matrix algebra

b)

Laplace transforms

c)

Jacobian

d)

Only constant term

Q1.10

Triple integral over a region gives:

a)

Area

b)

Length

c)

Volume

d)

Angle

Q.2 Solve both questions :

Q2.1

What is the difference between a vector space and a subspace? Illustrate with examples.

Q2.2

Define and explain rank, row space, and column space of a matrix with examples.

Q.3 Solve both questions :

Q3.1

Explain the Cayley-Hamilton Theorem and verify it for a 2x2 matrix.

Q3.2

Define and compute the Jacobian for transformation $ x = u + v $ of $ u - v $.

Q.4 Solve both questions :

Q4.1

Explain the Gauss-Jordan method for finding the inverse of a matrix with an example.

Q4.2

Define Hermitian, Skew-Hermitian, and Unitary matrices. Provide examples for each.

Q.5 Solve both questions :

Q5.1

Explain and prove the Rank-Nullity theorem.

Q5.2

Describe the method of finding eigenvalues and eigenvectors of a matrix.

Q.6 Solve both questions :

Q6.1

State and prove Rolle's Theorem with a graphical example.

Q6.2

Define Beta and Gamma functions. Derive the relation between them.

Q.7 Solve both questions :

Q7.1

Derive the Taylor series for a function of two variables.

Q7.2

Solve a maxima-ratum problem using second derivative test.

Q.8 Solve both questions :

Q8.1

Evaluate a double integral to find area under a curve.

Q8.2

Convert a double integral into polar coordinates and evaluate.

Q.9 Write short notes on any two of the following:

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