Engineering Mathematics -I - B. Tech. Ist Semester Examination, 2024

Which operation is not an elementary row operation?

1. Swapping two rows
2. Adding a multiple of one row to another
3. Multiplying a row by zero.
4. Multiplying a row by a non-zero constant

The rank of a matrix is:

1. Number of non-zero rows in echelon form
2. Number of columns
3. Number of rows
4. None above

The inverse of a matrix using Gauss-Jordan:

1. only determinant calculation
2. Reducing matrix to diagonal form
3. Augmenting with identity matrix and applying row operations
4. None of the above

A matrix is diagonalizable:

1. It has repeated eigenvalues only
2. It is symmetric
3. It has a full set of linearly independent eigenvectors
4. It is square

A matrix is similar to a diagonal matrix if:

1. It has complex entries
2. It has n linearly independent eigenvectors
3. It is invertible
4. None of the above

A function is Riemann integrable if:

1. It is continuous
2. It has infinite discontinuities
3. It is discontinuous everywhere
4. None of the above

Jacobian is:

1. A scalar
2. A vector
3. A matrix of partial derivatives
4. A function

Taylor's expansion for multivariable function includes:

1. Only first order terms
2. Only second order
3. All orders
4. Only constant term

Change of variables in double integral involves:

1. Matrix algebra
2. Laplace transforms
3. Jacobian
4. Only constant term

Triple integral over a region gives:

1. Area
2. Length
3. Volume
4. Angle

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

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

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

Define and compute the Jacobian for transformation \( x = u + v \) of \( u - v \).

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

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

Explain and prove the Rank-Nullity theorem.

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

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

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

Derive the Taylor series for a function of two variables.

Solve a maxima-ratum problem using second derivative test.

Evaluate a double integral to find area under a curve.

Convert a double integral into polar coordinates and evaluate.