Questions
Syllabus
Qns. By Syllabus

Semester

Subject

Year

Bachelors Level/First Year/Second Semester/Science csit/second semester/mathematics ii/syllabus wise questions

B.Sc Computer Science and Information Technology

Institute of Science and Technology, TU

Mathematics II (MTH168)

Year Asked: 2080, syllabus wise question

Determinants
1.
Evaluate the determinant of the matrix.

$\begin{bmatrix} 1 & -7 & 8 & 9 & -6 \\ 0 & 2 & -5 & 7 & 3 \\ 0 & 0 & 2 & 4 & -1 \\ 0 & 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & -1 & 0 \end{bmatrix}$
[5]
Eigenvalues and Eigen Vectors
1.
Find the eigenvalue of

$A = \begin{bmatrix} 7 & 3 \\ 3 & -1 \end{bmatrix}$
[5]
Groups and Subgroups
1.
Define group. Show that the set of integers is a group with respect to addition operation. [5]
Linear Equations in Linear Algebra
1.
Define system of linear equations. When a system of equations is consistent? Make echelon form to solve:

$-2a - 3b + 4c = 5$

$b - 2c = 4$

$a + 3b - c = 2$
[10]
Matrix Algebra
1.
Find AB by block multiplication of the matrices.

$A = \begin{bmatrix} 2 & -3 & -1 & 0 & -4 \\ 1 & -5 & -2 & 3 & -1\\ 0 & -4 & -2 & 7 & -1 \end{bmatrix},\ B = \begin{bmatrix} 6 & 4 \\ 2 & -1 \\ -3 & 7 \\ 1 & 3 \\ 5 & -3 \end{bmatrix}$
[10]
2.
Let A What value(s) of k, if any, will make AB = BA?

$A = \begin{bmatrix} 1 & 5 \\ -3 & 1 \end{bmatrix},\ B = \begin{bmatrix} 4 & -5 \\ 3 & k \end{bmatrix}$
[5]
Orthogonality and Least Squares
1.
Find the least square solution of Ax = c where

$A = \begin{bmatrix} 1 & -3 & 3 \\ 1 & 5 & 1 \\ 1 & 7 & 2 \end{bmatrix},\ c = \begin{bmatrix} 5 \\ -3 \\ 5 \end{bmatrix}$
and compute the associated least square error. [10+0]
2.
Find the equation y = $a_0 + a_1 x$ of the least squares line that best fits the data points (2,1), (5,2), (7,3), (8,3). [5]
Rings and Fields
1.
Define ring and show that set of positive integers with respect to addition and multiplication operation is not a ring. [5]
Transformation
1.
Define linear transformation with an example.

$let A = \begin{bmatrix} 1 & -3 \\ 3 & 5 \\ -1 & 7 \end{bmatrix},\ v = \begin{bmatrix} -2 \\ 1 \end{bmatrix},\ b = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix},\ x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix},\ T(x) = Ax$

$a. \text{find } T(v)$

$b. \text{Find } x \in \mathbb{R}^2 \text{ whose image under } T \text{ is } b$
[10]
Vector Space Continued
1.
Define null space of a matrix A. Let then show that v belongs to the null space matrix A.

$A = \begin{bmatrix} 1 & -3 & 2 \\ -5 & 9 & -1 \end{bmatrix},\ v = \begin{bmatrix} 5 \\ 3 \\ 2 \end{bmatrix}$
[5]
Vector Spaces
1.
Determine the column of the matrix A are linearly independent where

$A = \begin{bmatrix} 3 & -3 & 6 \\ 0 & 2 & 4 \\ 0 & 3 & 0 \end{bmatrix}$
[5]
2.
When two column vectors in $R^2$ are equal? Give an example. Compute u + 3v, u - 2v, where

$u = \begin{bmatrix} 1 \\ -3 \\ 2 \end{bmatrix},\quad v = \begin{bmatrix} 1 \\ -1 \\ 3 \end{bmatrix}$
[5]
3.
Show that the solutions of $y_{k+2} - 4y_{k+1} + 3y_k = 0$ are linearly independent. [5]
4.
Prove that the two vectors u and v are perpendicular to each other if and only if the line through u is perpendicular bisector of the line segment from -u to v. [5]