Questions
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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: 2080.1, syllabus wise question

Determinants
1.
Compute Det of A where

$A = \begin{bmatrix} 2 & -8 & 6 & 8 \\ 3 & -9 & 5 & 10 \\ -3 & 0 & 1 & -2 \\ 1 & -4 & 0 & 6 \end{bmatrix}$
[5]
Eigenvalues and Eigen Vectors
1.
Is \( \begin{bmatrix} 3 \\ 2 \end{bmatrix} \) an eigen vector of \( \begin{bmatrix} 5 & -3 \\ -4 & 9 \end{bmatrix} \)? If so, find eigenvalue. [5]
Groups and Subgroups
1.
Define group. Show that (Z, .) doesn't form a group. [5]
Linear Equations in Linear Algebra
1.
What is a system of linear equations? When the system is consistent? Find the condition on g, h, k that makes the system consistent.

$x_1 - 4x_2 + 7x_3 = g$

$3x_2 - 5x_3 = h$

$-2x_1 + 5x_2 - 9x_3 = k$
[10]
Matrix Algebra
1.
Find LU Factorization. Given the matrix:

$\begin{bmatrix} 2 & 3 & 4 \\ 4 & 5 & 10 \\ 4 & 8 & 2 \end{bmatrix}$
[5]
Orthogonality and Least Squares
1.
Find the least square solution of Ax = b where and compute the associated least square error.

$A = \begin{bmatrix} 1 & -3 & -3 \\ 1 & 5 & 1 \\ 1 & 7 & 2 \end{bmatrix},\ b = \begin{bmatrix} 5 \\ -3 \\ -5 \end{bmatrix}$
[10]
Rings and Fields
1.
Show that every field is an integral domain. [5]
Transformation
1.
define a transformation $T : \mathbb{R}^3 \to \mathbb{R}^2 \text{ by } T(x) = Ax$ then

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

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

$b. \text{Find } x \in \mathbb{R}^3 \text{ whose image under } T \text{ is } b$

$c. \text{Is } x \text{ unique?}$
[10]
Vector Space Continued
1.
Find the basis and dimension of Null A where A =

$A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 4 & 7 & 8 \end{bmatrix}$
[5]
Vector Spaces
1.
Are vectors $v_1$, $v_2$, and $v_3$ linearly independent? Justify.

$v_1 = \begin{bmatrix} 1 \\ 4 \\ 0 \end{bmatrix},\ v_2 = \begin{bmatrix} 10 \\ 2 \\ 1 \end{bmatrix},\ v_3 = \begin{bmatrix} -5 \\ 0 \\ 6 \end{bmatrix}$
[5]
2.
Show that $H = \{(a-3b, b-a, a, b) : a, b \in \mathbb{R}\}$ is a subspace of $\mathbb{R}^4$. [5]
3.
Let u = (1, -2, 2, 0). Find a unit vector of v in the same direction of u. [5]