Questions
Syllabus
Qns. By Syllabus

Semester

Subject

Year

Tribhuwan University

Institute of Science and Technology

2079

Bachelor Level / First Year / Second Semester / Science

B.Sc in Computer Science and Information Technology (MTH168)

(Mathematics II)

Full Marks: 60

Pass Marks: 24

Time: 3 Hours

Candidates are required to give their answers in their own words as for as practicable.

The figures in the margin indicate full marks.

Section A

Long Answers Questions

Attempt any TWO questions.
[2*10=20]
1.
Reduce the system of equations into echelon form and solve:

$x_1 - 2x_2 - x_3 + 3x_4 = 0$

$-2x_1 + 4x_2 + 5x_3 - 5x_4 = 3$

$3x_1 - 6x_2 - 6x_3 + 8x_4 = 2$
[10]
2.
Define linear transformation with an example. Let

$A = \begin{bmatrix} 1 & -3 \\ 3 & 5 \\ -1 & 7 \end{bmatrix},\ v = \begin{bmatrix} 2 \\ -1 \end{bmatrix},\ b = \begin{bmatrix} 3 \\ 2 \\ 4 \end{bmatrix},\ x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$

$\text{and define a transformation } \mathbf{T}: \mathbb{R}^2 \to \mathbb{R}^2 \text{ by } T(x) = Ax \text{ then}$

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

$b.\ \text{find } x \in \mathbb{R}^2 \text{ whose image under } T \text{ is } b$
[10]
3.
The economy whose consumption matrix C is and the final demand is 50 units for manufacturing, 30 units for agriculture and 20 units for service. Find the production level x that will satisfy this demand.

$C = \begin{bmatrix} 0.5 & 0.4 & 0.2 \\ 0.2 & 0.3 & 0.1 \\ 0.1 & 0.1 & 0.3 \end{bmatrix}$
[10]
4.
Find the equation y = $a_0$ + $a_1$ x of the least square line that best fits the data points (0, 1), (1, 1), (1, 1), (2, 2), (3, 2).[10]
Section B

Short Answers Questions

Attempt any Eight questions.
[8*5=40]
5.
When a linear system of equation is consistent? Find the values of h and k for which the system is consistent:

$2x_1 - x_2 = h$

$-6x_1 + 3x_2 = k$
[5]
6.
Determine the column of the matrix A are linearly independent, where

$A = \begin{bmatrix} -2 & 8 & -1 \\ 0 & 0 & 0 \\ 0 & -5 & 3 \end{bmatrix}$
[5]
7.
When two column vector in $\mathbb{R}^2$ are equal? Give an example. Compute $u + 3v$, $u - 2v$, where

$u = \begin{bmatrix} 1 \\ -3 \\ 2 \end{bmatrix},\ v = \begin{bmatrix} 1 \\ -1 \\ 3 \end{bmatrix}$
[5]
8.
The column of $I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ are $e_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $e_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$. Suppose T is a linear transformation from $\mathbb{R}^2$ into $\mathbb{R}^3$ such that

$T(e_1) = \begin{bmatrix} 5 \\ 1 \\ -2 \end{bmatrix},\ T(e_2) = \begin{bmatrix} 0 \\ -1 \\ 8 \end{bmatrix}$
find a formula for the image of an arbitrary $x$ in $\mathbb{R}^2$. That is, find $T(x)$ for $x$ in $\mathbb{R}^2$. [2.5+2.5]
9.
Find the eigenvalues of the matrix

$\begin{bmatrix} 6 & 3 & -8 \\ 0 & -2 & 0 \\ 1 & 0 & -3 \end{bmatrix}$
[5]
10.
Define null space of a matrix A. show that v is null of A.

$A = \begin{bmatrix} -1 & -3 & 2 \\ 5 & -9 & 1 \end{bmatrix},\ v = \begin{bmatrix} 5 \\ -3 \\ -2 \end{bmatrix}$
[5]
11.
Verify that $1k, -2^k, 3^k$ are linearly independent signals. [5]
12.
Evaluate the determinant of the matrix

$\begin{bmatrix} 5 & -7 & 2 & 2 \\ 0 & 3 & 0 & -4 \\ -5 & -8 & 0 & 3 \\ 0 & 5 & 0 & -6 \end{bmatrix}$
[5]
13.
Define unit vector. Find a unit vector of u = (0, -2, 2, -3) in the direction of u. [5]
14.
Define group. Show that the set of integers is not a group with respect to subtraction operation. [5]
15.
Define ring. Show that the set of positive integers with respect to addition and multiplication operation is not a ring. [5]