### Exam Year

Tribhuvan University

Institute of Science and Technology

2078

Bachelor Level / second-semester / Science

Computer Science and Information Technology( MTH163 )

Mathematics II

Full Marks: 80

Pass Marks: 32

Time: 3 Hours

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

The figures in the margin indicate full marks.

Group A

Attempts any THREE questions

1

Define system of linear equations. When a system of equation is consistent? Determine if the system -2x1 – 3x2 + 4x3 = 5, x2 – 2x3 = 4, x1 + 3x2 – x3 = 2 is consistent.

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 = $$\begin{bmatrix}x_1\\ x_2\end{bmatrix}$$

and define a transformation T:R2 → R2 by T(x) = Ax then

1. find T(v)
2. find x ∈ R2 whose image under T is b
3

Find the LU factorization of $$\begin{bmatrix}2 & 4 & -1 & 5 & -2\\ -4 & -5 & 3 & -8 & 1\\ 2 & -5 & -4 & 1 & 8\\ -6 & 0 & 7 & -3 & 1\end{bmatrix}$$

4

Find a least square solution of the inconsistent system Ax =  b for

A = $$\begin{bmatrix}-1 & 2\\ 2 & -3\\ -1 & 3\end{bmatrix}$$, b = $$\begin{bmatrix}4\\ 2\\ 1\end{bmatrix}$$

Group B

Attempts any EIGHT questions

5

Determine the column of the matrix A are linearly independent, where

$$A = \begin{bmatrix}0 & 1 & 4\\ 1 & 2 & -1\\ 5 & 8 & 0\end{bmatrix}$$

6

When two column vector in R2 are equal? Give an example. Computer u + 3v, u – 2v, where

u = $$\begin{bmatrix}1\\ -3\\ 2\end{bmatrix}$$, v = $$\begin{bmatrix}1\\ -1\\ 3\end{bmatrix}$$

7

Let A = $$\begin{bmatrix}0 & 1\\ -1 & 0\end{bmatrix}$$ and define T:R2 → R2 by T(x) = Ax, find the image under T of

$$u = \begin{bmatrix}1\\ -3\end{bmatrix}$$ and $$v = \begin{bmatrix}1\\ 5\end{bmatrix}$$

8

Find the eigen value of $$\begin{bmatrix}3 & 6 & -8\\ 0 & 0 & 6\\ 0 & 0 & 2\end{bmatrix}$$

9

Define null space of a matrix A. Let

A = $$\begin{bmatrix}-1 & -3 & 2\\ -5 & -9 & 1\end{bmatrix}$$, and v = $$\begin{bmatrix}5\\ -3\\ -2\end{bmatrix}$$

Then show that v is in the null A

10

Verify that 1k, (-2)k, 3k are linearly independent signals.

11

If A = $$\begin{bmatrix}7 & 2\\ -4 & 1\end{bmatrix}$$. find a formula for An, where A = PDP-1

P = $$\begin{bmatrix}1 & 1\\ -1 & -2\end{bmatrix}$$ and D = $$\begin{bmatrix}5 & 0\\ 0 & 3\end{bmatrix}$$

12

Find a unit vector v of u = (1, -2, 2, 3) in the direction of u.

13

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

14

Let an operation * be defined on Q+ by $$a + b = \frac{ab}{2}$$. Then show that Q+ forms a group.

15

Define ring and show that set of real numbers with respect to addition and multiplication operation is a ring.