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 -2x_{1} – 3x_{2} + 4x_{3} = 5, x_{2} – 2x_{3} = 4, x_{1} + 3x_{2} – x_{3} = 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:R ^{2} → R^{2} **by T(x) = Ax then

- find T(v)
- find x ∈ R
^{2}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 R^{2} 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:R ^{2} → R^{2}** 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 1^{k}, (-2)^{k}, 3^{k} are linearly independent signals.

11

If A = \(\begin{bmatrix}7 & 2\\ -4 & 1\end{bmatrix}\). find a formula for A^{n}, 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.

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