### Exam Year

Tribhuvan University

Institute of Science and Technology

2076

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 (3 x 10 = 30)

Attempt any THREE questions.

1

When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve the system of equations: x – 2y = 5, -x + y + 5z = 2, y + z = 0

2

What is the condition of a matrix to have an inverse? Find the inverse of the matrix $$A = \begin{bmatrix} 5 & 1 & 2\\ 1 & 0 & 3\\ 4 & -3 & 8 \end{bmatrix}$$  If it exists.

3

Find the least-square solution of Ax=b for $$A = \begin{bmatrix} 1 & -6\\ 1 & -2\\ 1 & 1\\ 1 & 7 \end{bmatrix}$$ and $$b = \begin{bmatrix}-1 \\ 2\\ 1\\ 6 \end{bmatrix}$$

4

Let T is a linear transformation. Find the standard matrix of T such that

1. T:R2 → R4 by T(e1) = (3, 1, 3, 1) and T(e2) = (-5, 2, 0, 0) where e1 = (1, 0) and e2 = (0, 1);
2. T:R2 → R4 rotates point as the origin through $$\frac{3\pi}{4}$$ radians counter clockwise.
3. T:R2 → R4 Is a vertical shear transformation that maps e1 into e1-2e2 but leaves vector e2 unchanged.

Group B (10 x 5 = 50)

Attempt any TEN questions.

5

For what value of h will y be in span {v1 , v2, v3} if $$v_1 = \begin{bmatrix}1\\ -1\\ -2\end{bmatrix}$$, $$v_2 = \begin{bmatrix}5\\ -4\\ -7\end{bmatrix}$$, $$v_3 = \begin{bmatrix}-3\\ 1\\ 0\end{bmatrix}$$ and $$y = \begin{bmatrix}-4\\ 3\\ h\end{bmatrix}$$

6

Let us define a linear transformation T:R2 → R2 by T(x) = $$\begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}\begin{bmatrix}x_1\\ x_2\end{bmatrix}$$ = $$\begin{bmatrix}-x_2\\ x_1\end{bmatrix}$$. Find the image under T of $$u = \begin{bmatrix}4\\ 1\end{bmatrix}$$, $$v = \begin{bmatrix}2\\ 3\end{bmatrix}$$ and u + v = $$\begin{bmatrix}6\\ 4\end{bmatrix}$$

7

Let $$A = \begin{bmatrix}2 & 5\\ -3 & 1\end{bmatrix}$$ and $$B = \begin{bmatrix}4 & -5\\ 3 & k\end{bmatrix}$$. Determine the value (s) of k if any will make AB = BA.

8

Define determinant. Compute the determinant without expanding $$\begin{bmatrix}-2 & 8 & -9\\ -1 & 7 & 0\\ 1 & -4 & 2\end{bmatrix}$$

9

Define null space . Find the basis for the null space of the matrix $$A = \begin{bmatrix}1 & 2 & 3\\ 2 & 3 & 4\end{bmatrix}$$

10

Let B = {b1, b2} and C = (c1, c2) be bases for a vector  V, and suppose b1 = -c1 + 4c2 and b2 = 5c1 – 3c2. Find the change of coordinate matrix for a vector space and find [x]c for x = 5b1 + 3b2.

11

Find the eigen values of the matrix $$\begin{bmatrix}6 & 5\\ -8 & -6\end{bmatrix}$$

12

Find the QR factorization of the matrix $$\begin{bmatrix}2 & 1\\ 3 & -1\end{bmatrix}$$

13

Define binary operation. Determine whether the binary operation * is associative or commutative or both where * is defined on Q by letting $$x * y = \frac{x+y}{3}$$

14

Show that the ring (Z4, +4, .4) is an integral domain.

15

Find the vector x determined by the coordinate vector $$[x]_{β} = \begin{bmatrix}-4\\ 8\\ 7\end{bmatrix}$$ where $$β = \left \{ \begin{bmatrix}-1\\ 2\\ 0\end{bmatrix}, \begin{bmatrix}3\\ -5\\ 2\end{bmatrix}, \begin{bmatrix}4\\ -7\\ 3\end{bmatrix} \right \}$$