Tribhuvan University

Institute of Science and Technology

2079

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**

**Attempt any THREE questions (3 x 10 = 30).**

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

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

The economy whose consumption matix C is

C = \(\begin{bmatrix}0.5 & 0.4 & 0.2\\ 0.2 & 0.3 & 0.1\\ 0.1 & 0.1 & 0.3\end{bmatrix}\)

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.

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).

**Group B**

**Attempt any TEN questions (10 x 5 = 50).**

5

When a linear system of equation is consistent? Find the values of h and k for which the system: 2x_{1} – x_{2} = h; -6x_{1} + 3x_{2} = k is consistent?

6

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

A = \(\begin{bmatrix}-2 & 8 & -1\\ 0 & 0 & 0\\ 0 & -5 & 3\end{bmatrix}\)

7

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 = \(\begin{bmatrix}1\\ -1\\ 3\end{bmatrix}\)

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 R^{2} into R^{3} such that

T(e_{1}) = \(\begin{bmatrix}5\\ 1\\ -2\end{bmatrix}\) and T(e_{2}) = \(\begin{bmatrix}0\\ -1\\ 8\end{bmatrix}\)

find a formula for the image of an arbitrary x in R^{2}. That is, find T(x) for x in R^{2}.

9

Find the eigenvalues of the matrix \(\begin{pmatrix}6 & 3 & -8\\ 0 & -2 & 0\\ 1 & 0 & -3\end{pmatrix}\)

10

Define null space of a matrix A. If

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 null of A.

11

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

12

Evaluate the determinant of the matrix

\(\begin{bmatrix}5 & -7 & 2 & 2\\ 0 & 3 & 0 & -4\\ -5 & -8 & 0 & 3\\ 0 & 5 & 0 & -6\end{bmatrix}\)

13

Define unit vector. Find a unit vector v of u = (0, -2, 2, -3) in the direction of u.

14

Define group. Show that the set of integers is not a group with respect to subtraction operation.

15

Define ring. Show that set of positive integers with respect to addition and multiplication operation is not a ring.

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