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:

x1 – 2x2 – x3 + 3x4 = 0

-2x1 + 4x2 + 5x3 – 5x4 = 3

3x1 – 6x2 – 6x3 + 8x4 = 2

2

Define linear transformation of a matrix A.

Let A = \(\begin{bmatrix}1 & -3\\ 3 & 5\\ -1 & 7\end{bmatrix}\), v = \(\begin{bmatrix}-2\\ 1\end{bmatrix}\), c = \(\begin{bmatrix}3\\ 2\\ 4\end{bmatrix}\), v = \(\begin{bmatrix}x_1\\ x_2\end{bmatrix}\)

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

  1. find T(v)
  2. find x ∈ R2 whose image under T is c.
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 = a0 + a1x 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: 2x1 – x2 = h; -6x1 + 3x2 = 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 R2 are equal? Give an example. Computer u + 3v, u – 2v, where

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

8

The column of I2 = \(\begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}\) are (e1) = \(\begin{bmatrix}1\\ 0\end{bmatrix}\), and e2 = \(\begin{bmatrix}0\\ 1\end{bmatrix}\). Suppose T is a linear transformation from R2 into R3 such that

T(e1) = \(\begin{bmatrix}5\\ 1\\ -2\end{bmatrix}\) and T(e2) = \(\begin{bmatrix}0\\ -1\\ 8\end{bmatrix}\)

find a formula for the image of an arbitrary x in R2. That is, find T(x) for x in R2.

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 = \(\begin{bmatrix}5\\ -3\\ -2\end{bmatrix}\)

11

Verify that 1k, (-2)k, 3k 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.