StudyNp | CSIT Second Semester – 2075 Question for Mathematics II

Tribhuwan University

Institute of Science and Technology

2075

Bachelor Level / First Year / Second Semester / Science

B.Sc in Computer Science and Information Technology (MTH168)

(Mathematics II)

Full Marks: 60

Pass Marks: 24

Time: 3 Hours

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

The figures in the margin indicate full marks.

Section A

Long Answers Questions

Attempt any TWO questions.

[2*10=20]

When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve:

$x + y + z = 4$

$x + 2y + 2z = 2$

$2x + 2y + z = 5$

[10]

What is the condition of a matrix to have an inverse? Find the inverse of the matrix

$A = \begin{bmatrix} 5 & 1 & 2 \\ 1 & 0 & 3 \\ 4 & -3 & 8 \end{bmatrix}$

[10]

Define linearly independent set of vectors with an example. Show that the vectors (1,4,3), (0,3,1) and (3,-5,4) are linearly independent. Do they form a basis? Justify.[10]

Find the least-square solution of Ax = b for

$A = \begin{bmatrix} 1 & 3 & 5 \\ 1 & 1 & 0 \\ 1 & 1 & 2 \\ 1 & 3 & 3 \end{bmatrix},\quad b = \begin{pmatrix} 3 \\ 5 \\ 7 \\ 3 \end{pmatrix}$

[10]

Section B

Short Answers Questions

Attempt any Eight questions.

[8*5=40]

Change into reduce echelon form of the matrix

$$\begin{pmatrix} 0 & 3 & -6 \\ 3 & -1 & 8 \\ 3 & -9 & 12 \end{pmatrix}$$

[5]

Define linear transformation with an example. Is a transformation T(x, y) = (3x + y, 5x + 7y, x+3y) linear? Justify. [5]

Let A and B be matrices. What value(s) of k if any will make AB = BA?

$A = \begin{bmatrix} -1 & -2 \\ 5 & 9 \end{bmatrix},\ B = \begin{bmatrix} 9 & 2 \\ k & -1 \end{bmatrix}$

[5]

Define determinant. Evaluate without expanding:

$\begin{bmatrix} 1 & 5 & -6 \\ -1 & -4 & 4 \\ -2 & -7 & 9 \end{bmatrix}$

[5]

Define subspace of a vector space. Let $ H = {\begin{bmatrix} s \\ t \\ O \end{bmatrix} : s, t \in \mathbb{R}}$, show that H is a subspace of $\mathbb{R}^3$. [5]

10.

Find the dimension of the null space and column space of $$A = \begin{bmatrix} -3 & 6 & -1 & 1 & -7 \\ 1 & -2 & 2 & 3 & -1 \\ 2 & -4 & 5 & 8 & -4 \end{bmatrix}$$

$A = \begin{bmatrix} -3 & 6 & -1 & 1 & -7 \\ 1 & -2 & 2 & 3 & -1 \\ 2 & -4 & 5 & 8 & -4 \end{bmatrix}$

[5]

11.

Find the eigenvalues of the matrix

$\begin{bmatrix} 6 & 3 & -8 \\ 0 & -2 & 0 \\ 1 & 0 & -3 \end{bmatrix}$

[5]

12.

Find LU factorization of the matrix

$\begin{bmatrix} 2 & 5 \\ 6 & -7 \end{bmatrix}$

[5]

13.

Define group. Show that the set of all integers, Z forms group under addition operation. [5]

14.

Define ring with an example. Compute the product in the given ring (-3,5), (2,-4) in $Z_4$ x $Z_{11}$. [5]

15.

State and prove the Pythagorean theorem of two vectors and verify this for u = (1, -1) and v = (1, 1). [5]