Questions
Syllabus
Qns. By Syllabus

Semester

Subject

Year

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]
1.
When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve:
x+y+z=4x + y + z = 4
x+2y+2z=2x + 2y + 2z = 2
2x+2y+z=52x + 2y + z = 5
[10]
2.
What is the condition of a matrix to have an inverse? Find the inverse of the matrix
A=[5121034−38]A = \begin{bmatrix} 5 & 1 & 2 \\ 1 & 0 & 3 \\ 4 & -3 & 8 \end{bmatrix}
[10]
3.
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]
4.
Find the least-square solution of Ax = b for
A=[135110112133],b=(3573)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]
5.
Change into reduce echelon form of the matrix
(03−63−183−912)\begin{pmatrix} 0 & 3 & -6 \\ 3 & -1 & 8 \\ 3 & -9 & 12 \end{pmatrix}
[5]
6.
Define linear transformation with an example. Is a transformation T(x, y) = (3x + y, 5x + 7y, x+3y) linear? Justify. [5]
7.
Let A and B be matrices. What value(s) of k if any will make AB = BA?
A=[−1−259], B=[92k−1]A = \begin{bmatrix} -1 & -2 \\ 5 & 9 \end{bmatrix},\ B = \begin{bmatrix} 9 & 2 \\ k & -1 \end{bmatrix}
[5]
8.
Define determinant. Evaluate without expanding:
[15−6−1−44−2−79]\begin{bmatrix} 1 & 5 & -6 \\ -1 & -4 & 4 \\ -2 & -7 & 9 \end{bmatrix}
[5]
9.
Define subspace of a vector space. Let H=[stO]:s,t∈RH = {\begin{bmatrix} s \\ t \\ O \end{bmatrix} : s, t \in \mathbb{R}}, show that H is a subspace of R3\mathbb{R}^3. [5]
10.
Find the dimension of the null space and column space of
A=[−36−11−71−223−12−458−4]A = \begin{bmatrix} -3 & 6 & -1 & 1 & -7 \\ 1 & -2 & 2 & 3 & -1 \\ 2 & -4 & 5 & 8 & -4 \end{bmatrix}
A=[−36−11−71−223−12−458−4]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
[63−80−2010−3]\begin{bmatrix} 6 & 3 & -8 \\ 0 & -2 & 0 \\ 1 & 0 & -3 \end{bmatrix}
[5]
12.
Find LU factorization of the matrix
[256−7]\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 Z4Z_4 x Z11Z_{11}. [5]
15.
State and prove the Pythagorean theorem of two vectors and verify this for u = (1, -1) and v = (1, 1). [5]