Questions
Syllabus
Qns. By Syllabus

Semester

Subject

Year

Tribhuwan University

Institute of Science and Technology

2078

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.
Define system of linear equations. When a system of equation is consistent? Determine if the system is consistent:
−2x1−3x2+4x3=5-2x_1 - 3x_2 + 4x_3 = 5
x2−2x3=4x_2 - 2x_3 = 4
x1+3x2−x3=2x_1 + 3x_2 - x_3 = 2
[10]
2.
Define linear transformation with an example. Let
A=[1−335−17], v=[2−1], b=[324], x=[x1x2]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:R2→R2T: \mathbb{R}^2 \to \mathbb{R}^2 by T(x)=AxT(x) = Ax then a. find T(v)T(v) b. find x∈R2x \in \mathbb{R}^2 whose image under TT is bb [10+0]
3.
Find the LU factorization of
[24−15−2−4−53−812−5−418−607−31]\begin{bmatrix} 2 & 4 & -1 & 5 & -2 \\ -4 & -5 & 3 & -8 & 1 \\ 2 & -5 & -4 & 1 & 8 \\ -6 & 0 & 7 & -3 & 1 \end{bmatrix}
[10]
4.
Find a least square solution of the inconsistent system Ax=bAx = b for
A=[−122−3−13], b=[421]A = \begin{bmatrix} -1 & 2 \\ 2 & -3 \\ -1 & 3 \end{bmatrix},\ b = \begin{bmatrix} 4 \\ 2 \\ 1 \end{bmatrix}
[10]
Section B

Short Answers Questions

Attempt any Eight questions.
[8*5=40]
5.
Determine the column of the matrix AA are linearly independent, where
A=[01412−1580]A = \begin{bmatrix} 0 & 1 & 4 \\ 1 & 2 & -1 \\ 5 & 8 & 0 \end{bmatrix}
[5]
6.
When two column vector in R2\mathbb{R}^2 are equal? Give an example. Compute u+3vu + 3v, u−2vu - 2v, where
u=[1−32], v=[1−13]u = \begin{bmatrix} 1 \\ -3 \\ 2 \end{bmatrix},\ v = \begin{bmatrix} 1 \\ -1 \\ 3 \end{bmatrix}
[5]
7.
Let A=[01−10]A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} and define T:R2→R2T: \mathbb{R}^2 \to \mathbb{R}^2 by T(x)=AxT(x) = Ax, find the image under TT of
u=[1−3], v=[15]u = \begin{bmatrix} 1 \\ -3 \end{bmatrix},\ v = \begin{bmatrix} 1 \\ 5 \end{bmatrix}
[5]
8.
Find the eigen value of
[36−8006002]\begin{bmatrix} 3 & 6 & -8 \\ 0 & 0 & 6 \\ 0 & 0 & 2 \end{bmatrix}
[5]
9.
Define null space of a matrix AA. Let
A=[−1−325−91], v=[5−3−2]A = \begin{bmatrix} -1 & -3 & 2 \\ 5 & -9 & 1 \end{bmatrix},\ v = \begin{bmatrix} 5 \\ -3 \\ -2 \end{bmatrix}
Then show that vv is in the null AA. [5+0]
10.
If A=[72−41]A = \begin{bmatrix} 7 & 2 \\ -4 & 1 \end{bmatrix}, find a formula for AnA^n, where A=PDP−1A = PDP^{-1}, P=[11−1−2]P = \begin{bmatrix} 1 & 1 \\ -1 & -2 \end{bmatrix} and D=[5003]D = \begin{bmatrix} 5 & 0 \\ 0 & 3 \end{bmatrix} [5]
11.
Find a unit vector vv of u=(1,−2,2,3)u = (1, -2, 2, 3) in the direction of uu. [5]
12.
Prove that the two vectors uu and vv are perpendicular to each other if and only if the line through uu is perpendicular bisector of the line segment from −u-u to vv. [5]
13.
Let an operation ∗* be defined on Q+\mathbb{Q}^+ by a∗b=ab2a * b = \frac{ab}{2}. Then show that Q+\mathbb{Q}^+ forms a group. [5]
14.
Define ring and show that set of real numbers with respect to addition and multiplication operation is a ring. [5]
15.
Verify that 1k,(−2k),3k1k, (-2^k), 3k are linearly independent signals. [5]