Questions
Syllabus
Qns. By Syllabus

Semester

Subject

Year

Bachelors Level/First Year/Second Semester/Science csit/second semester/mathematics ii/syllabus wise questions

B.Sc Computer Science and Information Technology

Institute of Science and Technology, TU

Mathematics II (MTH168)

Year Asked: 2080, syllabus wise question

Determinants
1.
Evaluate the determinant of the matrix.
[1−789−602−5730024−100150000−10]\begin{bmatrix} 1 & -7 & 8 & 9 & -6 \\ 0 & 2 & -5 & 7 & 3 \\ 0 & 0 & 2 & 4 & -1 \\ 0 & 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & -1 & 0 \end{bmatrix}
[5]
Eigenvalues and Eigen Vectors
1.
Find the eigenvalue of
A=[733−1]A = \begin{bmatrix} 7 & 3 \\ 3 & -1 \end{bmatrix}
[5]
Groups and Subgroups
1.
Define group. Show that the set of integers is a group with respect to addition operation. [5]
Linear Equations in Linear Algebra
1.
Define system of linear equations. When a system of equations is consistent? Make echelon form to solve:
−2a−3b+4c=5-2a - 3b + 4c = 5
b−2c=4b - 2c = 4
a+3b−c=2a + 3b - c = 2
[10]
Matrix Algebra
1.
Find AB by block multiplication of the matrices.
A=[2−3−10−41−5−23−10−4−27−1], B=[642−1−37135−3]A = \begin{bmatrix} 2 & -3 & -1 & 0 & -4 \\ 1 & -5 & -2 & 3 & -1\\ 0 & -4 & -2 & 7 & -1 \end{bmatrix},\ B = \begin{bmatrix} 6 & 4 \\ 2 & -1 \\ -3 & 7 \\ 1 & 3 \\ 5 & -3 \end{bmatrix}
[10]
2.
Let A What value(s) of k, if any, will make AB = BA?
A=[15−31], B=[4−53k]A = \begin{bmatrix} 1 & 5 \\ -3 & 1 \end{bmatrix},\ B = \begin{bmatrix} 4 & -5 \\ 3 & k \end{bmatrix}
[5]
Orthogonality and Least Squares
1.
Find the least square solution of Ax = c where
A=[1−33151172], c=[5−35]A = \begin{bmatrix} 1 & -3 & 3 \\ 1 & 5 & 1 \\ 1 & 7 & 2 \end{bmatrix},\ c = \begin{bmatrix} 5 \\ -3 \\ 5 \end{bmatrix}
and compute the associated least square error. [10+0]
2.
Find the equation y = a0+a1xa_0 + a_1 x of the least squares line that best fits the data points (2,1), (5,2), (7,3), (8,3). [5]
Rings and Fields
1.
Define ring and show that set of positive integers with respect to addition and multiplication operation is not a ring. [5]
Transformation
1.
Define linear transformation with an example.
letA=[1−335−17], v=[−21], b=[321], x=[x1x2], T(x)=Axlet A = \begin{bmatrix} 1 & -3 \\ 3 & 5 \\ -1 & 7 \end{bmatrix},\ v = \begin{bmatrix} -2 \\ 1 \end{bmatrix},\ b = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix},\ x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix},\ T(x) = Ax
a.find T(v)a. \text{find } T(v)
b.Find x∈R2 whose image under T is bb. \text{Find } x \in \mathbb{R}^2 \text{ whose image under } T \text{ is } b
[10]
Vector Space Continued
1.
Define null space of a matrix A. Let then show that v belongs to the null space matrix A.
A=[1−32−59−1], v=[532]A = \begin{bmatrix} 1 & -3 & 2 \\ -5 & 9 & -1 \end{bmatrix},\ v = \begin{bmatrix} 5 \\ 3 \\ 2 \end{bmatrix}
[5]
Vector Spaces
1.
Determine the column of the matrix A are linearly independent where
A=[3−36024030]A = \begin{bmatrix} 3 & -3 & 6 \\ 0 & 2 & 4 \\ 0 & 3 & 0 \end{bmatrix}
[5]
2.
When two column vectors in R2R^2 are equal? Give an example. Compute u + 3v, u - 2v, where
u=[1−32],v=[1−13]u = \begin{bmatrix} 1 \\ -3 \\ 2 \end{bmatrix},\quad v = \begin{bmatrix} 1 \\ -1 \\ 3 \end{bmatrix}
[5]
3.
Show that the solutions of yk+2−4yk+1+3yk=0y_{k+2} - 4y_{k+1} + 3y_k = 0 are linearly independent. [5]
4.
Prove that the two vectors u and v are perpendicular to each other if and only if the line through u is perpendicular bisector of the line segment from -u to v. [5]