TRIBHUVAN UNIVERSITY
INSTITUTE OF SCIENCE AND TECHNOLOGY
2079 B.S.
Candidates are required to give their answers in their own words as far as practicable.
Group A
Long Answer QuestionsAttempt any Two questions.
- 1Reduce the system of equation into echelon form and solve: $x_{1}-2x_{2}-x_{3}+3x_{4}=0$, $-2x_{1}+4x_{2}+5x_{3}-5x_{4}=3$ $3x_{1}-6x_{2}+8x_{4}=2$
- 2Define linear transformation of a matrix A. let $A=\begin{bmatrix} 1 & -3 \\ 3 & 5 \\ -1 & 7 \end{bmatrix}$, $v=\begin{bmatrix} -2 \\ 1 \end{bmatrix}$, $C=\begin{bmatrix} 3 \\ 2 \\ 4 \end{bmatrix}$, $x=\begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}$, and define a transformation $T: \mathbb{R}^{2} \to \mathbb{R}^{3}$ by $T(x)=Ax$ then Find $T(v)$ Find $x \in \mathbb{R}^{2}$ whose consumption matrix C. The economy whose consumption matrix 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 then will satisfy this demand.
- 3Find the equation $y=a_{0}+a_{1}x$ of the least squares line best fits the data points: (0,1), (1,1), (2,2), (3,2).
Group B
Short Answer QuestionsAttempt any Eight questions.
- 4When a linear system of equation is consistent? Find the values of values of h and k for which the system: $2x_{1}-x_{2}=h$; $-6x_{1}+3x_{2}=k$ is consistent?
- 5Determine the column of the matrix A are linearly independent, where $A=\begin{bmatrix} -2 & 8 & -1 \\ 0 & 0 & 0 \\ 0 & -5 & 3 \end{bmatrix}$.
- 6When two column vectors in $\mathbb{R}^{2}$ are equal? Give an example. Compute $a+3b$, $a-2b$, where $a=\begin{bmatrix} 1 \\ -3 \\ 2 \end{bmatrix}$, $b=\begin{bmatrix} 0 \\ -1 \\ 3 \end{bmatrix}$
- 7The column of $I_{2} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ are $e_{1}=\begin{bmatrix} 1 \\ 0 \end{bmatrix}$, and $e_{2}=\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ suppose T is a linear transformation from $\mathbb{R}^{2}$ into $\mathbb{R}^{3}$ such that $T(e_{1})=\begin{bmatrix} 5 \\ 1 \\ -2 \end{bmatrix}$, and $T(e_{2})=\begin{bmatrix} 0 \\ -1 \\ 8 \end{bmatrix}$ find a formula for the image of an arbitrary X in $\mathbb{R}^{2}$. that is find $T(x)$ for x in $\mathbb{R}^{2}$
- 8Find the eigenvalues of $\begin{bmatrix} 3 & 6 & -4 \\ 0 & 0 & -6 \\ 0 & 0 & -2 \end{bmatrix}$.
- 9Define null space of a matrix A. if $A=\begin{bmatrix} -1 & -3 & 2 \\ -5 & -9 & 1 \end{bmatrix}$ and $v=\begin{bmatrix} 5 \\ -3 \\ -2 \end{bmatrix}$. then v belongs null space of matrix A?
- 10Verify that $1^{k}$, $(-2)^{k}$, $3^{k}$ are linearly independent signals.
- 11Evaluate the determinant of the matrix $\begin{bmatrix} 5 & -3 & 2 & 2 \\ 0 & 3 & 0 & -4 \\ -5 & 0 & -8 & 5 \\ 0 & 3 & 0 & -6 \end{bmatrix}$.
- 12Define unit vector. Find a unit vector v of $u= (0,-2,2,-3)$ in the direction of u.
- 13Define group. Show that the set of integers is not a group with respect to subtraction operation.
- 14Define ring. Show that set of positive integers with respect to addition and multiplication operation is not ring.