Which of the following order pair would minimize the objective function of the linear programming problem; z = x + 5 y subject to x \geq 2 , y \geq 0 ?

• A) (0,0)
• B) (0,3)
• C) (2,0)
• D) (2,3)

When we have \geq type constraints, then we convert it to equality constraints by introducing _______ variable(s) and solve it by __________ method.

• A) artificial and slack, Big M or Two phase
• B) artificial, Two phase
• C) artificial, Big M
• D) slack, Big M

For Z = 4 x_1 + x_2 subject to 3 x_1 + 4 x_2 \geq 20, x_1 + 5 x_2 \geq 15, we have ___________ variables.

• A) two slack
• B) two artificial
• C) three objective
• D) both a and b

Minimizing\,\,\,Z = 4{x_1} + 2{x_2} + {x_3}\,\,is\,\,equivalent\,\,to\,\,\max imizing\,\,Z = \_\_\_\_\_\_\_\_\_\_\_\_\_.

• A) - 4{x_1} - 2{x_2} - {x_3}
• B) - 4{x_1} - 2{x_2} + {x_3}
• C) - 4{x_1} + 2{x_2} + {x_3}
• D) - 4{x_1} + 2{x_2} - {x_3}

Which of the following order pair would minimize the objective function of the linear programming problem; \(z = x + 5 y \) subject to \(x \geq 2 , y \geq 0\) ?

• A) (2,3)
• B) (0,0)
• C) (2,0)
• D) (0,3)

For \( Z = 4 x_1 + x_2 \) subject to \( 3 x_1 + 4 x_2 \geq 20, x_1 + 5 x_2 \geq 15 \), we have ___________ variables.

• A) both a and b
• B) two slack
• C) three objective
• D) two artificial

\[{\text{Minimizing}}\,\,{\text{any}}\,\,{\text{function}}\,f({x_1},{x_2},{x_3},......,{x_n})\,{\text{subject}}\,{\text{to}}\,{\text{the}}\,{\text{set}}\,{\text{of}}\,{\text{constraints}}\,\,{\text{is}}\,\,{\text{equivalent}}\,{\text{to}}\,{\text{maximizing _________ subject}}\,{\text{to}}\,{\text{the}}\,{\text{same}}\,{\text{set}}\,{\text{of}}\,{\text{constraints}}{\text{.}}\]

• A) \[f( - {x_1},{x_2},{x_3},......,{x_n})\]
• B) \[f( - {x_1}, - {x_2}, - {x_3},......, - {x_n})\]
• C) \[ - f({x_1},{x_2},{x_3},......,{x_n})\]
• D) \[f({x_1}, - {x_2},{x_3},......,{x_n})\]

\[Minimizing\,\,\,Z = 4{x_1} + 2{x_2} + {x_3}\,\,is\,\,equivalent\,\,to\,\,\max imizing\,\,Z = \_\_\_\_\_\_\_\_\_\_\_\_\_.\]

• A) \[ - 4{x_1} - 2{x_2} + {x_3}\]
• B) \[ - 4{x_1} + 2{x_2} - {x_3}\]
• C) \[ - 4{x_1} + 2{x_2} + {x_3}\]
• D) \[ - 4{x_1} - 2{x_2} - {x_3}\]

The inequality \(3x - 2y \geq 14\) is equivalent to______.

• A) \(3x - 2y < - 14\)
• B) \(3x - 2y \leq 14\)
• C) \( - 3x + 2y \leq - 14\)
• D) \( - 3x + 2y > 14\)

The inequality 3x - 2y \geq 14 is equivalent to______.

• A) 3x - 2y \leq 14
• B) 3x - 2y < - 14
• C) - 3x + 2y > 14
• D) - 3x + 2y \leq - 14

When we have \( \geq \) type constraints, then we convert it to equality constraints by introducing _______ variable(s) and solve it by __________ method.

• A) artificial and slack, Big M or Two phase
• B) slack, Big M
• C) artificial, Two phase
• D) artificial, Big M

For \( Z = 4 x_1 + x_2\) subject to \( 3 x_1 + 4 x_2 \geq 20, x_1 + 5 x_ 2 \geq 15 \), we have ___________ variables.

• A) two slack
• B) one artificial
• C) three objective
• D) six

If the total demand is equal to total supply as per requirement of a balanced transportation problem i-e “a_1+a_2+ \cdots +a_n = b_1+b_2+\cdots+b_n” then which of the following is true?

• A) “a_1=b_1, a_2=b_2,\cdots, a_n=b_n” is necessarily implied
• B) “a_1=b_1, a_2 =b_2,\cdots, a_n=b_n” is NOT necessarily implied
• C)
• D)

In a Transportation problem, which of the following are taken as the coefficients of unknown “\(x_{ij}\)'s” ?

• A) Ones
• B) Zeros
• C) Zeros or ones
• D) Any arbitrary non-negative integer

In a Transportation problem, which of the following are taken as the coefficients of unknown “x_{ij}'s” ?

• A) Zeros or ones
• B) Ones
• C) Any arbitrary non-negative integer
• D) Zeros

Which values of “\(a_{ij}\)'s” in the standard form of a linear programming problem will transform it in standard Transportation problem?

• A) \(a_{ij} = 0\) and 1 for all \(i =1,2,\cdots,m; j = 1,2,\cdots,n\)
• B) \(a_{ij} = 0\) for all \(i =1,2,\cdots,m; j = 1,2,\cdots,n\)
• C) \(a_{ij} = 1\) for all \(i =1,2,\cdots,m; j = 1,2,\cdots,n\)
• D) \(a_{ij} = 0\) or 1 for all \(i =1,2,\cdots,m; j = 1,2,\cdots,n\)

If the total demand is equal to total supply as per requirement of a balanced transportation problem i-e “\(a_1+a_2+ \cdots +a_n = b_1+b_2+\cdots+b_n\)” then which of the following is true?

• A) “\(a_1=b_1, a_2=b_2,\cdots, a_n=b_n\)” is necessarily implied
• B) “\(a_1=b_1, a_2 =b_2,\cdots, a_n=b_n\)” is NOT necessarily implied
• C)
• D)

Which values of “a_{ij}'s” in the standard form of a linear programming problem will transform it in standard Transportation problem?

• A) a_{ij} = 0 or 1 for all i =1,2,\cdots,m; j = 1,2,\cdots,n
• B) a_{ij} = 0 and 1 for all i =1,2,\cdots,m; j = 1,2,\cdots,n
• C) a_{ij} = 1 for all i =1,2,\cdots,m; j = 1,2,\cdots,n
• D) a_{ij} = 0 for all i =1,2,\cdots,m; j = 1,2,\cdots,n

Let $T:D\left( T\right) \rightarrow Y$ be a bounded linear operator from normed space $X$ to Banach space $Y$, and $\forall x\in \overline{D\left( T\right) }~\exists ~$a sequence $\left\{ x_{n}\right\}$ in $D(T)$ such that $x_{n}\rightarrow x,$then$~\left\Vert Tx_{m}-Tx_{n}\right\Vert \Longrightarrow$

• A) $\left\{ Tx_{n}\right\} ~$is a Cauchy Sequence in $Y$
• B) $\left\{ Tx_{n}\right\} ~$is a Cauchy Sequence in $\overline{Y}$
• C) $\left\{ Tx_{n}\right\} ~$is a Cauchy Sequence in $\overline{R(T)}$
• D) $\left\{ Tx_{n}\right\} ~$is not a Cauchy Sequence in $Y$

If $T$ is a bounded linear operator on a normed space $X$, then for the Null space $N(T)$;

• A) $N(T)\subseteq \overline{N(T)}$
• B) $N(T)\nsubseteq \overline{N(T)}$
• C)
• D)