MTH631 Quiz 3 Set 2

Q1. A full adder circuit takes ______________ bit(s) as input and returns ____________ bit(s) as output.

1. 2, 1
2. 2, 2
3. 3, 1
4. 3, 2

Correct answer: 3, 2

Q2. In order to build an XOR gate using NAND gates only, ______________ gate(s) are required.

1. 1
2. 4
3. 2
4. 3

Correct answer: 4

Q3. Which of the following expressions represents S output in a full adder?

1. ( {A B} ) {C_{in}}
2. ( {A {C_{in}}} ) B
3. ( {A + B} ) {C_{in}}
4. A B {C_{in}}

Correct answer: A B {C_{in}}

Q4. Which of the following is an invalid BCD code?

1. 1101
2. 1001
3. 0101
4. 0011

Correct answer: 1101

Q5. When implementing a half adder by using gates, which of the following gates is used to give Sum (S) as an output?

1. XOR
2. NOR
3. OR
4. AND

Correct answer: XOR

Q6. {\text{In }}{\mathbb{R}^n},{\text{which of the following is true about }}\phi = \left\{ {} \right\}{\text{ and }}A = \left\{ {\left( {{a_1},{a_2}, \ldots ,{a_n}} \right),{a_i} \in \mathbb{R},1 \leqslant i \leqslant n,i \in \mathbb{N}} \right\}?

1. {{ is Connected and }}A{{ is Disconnected}}
2. {{ is Disconnected and }}A{{ is Connected}}
3. {{Both are Connected}}
4. {{Both are Disconnected}}

Correct answer: {{Both are Connected}}

Q7. {\text{Set of isolated point(s) of the complement of set }}\left\{ {\left( {x,y} \right): - n < x,y < n,\left( {x,y} \right) \ne \left( {0,0} \right),n \in \mathbb{N}} \right\}{\text{ in }}{\mathbb{R}^{\text{2}}},{\text{is}} - - - - .

1. \{ {( {x,y} ) {{R}^2}:x = y = | n |,n {N}} \} \{ {( {0,0} )} \}
2. \{ {( {x,y} ) {{R}^2}:x = y = | n |,n {N}} \}
3. \{ {( {0,0} )} \}
4. \{ {( {x,y} ) {{R}^2}:x = y = | n |,n {N}} \} \{ {( {0,0} )} \}

Correct answer: \{ {( {0,0} )} \}

Q8. \begin{gathered} {\text{For a non empty closed and bounded subset }}S{\text{ in }}{\mathbb{R}^n},{\text{ if }}{\rm H}{\text{ is the collection open sets such that }} \hfill \\ S \subset \cup \left\{ {H:H \in {\rm H}} \right\},{\text{then by Heine - Borel theorem,}} \hfill \\\ \end{gathered}

1. S _{ } \{ {{H_ }:{H_ } { H}} \}
2. S _{ = 1}^ \{ {{H_ }:{H_ } { H}} \}
3. S _{ = 1}^n \{ {{H_ }:{H_ } { H}} \}
4. {{All above are equally valid}}

Correct answer: S _{ = 1}^n \{ {{H_ }:{H_ } { H}} \}

Q9. The limit of the function f(x) = \frac{{xy}}{{{x^2} + {y^2}}} by letting \left( {x,{\rm{ }}y} \right)approach \left( {0,{\rm{ }}0} \right)along the line y{\rm{ }} = {\rm{ }}x is ________.

1. - {1}{2}
2. Undefined
3. Finite
4. {1}{2}

Correct answer: {1}{2}

Q10. {\text{A compact set in }}{\mathbb{R}^n}{\text{ is - - - - - - - - - - }}{\text{.}}

1. {{open and unbounded}}
2. {{closed and unbounded}}
3. {{open and bounded}}
4. {{closed and bounded}}

Correct answer: {{closed and bounded}}

Q11. {\text{In }}{\mathbb{R}^n},\,{\text{the interior }}{S^0}{\text{ of a non - empty set S is - - - - - - }}{\text{.}}

1. {{open - disconnected}}
2. {{open - connected (region)}}

Correct answer: {{open - disconnected}}

Q12. {\text{The set }}\left\{ {\left( {x,y} \right): - n < x,y < n,\left( {x,y} \right) \ne \left( {0,0} \right),n \in \mathbb{N}} \right\}{\text{ is - - - - - - in }}{\mathbb{R}^{\text{2}}}.

1. neither open nor closed
2. both open or closed
3. closed
4. open

Correct answer: open

Q13. \begin{gathered} {\text{In }}{\mathbb{R}^n}{\text{, if }}{S_1},{S_2},{S_3} \ldots \,\,{\text{are non - empty closed subsets such that }}{S_1} \supset {S_2} \supset \cdots \; \supset {S_r} \supset \cdots {\text{ and }} \hfill \\ {\text{sup}}\left\{ {\left| {{X_r} - {Y_r}} \right|:{X_r},{Y_r} \in {S_r},r \geqslant 1} \right\} \to 0{\text{ as }}r \to \infty {\text{,then the order of set }}\mathop \cap \limits_{r = 1}^\infty {S_r} = - - - . \hfill \\\ \end{gathered}

1. {{zero}}
2. {{one}}
3. {{finite but more than one}}
4. {infinite}

Correct answer: {{one}}

Q14. A set “ S ” is polygonally connected if, ------- pair of points in S can be connected by a polygonal path lying -------- in “ S ”.

1. some, entirely
2. every, partially
3. some, partially
4. every , entirely

Correct answer: every , entirely

Q15. If \mathop {\lim }\limits_{X \to {X_0}} f\left( X \right) exists, then it is _____.

1. 0
2. none of these
3. 1
4. unique

Correct answer: unique