1
GATE CSE 2016 Set 2
+2
-0.6
Consider the following two statements:

$$\,\,\,\,\,\,\,{\rm I}.\,\,\,\,\,$$ If all states of an $$NFA$$ are accepting states then the language accepted by the
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$NFA$$ is $$\sum {^ * } .$$
$$\,\,\,\,\,{\rm I}{\rm I}.\,\,\,\,\,$$ There exists a regular language $$A$$ such that for all languages $$B,A \cap B$$ is
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ regular.

Which one of the following is CORRECT?

A
Only $${\rm I}$$ is true
B
Only $${\rm II}$$ is true
C
Both $${\rm I}$$ and $${\rm II}$$ are true
D
Both $${\rm I}$$ and $${\rm II}$$ are false
2
GATE CSE 2016 Set 2
+2
-0.6
Consider the following languages: \eqalign{ & {L_1} = \left\{ {{a^n}{b^m}{c^{n + m}}:m,n \ge 1} \right\} \cr & {L_2} = \left\{ {{a^n}{b^n}{c^{2n}}:n \ge 1} \right\} \cr}\$

Which one of the following is TRUE?

A
Both $${L_1}$$ and $${L_2}$$ are context-free.
B
$${L_1}$$ is context-free while $${L_2}$$ is not context-free.
C
$${L_2}$$ is context-free while $${L_1}$$ is not context-free.
D
Neither $${L_1}$$ nor $${L_2}$$ is context-free.
3
GATE CSE 2016 Set 2
+2
-0.6
Consider the following languages.

$$\,\,\,\,\,\,\,\,\,\,\,\,$$ $${L_1} = \left\{ {\left\langle M \right\rangle |M} \right.$$ takes at least $$2016$$ steps on some input $$\left. \, \right\},$$
$$\,\,\,\,\,\,\,\,\,\,\,\,$$ $${L_2} = \left\{ {\left\langle M \right\rangle |M} \right.$$ takes at least $$2016$$ steps on all inputs $$\left. \, \right\}$$ and
$$\,\,\,\,\,\,\,\,\,\,\,\,$$ $${L_3} = \left\{ {\left\langle M \right\rangle |M} \right.$$ accepts $$\left. \varepsilon \right\},$$

where for each Turing machine $${M,\left\langle M \right\rangle }$$ denotes a specific encoding of $$M.$$ Which one of the following is TRUE?
A
$${L_1}$$ is recursive and $${L_2},$$$${L_3}$$ are not recursive
B
$${L_2}$$ is recursive and $${L_1},$$$${L_3}$$ are not recursive
C
$${L_1},$$$${L_2}$$ are recursive and $${L_3}$$ is not recursive
D
$${L_{1,}}$$$${L_{2,}}$$$${L_{3}}$$ are recursive
4
GATE CSE 2016 Set 2
+1
-0.3
The man who is now Municipal Commissioner worked as ____________________.
A
the security guard at a university
B
a security guard at the university
C
a security guard at university
D
the security guard at the university
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