1
GATE CSE 2001
+2
-0.6
Consider the following languages:
$${L_1} = \left\{ {w\,w\left| {w \in {{\left\{ {a,\,b} \right\}}^ * }} \right.} \right\}$$
$${L_2} = \left\{ {w\,{w^R}\left| {w \in {{\left\{ {a,\,b} \right\}}^ * },} \right.{w^R}\,\,} \right.$$ is the reverse of $$\left. w \right\}$$
$${L_3} = \left\{ {{0^{2i}}\left| {i\,\,} \right.} \right.$$ is an integer$$\left. \, \right\}$$
$${L_4} = \left\{ {{0^{{i^2}}}\left| {i\,\,} \right.} \right.$$ is an integer$$\left. \, \right\}$$

Which of the languages are regular?

A
Only $${L_1}$$ and $${L_2}$$
B
Only $${L_2},$$ $${L_3}$$ and $${L_4}$$
C
Only $${L_3}$$ and $${L_4}$$
D
Only $${L_3}$$
2
GATE CSE 2000
+2
-0.6
What can be said about a regular language $$L$$ over $$\left\{ a \right\}$$ whose minimal finite state automation has two states?
A
Must be $$\left\{ {{a^n}\left| n \right.\,\,} \right.$$ is odd $$\left. \, \right\}$$
B
Must be $$\left\{ {{a^n}\left| n \right.\,\,} \right.$$ is even $$\left. \, \right\}$$
C
Must be $$\left\{ {{a^n}\left| {n \ge } \right.\,\,} \right.0\left. \, \right\}$$
D
Either $$L$$ must be $$\left\{ {{a^n}\left| n \right.\,\,} \right.$$ is odd$$\left. \, \right\}\,\,$$ or $$L$$ must be $$\left\{ {{a^n}\left| n \right.} \right.$$ is even$$\left. \, \right\}$$
3
GATE CSE 1998
+2
-0.6
Let $$L$$ be the set of all binary strings whose last two symbols are the same. The number of states in the minimum state deterministic finite-state automaton accepting $$L$$ is
A
$$2$$
B
$$5$$
C
$$8$$
D
$$3$$
4
GATE CSE 1997
+2
-0.6
Which one of the following regular expressions over $$\left\{ {0,\,\,1} \right\}$$ denotes the set of all strings not containing $$100$$ as substring?
A
$${0^ * }{\left( {1 + 0} \right)^ * }$$
B
$${0^ * }\,\,{1010^ * }$$
C
$${0^ * }\,\,{1^ * }\,\,{01^ * }$$
D
$${0^ * }{\left( {10 + 1} \right)^ * }$$
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