1
GATE CSE 2003
+2
-0.6
A 1- input, 2- output synchronous sequential circuit behaves as follows.

Let $${z_k},\,{n_k}$$ denote the number of $$0’s$$ and $$1’s$$ respectively in initial $$k$$ bits of the input

$$\left({{z_k} + {n_k} = k} \right).$$ The circuit outputs $$00$$ until one of the following conditions holds.

$$* \,\,\,\,\,$$ $${z_k} = {n_k} + 2.\,\,\,$$ In this case, the output at the $$k$$-th and all subsequent clock ticks is $$10.$$

$$* \,\,\,\,\,$$ $${n_k} = {z_k} + 2.\,\,\,$$ In this case, the output at the $$k$$-th and all subsequent clock ticks is $$01.$$

What is the minimum number of states required in the state transition graph of the above circuit?

A
$$5$$
B
$$6$$
C
$$7$$
D
$$8$$
2
GATE CSE 2003
+2
-0.6
Consider the $$ALU$$ shown below If the operands are in $$2's$$ complement representation, which of the following operations can be performed by suitably setting the control lines $$K$$ and $${C_0}$$ only ( + and - denote addition and subtraction respectively)?

A
$$A+B,$$ and $$A-B,$$ but not $$A+1$$
B
$$A+B,$$ and $$A+1,$$ but not $$A-B$$
C
$$A+B,$$ but not $$A-B,$$ or $$A+1$$
D
$$A+B,$$ and $$A-B,$$ and $$A+1$$
3
GATE CSE 2003
+2
-0.6
The following resolution rule is used in logic programming. Derive clause $$\left( {P \vee Q} \right)$$ from clauses $$\left( {P \vee R} \right)$$, $$\left( {Q \vee \neg R} \right)$$.

Which of the following statements related to this rule is FALSE?

A
$$\left( {\left( {P \vee R} \right) \wedge \left( {Q \vee \neg R} \right)} \right) \Rightarrow \left( {P \vee Q} \right)$$ is logically valid
B
$$\left( {P \vee Q} \right) \Rightarrow \left( {\left( {P \vee R} \right) \wedge \left( {Q \vee \neg R} \right)} \right)$$ is logically valid
C
$$\left( {P \vee Q} \right)$$ is satisfiable if and only if $${\left( {P \vee R} \right) \wedge \left( {Q \vee \neg R} \right)}$$ is satisfiable
D
$$\left( {P \vee Q} \right) \Rightarrow$$ FALSE if and only if both $$P$$ and $$Q$$ are unsatisfiable
4
GATE CSE 2003
+1
-0.3
Let P(E) denote the probability of the event E. Given P(A) = 1, P(B) = $${\raise0.5ex\hbox{1} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{2}}$$, the values of $$P\,(A\,\left| {B) \,} \right.$$ and $$P\,(B\,\left| {A) \,} \right.$$ respectively are
A
$${\raise0.5ex\hbox{1} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{4}},\,{\raise0.5ex\hbox{1} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{2}}$$
B
$${\raise0.5ex\hbox{1} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{2}},\,{\raise0.5ex\hbox{1} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{4}}$$
C
$${\raise0.5ex\hbox{1} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{2}},\,1$$
D
$$1,\,\,{\raise0.5ex\hbox{1} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{2}}$$
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