1
GATE CSE 2006
MCQ (Single Correct Answer)
+2
-0.6
Consider a Boolean function $$f(w, x, y, z).$$ Suppose that exactly one of its inputs is allowed to change at a time. If the function happens to be true for two input vectors $${i_1} = < {w_1},{x_1},{y_1},{z_1} > $$ and $${i_2} = < {w_2},{x_2},{y_2},{z_2} > ,$$ we would like the function to remain true as the input changes from $${i_1}$$ to $${i_2}$$ ($${i_1}$$ and $${i_2}$$ differ in exactly one bit position), without becoming false momentarily. Let $$f\left( {w,x,y,z} \right) = \sum {\left( {5,7,11,12,13,15} \right)} .$$ Which of the following cube covers of $$f$$ will entire that the required property is satisfied?
A
$$\overline w xz,\,wx\overline y ,\,x\overline y z,\,xyz,wyz$$
B
$$wxy,\,\overline w xz,\,wyz$$
C
$$wx\overline {yz} ,\,xz,\,w\overline x yz$$
D
$$wzy,\,wyz,\,wxz,\,\overline w xz,\,x\overline y z,\,xyz$$
2
GATE CSE 2006
MCQ (Single Correct Answer)
+2
-0.6
Given a set of elements N = {1, 2, ....., n} and two arbitrary subsets $$A\, \subseteq \,N\,$$ and $$B\, \subseteq \,N\,$$, how many of the n! permutations $$\pi $$ from N to N satisfy $$\min \,\left( {\pi \,\left( A \right)} \right) = \min \,\left( {\pi \,\left( B \right)} \right)$$, where min (S) is the smallest integer in the set of integers S, and $${\pi \,\left( S \right)}$$ is the set of integers obtained by applying permutation $${\pi}$$ to each element of S?
A
$$\left( {n - \left| {A\, \cup \,B} \right|} \right)\,\left| A \right|\,\left| B \right|$$
B
$$\left( {{{\left| A \right|}^2} + {{\left| B \right|}^2}} \right)\,{n^2}$$
C
$$n!{{\left| {A\, \cap \,B} \right|} \over {\left| {A\, \cup B} \right|}}$$
D
$$\,{{{{\left| {A\, \cap \,B} \right|}^2}} \over {\left( {\matrix{ n \cr {\left| {A\, \cup \,B} \right|} \cr } } \right)}}$$
3
GATE CSE 2006
MCQ (Single Correct Answer)
+2
-0.6
What are the eigen values of the matrix $$P$$ given below? $$$P = \left( {\matrix{ a & 1 & 0 \cr 1 & a & 1 \cr 0 & 1 & a \cr } } \right)$$$
A
$$a,a - \sqrt {2,} a + \sqrt 2 $$
B
$$a,a,a$$
C
$$0,a,2a$$
D
$$ - a,2a,2a$$
4
GATE CSE 2006
MCQ (Single Correct Answer)
+2
-0.6
$$F$$ is an $$n$$ $$x$$ $$n$$ real matrix. $$b$$ is an $$n$$ $$x$$ $$1$$ real vector. Suppose there are two $$n$$ $$x$$ $$1$$ vectors, $$u$$ and $$v$$ such that $$u \ne v$$, and $$Fu = b,\,\,\,\,Fv = b$$

Which one of the following statements is false?

A
Dererminant of $$F$$ is zero
B
There are an infinite number of solutions to $$Fx$$ $$=$$ $$b$$
C
There is an $$x \ne 0$$ such that $$Fx = 0$$
D
$$F$$ must have two identical rows
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