1
GATE CSE 1996
+1
-0.3
Suppose $$X$$ and $$Y$$ are sets and $$\left| X \right|$$ and $$\left| Y \right|$$ are their respective cardinalities. It is given that there are exactly 97 functions from $$X$$ to $$Y$$. From this one can conclude that
A
$$\left| X \right| = 1,\,\,\,\,\,\,\,\,\,\left| Y \right| = 97$$
B
$$\left| X \right| = 97,\,\,\,\,\,\,\,\,\,\left| Y \right| = 1$$
C
$$\left| X \right| = 97,\,\,\,\,\,\,\,\,\,\left| Y \right| = 97$$
D
None of the above
2
GATE CSE 1996
+1
-0.3
Which of the following statements is false?
A
The set of rational numbers is an abelian group under addition.
B
The set of integers is an abelian group under addition.
C
The set of rational numbers from an abelian group under multiplication.
D
The set of real numbers excluding zero is an abelian group under multiplication.
3
GATE CSE 1996
+1
-0.3
Let AX = B be a system of linear equations where A is an m x n matrix and B is a $$m\,\, \times \,\,1$$ column vector and X is a n x 1 column vector of unknowns. Which of the following is false?
A
The system has a solution if and only if, both A and the augmented matrix [A B] have the same rank.
B
If m < n and B is the zero vector, then the system has infinitely many solutions.
C
If m = n and B is a non zero vector, then the system has a unique solution.
D
The system will have only a trivial solution when m = n, B is the zero vector and rank (A) = n
4
GATE CSE 1996
+2
-0.6
The recurrence relation $$\,\,\,\,\,$$ $$T\left( 1 \right) = 2$$
$$T\left( n \right) = 3T\left( {{n \over 4}} \right) + n$$ has the solution $$T(n)$$ equal to
A
$$O(n)$$
B
$$O$$ (log n)
C
$$O\left( {{n^{3/4}}} \right)$$
D
None of the above
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