1
GATE CSE 2003
+1
-0.3
$$m$$ identical balls are to be placed in $$n$$ distinct bags. You are given that $$m \ge kn$$, where $$k$$ is a natural number $$\ge 1$$. In how many ways can the balls be placed in the bags if each bag must contain at least $$k$$ balls?
A
$$\left( {\matrix{ {m - k} \cr {n - 1} \cr } } \right)$$
B
$$\left( {\matrix{ {m - kn + n - 1} \cr {n - 1} \cr } } \right)$$
C
$$\left( {\matrix{ {m - 1} \cr {n - k} \cr } } \right)$$
D
$$\left( {\matrix{ {m - kn + n + k - 2} \cr {n - k} \cr } } \right)$$
2
GATE CSE 2003
+1
-0.3
Let $$G$$ be an arbitrary graph with $$n$$ nodes and $$k$$ components. If a vertex is removed from $$G$$, the number of components in the resultant graph must necessarily lie between
A
$$k$$ and $$n$$
B
$$k - 1$$ and $$k + 1$$
C
$$k - 1$$ and $$n - 1$$
D
$$k + 1$$ and $$n -k$$
3
GATE CSE 2003
+2
-0.6
Consider the following system of linear equations $$\left[ {\matrix{ 2 & 1 & { - 4} \cr 4 & 3 & { - 12} \cr 1 & 2 & { - 8} \cr } } \right]\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ \alpha \cr 5 \cr 7 \cr } } \right]$$\$

Notice that the second and the third columns of the coefficient matrix are linearly dependent. For how many values of $$\alpha$$, does this system of equations have infinitely many solutions?

A
$$0$$
B
$$1$$
C
$$2$$
D
infinitely many
4
GATE CSE 2003
+2
-0.6
$$A$$ graph $$G$$ $$=$$ $$(V, E)$$ satisfies $$\left| E \right| \le \,3\left| V \right| - 6.$$ The min-degree of $$G$$ is defined as $$\mathop {\min }\limits_{v \in V} \left\{ {{{\mathop{\rm d}\nolimits} ^ \circ }egree\left( v \right)} \right\}$$. Therefore, min-degree of $$G$$ cannot be
A
$$3$$
B
$$4$$
C
$$5$$
D
$$6$$
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