Suppose that a robot is placed on the Cartesian plane. At each step it is allowed to move either one unit up or one unit right, i.e., if it is at $$(i, j)$$ then it can move to either $$(i+1, j)$$ or $$(i, j+1)$$

Suppose that the robot is not allowed to traverse the line segment from $$(4, 4)$$ to $$(5,4)$$. With this constraint, how many distinct path are there for the robot to reach $$(10, 10)$$ starting from $$(0,0)$$?

Let Graph$$(x)$$ be a predicate which denotes that $$x$$ is a graph. Let Connected$$(x)$$ be a predicate which denotes that $$x$$ is connected. Which of the following first order logic sentences DOES NOT represent the statement: $$Not\,\,\,every\,\,\,graph\,\,\,is\,\,\,connected?$$

Which of the following graphs has an Eulerian circuit?

Let $$A$$ be $$a$$ $$4$$ $$x$$ $$4$$ matrix with eigen values $$-5$$, $$-2, 1, 4$$.

Which of the following is an eigen value of $$\left[ {\matrix{ {\rm A} & {\rm I} \cr {\rm I} & {\rm A} \cr } } \right]$$, where $$I$$ is the $$4$$ $$x$$ $$4$$ identity matrix?