1
GATE CSE 2016 Set 2
+1
-0.3
Consider the system, each consisting of m linear equations in $$n$$ variables.
$$I.$$ $$\,\,\,$$ If $$m < n,$$ then all such system have a solution
$$II.$$ $$\,\,\,$$ If $$m > n,$$ then none of these systems has a solution
$$III.$$ $$\,\,\,$$ If $$m = n,$$ then there exists a system which has a solution

Which one of the following is CORRECT?

A
$$I$$ , $$II$$ and $$III$$ are true
B
Only $$II$$ and $$III$$ are true
C
Only $$III$$ is true
D
None of them is true
2
GATE CSE 2016 Set 1
Numerical
+1
-0
Two eigenvalues of a $$3 \times 3$$ real matrix $$P$$ are $$\left( {2 + \sqrt { - 1} } \right)$$ and $$3.$$ The determinant of $$P$$ is _______.
3
GATE CSE 2016 Set 1
+1
-0.3
Let $${a_n}$$ be the number of $$n$$-bit strings that do NOT contain two consecutive $$1s.$$ Which one of the following is the recurrence relation for $${a_n}$$?
A
$${a_n} = {a_{n - 1}} + 2{a_{n - 2}}$$
B
$${a_n} = {a_{n - 1}} + {a_{n - 2}}$$
C
$${a_n} = 2{a_{n - 1}} + {a_{n - 2}}$$
D
$${a_n} = 2{a_{n - 1}} + 2{a_{n - 2}}$$
4
GATE CSE 2015 Set 1
Numerical
+1
-0
In the LU decomposition of the matrix $$\left[ {\matrix{ 2 & 2 \cr 4 & 9 \cr } } \right]$$, if the diagonal elements of U are both 1, then the lower diagonal entry $${l_{22}}$$ of L is ________.