1
GATE CSE 2018
+2
-0.6
Which one of the following is a closed form expression for the generating function of the sequence $$\left\{ {{a_n}} \right\},$$ where $${a_n} = 2n + 3$$ for all $$n = 0,1,2,....?$$
A
$${3 \over {{{\left( {1 - x} \right)}^2}}}$$
B
$${{3x} \over {{{\left( {1 - x} \right)}^2}}}$$
C
$${\left( {1 - x} \right)}$$
D
$${{3 - x} \over {{{\left( {1 - x} \right)}^2}}}$$
2
GATE CSE 2018
+2
-0.6
Consider a matrix P whose only eigenvectors are the multiples of $$\left[ {\matrix{ 1 \cr 4 \cr } } \right].$$

Consider the following statements.

$$\left( {\rm I} \right)$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$P$$ does not have an inverse
$$\left( {\rm II} \right)$$ $$\,\,\,\,\,\,\,\,\,\,\,$$ $$P$$ has a repeated eigenvalue
$$\left( {\rm III} \right)$$ $$\,\,\,\,\,\,\,\,\,$$ $$P$$ cannot be diagonalized

Which one of the following options is correct?

A
Only $${\rm I}$$ and $${\rm III}$$ are necessarily true
B
Only $${\rm II}$$ is necessarily true
C
Only $${\rm I}$$ and $${\rm II}$$ are necessarily true
D
Only $${\rm II}$$ and $${\rm III}$$ are necessarily true
3
GATE CSE 2017 Set 1
+2
-0.6
Let $$A$$ be $$n\,\, \times \,\,n$$ real valued square symmetric matrix of rank $$2$$ with $$\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {A_{ij}^2 = 50.} }$$
Consider the following statements.
$$(I)$$ One eigenvalue must be in $$\left[ { - 5,5} \right]$$
$$(II)$$ The eigenvalue with the largest magnitude must be strictly greater than $$5$$
Which of the above statements about engenvalues of $$A$$ is/are necessarily correct?
A
Both $$(I)$$ and $$(II)$$
B
$$(I)$$ only
C
$$(II)$$ only
D
Neither $$(I)$$ nor $$(II)$$
4
GATE CSE 2017 Set 2
Numerical
+2
-0
If the characteristic polynomial of a $$3 \times 3$$ matrix $$M$$ over $$R$$(the set of real numbers) is $${\lambda ^3} - 4{\lambda ^2} + a\lambda + 30.\,a \in R,$$ and one eigenvalue of $$M$$ is $$2,$$ then the largest among the absolute values of the eigenvalues of $$M$$ is ________.