Let $T$ be a binary search tree with 15 nodes. The minimum and maximum possible heights of $T$ are :

Note : The height of a tree with a single node is 0.

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?

The value of $$\mathop {\lim }\limits_{x \to 1} {{{x^7} - 2{x^5} + 1} \over {{x^3} - 3{x^2} + 2}}.$$