Consider the following functions from positive integers to real numbers :

$10, \sqrt{n}, n, \log _2 n, \frac{100}{n}$.

The CORRECT arrangement of the above functions in increasing order of asymptotic complexity is :

A sender S sends a message m to receiver R, which is digitally signed by S with its private key. In this scenario, one or more of the following security violations can take place.

I. S can launch a birthday attack to replace m with a fraudulent message.
II. A third party attacker can launch a birthday attack to replace m with a fraudulent message.
III. R can launch a birthday attack to replace m with a fraudulent message.

Which of the following are possible security violations?

Let $${c_1},.....,\,\,{c_n}$$ be scalars, not all zero, such that $$\sum\limits_{i = 1}^n {{c_i}{a_i} = 0} $$ where $${{a_i}}$$ are column vectors in $${R^{11}}.$$ Consider the set of linear equations $$AX=b$$

Where $$A = \left[ {{a_1},.....,\,\,{a_n}} \right]$$ and $$b = \sum\limits_{i = 1}^n {{a_i}.} $$
The set of equations has

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