Let $p,q$ and $r$ be distinct positive integers satisfying $p<q<r$ and $p+q+r=k$. What is the smallest value of $k$ that does not determine $p,q,r$ uniquely?

Mala while teaching her class on functions gives her students a question.

According to the question the functions are $f\left(x\right)=-x, g\left(x\right)=x$. She also provides her students with the following functions.

$f\left(x,y\right)=x-y$ and $g\left(x,y\right)=x+y$

Since she wants to test the grasp of her students on functions she ask them simple questions "Which of the following is not true?" and provides her student with the following options. None of her students were able to answer the question in single attempt. Can you answer her question?

We have three inequalities as:

(i) $2^a>a$

(ii) $2^a>2a+1$

(iii) $2^a>a^2$

For what natural numbers $n$ are all the three inequalities satisfied?

If $f\left(a\right)=a^2-\frac{1}{a^2}$ and $g\left(a\right)=\frac{1}{\sqrt{f\left(a\right)-4}}$, then the real domain for all values of $\text{'}a\text{'}$ such that $f\left(a\right)$ and $g\left(a\right)$ are both real and defined is respectively by the inequality: