In a quadratic function, the value of the product of the roots $$\left( {\alpha ,\beta } \right)$$ is $$4.$$ Find the value of $$${{{\alpha ^n} + {\beta ^n}} \over {{\alpha ^{ - n}} + {\beta ^{ - n}}}}$$$

Among $$150$$ faculty members in an institute, $$55$$ are connected with each other through Facebook^® and $$85$$ are connected through WhatsApp^®. $$30$$ faculty members do not have Facebook^® or whatsApp^® accounts. The number of faculty members connected only through Facebook^® accounts is ______________.

Computers were invented for performing only high-end useful computations. However, it is no understatement that they have taken over our world today. The internet, for example, is ubiquitous. Many believe that the internet itself is an unintended consequence of the original invention. With the advent of mobile computing on our phones, a whole new dimension is now enabled. One is left wondering if all these developments are good or, more importantly, required.

Which of the statement(s) below is/are logically valid and can be inferred from the above paragraph?

$$\,\,\,\,\,\,\,\,\,$$$$(i)$$ $$\,\,\,\,\,\,\,\,\,$$ The author believes that computers are not good for us.
$$\,\,\,\,\,\,\,$$ $$(ii)$$ $$\,\,\,\,\,\,\,\,\,$$ Mobile computers and the internet are both intended inventions

Choose the correct expression for $$f(x)$$ given in the graph.