Let $$G$$ be a finite group on $$84$$ elements. The size of a largest possible proper subgroup of $$G$$ is ________.

Consider the first-order logic sentence
$$\varphi \equiv \,\,\,\,\,\,\,\exists s\exists t\exists u\forall v\forall w$$ $$\forall x\forall y\psi \left( {s,t,u,v,w,x,y} \right)$$
where $$\psi $$ $$(𝑠,𝑡, 𝑢, 𝑣, 𝑤, 𝑥, 𝑦)$$ is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose $$\varphi $$ has a model with a universe containing $$7$$ elements.

Which one of the following statements is necessarily true?

Consider a storage disk with $$4$$ platters (numbered as $$0, 1, 2$$ and $$3$$), $$200$$ cylinders (numbered as $$0, 1,$$ … , $$199$$), and $$256$$ sectors per track (numbered as $$0, 1, … , 255$$). The following $$6$$ disk requests of the form [sector number, cylinder number, platter number] are received by the disk controller at the same time: $$[120, 72, 2] , [180, 134, 1] , [60, 20, 0] , [212, 86, 3] , [56, 116, 2] , [118, 16, 1]$$

Currently the head is positioned at sector number $$100$$ of cylinder $$80,$$ and is moving towards higher cylinder numbers. The average power dissipation in moving the head over $$100$$ cylinders is $$20$$ milliwatts and for reversing the direction of the head movement once is $$15$$ milliwatts. Power dissipation associated with rotational latency and switching of head between different platters is negligible.

The total power consumption in milliwatts to satisfy all of the above disk requests using the Shortest Seek Time First disk scheduling algorithm is _______.

Consider a system with $$3$$ processes that share $$4$$ instances of the same resource type. Each process can request a maximum of $$K$$ instances. Resource instances can be requested and released only one at a time. The largest value of $$K$$ that will always avoid deadlock is ____.