Design a digital system with three 16-bit registers AR, BR, and CR and 16-bit data input IN to perform the following operations assuming a two's complement representation and ignoring overflow: a) Transfer two 16-bit signed numbers to AR and BR on successive clock cycles after a go signal G is enabled. b) If the number in AR is positive but nonzero, multiply the contents of AR by two and transfer the result to register CR. c) If the number in AR is negative, multiply the contents of BR by two and transfer the result to register CR. d) If the number in AR is zero, reset register CR to 0.

Suppose there is a relation r(a, b, c) with a b+-tree index with search keys (a, b).1. what is the worst-case cost of finding records satisfying 10 < a < 50 using this index, in terms of the number of records n1, retrieved and the height h of the tree? 2. what is the worst-case cost of finding records satisfying 10 < a < 50 and 5 < b < 10 using this index, in terms of the number of records n2 that satisfy this selection, as well as n1 and h defined above? 3. under what conditions on n1 and n2, would the index be an efficient way of finding records satisfying the condition from part (2)?