$($a$)($i$)$ Write down the $16-$bit and $32-$bit $1\text{'}$s complement and $2\text{'}$s complement in

hexadecimal of $-23.($total $4$ values$)[4]$

$($ii$)$ What is the rule for sign extension from $m-$bit $2\text{'}$s complement number to

$n-$bit $2\text{'}$s complement number, where $n>m.$ Show that the algorithm is

correct. $6$

$($b$)$ A machine uses $36-$bit floating number representation as follows.

$S,10-$bit exponent $(E),25-$bit significand $(M)$

The value represented is given by $(-1{)}^{S}1.M{2}^{E-b},$ where $b$ is the bias in excess

representation.

$($i$)$ What is the value of $b?[1]$

$($ii$)$ Write down the bit pattern corresponding to the value $-26\mathrm{.}5625[3]$

$($iii$)$ Write down the value corresponding to the bit pattern c$07780000[3]$

$($iv$)$ In this representation scheme, we cannot represent the value $0.$ Suggest a

way how this can be solved. $[2]$

$($v$)$ Another designer uses the same bit allocation for sign, exponent, and sig$-$

nificand, but uses the following formula for the value:

$V=(-1{)}^{S}1.M{4}^{E-b}$

Write down one advantage and one disadvantage of this scheme. $[3]$