$1.$ a$)$ Assume that your car's odometer shows the mileage in octal. If the current reading of the odometer is $24,516,$ how many miles $($in decimals$)$ has the car been driven? What will be the new reading if the car is driven $23($decimal$)$ miles?

b$)$ What will be the current odometer reading in the problem given earlier, if the odometer uses a $5-$ digit hexadecimal representation? What will be the new reading after driving $23($decimal$)$ miles?

$2.$ Find the $1$s and $2$s complements of the following binary numbers:

a$)10010$ b$)110010$ c$)0010101$ d$)10110\mathrm{.}0101$ e$)1102\mathrm{.}1100$ f$)111010\mathrm{.}0011$

$3.$ Determine the X$+$Y$,$ X $-$ Y$,$ X x Y$,$ and X$/$Y in each of the following sets of binary numbers:

a$)$ X $=1101010$ & Y $=10111$ b$)$ X $=101101$ & Y $=1111$ c$)$ X $-1011$ & Y $=1111$

d$)$ X $=110\mathrm{.}11$ & Y $=10\mathrm{.}11$ e$)$ X $=1110\mathrm{.}101$ & Y $=1011\mathrm{.}10$

$4.$ Determine the X$+$Y$,$ X $-$ Y$,$ X x Y$,$ and X$/$Y in each of the following sets of hexadecimal numbers:

a$)$ X$=1$CF & Y $=$ B$6$ b$)$ X $=1$B$59$A & Y $=$ C$23$ c$)$ X$=$ B$6$ & Y $=1$CF d$)$ X $=234$F$\mathrm{.}16$ & Y $=456$E

$5.$ If $(130)$x $=(20)10,$ find the value of x $($x is a positive integer$).$

$6.$ Represent the following numbers in the IEEE standard floating $-$ point format:

a$)(11010\mathrm{.}010)2$ b$)(432\mathrm{.}26)10$ c$)(10100112\mathrm{.}1001)2$

$7.$ A $16-$ bit register contains the following: $0100100101010111$

Interpret the contents as $-$

a$)$ a BCD number b$)$ a binary number c$)$ an excess $-3$ number d$)$ two ASCII characters

$8.$ Assume that you have a $64-$ bit computer. That means, all data items are represented. in a $64-$ bit field

a$)$ What is the decimal rage of value that can be represented using fixed$-$ point and IEEE standard floating$-$point representation?

b$)$ Which one has a larger range?

c$)$ Why do we need to use the one with a smaller range?