GATE CS Number Representations (Fixed and Floating Point) questions

Consider the $8$ -bit signed integers $X, Y$ and $Z$ represented using the sign-magnitude form. The binary representations of $X$ and $Y$ are as follows: $$X: 10110100 \quad Y: 01001100$$ Which of the following operations to compute $Z$ result(s) in an arithmetic overflow?

Consider the real valued variables $X, Y$ and $Z$ represented using the IEEE $754$ singleprecision floating-point format. The binary representations of $X$ and $Y$ in hexadecimal notation are as follows: $$X: 35 \mathrm{C} 00000 \quad Y: 34 \mathrm{A} 00000$$ Let $Z=X+Y$. Which one of the following is the binary representation of $Z$, in …

In a system, numbers are represented using $4$-bit two's complement form. Consider four numbers $N 1=1011, N 2=1101, N 3=1010$ and $N 4=1001$ in the system. Which of the following operations will result in arithmetic overflow?

The $32$-bit $\text{IEEE}$ $754$ single precision representation of a number is $0\text{xC}2710000$. The number in decimal representation is $\_\_\_\_$. (rounded off to two decimal places)

The number $-6$ can be represented as $1010$ in $4$-bit $2$'s complement representation. Which of the following is/are CORRECT $2$ 's complement representation(s) of $-6$?

Three floating point numbers $X, Y$, and $Z$ are stored in three registers $\mathrm{R}_{\mathrm{X}}, \mathrm{R}_{\mathrm{Y}}$, and $\mathrm{R}_{\mathrm{Z}}$, respectively in $IEEE$ $754$ single precision format as given below in hexadecimal: …

Consider a system that uses $5$ bits for representing signed integers in $2$ 's complement format. In this system, two integers $A$ and $B$ are represented as $A$=$01010$ and $B$=$11010$. Which one of the following operations will result in either an arithmetic overflow or an arithmetic underflow?

Which of the following is/are EQUAL to $224$ in radix - $5$ (i.e., base - $5$) notation?

A particular number is written as $132$ in radix-$4$ representation. The same number in radix-$5$ representation is _____________.

Consider the $\textsf{IEEE-754}$ single precision floating point numbers $\text{P} = \textsf{0xC1800000}$ and $\text{Q} = \textsf{0x3F5C2EF4}.$ Which one of the following corresponds to the product of these numbers $(\text{i.e., P} \times \mathrm{Q} ),$ represented in the $\textsf{IEEE-754}$ single precision format?

Let $\text{R1}$ and $\text{R2}$ be two $4 – \text{bit}$ registers that store numbers in $2\text{’s}$ complement form. For the operation $\text{R1 + R2},$ which one of the following values of $\text{R1}$ and $\text{R2}$ gives an arithmetic overflow?

Consider three floating point numbers $\textit{A, B}$ and $\textit{C}$ stored in registers $\text{R}_{\text{A}}, \text{R}_{\text{B}}$ and $\text{R}_{\text{C}},$ respectively as per $\textsf{IEEE-754}$ single precision floating point format. The $\text{32-bit}$ content stored in these registers $\text{(in hexadecimal form)}$ are as follows. …

Let the representation of a number in base $3$ be $210$. What is the hexadecimal representation of the number?

Consider the following representation of a number in $\text{IEEE 754}$ single-precision floating point format with a bias of $127$.$$S: 1\quad\quad E:\; 10000001\quad\quad F:\;11110000000000000000000$$ Here $S, \;E$ and $F$ denote the sign, exponent, and fraction components of the floating point representation. The decimal value corresponding to the …

The format of the single-precision floating point representation of a real number as per the $\text{IEEE 754}$ standard is as follows: $$\begin{array}{|c|c|c|} \hline \text{sign} & \text{exponent} & \text{mantissa} \\ \hline \end{array}$$ Which one of the following choices is correct with respect to the smallest normalized positive number …

If $x$ and $y$ are two decimal digits and $(0.1101)_2 = (0.8xy5)_{10}$, the decimal value of $x+y$ is ___________

If the numerical value of a $2$-byte unsigned integer on a little endian computer is $255$ more than that on a big endian computer, which of the following choices represent(s) the unsigned integer on a little endian computer?

Consider three registers $R1$, $R2$, and $R3$ that store numbers in $\textsf{IEEE-754}$ single precision floating point format. Assume that $R1$ and $R2$ contain the values (in hexadecimal notation) $\textsf{0x42200000}$ and $\textsf{0xC1200000},$ respectively. If $R3=\frac{R1}{R2}$, what is the value stored in $R3$?

In $16$-bit $2$’s complement representation, the decimal number $-28$ is:

Consider $Z=X-Y$ where $X, Y$ and Z are all in sign-magnitude form. X and Y are each represented in $n$ bits. To avoid overflow, the representation of $Z$ would require a minimum of: