GATE EC Number Representations questions

What is the $10^{\prime} s$ complement of $(47)_{10}?$

In a number system of base $r$, the equation $x^{2}-12 x+37=0$ has $x=8$ as one of its solutions. The value of $r$ is $\_\_\_\_\_\_\_$.

​​​​​​​If $(1235)_{x}\:=\:(3033)_{y}$, where $x$ and $y$ indicate the bases of the corresponding numbers, then

Two sequences $x_{1}[n]$ and $x_{2}[n]$ have the same energy. Suppose $x_{1}[n]=\alpha\: 0.5 ^{n}\:u[n],$ where $\alpha$ is a positive real number and $u[n]$is the unit step sequence. Assume ݊$$x_{2}[n]= \begin{cases} \sqrt{1.5}& \text{for } n = 0,1 \\ 0 &\text{otherwise.} \end{cases}$$ Then the value of $\alpha$ is _________.

The number of bytes required to represent the decimal number $1856357$ in packed BCD (Binary Coded Decimal) form is ___________.

$\mathrm{X}=01110$ and $\mathrm{Y}=11001$ are two $5$-bit binary numbers represented in two's complement format. The sum of $\mathrm{X}$ and $\mathrm{Y}$ represented in two's complement format using $6$ bits is

A new Binary Coded Pentary (BCP) number system is proposed in which every digit of a base-$5$ number is represented by tis corresponding $3$-bit binary code. For example, the base-$5$ number $24$ will be represented by its BCP code $010100.$ In this numbering system, the BCP code $100010011001$ corresponds to the following number in base-$5$ …

Decimal $43$ in Hexadecimal and $\text{BCD}$ number system is respectively

The range of signed decimal numbers that can be represented by $6$-bite $1$'s complement number is

A digital system is required to amplify a binary-encoded audio signal. The user should be able to control the gain of the amplifier from a minimum to a maximum in $100$ increments. The minimum number of bits required to encode, in straight binary, is

$11001, 1001$ and $111001$ correspond to the $2$'s complement representation of which one of the following sets of number ?

$4$-bit $2$'s complement representation of a decimal number is $1000.$ The number is

The $2$'s complement representation of $-17$ is

An equivalent $2^{\prime} s$ complement representation of the $2$'s complement number $1101$ is

A signed integer has been stored in a byte using the $2$'s complement format. We wish to store the same integer in a $16$ bit word. We should

$2$'s complement representation of a $16$ – bit number (one sign bit and $15$ magnitude bits) is $\text{FFFI}$. Its magnitude in decimal representation is