GATE EC 1998 Set 1 Q2.4 — Number Representations

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