GATE EC 2026 Set 1 Q35 — Number Representations

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-$5$ …