Q174 / 270 · NDA 2017 (Phase 1) · Mathematics

In the binary equation $(1 \mathrm{p} 101)_2+(10 \mathrm{ql})_2=(100 \mathrm{r} 00)_2$ where $\mathrm{p}, \mathrm{q}$ and r are binary digits, what are the possible values of $p, q$ and $r$ respectively?

  • Correct answer
    A
    0,1,0
  • B
    1,1,0
  • C
    0,0,1
  • D
    1,0,1

Solution

$\begin{aligned} & (1 \mathrm{p} 101)_2+(10 \mathrm{q} 1)_2=(100 \mathrm{r} 00)_2 \\ & \Rightarrow\left(1 \times 2^4+\mathrm{p} \times 2^3+1 \times 2^2+0 \times 2^1+1 \times 2^0\right) \\ & +\left(1 \times 2^3+0 \times 2^2+\mathrm{q} \times 2^1+1 \times 2^0\right) \\ & =1 \times 2^5+0+0+\mathrm{r} \times 2^2+0+0\end{aligned}$ $\begin{aligned} & \Rightarrow 16+8 p+4+1+8+2 q+1=32+4 r \\ & \Rightarrow 30+8 p+2 q=32+4 r \\ & \Rightarrow 8 p+2 q=2+4 r \end{aligned}$ from options, substitute $\mathrm{p}=0, \mathrm{q}=1, \mathrm{r}=0$ we get $0+2(1)=2+0 \Rightarrow 2=2 \text {. }$

Hence, the correct answer is option: 'A'.

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