$\left( {7 - \lambda } \right)\left( {31 - \lambda } \right) < 0$ (since centres lie opposite side)
$\lambda \in \left( {7,13} \right)\,\,\,\,\,\,\,\,\,\,\,.......\left( 1 \right)$
$\left| {\frac{{7 - \lambda }}{5}} \right| \ge 1$ and $\left| {\frac{{31 - \lambda }}{5}} \right| \ge 2$
$\left| {7 - \lambda } \right| \ge 5\,$ and $\,\,\left| {31 - \lambda } \right| \ge 10\,$
$\lambda \le 2\,\,$ or $\lambda \ge 12\,\,\,\,\,\,\,\,\,......\left( 2 \right)$
and $\lambda \le 21\,\,$ or $\lambda \ge 41\,\,\,\,\,\,\,\,\,\,\,.....\left( 3 \right)$
$\left( 1 \right) \cap \left( 2 \right) \cap \left( 3 \right)$
$\lambda \in \left[ {12,21} \right]$
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$\mathrm{A}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{Z} \times \mathbb{Z}:(\mathrm{x}-2)^{2}+\mathrm{y}^{2} \leq 4\right\}$
$\mathrm{B}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{Z} \times \mathbb{Z}: \mathrm{x}^{2}+\mathrm{y}^{2} \leq 4\right\} \text { and }$
$\mathrm{C}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{Z} \times \mathbb{Z}:(\mathrm{x}-2)^{2}+(\mathrm{y}-2)^{2} \leq 4\right\}$
If the total number of relation from $\mathrm{A} \cap \mathrm{B}$ to $\mathrm{A} \cap \mathrm{C}$ is $2^{\mathrm{p}}$, then the value of $\mathrm{p}$ is :