To solve this problem, we need to compare the acceleration due to gravity at a height above the Earth's surface and at a depth below it.$\\$The formula for acceleration due to gravity at a height $h$ above the Earth's surface is:$\\ g_h = g \left(1 - \frac{2h}{R}\right) \\$The formula for acceleration due to gravity at a depth $d$ below the Earth's surface is:$\\ g_d = g \left(1 - \frac{d}{R}\right) \\$Given that the acceleration due to gravity at a height of 1 km is the same as at a depth $d,$ we equate the two expressions:$\\ g \left(1 - \frac{2 \times 1}{R}\right) = g \left(1 - \frac{d}{R}\right) \\$Cancel $g$ from both sides:$\\ 1 - \frac{2}{R} = 1 - \frac{d}{R} \\$Rearrange to solve for $d: \\ \frac{d}{R} = \frac{2}{R} \\ d = 2 \\$Thus, the depth $d$ is $2$ km.$\\$Therefore, the correct option is Option 3.