GATE 2024 IN GA Q.10: solution and answer

Official answer: ASource: Official answer key, GATE 2024 IN (IISc Bengaluru)

1. A body diagonal joins two opposite vertices, say $P$ (top) and $Q$ (bottom). Stand the cube on $Q$ so that $PQ$ is vertical. With coordinates, take $Q=(0,0,0)$ and $P=(1,1,1)$, so the axis is the line through $(1,1,1)$.

2. Height along the axis is proportional to $x+y+z$. The three neighbours of $P$, $(0,1,1)$, $(1,0,1)$ and $(1,1,0)$, all have $x+y+z=2$, so they sit at height $\frac{2}{3}$ of the diagonal (the upper level). The three neighbours of $Q$, $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$, have $x+y+z=1$, so they sit at height $\frac{1}{3}$ (the lower level). No other vertex is at either height.

3. Seen from above, each level is an equilateral triangle, so its three vertices are $\frac{360^{\circ }}{3}=120^{\circ }$ apart around the axis. The lower triangle points the opposite way to the upper one: the two are offset by $60^{\circ }$, and together they make the familiar zig-zag ring of 6 vertices.

4. Turning about the vertical axis never changes a point's height. So if the cube is to look exactly the same, the three upper-level vertices (the only vertices at height $\frac{2}{3}$) must land on one another. Because they are $120^{\circ }$ apart, the angle of rotation must be a multiple of $120^{\circ }$.

5. $120^{\circ }$ works. The rotation by $120^{\circ }$ about this diagonal is the cyclic swap $(x,y,z)\to (z,x,y)$. It sends every corner of the cube to another corner, for example $(1,0,0)\to (0,1,0)\to (0,0,1)\to (1,0,0)$, so the cube lands exactly on itself and the view does not change.

6. $60^{\circ }$ fails. It turns an upper-level vertex to the direction of a lower-level vertex while keeping it at height $\frac{2}{3}$, and there is no corner there. $180^{\circ }=120^{\circ }+60^{\circ }$ fails for the same reason. $90^{\circ }$ fails because it is not a multiple of $120^{\circ }$: the three edges at $P$, which are $120^{\circ }$ apart, would not land on one another.

7. So the body diagonal is a 3-fold symmetry axis of the cube, and the smallest rotation that leaves the view unchanged is $\frac{360^{\circ }}{3}=120^{\circ }$. Answer: (A).