GATE 2025 EE GA Q.9: solution and answer

GATE 2025 · EEGA Q.9 · 2 marks · MCQ

Spheres of unit diameter are centered at $(l, m, n)$ where $l$ , $m$ , and $n$ take every possible integer values.

The distance between two spheres is computed from the center of one sphere to the center of the other sphere. For a given sphere, $x$ is the distance to its nearest sphere and $y$ is the distance to its next nearest sphere. The value of $\frac{y}{x}$ is:

(A)
$22$
(B)
$\frac{1}{2}$
(C)
$2$
(D)
$2$

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Answer and solution

ANSWER

$\frac{y}{x} = 2$ , which is option (C).

Official answer: CSource: Official answer key, GATE 2025 EE (IIT Roorkee)

The centres $(l, m, n)$ , with $l$ , $m$ , $n$ all integers, form a cubic lattice of side $1$ . Every sphere sees the same pattern of neighbours, so we can take the sphere centred at the origin $(0, 0, 0)$ .
By the 3D distance formula, the distance from the origin to the centre $(l, m, n)$ is $l^{2} + m^{2} + n^{2}$ , where $l, m, n$ are integers and not all zero.
The smallest possible value of $l^{2} + m^{2} + n^{2}$ is $1$ , for example at $(1, 0, 0)$ , the next centre along one axis (one cube edge away). So $x = 1 = 1$ . Because the diameter is $1$ , these spheres just touch.
The next possible value of $l^{2} + m^{2} + n^{2}$ is $2$ , for example at $(1, 1, 0)$ , the opposite corner of a face of the unit cube (along the face diagonal). No integer point gives a value between $1$ and $2$ , so $y = 2$ .
$\frac{y}{x} = \frac{2}{1} = 2$ This is option (C).

KEY IDEA

Integer centres form a unit cubic lattice, and the squared distance to any other centre is $l^{2} + m^{2} + n^{2}$ . The nearest centre gives the smallest value of that sum ( $1$ , along an edge) and the next nearest gives the next value ( $2$ , along a face diagonal). So $x = 1$ , $y = 2$ , and $\frac{y}{x} = 2$ .

WATCH OUT FOR

Taking the next nearest centre as the second one along the same axis, $(2, 0, 0)$ , which gives $y = 2$ and option (D). The face-diagonal centre $(1, 1, 0)$ at $2$ is closer. Other slips are jumping to the body diagonal $(1, 1, 1)$ at $3$ , which is the third-nearest, or inverting the ratio to get $\frac{1}{2}$ (option B).

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GATE 2025 EE GA Q.9: solution and answer

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