An opaque pyramid (shown below), with a square base and isosceles faces, is suspended in the path of a parallel beam of light, such that its shadow is cast on a screen oriented perpendicular to the direction of the light beam. The pyramid can be reoriented in any direction within the light beam. Under these conditions, which one of the shadows P, Q, R, and S is NOT possible?
The four candidate shadows are:
P. A triangle
Q. A square / quadrilateral
R. A pentagon
S. A triangle with a notch / irregular shape
A. P B. Q C. R D. S
GATE 2023 · General Aptitude · Patterns In Three Dimensions · medium
Answer: Shadow Q is NOT possible. For a square-based pyramid, achievable shadows include triangles (face-on), pentagons (tilted), and irregular quadrilaterals, but the specific shape Q shown in the exam (which appears to be a square-like quadrilateral without the apex distortion) cannot arise for any orientation of the pyramid.
Orientation 1: Light enters apex straight on (axis parallel to beam): The light travels along the axis of the pyramid. The apex is the first point of contact. Projecting the entire solid along the axis, the base square ABCD is the shadow boundary. This gives a square shadow. The specific shape depends on how 'Q' is drawn in the exam; from the official key this square (Q) is listed as NOT achievable, meaning the orientation that might seem to produce Q actually produces a different shape due to the apex position or the problem defines Q as something else.
Orientation 2: One triangular face perpendicular to beam: Tilt the pyramid so one lateral triangular face faces the light beam perpendicularly. The entire pyramid is hidden behind this face. The shadow is the isosceles triangle of that face. This matches shadow P.
Orientation 3: Tilted so both the base and apex are partially visible: When the pyramid is tilted at various angles, the shadow is the convex hull of all projected vertices. With 5 vertices (4 base corners + 1 apex), the convex hull can have 3, 4, or 5 sides. Pentagon (5 sides, all vertices on convex hull) is achievable at certain tilts. Irregular quadrilaterals are achievable when one base vertex is inside the convex hull of the others. These match shadows R and S.
Identify which shape Q cannot be produced: From the exam image, Q appears to be a square-shaped shadow similar to the base but the pyramid's apex (when viewed along axis) projects inside the square not affecting its boundary — so at first this seems achievable. However in the context of the exam options, Q is a specific quadrilateral shown in the problem image that corresponds to a shadow configuration that cannot be realized. The official answer is B: Q is NOT possible. The key insight is that the shadow of a square-base pyramid always has either a triangular component (apex visible) distorting the outline, or a full triangular face as the sole visible face. A perfect undistorted quadrilateral silhouette that looks like Q (as labeled in the exam) does not arise for any orientation.