GATE 2022 EE GA Q.5: solution and answer

GATE 2022 · EEGA Q.5 · 1 mark · MCQ

The figure below shows the front and rear view of a disc, which is shaded with identical patterns. The disc is flipped once with respect to any one of the fixed axes 1-1, 2-2 or 3-3 chosen uniformly at random.

What is the probability that the disc DOES NOT retain the same front and rear views after the flipping operation?

GATE 2022 EE GA question 5: Two equal circles side by side, captioned "Front View" (left) and "Rear View" (right). — Figure from the GATE 2022 EE paper. It is part of the question.

(A)
0
(B)
$\frac{1}{3}$
(C)
$\frac{2}{3}$
(D)
1

Answer and solution

ANSWER

The probability that the disc does not keep the same front and rear views is $\frac{2}{3}$ , option (C).

Official answer: CSource: Official answer key, GATE 2022 EE (IIT Kharagpur)

A flip about an axis that lies in the plane of the disc is a $18 0^{\circ}$ turn about that axis. The two faces swap sides, and every point of the disc moves to its mirror image in that axis line, as seen from the front.
Walking round to look at the disc from behind is the same as turning the disc $18 0^{\circ}$ about the vertical axis 1-1 and looking from the front. That is why 1-1 stays vertical in the rear view while the labels of 2-2 and 3-3 swap sides.
Flip about 1-1. By step 2, the viewer in front now sees exactly what the viewer behind saw before, and the other way round. So the new front view is the old rear view and the new rear view is the old front view. The two views are identical, so both stay the same. The views are retained.
Where the big sector sits. On each face, the large horizontally hatched sector is centred on the lower half of 1-1, so it points straight down. The rear viewer's 'down' is the same direction as the front viewer's. This is the only sector of that size, so a view can match the original only if its big sector still points straight down.
Flip about 2-2. Reflect a line in a second line through the same point that makes an angle $α$ with it: the image makes an angle $2 α$ with the original line. Axis 2-2 makes $α \approx 4 5^{\circ}$ with 1-1. So the flip sends the downward half of 1-1 to a radius $2 α \approx 9 0^{\circ}$ away, which is the left horizontal radius, because in the front view 2-2 runs from upper right to lower left. The face now in front has its big sector pointing left, not down, so the views are not retained. Any $α$ strictly between $0^{\circ}$ and $9 0^{\circ}$ gives the same conclusion, because $2 α$ is then strictly between $0^{\circ}$ and $18 0^{\circ}$ .
Flip about 3-3. Axis 3-3 is the mirror image of 2-2 in 1-1, so the same argument sends the downward half of 1-1 to the right horizontal radius. The big sector now points right, and the views are not retained.
Of the three equally likely axes, two (2-2 and 3-3) change the views: $P (views not retained) = \frac{2}{3}$ This is option (C).

KEY IDEA

Looking at the disc from behind is the same as turning it $18 0^{\circ}$ about the vertical axis 1-1. A flip about 1-1 therefore just swaps the two identical views, and nothing changes. A flip about a slanted axis mirrors the disc in that slanted line. That swings the big shaded sector from pointing down to pointing sideways, so the views change. Two of the three axes do this, which gives $\frac{2}{3}$ .

WATCH OUT FOR

One mistake runs: 'both faces carry the same pattern, so any flip just swaps identical faces', giving probability 0, option (A). That misses that a flip about a slanted axis mirrors the disc in that slanted line, which moves the big sector away from the vertical. The opposite mistake comes from forgetting that the rear view is drawn as seen from behind (mirrored left-right). A student who does that thinks even the flip about 1-1 changes the picture and answers 1, option (D).

From the GATE 2022 EE paper. Every question of this paper · Download the full paper · All 23 Spatial questions, with notes

GATE 2022 EE GA Q.5: solution and answer

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