GATE 2024 DA Q.53: solution and answer

Official answer: A, DSource: Official answer key, GATE 2024 DA (IISc Bengaluru)

1. Linear separability means one straight line (a hyperplane in 2-D) puts every circle on one side and every square on the other. Check each panel in turn.

2. (i) The circles are at $(1,3),(2,3),(1,2)$, all with $y\ge 2$. The squares are at $(2,1),(3,1),(2,0.3)$, all with $y\le 1$. The horizontal line $y=1.5$ separates them, so (i) is linearly separable.

3. (ii) The circle at $(2,2)$ is the centre of the four squares $(1,2),(3,2),(2,3),(2,1)$. It is the midpoint of the squares $(1,2)$ and $(3,2)$, so it lies in the convex hull of the squares. A line with all squares on one side puts their whole hull, and so the circle, on that side too. So (ii) is not linearly separable.

4. (iii) This is the XOR layout. The squares are at $(1,3),(3,1)$ and the circles at $(3,3),(1,1)$. The segment joining the two squares and the segment joining the two circles both have midpoint $(2,2)$, so they cross and the convex hulls intersect. So (iii) is not linearly separable.

5. (iv) The squares are at $(1,2.2),(2,2.6),(1,1.2)$ and the circles at $(2,0.4),(3,1.5),(3,0.5)$. Take the line $y=0.5x+0.3$. At $x=1,2,1$ its heights are $0.8,1.3,0.8$, and the squares' heights $2.2,2.6,1.2$ are all above these. At $x=2,3,3$ its heights are $1.3,1.8,1.8$, and the circles' heights $0.4,1.5,0.5$ are all below these. The classes are separated, so (iv) is linearly separable.

6. True statements: (i) and (iv), so options A and D.