GATE 2023 EC GA Q.5: solution and answer

Official answer: CSource: Official answer key, GATE 2023 EC (IIT Kanpur)

1. Sort the lines by direction. Horizontal and vertical: only the four sides of the outer square. Slanting one way ($\nearrow$): the diagonal $DB$ and the diamond sides $SP$ and $RQ$. Slanting the other way ($\searrow$): the diagonal $AC$ and the diamond sides $SR$ and $PQ$.

2. A rectangle needs every corner to be a right angle. The horizontal and vertical lines meet each other at $90^{\circ }$ but meet the slanting lines at $45^{\circ }$, so a rectangle cannot mix the two kinds of line. The only rectangle made from horizontal and vertical lines is the outer square $ABCD$: 1.

3. The six slanting lines form a tilted grid: 3 parallel lines one way ($SP$, $DB$, $RQ$) and 3 parallel lines the other way ($SR$, $AC$, $PQ$), and the two families are perpendicular. Each line of one family crosses every line of the other family at a point that is actually drawn (the 9 crossings are $S$, $P$, $Q$, $R$, the centre $O$ and the four midpoints of the diamond's sides). So the diamond is a $2\times 2$ grid of small squares.

4. Rule: a grid formed by $m$ parallel lines crossing $n$ perpendicular parallel lines, with every crossing drawn, contains $\left(\genfrac{}{}{0ex}{}{m}{2}\right)\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ rectangles, one for each choice of two lines from each family.

5. Tilted rectangles: $\left(\genfrac{}{}{0ex}{}{3}{2}\right)\times \left(\genfrac{}{}{0ex}{}{3}{2}\right)=3\times 3=9$. These are the 4 small squares, the 4 rectangles made of two small squares side by side (2 in each direction), and the diamond $PQRS$ itself: $4+4+1=9$.

6. Outside the diamond, the diagonals only cut the four corner triangles in two and have no parallel partner there, so they form no further rectangle.

7. Total $=1+9=10$, which is option (C).