GATE 2023 EC GA Q.10: solution and answer

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

1. Folding so that a long edge (100 cm) meets the opposite long edge makes the fold line parallel to the long edges. It halves the short side and leaves the 100 cm length unchanged. Over the 5 folds the width goes $32\to 16\to 8\to 4\to 2\to 1$ cm, which gives the stated final size of $100\text{cm}\times 1\text{cm}$.

2. Each fold doubles the number of layers, so after $n$ folds there are $2^n$ layers. Here $2^5=32$ layers, each $1\text{cm}$ wide, and $32\times 1\text{cm}=32\text{cm}$ is the original width, so the numbers agree.

3. When the sheet is unfolded, it is divided into 32 strips of $100\text{cm}\times 1\text{cm}$, all parallel to the long edge. Adjacent strips were joined at a fold, so every boundary between two adjacent strips is a crease.

4. Cutting a width into $k$ equal parts takes $k-1$ dividing lines (fence-post rule). That gives $32-1=31$ creases, one at every whole centimetre from $1\text{cm}$ to $31\text{cm}$ across the width.

5. Cross-check fold by fold. Before fold $k$ there are $2^{k-1}$ layers, and the fold puts one new crease in each layer, each at a new position (the middle of each current strip). Total: $1+2+4+8+16=2^5-1=31$.

6. The answer is 31, option (C).