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    Paper Folding, Cutting and Unfolding PYQs for GATE CS

    Solve 3+ Paper Folding, Cutting and Unfolding previous year questions for GATE CS with answers and detailed solutions. Free sample questions below.

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    Question 1
    2025 Slot Set1 PYQ
    Level 3: Exam Standard
    A square paper, shown in figure (I), is folded along the dotted lines as shown in the figures (II) and (III). Then a few cuts are made as shown in figure (IV). Which one of the following patterns will be obtained when the paper is unfolded?

    Note: The figures shown are representative.

    (I)(II)(III)(IV)
    Question 2
    2024 Slot Set1 PYQ
    Level 3: Exam Standard

    A rectangular paper of is folded 3 times. Each fold is made along the line of symmetry, which is perpendicular to its long edge. The perimeter of the final folded sheet (in cm) is

    Question 3
    2021 Slot Set2 PYQ
    Level 3: Exam Standard

    A transparent square sheet shown above is folded along the dotted line. The folded sheet will look like ________.
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    Paper Folding, Cutting and Unfolding PYQs for GATE CS

    Solve 3+ Paper Folding, Cutting and Unfolding previous year questions for GATE CS with answers and detailed solutions. Free sample questions below.

    Chapter Roadmap: Paper Folding, Cutting and Unfolding

    1. Paper Folding and Final Dimensions

    Master how sequential folds alter the length, width, and perimeter of a given shape. This is the foundational step for all spatial transformation problems.

    Foundational

    2. Cut Patterns and Unfolded Figures

    Visualize the symmetry and exact placement of holes or cuts when a folded paper is completely unfolded.

    High Weightage

    Topic Hero: The Mechanics of Paper Folding

    L/2 L/2

    Core Intuition

    Folding a flat shape along a line of symmetry is a geometric transformation that halves the dimension perpendicular to the fold line.

    • Parallel Dimension: Remains completely unchanged.
    • Perpendicular Dimension: Is exactly halved.
    • Thickness: Doubles with each fold, but is ignored for perimeter and area calculations of the visible face.
    Visualizing the Axis: If a fold is described as "perpendicular to the long edge", the fold line itself runs parallel to the short edge. Consequently, it is the long edge that gets bisected.

    Paper Folding, Cutting and Unfolding: Solved Questions with Step-by-Step Explanations (3 Problems)

    Question 1 · Spatial Aptitude · 2025_Set1 MCQ
    A square paper, shown in figure (I), is folded along the dotted lines as shown in the figures (II) and (III). Then a few cuts are made as shown in figure (IV). Which one of the following patterns will be obtained when the paper is unfolded?

    Note: The figures shown are representative.

    (I)(II)(III)(IV)
    1. A.
    2. B.
    3. C.
    4. D.
    Correct Answer:

    C

    Step-by-Step Solution

    Key idea: Reverse unfolding using the reflection principle. We start from the final cut state and unfold step-by-step, mirroring the cuts across the fold lines.

    Step 1: Analyze the final folded state (Figure IV). It is a right-angled triangle. The cuts are:

    • A rectangular hole inside the triangle.
    • A triangular notch on the top edge (hypotenuse of this small triangle).
    • A rectangular notch on the left edge (a leg of this small triangle).

    Step 2: Understand the fold mapping.

    • The first fold was along the diagonal of the original square.
    • The second fold was along the altitude to the hypotenuse, creating the small triangle in Figure IV.
    • The hypotenuse of the small triangle in Figure IV corresponds to the outer perimeter of the original square.
    • The legs of the small triangle correspond to the diagonal fold lines of the original square.

    Step 3: Unfold and reflect.

    • The rectangular hole is offset from the center (the right-angle vertex). Since the paper has 4 layers here, unfolding will replicate this hole 4 times, symmetrically placed in the four quadrants. The hole's sides are parallel to the hypotenuse (original square's sides), so the unfolded holes will be upright squares/rectangles, not rotated diamonds.
    • The triangular notch on the hypotenuse (perimeter) will be replicated on all four sides of the original square. Since the cut removes material from the edge, the unfolded representation must show a notch pointing inward (missing material from the boundary).

    Step 4: Evaluate the options.

    • Option A shows notches pointing outward (outside the boundary), which is an incorrect representation of a removed edge piece.
    • Option B shows diamond-shaped holes, which implies the cut was rotated 45 degrees relative to the sides. This is incorrect.
    • Option C correctly shows upright square holes and inward-pointing triangular notches on all four edges, matching the reflection of the perimeter cut.
    • Option D shows a single central hole, which would only occur if the cut was made exactly at the right-angle vertex (the center of the original square).

    Answer: C

    Question 2 · Spatial Aptitude · 2024_Set1 MCQ

    A rectangular paper of is folded 3 times. Each fold is made along the line of symmetry, which is perpendicular to its long edge. The perimeter of the final folded sheet (in cm) is

    1. A.

    2. B.

    3. C.

    4. D.

    Correct Answer:

    A

    Step-by-Step Solution

    Key idea: Tracking dimensions through sequential folds by dynamically identifying the "long edge" at each step.

    Initial Dimensions: Length cm, Width cm.

    Rule: "Each fold is made along the line of symmetry, which is perpendicular to its long edge." This means we always halve the current longest dimension.

    Step 1: First fold.

    Current dimensions: .

    Long edge is cm.

    Fold perpendicular to the cm edge halves it.

    New dimensions: cm.

    Step 2: Second fold.

    Current dimensions: cm.

    Long edge is now cm.

    Fold perpendicular to the cm edge halves it.

    New dimensions: cm.

    Step 3: Third fold.

    Current dimensions: cm.

    Long edge is now cm.

    Fold perpendicular to the cm edge halves it.

    New dimensions: cm.

    Step 4: Calculate the final perimeter.

    Final dimensions are cm and cm.

    Perimeter

    cm.

    Common Trap: Assuming the fold is always perpendicular to the original cm edge. If you did that, you would get cm, leading to a perimeter of cm (Option D). The phrase "its long edge" requires re-evaluating the longest side after each fold.

    Answer: A

    Question 3 · Spatial Aptitude · 2021_Set2 MCQ

    A transparent square sheet shown above is folded along the dotted line. The folded sheet will look like ________.
    1. A.
    2. B.
    3. C.
    4. D.
    Correct Answer:

    A

    Step-by-Step Solution

    Key idea: Folding a transparent sheet reflects the drawn patterns across the fold line, and because it is transparent, both the original and mirrored patterns are visible simultaneously (superposition).

    Step 1: Identify the fold line and direction.

    The dotted line is vertical, passing through the center (). The options show the right half remaining, implying the left half is folded over onto the right half.

    Step 2: Reflect the left-side patterns onto the right side.

    Original patterns on the LEFT ():

    • A curve segment in the bottom-left quadrant.
    • Two vertical line segments.

    Original patterns on the RIGHT ():

    • A curve segment in the bottom-right quadrant.
    • Two slanted/horizontal line segments.

    Step 3: Analyze the superposition on the right half.

    • The left curve mirrors across to perfectly overlap with the existing right curve (they form a continuous symmetric wave).
    • The two vertical lines on the left will mirror to become two vertical lines on the right side.
    • The original right-side patterns (two slanted lines) remain unchanged and visible through the transparent sheet.

    Step 4: Evaluate the options.

    • Option A shows both the two vertical lines (mirrored from the left) and the two slanted lines (original on the right). This matches our superposition analysis.
    • Option B distorts the curves and misses the line reflections.
    • Option C only shows the slanted lines, ignoring the mirrored vertical lines from the left half.
    • Option D shows a connected loop, which does not result from reflecting disjoint vertical lines.

    Answer: A

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