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    Symmetry, Rotation and Reflection PYQs for GATE CS

    Solve 3+ Symmetry, Rotation and Reflection previous year questions for GATE CS with answers and detailed solutions. Free sample questions below.

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    Question 1
    2024 Slot Set1 PYQ
    Level 3: Exam Standard
    The least number of squares to be added in the figure to make AB a line of symmetry is

    A B
    Question 2
    2023 PYQ
    Level 3: Exam Standard
    Which one of the options best describes the transformation of the 2-dimensional figure P to Q, and then to R, as shown?

    POperation 1QOperation 2R
    Question 3
    2021 Slot Set1 PYQ
    Level 3: Exam Standard

    A circular sheet of paper is folded along the lines in the directions shown. The paper, after being punched in the final folded state as shown and unfolded in the reverse order of folding, will look like _______.
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    Symmetry, Rotation and Reflection PYQs for GATE CS

    Solve 3+ Symmetry, Rotation and Reflection previous year questions for GATE CS with answers and detailed solutions. Free sample questions below.

    Chapter Roadmap: Symmetry, Rotation and Reflection

    1. Line Symmetry and Symmetric Completion
    Current Topic: Master mirror images and completing half-drawn figures.
    2. Rotation and Reflection Transformations
    Next: Tracking shape changes through turns and flips.

    What is Line Symmetry?

    A figure has line symmetry (or reflectional symmetry) if there exists a straight line that divides the figure into two identical halves that are mirror images of each other.

    • This dividing line is called the line of symmetry or axis of symmetry.
    • The Folding Test: If you were to fold the figure along this line, the two halves would perfectly overlap, with all edges and vertices matching exactly.

    Symmetry, Rotation and Reflection: Solved Questions with Step-by-Step Explanations (3 Problems)

    Question 1 · Spatial Aptitude · 2024_Set1 MCQ
    The least number of squares to be added in the figure to make AB a line of symmetry is

    A B
    1. A.

    2. B.

    3. C.

    4. D.

    Correct Answer:

    C

    Step-by-Step Solution

    Key idea: This is a line symmetry completion problem. We need to identify which squares are missing on one side of the axis AB so that they mirror the squares present on the other side.

    Step 1: Identify the axis of symmetry. The line AB is horizontal.

    Step 2: Analyze the existing squares relative to the axis.

    Let's define the grid coordinates relative to the axis AB (y=0).

    Existing squares below the axis (y < 0):

    1. At x=150, y=-40 (immediately below the axis, left side) -> Mirror should be at x=150, y=+40.
    2. At x=190, y=-80 (two steps below) -> Mirror should be at x=190, y=+80.

    Existing squares above the axis (y > 0):

    1. At x=190, y=+40 (immediately above) -> Mirror should be at x=190, y=-40.

    Existing squares on the right side:

    1. At x=300, y=-40 -> Mirror at x=300, y=+40.
    2. At x=340, y=-40 -> Mirror at x=340, y=+40.
    3. At x=340, y=-80 -> Mirror at x=340, y=+80.

    Step 3: Count the missing mirrors.

    • For square at (150, -40), we need a square at (150, +40). (1 square)
    • For square at (190, -80), we need a square at (190, +80). (1 square)
    • For square at (190, +40), we need a square at (190, -40). (1 square)
    • For square at (300, -40), we need a square at (300, +40). (1 square)
    • For square at (340, -40), we need a square at (340, +40). (1 square)
    • For square at (340, -80), we need a square at (340, +80). (1 square)

    Total missing squares = 6? Wait, let me re-examine the image carefully.

    Let's look at the SVG coordinates provided in the prompt:

    • Rect 1: x=190, y=50 (Above axis y=90? No, axis is y=90. Height 40. So y ranges 50-90. It touches the axis from above.)
    • Rect 2: x=150, y=90 (Touches axis from below. y ranges 90-130.)
    • Rect 3: x=190, y=130 (Below Rect 1? No, y=130-170. It is 2 units down from axis if unit is 40.)
    • Rect 4: x=300, y=90 (Touches axis from below. y ranges 90-130.)
    • Rect 5: x=340, y=90 (Touches axis from below. y ranges 90-130.)
    • Rect 6: x=340, y=130 (Below Rect 5. y ranges 130-170.)

    Axis is y=90.

    Squares Present:

    1. Left Side, Above: (190, 50-90). Let's call this position L-Up-1.
    2. Left Side, Below: (150, 90-130). Position L-Down-1.
    3. Left Side, Below: (190, 130-170). Position L-Down-2.
    4. Right Side, Below: (300, 90-130). Position R-Down-1.
    5. Right Side, Below: (340, 90-130). Position R-Down-2.
    6. Right Side, Below: (340, 130-170). Position R-Down-3.

    Required Mirrors for Symmetry about y=90:

    1. Mirror of L-Up-1 (190, 50-90) is (190, 90-130). Is it present? No. -> Add 1.
    2. Mirror of L-Down-1 (150, 90-130) is (150, 50-90). Is it present? No. -> Add 1.
    3. Mirror of L-Down-2 (190, 130-170) is (190, 10-50). Is it present? No. -> Add 1.
    4. Mirror of R-Down-1 (300, 90-130) is (300, 50-90). Is it present? No. -> Add 1.
    5. Mirror of R-Down-2 (340, 90-130) is (340, 50-90). Is it present? No. -> Add 1.
    6. Mirror of R-Down-3 (340, 130-170) is (340, 10-50). Is it present? No. -> Add 1.

    Total squares to add = 6.

    Let me re-read the options. 6, 4, 5, 7.

    My count is 6. Option A is 6.

    Let me double check if any squares are already symmetric.

    None of the existing squares have a partner across the line y=90.

    Therefore, every existing square needs a mirror image added.

    There are 6 existing squares.

    So we must add 6 squares.

    Answer: A

    Wait, looking at the previous thought process in similar problems, sometimes "added" implies completing the pattern. If I add 6 squares, do I create new asymmetries? No, because I am adding the exact mirrors.

    Let's re-verify the positions.

    Left cluster:

    • One above axis at col 2 (x=190).
    • Two below axis at col 1 (x=150) and col 2 (x=190).

    Right cluster:

    • Three below axis at col 3 (x=300) and col 4 (x=340). Specifically, one at col 3, two at col 4.

    Mirrors needed:

    • For Left Above (col 2): Need Left Below (col 2). But Left Below col 2 is occupied by a square at y=130-170. The mirror of y=50-90 is y=90-130. The square at 190,130-170 is NOT the mirror. The mirror slot (190, 90-130) is empty. So add 1.
    • For Left Below (150, 90-130): Mirror is (150, 50-90). Empty. Add 1.
    • For Left Below (190, 130-170): Mirror is (190, 10-50). Empty. Add 1.
    • For Right Below (300, 90-130): Mirror is (300, 50-90). Empty. Add 1.
    • For Right Below (340, 90-130): Mirror is (340, 50-90). Empty. Add 1.
    • For Right Below (340, 130-170): Mirror is (340, 10-50). Empty. Add 1.

    Total 6.

    Answer: A

    Question 2 · Spatial Aptitude · 2023 MCQ
    Which one of the options best describes the transformation of the 2-dimensional figure P to Q, and then to R, as shown?

    POperation 1QOperation 2R
    1. A. Operation 1: A clockwise rotation by about an axis perpendicular to the plane of the figure

      Operation 2: A reflection along a horizontal line
    2. B. Operation 1: A counter clockwise rotation by about an axis perpendicular to the plane of the figure

      Operation 2: A reflection along a horizontal line
    3. C. Operation 1: A clockwise rotation by about an axis perpendicular to the plane of the figure

      Operation 2: A reflection along a vertical line
    4. D. Operation 1: A counter clockwise rotation by about an axis perpendicular to the plane of the figure

      Operation 2: A reflection along a vertical line
    Correct Answer:

    A

    Step-by-Step Solution

    Key idea: This is a sequential transformation identification problem. We need to determine the geometric operation that transforms P to Q, and then Q to R.

    Step 1: Analyze Transformation P to Q.

    Compare figure P and figure Q.

    • Figure P is an irregular polygon.
    • Figure Q appears to be rotated relative to P.
    • Let's track a specific vertex. The top-most vertex of P moves to the right-most position in Q?
    • Visually, P looks like it has been rotated 90 degrees Clockwise.
    • Check orientation: The sequence of vertices is preserved (no reflection).
    • Conclusion: Operation 1 is a 90-degree Clockwise Rotation.

    Step 2: Analyze Transformation Q to R.

    Compare figure Q and figure R.

    • Figure Q is the rotated P.
    • Figure R is a mirror image of Q?
    • Let's check for reflection.
    • If we reflect Q across a horizontal line, does it match R?
    • Top of Q becomes Bottom of R. Left of Q becomes Left of R?
    • Let's look at the options.
    • Option A says: Op 2 is Reflection along a horizontal line.
    • Option B says: Op 2 is Reflection along a horizontal line.
    • Option C says: Op 2 is Reflection along a vertical line.
    • Option D says: Op 2 is Reflection along a vertical line.

    Let's verify the axis.

    In Q, the "pointy" part is to the right. In R, the "pointy" part is to the right? No, R looks like Q flipped upside down.

    If Q is flipped upside down (Horizontal Axis Reflection), the top becomes bottom.

    Does R look like Q upside down? Yes.

    Therefore:

    Operation 1: 90-degree Clockwise Rotation.

    Operation 2: Reflection along a horizontal line.

    This matches Option A.

    Answer: A

    Question 3 · Spatial Aptitude · 2021_Set1 MCQ

    A circular sheet of paper is folded along the lines in the directions shown. The paper, after being punched in the final folded state as shown and unfolded in the reverse order of folding, will look like _______.
    1. A.
    2. B.
    3. C.
    4. D.
    Correct Answer:

    A

    Step-by-Step Solution

    Key idea: This is a paper folding and punching problem. We must reverse the folding steps to determine the location and orientation of the punched holes on the unfolded sheet.

    Step 1: Analyze the final folded state and the punch.

    The final state is a quarter-circle sector (top-right quadrant of the original circle, if we assume standard folding).

    The punch consists of:

    • A small circular hole near the center (the corner of the sector).
    • A rectangular notch on the curved edge.
    • A rectangular notch on the straight vertical edge.
    • A rectangular notch on the straight horizontal edge.

    Actually, looking at the SVG for the final state:

    • It's a quarter circle.
    • Punches:
    1. A small rectangle on the vertical radius.
    2. A small rectangle on the horizontal radius.
    3. A larger rectangle-like shape on the arc? No, it looks like a hole inside.

    Let's look at the options to infer the punch pattern.

    The options show:

    • A central circular hole.
    • Four rectangular holes arranged symmetrically.

    Let's trace the folds backwards.

    Fold 1: Vertical fold (Left over Right? Or Right over Left?). The arrow shows the left side folding to the right. So we have a semi-circle (Right half).

    Fold 2: Horizontal fold (Bottom over Top? Or Top over Bottom?). The arrow shows the bottom folding up. So we have a quarter-circle (Top-Right quadrant).

    The punch is made in this Top-Right quadrant.

    Unfold Step 1 (Reverse Horizontal Fold):

    Reflect the punch pattern across the horizontal axis (the bottom edge of the current quarter circle).

    • The Top-Right quadrant becomes the Right Half (Top-Right and Bottom-Right).
    • Any hole in the Top-Right will have a mirror image in the Bottom-Right.

    Unfold Step 2 (Reverse Vertical Fold):

    Reflect the entire Right Half pattern across the vertical axis (the left edge of the semi-circle).

    • The Right Half becomes the Full Circle.
    • Any hole in the Right Half will have a mirror image in the Left Half.

    Analysis of Holes:

    1. Central Hole: The punch near the corner (center of circle) reflects to itself or creates a cluster. In the options, there is a single central circular hole. This implies the punch was at the center or created a symmetric pattern that merges.
    2. Rectangular Holes:
    • In the final quadrant, there appear to be punches on the edges.
    • Option A shows 4 rectangular holes: Top, Bottom, Left, Right.
    • This symmetry corresponds to a single punch in the quadrant that gets reflected 3 times.

    Let's look at the specific shape in the final fold SVG:

    • There is a small rectangular cut on the vertical edge.
    • There is a small rectangular cut on the horizontal edge.
    • There is a cut on the arc?

    Comparing with Option A:

    • Option A has rectangles at 12, 6, 3, 9 o'clock positions.
    • If we punch a rectangle at the midpoint of the vertical radius in the folded state, unfolding horizontally creates a pair at 3 o'clock (if it was on the edge) or similar.

    Let's look at the "notch" shapes in the final fold.

    • One notch on the vertical straight edge.
    • One notch on the horizontal straight edge.
    • One notch on the curved edge?

    If we unfold a notch on the vertical edge of the Top-Right quadrant:

    • Reflect across Horizontal: It stays in the Right Half (Top-Right and Bottom-Right? No, the vertical edge is the axis of the first fold? No, the vertical edge is the center of the original circle? No.

    Let's re-evaluate the fold axes.

    1. Circle folded vertically. Axis is vertical diameter. Result: Semi-circle.
    2. Semi-circle folded horizontally. Axis is horizontal diameter. Result: Quarter-circle.

    The straight edges of the final quarter-circle are the radii along the axes.

    • Vertical straight edge: Part of the vertical diameter.
    • Horizontal straight edge: Part of the horizontal diameter.

    Punches in the final state:

    1. A punch on the vertical straight edge. When unfolded horizontally (Step 1), this punch is on the axis of reflection? No, the horizontal fold axis is the horizontal edge. The vertical edge is perpendicular to it. So the punch on the vertical edge reflects to a punch on the vertical edge in the lower quadrant. So we have two punches on the vertical radius (one up, one down).

    Then unfold vertically (Step 2). The vertical radius is the axis of reflection. Punches on the axis reflect to themselves? Or do they split? If it's a hole on the edge, it becomes a symmetric hole centered on the axis.

    1. A punch on the horizontal straight edge. When unfolded horizontally, this is on the axis. It becomes a symmetric hole centered on the horizontal axis.

    Option A shows:

    • Rectangles at Top and Bottom (on vertical axis).
    • Rectangles at Left and Right (on horizontal axis).

    This matches perfectly if there were punches on the straight edges of the folded quadrant.

    Answer: A

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