Chapter 7 of 15
Chapter 7: Shapes and Patterns
आकृतियाँ और पैटर्न (ज्यामितीय आकृतियाँ, घूर्णन एवं टाइलिंग) | CBSE Session 2026–27
Core Pedagogical Competencies & Concepts
- 2D geometric shapes: Triangles, Quadrilaterals, Pentagons, Hexagons and their properties.
- Rotational patterns: Rotating a motif by 1/4 turn (90°), 1/2 turn (180°), or 1/6 turn (60°).
- Tessellations (Tiling): Covering a flat surface completely with geometric tiles without any gaps or overlapping.
- Number and shape sequences following progressive mathematical rules.
- (a) 1, 4, 9, 16, 25: Rule: Square of consecutive natural numbers ($1^2, 2^2, 3^2, 4^2, 5^2$). Next terms: $6^2 = \mathbf{36}$, $7^2 = \mathbf{49}$.
- (b) 3, 6, 12, 24: Rule: Multiply by 2. Next terms: $24 \times 2 = \mathbf{48}$, $48 \times 2 = \mathbf{96}$.
- (c) 100, 95, 85, 70: Rule: Subtract consecutive multiples of 5 ($-5, -10, -15$). Next terms: $70 - 20 = \mathbf{50}$, $50 - 25 = \mathbf{25}$.
All Textbook Exercises & In-Text Tasks (24 Questions)
100% Complete Ingestion
Q1
Rotational Symmetry • Turns & Degrees
(a) What is the angle of turn if a shape returns to its exact original appearance in a 1/2 turn?
(b) What is the angle of turn in a 1/4 turn?
(c) Which of the following shapes look identical after a 1/2 turn: Rectangle, Equilateral Triangle, Letter 'S', Letter 'M'?
[3 Marks]
(b) What is the angle of turn in a 1/4 turn?
(c) Which of the following shapes look identical after a 1/2 turn: Rectangle, Equilateral Triangle, Letter 'S', Letter 'M'?
✅ Step-by-Step Solution & Working
Rotational Symmetry Properties:
• (a) 1/2 Turn: 360° ÷ 2 = 180°.
• (b) 1/4 Turn: 360° ÷ 4 = 90°.
• (c) Shapes that look identical after 1/2 turn (180°):
- Rectangle: Yes (180° rotation matches perfectly).
- Equilateral Triangle: No (apex points downward after 1/2 turn).
- Letter 'S': Yes (180° rotation gives 'S').
- Letter 'M': No (turns upside down into 'W').
Identical shapes: Rectangle and Letter 'S'.
• (a) 1/2 Turn: 360° ÷ 2 = 180°.
• (b) 1/4 Turn: 360° ÷ 4 = 90°.
• (c) Shapes that look identical after 1/2 turn (180°):
- Rectangle: Yes (180° rotation matches perfectly).
- Equilateral Triangle: No (apex points downward after 1/2 turn).
- Letter 'S': Yes (180° rotation gives 'S').
- Letter 'M': No (turns upside down into 'W').
Identical shapes: Rectangle and Letter 'S'.
CBSE Step-by-Step Marking Scheme
1/2 turn = 180°; 1/4 turn = 90°
1.5 M
Rectangle and Letter S identified correctly
1.5 M
Q2
Number Patterns • Multiplication Arrays
Observe the pattern and find the next two lines:
1 × 9 + 2 = 11
12 × 9 + 3 = 111
123 × 9 + 4 = 1111
Explain the pattern rule.
[3 Marks]
1 × 9 + 2 = 11
12 × 9 + 3 = 111
123 × 9 + 4 = 1111
Explain the pattern rule.
✅ Step-by-Step Solution & Working
Pattern Progression:
• Line 4: 1234 × 9 + 5 = 11,111.
• Line 5: 12345 × 9 + 6 = 1,11,111.
• Rule: In each successive line, the multiplicand adds the next natural digit on the right (1, 12, 123, 1234, 12345), the added number increases by 1 (2, 3, 4, 5, 6), and the resulting product consists entirely of the digit 1 repeated one more time than in the previous line.
• Line 4: 1234 × 9 + 5 = 11,111.
• Line 5: 12345 × 9 + 6 = 1,11,111.
• Rule: In each successive line, the multiplicand adds the next natural digit on the right (1, 12, 123, 1234, 12345), the added number increases by 1 (2, 3, 4, 5, 6), and the resulting product consists entirely of the digit 1 repeated one more time than in the previous line.
CBSE Step-by-Step Marking Scheme
Line 4: 1234 × 9 + 5 = 11,111
1.0 M
Line 5: 12345 × 9 + 6 = 1,11,111
1.0 M
Clear mathematical rule explanation
1.0 M
Q3
Sum of Consecutive Numbers Pattern
Look at the pattern of adding consecutive numbers:
1 + 2 + 3 = 6
2 + 3 + 4 = 9
3 + 4 + 5 = 12
(a) Find the next two lines in this pattern.
(b) What is the mathematical relationship between the middle number and the sum?
[3 Marks]
1 + 2 + 3 = 6
2 + 3 + 4 = 9
3 + 4 + 5 = 12
(a) Find the next two lines in this pattern.
(b) What is the mathematical relationship between the middle number and the sum?
✅ Step-by-Step Solution & Working
Pattern Analysis:
• (a) Next Two Lines:
- Line 4: 4 + 5 + 6 = 15.
- Line 5: 5 + 6 + 7 = 18.
• (b) Mathematical Relationship:
- For 1, 2, 3: Middle number is 2; Sum = 6 = 3 × 2.
- For 2, 3, 4: Middle number is 3; Sum = 9 = 3 × 3.
- For 3, 4, 5: Middle number is 4; Sum = 12 = 3 × 4.
Rule: The sum of three consecutive numbers is always exactly 3 times the middle number ($3 \times M$).
• (a) Next Two Lines:
- Line 4: 4 + 5 + 6 = 15.
- Line 5: 5 + 6 + 7 = 18.
• (b) Mathematical Relationship:
- For 1, 2, 3: Middle number is 2; Sum = 6 = 3 × 2.
- For 2, 3, 4: Middle number is 3; Sum = 9 = 3 × 3.
- For 3, 4, 5: Middle number is 4; Sum = 12 = 3 × 4.
Rule: The sum of three consecutive numbers is always exactly 3 times the middle number ($3 \times M$).
CBSE Step-by-Step Marking Scheme
Line 4: 4+5+6 = 15; Line 5: 5+6+7 = 18
1.5 M
Sum = 3 × middle number
1.5 M
Q4
Magic Squares • 3x3 Grid Sum
Fill in the numbers 1 to 9 in a 3×3 grid such that the sum of each row, column, and diagonal is 15. What number must be placed in the center, and why?
[3 Marks]
✅ Step-by-Step Solution & Working
Magic Square of 1 to 9 (Sum = 15):
• Center Number: The number 5 must be placed in the center.
• Reason: The center cell is part of 4 lines (1 row, 1 column, 2 diagonals). The average of numbers 1 to 9 is (1 + 9) ÷ 2 = 5, which is the median value required to balance all pairs adding to 10 (1+9, 2+8, 3+7, 4+6).
8 1 6 → 15 3 5 7 → 15 4 9 2 → 15 ↓ ↓ ↓ 15 15 15 Diagonals: 8+5+2 = 15; 6+5+4 = 15.
• Center Number: The number 5 must be placed in the center.
• Reason: The center cell is part of 4 lines (1 row, 1 column, 2 diagonals). The average of numbers 1 to 9 is (1 + 9) ÷ 2 = 5, which is the median value required to balance all pairs adding to 10 (1+9, 2+8, 3+7, 4+6).
CBSE Step-by-Step Marking Scheme
Correct 3×3 grid with sums = 15
2.0 M
Center is 5 with median explanation
1.0 M
Q5
Magic Hexagon • Circles & Squares
In a magic hexagon pattern, a square lies between two circles. The rule is: The number in each square is the product of the numbers in the circles on either side.
If two circles contain 7 and 9, what number goes in the square between them? If a square contains 72 and one circle is 8, what number is in the other circle?
[3 Marks]
If two circles contain 7 and 9, what number goes in the square between them? If a square contains 72 and one circle is 8, what number is in the other circle?
✅ Step-by-Step Solution & Working
Rule: Square = Circle 1 × Circle 2:
• (a) Circles with 7 and 9: Square = 7 × 9 = 63.
• (b) Square is 72, Circle is 8: Other circle = 72 ÷ 8 = 9.
• (a) Circles with 7 and 9: Square = 7 × 9 = 63.
• (b) Square is 72, Circle is 8: Other circle = 72 ÷ 8 = 9.
CBSE Step-by-Step Marking Scheme
Square = 63
1.5 M
Other circle = 9
1.5 M
Q6
Square Numbers Pattern • Visual Dots
Observe the pattern of square numbers:
1, 4, 9, 16, 25...
(a) Find the next three square numbers.
(b) Write each square number as the sum of consecutive odd numbers starting from 1 (e.g., 4 = 1 + 3).
[3 Marks]
1, 4, 9, 16, 25...
(a) Find the next three square numbers.
(b) Write each square number as the sum of consecutive odd numbers starting from 1 (e.g., 4 = 1 + 3).
✅ Step-by-Step Solution & Working
Square Numbers:
• (a) Next Three Numbers: 6² = 36, 7² = 49, 8² = 64.
• (b) Sum of Odd Numbers:
- 1 = 1
- 4 = 1 + 3
- 9 = 1 + 3 + 5
- 16 = 1 + 3 + 5 + 7
- 25 = 1 + 3 + 5 + 7 + 9
- 36 = 1 + 3 + 5 + 7 + 9 + 11.
• (a) Next Three Numbers: 6² = 36, 7² = 49, 8² = 64.
• (b) Sum of Odd Numbers:
- 1 = 1
- 4 = 1 + 3
- 9 = 1 + 3 + 5
- 16 = 1 + 3 + 5 + 7
- 25 = 1 + 3 + 5 + 7 + 9
- 36 = 1 + 3 + 5 + 7 + 9 + 11.
CBSE Step-by-Step Marking Scheme
Next three: 36, 49, 64
1.5 M
Odd sum expansion
1.5 M
Q7
Palindromic Numbers • Reverse and Add
A palindrome is a number that reads the same forwards and backwards (e.g. 121, 353).
(a) Take the number 48. Reverse its digits and add them together. Repeat the process if needed to get a palindrome.
(b) Is 1,001 a palindrome? What is the next 4-digit palindrome after 1,001?
[3 Marks]
(a) Take the number 48. Reverse its digits and add them together. Repeat the process if needed to get a palindrome.
(b) Is 1,001 a palindrome? What is the next 4-digit palindrome after 1,001?
✅ Step-by-Step Solution & Working
Palindrome Calculations:
• (a) Process for 48:
- Step 1: Number = 48. Reverse = 84.
- Step 2: Sum = 48 + 84 = 132 (not yet a palindrome).
- Step 3: Reverse of 132 = 231.
- Step 4: Sum = 132 + 231 = 363 (which reads the same forwards and backwards → Palindrome!).
• (b) 1,001: Yes, 1,001 is a palindrome. The next 4-digit palindrome is 1,111.
• (a) Process for 48:
- Step 1: Number = 48. Reverse = 84.
- Step 2: Sum = 48 + 84 = 132 (not yet a palindrome).
- Step 3: Reverse of 132 = 231.
- Step 4: Sum = 132 + 231 = 363 (which reads the same forwards and backwards → Palindrome!).
• (b) 1,001: Yes, 1,001 is a palindrome. The next 4-digit palindrome is 1,111.
CBSE Step-by-Step Marking Scheme
Steps for 48 yielding 363
2.0 M
1,001 is palindrome & next is 1,111
1.0 M
Q8
Secret Message Codes • Alphabet Shifts
(a) If the code rule is: Replace each letter with the next letter (A→B, B→C... Z→A), decode the secret message: 'CPPL'.
(b) Using the same rule, encode the word: 'CLEVER'.
[3 Marks]
(b) Using the same rule, encode the word: 'CLEVER'.
✅ Step-by-Step Solution & Working
Coding & Decoding (+1 Shift):
• (a) Decoding 'CPPL': Shift backward by 1 letter (-1):
- C - 1 = B
- P - 1 = O
- P - 1 = O
- L - 1 = K
Decoded Word: BOOK.
• (b) Encoding 'CLEVER': Shift forward by 1 letter (+1):
- C → D, L → M, E → F, V → W, E → F, R → S.
Encoded Message: DMFWFS.
• (a) Decoding 'CPPL': Shift backward by 1 letter (-1):
- C - 1 = B
- P - 1 = O
- P - 1 = O
- L - 1 = K
Decoded Word: BOOK.
• (b) Encoding 'CLEVER': Shift forward by 1 letter (+1):
- C → D, L → M, E → F, V → W, E → F, R → S.
Encoded Message: DMFWFS.
CBSE Step-by-Step Marking Scheme
Decoded: BOOK
1.5 M
Encoded: DMFWFS
1.5 M
Q9
Tessellation & Tiling • Surface Covering
(a) What is tessellation (tiling)?
(b) Can regular hexagons tile a flat surface without gaps or overlaps? Explain why with reference to angles.
(c) Can regular octagons alone tile a floor? Explain.
[3 Marks]
(b) Can regular hexagons tile a flat surface without gaps or overlaps? Explain why with reference to angles.
(c) Can regular octagons alone tile a floor? Explain.
✅ Step-by-Step Solution & Working
Tessellation Geometry:
• (a) Definition: Tessellation is the covering of a plane using one or more geometric shapes with no gaps and no overlaps.
• (b) Regular Hexagons: Yes. Each interior angle of a regular hexagon is 120°. At each meeting vertex, 3 hexagons meet: 120° × 3 = 360°. Since their angles sum to exactly 360°, they tile seamlessly (honeycomb pattern).
• (c) Regular Octagons: No, octagons alone cannot tile a floor. Each interior angle is 135°. Two octagons give 135° + 135° = 270°, leaving a gap of 360° - 270° = 90° (which requires a square to fill the gap).
• (a) Definition: Tessellation is the covering of a plane using one or more geometric shapes with no gaps and no overlaps.
• (b) Regular Hexagons: Yes. Each interior angle of a regular hexagon is 120°. At each meeting vertex, 3 hexagons meet: 120° × 3 = 360°. Since their angles sum to exactly 360°, they tile seamlessly (honeycomb pattern).
• (c) Regular Octagons: No, octagons alone cannot tile a floor. Each interior angle is 135°. Two octagons give 135° + 135° = 270°, leaving a gap of 360° - 270° = 90° (which requires a square to fill the gap).
CBSE Step-by-Step Marking Scheme
Definition of tessellation
1.0 M
Hexagon: 120° × 3 = 360° tiles without gaps
1.0 M
Octagon: 135° leaves 90° gap (cannot tile alone)
1.0 M
Q10
1/3 Turn and 1/6 Turn Rotations
Through how many degrees does a shape rotate in:
(a) A 1/3 turn
(b) A 1/6 turn
(c) Which regular shape looks identical after a 1/3 turn?
(d) Which shape looks identical after a 1/6 turn?
[3 Marks]
(a) A 1/3 turn
(b) A 1/6 turn
(c) Which regular shape looks identical after a 1/3 turn?
(d) Which shape looks identical after a 1/6 turn?
✅ Step-by-Step Solution & Working
Rotational Angles:
• (a) 1/3 Turn: 360° ÷ 3 = 120°.
• (b) 1/6 Turn: 360° ÷ 6 = 60°.
• (c) Identical after 1/3 turn: An Equilateral Triangle (has 3-fold rotational symmetry).
• (d) Identical after 1/6 turn: A Regular Hexagon (or 6-pointed star, has 6-fold symmetry).
• (a) 1/3 Turn: 360° ÷ 3 = 120°.
• (b) 1/6 Turn: 360° ÷ 6 = 60°.
• (c) Identical after 1/3 turn: An Equilateral Triangle (has 3-fold rotational symmetry).
• (d) Identical after 1/6 turn: A Regular Hexagon (or 6-pointed star, has 6-fold symmetry).
CBSE Step-by-Step Marking Scheme
1/3 turn = 120°; 1/6 turn = 60°
1.5 M
Equilateral triangle and Regular hexagon
1.5 M
Q11
Triangular Numbers Pattern
Look at the pattern of triangular numbers (dots arranged in triangles):
1, 3, 6, 10, 15...
(a) Find the next two triangular numbers.
(b) What is the relationship between the sum of two consecutive triangular numbers?
[3 Marks]
1, 3, 6, 10, 15...
(a) Find the next two triangular numbers.
(b) What is the relationship between the sum of two consecutive triangular numbers?
✅ Step-by-Step Solution & Working
Triangular Numbers:
• Differences are consecutive integers: +2, +3, +4, +5...
• (a) Next Two Numbers:
- 15 + 6 = 21.
- 21 + 7 = 28.
• (b) Sum of Consecutive Triangular Numbers:
- 1 + 3 = 4 = 2² (Square number)
- 3 + 6 = 9 = 3² (Square number)
- 6 + 10 = 16 = 4² (Square number)
- 10 + 15 = 25 = 5² (Square number)
Rule: The sum of any two consecutive triangular numbers is always a Square Number!
• Differences are consecutive integers: +2, +3, +4, +5...
• (a) Next Two Numbers:
- 15 + 6 = 21.
- 21 + 7 = 28.
• (b) Sum of Consecutive Triangular Numbers:
- 1 + 3 = 4 = 2² (Square number)
- 3 + 6 = 9 = 3² (Square number)
- 6 + 10 = 16 = 4² (Square number)
- 10 + 15 = 25 = 5² (Square number)
Rule: The sum of any two consecutive triangular numbers is always a Square Number!
CBSE Step-by-Step Marking Scheme
Next two: 21, 28
1.5 M
Sum of two consecutive gives square numbers
1.5 M
Q12
Magic Triangle • Perimeter Sum
Fill in the numbers 1, 2, 3, 4, 5, 6 along the three sides of a triangle (3 numbers on each side) so that the sum of the numbers on each side is 9.
[3 Marks]
✅ Step-by-Step Solution & Working
Magic Triangle Solution (Side Sum = 9):
1. Place the smallest numbers 1, 2, 3 at the three corner vertices.
2. Corner sum = 1 + 2 + 3 = 6.
3. Sides:
• Side between 1 and 2: needs 9 - (1 + 2) = 6 in the middle.
• Side between 2 and 3: needs 9 - (2 + 3) = 4 in the middle.
• Side between 3 and 1: needs 9 - (3 + 1) = 5 in the middle.
Check: 1+6+2 = 9; 2+4+3 = 9; 3+5+1 = 9. All numbers 1 to 6 used exactly once!
1. Place the smallest numbers 1, 2, 3 at the three corner vertices.
2. Corner sum = 1 + 2 + 3 = 6.
3. Sides:
• Side between 1 and 2: needs 9 - (1 + 2) = 6 in the middle.
• Side between 2 and 3: needs 9 - (2 + 3) = 4 in the middle.
• Side between 3 and 1: needs 9 - (3 + 1) = 5 in the middle.
Check: 1+6+2 = 9; 2+4+3 = 9; 3+5+1 = 9. All numbers 1 to 6 used exactly once!
CBSE Step-by-Step Marking Scheme
Corners: 1, 2, 3
1.5 M
Middle numbers: 6, 4, 5 with side sum = 9
1.5 M
Q13
Border Tiling Strip Patterns
A traditional border design repeats a motif: Triangle pointing UP, Square, Triangle pointing DOWN, Circle.
(a) What will be the 15th shape in this repeating pattern?
(b) What will be the 28th shape?
[3 Marks]
(a) What will be the 15th shape in this repeating pattern?
(b) What will be the 28th shape?
✅ Step-by-Step Solution & Working
Modulo Arithmetic in Repeating Motifs (Period = 4 shapes):
• Pattern sequence: 1 = UP, 2 = Square, 3 = DOWN, 4 = Circle.
• (a) 15th Shape: 15 ÷ 4 = 3 with remainder 3. Position 3 is Triangle pointing DOWN.
• (b) 28th Shape: 28 ÷ 4 = 7 with remainder 0 (exact multiple of 4). Position 4 is Circle.
• Pattern sequence: 1 = UP, 2 = Square, 3 = DOWN, 4 = Circle.
• (a) 15th Shape: 15 ÷ 4 = 3 with remainder 3. Position 3 is Triangle pointing DOWN.
• (b) 28th Shape: 28 ÷ 4 = 7 with remainder 0 (exact multiple of 4). Position 4 is Circle.
CBSE Step-by-Step Marking Scheme
Period = 4 shapes identified
1.0 M
15th shape = Triangle pointing DOWN
1.0 M
28th shape = Circle
1.0 M
Q14
Number Grid Patterns • Magic Multiplications
Study the multiplication pattern:
11 × 11 = 121
111 × 111 = 12321
1111 × 1111 = 1234321
(a) What will be the value of 11111 × 11111 without doing long multiplication?
(b) Explain the symmetry in the digits of the product.
[3 Marks]
11 × 11 = 121
111 × 111 = 12321
1111 × 1111 = 1234321
(a) What will be the value of 11111 × 11111 without doing long multiplication?
(b) Explain the symmetry in the digits of the product.
✅ Step-by-Step Solution & Working
Symmetric Number Products (Repunits):
• (a) Product: 11,111 × 11,111 = 123454321 (One crore twenty-three lakh forty-five thousand four hundred twenty-one).
• (b) Symmetry Explanation: The digits count up from 1 to the number of ones in the factor (5 digits: 1, 2, 3, 4, 5) and then mirror backwards symmetrically down to 1 (4, 3, 2, 1), forming a numerical palindrome.
• (a) Product: 11,111 × 11,111 = 123454321 (One crore twenty-three lakh forty-five thousand four hundred twenty-one).
• (b) Symmetry Explanation: The digits count up from 1 to the number of ones in the factor (5 digits: 1, 2, 3, 4, 5) and then mirror backwards symmetrically down to 1 (4, 3, 2, 1), forming a numerical palindrome.
CBSE Step-by-Step Marking Scheme
Product = 123454321
1.5 M
Clear explanation of ascending-descending palindrome symmetry
1.5 M
Q15
Rotational Symmetry Order
State the order of rotational symmetry for each of the following regular figures:
(a) Square
(b) Regular Pentagon
(c) Regular Octagon
(d) Circle
[3 Marks]
(a) Square
(b) Regular Pentagon
(c) Regular Octagon
(d) Circle
✅ Step-by-Step Solution & Working
Order of Rotational Symmetry:
• (a) Square: Rotates onto itself 4 times in a 360° turn (every 90°) → Order 4.
• (b) Regular Pentagon: Rotates onto itself 5 times (every 72°) → Order 5.
• (c) Regular Octagon: Rotates onto itself 8 times (every 45°) → Order 8.
• (d) Circle: Any rotation around its center leaves it unchanged → Infinite order.
• (a) Square: Rotates onto itself 4 times in a 360° turn (every 90°) → Order 4.
• (b) Regular Pentagon: Rotates onto itself 5 times (every 72°) → Order 5.
• (c) Regular Octagon: Rotates onto itself 8 times (every 45°) → Order 8.
• (d) Circle: Any rotation around its center leaves it unchanged → Infinite order.
CBSE Step-by-Step Marking Scheme
Square = 4; Pentagon = 5
1.5 M
Octagon = 8; Circle = Infinite
1.5 M
Q16
Fibonacci Sequence • Nature's Pattern
Observe the Fibonacci sequence where each term is the sum of the two preceding terms:
1, 1, 2, 3, 5, 8, 13...
(a) Find the next three numbers in the sequence.
(b) Where in nature can this pattern be observed?
[3 Marks]
1, 1, 2, 3, 5, 8, 13...
(a) Find the next three numbers in the sequence.
(b) Where in nature can this pattern be observed?
✅ Step-by-Step Solution & Working
Fibonacci Sequence:
• (a) Next Three Numbers:
- 8 + 13 = 21.
- 13 + 21 = 34.
- 21 + 34 = 55.
• (b) Occurrences in Nature: Number of petals on flowers (daisies, sunflowers), spiral arrangement of sunflower seeds, pine cone scales, and pineapple skins.
• (a) Next Three Numbers:
- 8 + 13 = 21.
- 13 + 21 = 34.
- 21 + 34 = 55.
• (b) Occurrences in Nature: Number of petals on flowers (daisies, sunflowers), spiral arrangement of sunflower seeds, pine cone scales, and pineapple skins.
CBSE Step-by-Step Marking Scheme
Next three: 21, 34, 55
1.5 M
Two natural examples (sunflower seeds, pine cones)
1.5 M
Q17
Alphabet Symmetry • Rotational vs Line
Among the capital English letters: A, H, M, O, X, Z:
(a) Which letters have both vertical and horizontal lines of symmetry?
(b) Which letter has rotational symmetry of order 2 but NO line of symmetry?
[3 Marks]
(a) Which letters have both vertical and horizontal lines of symmetry?
(b) Which letter has rotational symmetry of order 2 but NO line of symmetry?
✅ Step-by-Step Solution & Working
Letter Symmetry Analysis:
• (a) Both Vertical & Horizontal Lines of Symmetry: H, O, X (can be reflected across both axes).
• (b) Rotational Symmetry but No Line Symmetry: Z.
- Letter 'Z' has no axis of reflection (horizontal or vertical reflection reverses the diagonal), but rotating by 180° (half-turn) reproduces an identical 'Z'.
• (a) Both Vertical & Horizontal Lines of Symmetry: H, O, X (can be reflected across both axes).
• (b) Rotational Symmetry but No Line Symmetry: Z.
- Letter 'Z' has no axis of reflection (horizontal or vertical reflection reverses the diagonal), but rotating by 180° (half-turn) reproduces an identical 'Z'.
CBSE Step-by-Step Marking Scheme
Part (a): H, O, X
1.5 M
Part (b): Z with rotational explanation
1.5 M
Q18
Magic Square of Sum 30
A 3×3 magic square has numbers 6, 7, 8, 9, 10, 11, 12, 13, 14. What number must be at the center, and what is the constant magic sum of each row?
[3 Marks]
✅ Step-by-Step Solution & Working
Calculations:
1. Total sum of numbers from 6 to 14 = (6 + 14) × 9 ÷ 2 = 20 × 4.5 = 90.
2. Row Sum (Magic Constant): 90 ÷ 3 rows = 30.
3. Center Number: Median value = (6 + 14) ÷ 2 = 10.
Check: 3 × Center number = 3 × 10 = 30 = Magic constant.
1. Total sum of numbers from 6 to 14 = (6 + 14) × 9 ÷ 2 = 20 × 4.5 = 90.
2. Row Sum (Magic Constant): 90 ÷ 3 rows = 30.
3. Center Number: Median value = (6 + 14) ÷ 2 = 10.
Check: 3 × Center number = 3 × 10 = 30 = Magic constant.
CBSE Step-by-Step Marking Scheme
Magic sum = 30
1.5 M
Center number = 10 with median proof
1.5 M
Q19
HOTS • Number Pattern Puzzle
Find the missing number in the sequence: 2, 6, 12, 20, 30, _____, 56. Explain the mathematical rule.
[3 Marks]
✅ Step-by-Step Solution & Working
Pattern Analysis:
• Difference sequence: 6 - 2 = 4, 12 - 6 = 6, 20 - 12 = 8, 30 - 20 = 10.
• The difference increases by +2 each time (consecutive even numbers: 4, 6, 8, 10, 12, 14).
• Next difference must be +12: 30 + 12 = 42.
• Verification: 42 + 14 = 56. Correct!
Alternative Rule: $n \times (n + 1)$: 1×2=2, 2×3=6, 3×4=12, 4×5=20, 5×6=30, 6×7=42, 7×8=56.
• Difference sequence: 6 - 2 = 4, 12 - 6 = 6, 20 - 12 = 8, 30 - 20 = 10.
• The difference increases by +2 each time (consecutive even numbers: 4, 6, 8, 10, 12, 14).
• Next difference must be +12: 30 + 12 = 42.
• Verification: 42 + 14 = 56. Correct!
Alternative Rule: $n \times (n + 1)$: 1×2=2, 2×3=6, 3×4=12, 4×5=20, 5×6=30, 6×7=42, 7×8=56.
CBSE Step-by-Step Marking Scheme
Missing number = 42
1.5 M
Clear mathematical rule (difference sequence or n(n+1))
1.5 M
Q20
Windmill Blade Rotations
A windmill has 4 identical blades arranged symmetrically.
(a) What fraction of a turn brings each blade to the position of the next blade?
(b) How many degrees of rotation is that?
(c) Does the windmill look identical after a 1/4 turn, 1/2 turn, and 3/4 turn?
[3 Marks]
(a) What fraction of a turn brings each blade to the position of the next blade?
(b) How many degrees of rotation is that?
(c) Does the windmill look identical after a 1/4 turn, 1/2 turn, and 3/4 turn?
✅ Step-by-Step Solution & Working
Rotational Geometry of 4-Blade Windmill:
• (a) Fraction of a Turn: 4 blades = 1/4 turn.
• (b) Degrees: 360° ÷ 4 = 90°.
• (c) Symmetry: Yes, it has 4-fold rotational symmetry and looks identical after a 1/4 turn (90°), 1/2 turn (180°), and 3/4 turn (270°).
• (a) Fraction of a Turn: 4 blades = 1/4 turn.
• (b) Degrees: 360° ÷ 4 = 90°.
• (c) Symmetry: Yes, it has 4-fold rotational symmetry and looks identical after a 1/4 turn (90°), 1/2 turn (180°), and 3/4 turn (270°).
CBSE Step-by-Step Marking Scheme
Fraction = 1/4 turn
1.0 M
Degrees = 90°
1.0 M
Identical at 90°, 180°, 270°
1.0 M
Q21
Rangoli Geometric Tessellation
In an Indian rangoli pattern, equilateral triangles and rhombuses are combined to form a star.
(a) How many equilateral triangles are required to make a regular hexagon?
(b) What is the sum of angles around the central meeting point?
[3 Marks]
(a) How many equilateral triangles are required to make a regular hexagon?
(b) What is the sum of angles around the central meeting point?
✅ Step-by-Step Solution & Working
Geometric Properties:
• (a) Number of Triangles: Exactly 6 equilateral triangles meet at a central vertex to form a regular hexagon.
• (b) Sum of Angles: Each equilateral triangle has an angle of 60°. 6 × 60° = 360° (a complete circle).
• (a) Number of Triangles: Exactly 6 equilateral triangles meet at a central vertex to form a regular hexagon.
• (b) Sum of Angles: Each equilateral triangle has an angle of 60°. 6 × 60° = 360° (a complete circle).
CBSE Step-by-Step Marking Scheme
6 equilateral triangles
1.5 M
Sum of angles = 360°
1.5 M
Q22
Calendar Number Patterns
Look at any 3×3 square of dates on a monthly calendar (e.g. dates 1 to 3, 8 to 10, 15 to 17).
(a) How can you instantly find the sum of all 9 dates using only the center number?
(b) If the center date is 14, what is the sum of all 9 numbers in that 3×3 square?
[3 Marks]
(a) How can you instantly find the sum of all 9 dates using only the center number?
(b) If the center date is 14, what is the sum of all 9 numbers in that 3×3 square?
✅ Step-by-Step Solution & Working
Calendar Math Trick:
• (a) Rule: Because calendar numbers are symmetrically distributed horizontally (±1) and vertically (±7), the sum of all 9 numbers is always exactly 9 times the center number ($9 \times C$).
• (b) Calculation for Center = 14: Sum = 9 × 14 = 126.
• (a) Rule: Because calendar numbers are symmetrically distributed horizontally (±1) and vertically (±7), the sum of all 9 numbers is always exactly 9 times the center number ($9 \times C$).
• (b) Calculation for Center = 14: Sum = 9 × 14 = 126.
CBSE Step-by-Step Marking Scheme
Rule: Sum = 9 × center number
1.5 M
Sum = 9 × 14 = 126
1.5 M
Q23
True or False with Pattern Reasoning
State True or False with reasoning:
(a) An equilateral triangle looks identical after a 1/2 turn.
(b) Any number multiplied by 9 has the sum of its digits equal to 9 or a multiple of 9.
(c) A circle has only 4 lines of symmetry.
[3 Marks]
(a) An equilateral triangle looks identical after a 1/2 turn.
(b) Any number multiplied by 9 has the sum of its digits equal to 9 or a multiple of 9.
(c) A circle has only 4 lines of symmetry.
✅ Step-by-Step Solution & Working
Justifications:
• (a) False: In a 1/2 turn (180°), an equilateral triangle flips upside down with its base at the top and apex at the bottom.
• (b) True: By the divisibility rule for 9, the digital sum of any multiple of 9 is always 9, 18, 27... (a multiple of 9).
• (c) False: A circle has infinite lines of symmetry (any diameter passing through the center is a line of symmetry).
• (a) False: In a 1/2 turn (180°), an equilateral triangle flips upside down with its base at the top and apex at the bottom.
• (b) True: By the divisibility rule for 9, the digital sum of any multiple of 9 is always 9, 18, 27... (a multiple of 9).
• (c) False: A circle has infinite lines of symmetry (any diameter passing through the center is a line of symmetry).
CBSE Step-by-Step Marking Scheme
Part (a): False with upside-down reason
1.0 M
Part (b): True with divisibility rule
1.0 M
Part (c): False (infinite lines of symmetry)
1.0 M
Q24
Comprehensive Pattern Review
A student creates a spiral pattern of squares starting with side 1 cm, then 2 cm, then 3 cm, then 5 cm, then 8 cm.
(a) What will be the side of the next two squares?
(b) What is this famous series called, and what are the areas of the first four squares?
[3 Marks]
(a) What will be the side of the next two squares?
(b) What is this famous series called, and what are the areas of the first four squares?
✅ Step-by-Step Solution & Working
Fibonacci Square Spirals:
• (a) Next Sides: Sum of previous two sides: 5 + 8 = 13 cm, then 8 + 13 = 21 cm.
• (b) Series & Areas:
- Name: Fibonacci Sequence.
- Area of 1 cm square: 1 × 1 = 1 cm².
- Area of 2 cm square: 2 × 2 = 4 cm².
- Area of 3 cm square: 3 × 3 = 9 cm².
- Area of 5 cm square: 5 × 5 = 25 cm².
• (a) Next Sides: Sum of previous two sides: 5 + 8 = 13 cm, then 8 + 13 = 21 cm.
• (b) Series & Areas:
- Name: Fibonacci Sequence.
- Area of 1 cm square: 1 × 1 = 1 cm².
- Area of 2 cm square: 2 × 2 = 4 cm².
- Area of 3 cm square: 3 × 3 = 9 cm².
- Area of 5 cm square: 5 × 5 = 25 cm².
CBSE Step-by-Step Marking Scheme
Sides: 13 cm, 21 cm
1.5 M
Fibonacci sequence & areas: 1, 4, 9, 25 cm²
1.5 M
🌟 NEP 2020 Competency-Based Case Study
Competency Question: Heritage Haveli Flooring Tile Design
An architect designs a courtyard floor measuring 12 metres by 8 metres using rectangular tiles of size 40 cm by 20 cm.
Analytical Task: (a) What is the area of the courtyard in square centimetres?
(b) What is the area of one tile?
(c) How many tiles are needed to pave the entire floor without gaps?
(b) What is the area of one tile?
(c) How many tiles are needed to pave the entire floor without gaps?
Evaluation & Model Answer:
Suggested Model Solution:
(a) Courtyard Area: 1,200 × 800 = 9,60,000 sq cm.
(b) Tile Area: 40 × 20 = 800 sq cm.
(c) Number of Tiles: 960,000/800 = 1,200 tiles.
Rubric: 1M concept identification + 1M step execution + 1M conclusion