Which of the following numbers CANNOT be expressed as a sum of consecutive natural numbers?
(a) 14
(b) 18
(c) 16
(d) 20
Correct Answer: (c) 16
Explanation: 16 is a power of 2 (2⁴). Powers of 2 cannot be expressed as sums of consecutive natural numbers because they have only one odd divisor (1).
Logical Reasoning
Pattern Recognition
2
For any 4 consecutive numbers, all expressions formed with '+' and '–' signs yield:
(a) Odd numbers
(b) Even numbers
(c) Prime numbers
(d) Multiples of 3
Correct Answer: (b) Even numbers
Explanation: Changing a sign changes value by an even number. All 8 expressions have same parity, starting with even result.
Analytical Thinking
Mathematical Proof
3
The sum of two even numbers is divisible by 4 if:
(a) Both are multiples of 4
(b) Both leave remainder 2 when divided by 4
(c) Either (a) or (b)
(d) None of these
Correct Answer: (c) Either (a) or (b)
Explanation: Even numbers are either 4k or 4k+2. Sum of two same type: 4k+4m=4(k+m) or (4k+2)+(4m+2)=4(k+m+1).
Algebraic Thinking
Pattern Recognition
4
If a number is divisible by 12, it must be divisible by:
(a) 24
(b) 36
(c) 8
(d) 4
Correct Answer: (d) 4
Explanation: Since 12 = 4 × 3, any multiple of 12 contains factor 4. Not necessarily divisible by 24, 36, or 8.
Divisibility Rules
5
The remainder when 427 is divided by 9 is:
(a) 4
(b) 5
(c) 6
(d) 7
Correct Answer: (a) 4
Explanation: Digital root: 4+2+7=13 → 1+3=4. So remainder is 4.
Mental Calculation
Divisibility Rules
6
Which number is divisible by 11?
(a) 158
(b) 841
(c) 462
(d) 5529
Correct Answer: (c) 462
Explanation: Alternating sum: (4+2)-6=0, divisible by 11.
Applying Rules
7
The digital root of 489710 is:
(a) 2
(b) 3
(c) 4
(d) 5
Correct Answer: (a) 2
Explanation: 4+8+9+7+1+0=29 → 2+9=11 → 1+1=2.
Computational Skills
8
In cryptarithm AB × 5 = BC, what must A be?
(a) 1
(b) 2
(c) 3
(d) 4
Correct Answer: (a) 1
Explanation: If A ≥ 2, product would be 3-digit. BC is 2-digit, so A must be 1.
Logical Reasoning
9
If a number leaves remainder 3 when divided by 5, it can be represented as:
Explanation: 6 = 2 × 3. Since 2 and 3 are co-prime, divisible by 6 iff divisible by both.
Understanding Rules
Assertion & Reasoning (20 Questions)
20 Qs × 1 mark = 20 marks
1
Assertion (A): All odd numbers can be expressed as sums of two consecutive numbers. Reason (R): Odd numbers are of the form 2n+1, which equals n + (n+1).
(a) Both A and R are true and R explains A
(b) Both A and R are true but R does not explain A
(c) A is true but R is false
(d) A is false but R is true
Correct Answer: (a) Both A and R are true and R explains A
Explanation: A is true (e.g., 7=3+4, 9=4+5). R provides algebraic proof: 2n+1 = n + (n+1).
Logical Reasoning
Mathematical Proof
2
Assertion (A): For 4 consecutive numbers, all '+'/'–' expressions give even results. Reason (R): Changing a sign changes value by an even number.
(a) Both A and R are true and R explains A
(b) Both A and R are true but R does not explain A
(c) A is true but R is false
(d) A is false but R is true
Correct Answer: (a) Both A and R are true and R explains A
Explanation: A is verified. R explains why: switching +b to -b changes total by 2b (even), so parity remains same.
Mathematical Reasoning
True/False (10 Questions)
10 Qs × 1 mark = 10 marks
1
Every even number can be expressed as a sum of consecutive numbers.
True
False
Correct Answer: False
Explanation: Powers of 2 (like 2, 4, 8, 16) cannot be expressed as sums of consecutive natural numbers.
Counterexample Reasoning
2
Digital root of a multiple of 9 is always 9.
True
False
Correct Answer: True
Explanation: Multiples of 9 have digit sums divisible by 9, repeated summing yields 9.
Understanding Rules
Short Answer I (15 Questions - 2 Marks Each)
15 Qs × 2 marks = 30 marks
1
Express 15 as sums of consecutive numbers in two ways. 2 marks
Answer: 15 = 7 + 8 OR 15 = 4 + 5 + 6
Explanation: 15 can be expressed as sum of 2 consecutive numbers (7+8) or 3 consecutive numbers (4+5+6).
Number Decomposition
2
Show that for 4 consecutive numbers, a ± b ± c ± d is always even. 2 marks
Answer: Changing signs changes value by even amounts. Starting expression gives even result, so all do.
Explanation: Let numbers be n, n+1, n+2, n+3. Changing +b to -b changes total by 2b (even).
Algebraic Proof
Short Answer II (10 Questions - 3 Marks Each)
10 Qs × 3 marks = 30 marks
1
Prove: If a number is divisible by both 9 and 4, it is divisible by 36. 3 marks
Answer: Since 9 and 4 are co-prime, their LCM is 36. Divisibility by both implies divisibility by LCM.
Explanation: Let N be divisible by 9 and 4. Then N contains factors 3² and 2², so contains factor 2²×3²=36.
Compass: Used to draw circles and arcs of a given radius.
Ruler: Used to draw straight lines and measure lengths.
A curve is any shape that can be drawn on paper, including straight lines, circles, and other figures.
2. Circle and Its Parts
Center (P): The fixed point from which all points on the circle are equidistant.
Radius: The distance from the center to any point on the circle.
All points on a circle are at the same distance from the center.
3. Squares and Rectangles
Rectangle Properties:
Opposite sides are equal.
All angles are 90°.
Square Properties:
All sides are equal.
All angles are 90°.
A square is a special type of rectangle.
Naming of quadrilaterals follows the order of corners around the shape.
4. Constructing Squares and Rectangles
A square can be constructed using a compass and ruler given its side length.
A rectangle can be constructed given:
Lengths of two adjacent sides, or
One side and the diagonal length.
5. Diagonals of Rectangles and Squares
Diagonals of a rectangle are equal in length.
In a square, diagonals also bisect the angles into two equal parts (45° each).
6. Points Equidistant from Two Given Points
Using a compass, we can find points that are equidistant from two given points by drawing arcs or circles of equal radius from both points.
7. Practical Construction Tips
Draw a rough diagram before starting construction.
Use a compass to transfer lengths without a ruler.
Light construction lines can help in locating points accurately.
❓ QUESTION BANK
A. Multiple Choice Questions (20 Questions)
In a rectangle, how many pairs of opposite sides are equal? a) 1 b) 2 c) 3 d) 4
If all sides of a quadrilateral are equal and all angles are 90°, it is a: a) Rectangle b) Rhombus c) Square d) Parallelogram
How many right angles does a square have? a) 1 b) 2 c) 3 d) 4
Which of the following is not a valid name for a rectangle with vertices A, B, C, D? a) ABCD b) BCDA c) ACBD d) CDAB
Which property is true for both squares and rectangles? a) All sides equal b) Diagonals equal c) All angles 90° d) Opposite sides parallel
If a diagonal of a rectangle divides an angle into 60° and 30°, the other angles are: a) 60° and 30° b) 90° each c) 120° and 60° d) 45° each
To construct a square of side 5 cm, which step comes first? a) Draw a perpendicular b) Draw a line of 5 cm c) Draw a diagonal d) Draw a circle
A rectangle with sides 4 cm and 6 cm has a perimeter of: a) 10 cm b) 20 cm c) 24 cm d) 30 cm
Which shape can be divided into two identical squares? a) Any rectangle b) Rectangle with sides in ratio 1:2 c) Rectangle with sides in ratio 2:3 d) Only a square
How many arcs are drawn to locate a point equidistant from two given points? a) 1 b) 2 c) 3 d) 4
In a square, each diagonal divides the opposite angles into: a) Two equal parts b) Unequal parts c) 60° and 30° d) 90° each
Which of these is a curve? a) Only circle b) Only straight line c) Both straight line and circle d) Only triangle
If AB = 8 cm in a rectangle, then CD = a) 4 cm b) 8 cm c) 12 cm d) 16 cm
A rotated square is still a square because: a) Sides change b) Angles change c) Both sides and angles remain same d) It becomes a rectangle
Which instrument is used to draw a perpendicular line? a) Only compass b) Only ruler c) Compass and ruler d) Protractor
A rectangle with one side 5 cm and diagonal 7 cm will have the other side approximately: a) 4.9 cm b) 5 cm c) 6 cm d) 8.6 cm
The number of diagonals in a rectangle is: a) 1 b) 2 c) 3 d) 4
B. Assertion & Reasoning (20 Questions)
Directions: Choose the correct option:
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true
Assertion: All squares are rectangles. Reason: A square has all angles 90° and opposite sides equal.
Assertion: A compass can be used to draw a circle of radius 4 cm. Reason: A compass has a pencil and a pointed tip.
Assertion: A rotated square remains a square. Reason: Rotation changes the side lengths.
Assertion: Diagonals of a rectangle are equal. Reason: Diagonals of a square are not equal.
Assertion: A rectangle can be named in 8 different ways. Reason: The vertices can be taken in any order.
Assertion: To find a point equidistant from two points, we draw two arcs. Reason: The intersection of two circles gives points at equal distance from both centers.
Assertion: A square can be divided into two identical rectangles. Reason: A square has all sides equal.
Assertion: In a rectangle, if one diagonal divides an angle into 45° and 45°, the rectangle is a square. Reason: In a square, diagonals bisect the angles equally.
Assertion: A circle is a curve. Reason: A straight line is also a curve.
Assertion: A rhombus with all angles 90° is a square. Reason: A rhombus has all sides equal.
Assertion: Using a compass, we can transfer lengths without a ruler. Reason: A compass can measure angles.
Assertion: A rectangle with sides 3 cm and 4 cm has a diagonal of 5 cm. Reason: For a rectangle, diagonal = √(length² + breadth²).
Assertion: In the “House” construction, point A is found using two circles. Reason: Point A is 5 cm from both B and C.
Assertion: A quadrilateral with all sides equal is a square. Reason: A rhombus also has all sides equal.
Assertion: A rectangle cannot be divided into 3 identical squares if the sides are not in ratio 1:3. Reason: For 3 identical squares, the longer side must be 3 times the shorter side.
A square satisfies the following two properties: Assertion (A): S1) All the sides are equal,.
Reason (R): and S2) All the angles are 90°.
Assertion: A square of side 6 cm has a diagonal of about 8.5 cm. Reason: Diagonal of a square = side × √2.
Assertion: In a rectangle, diagonals bisect each other. Reason: Diagonals of a rectangle are perpendicular.
Assertion: To construct a rectangle given one side and a diagonal, we use a circle. Reason: The third vertex lies on the intersection of a circle and a perpendicular line.
Assertion: A rotated rectangle is still a rectangle. Reason: Rotation does not change lengths and angles.
C. True/False (10 Questions)
A compass can only draw full circles, not arcs.
All rectangles are squares.
A square has 4 lines of symmetry.
The diagonals of a square are equal.
A quadrilateral with all angles 90° must be a square.
A rectangle can be constructed if only one side is known.
In a rectangle, opposite sides are parallel.
A rhombus is a square if one angle is 90°.
D. Short Answer Type I (2 Marks each – 15 Questions)
Write two properties of a rectangle.
Draw a rough sketch of a square PQRS of side 5 cm.
How many different ways can you name a rectangle with vertices W, X, Y, Z?
If a rectangle has length 8 cm and breadth 6 cm, what is the length of its diagonal?
How do you draw a perpendicular to a line using a compass?
What is the shape of the curve obtained by keeping the compass tip fixed and moving the pencil?
Write one similarity and one difference between a square and a rectangle.
In a rectangle ABCD, if AB = 7 cm and BC = 5 cm, what are the lengths of CD and AD?
What is the minimum number of measurements needed to construct a square?
If a diagonal of a square is 10 cm, what is the side length?
Can a rectangle be divided into two identical squares? If yes, give an example of side lengths.
What is the purpose of drawing light construction lines?
E. Short Answer Type II (3 Marks each – 10 Questions)
Construct a rectangle with sides 5 cm and 3 cm. Verify its properties.
Explain how to locate a point that is 4 cm from point P and 4 cm from point Q.
Draw a square of side 4 cm without using a protractor.
Divide a rectangle of sides 9 cm and 3 cm into three identical squares. Show construction steps.
In a rectangle, one diagonal divides an angle into 55° and 35°. What are the other angles?
Construct a rectangle with one side 6 cm and diagonal 10 cm.
How will you draw the “Wavy Wave” pattern using a compass?
Using a compass, bisect a line segment of length 8 cm.
F. Long Answer Type (5 Marks each – 10 Questions)
Construct a square of side 6 cm. Measure its diagonals and verify they are equal.
Construct a rectangle ABCD with AB = 8 cm and BC = 5 cm. Draw its diagonals and measure the angles they make with the sides.
Construct the “House” figure with all sides 5 cm. Show all construction arcs.
Construct a rectangle that can be divided into two identical squares. Explain your steps.
Construct a rectangle with one side 7 cm and a diagonal 9 cm. Verify rectangle properties.
Draw a square with 8 cm side. Inside it, draw a circle touching all four sides.
Construct a rectangle where one diagonal divides opposite angles into 60° and 30°.
Construct a “Wavy Wave” with central line 10 cm and half-circle waves.
Draw two identical “Eyes” using compass construction.
Construct a square with a circular hole at the center such that the circle touches all sides.
G. Case-Based Questions (5 Cases, each with 4 Sub-Questions)
Case 1: A rectangle has vertices A, B, C, D. AB = 6 cm, BC = 4 cm. Diagonals AC and BD intersect at O.
What is the length of CD? a) 4 cm b) 6 cm c) 10 cm d) 8 cm
What is the length of diagonal AC? a) 7.2 cm b) 10 cm c) 8.5 cm d) 9 cm
If ∠CAB = 30°, then ∠ACB = a) 30° b) 60° c) 90° d) 120°
How many pairs of equal triangles are formed by the diagonals? a) 2 b) 4 c) 6 d) 8
Case 2: A square sheet of side 10 cm is rotated to look like a diamond.
Is it still a square? a) Yes b) No
What is the length of each side after rotation? a) Changes b) Remains 10 cm c) Becomes 5 cm d) Doubles
What is the angle between two adjacent sides after rotation? a) 60° b) 90° c) 120° d) 45°
How many lines of symmetry does it have now? a) 1 b) 2 c) 4 d) 0
Case 3: In the “House” construction, all edges are 5 cm.
How many arcs are needed to locate point A? a) 1 b) 2 c) 3 d) 4
What is the shape of the roof? a) Triangle b) Square c) Circular arc d) Rectangle
Which tool is essential for this construction? a) Protractor b) Compass c) Set-square d) Divider
The base BC is of length: a) 5 cm b) 10 cm c) 15 cm d) 20 cm
Case 4: A rectangle is divided into 3 identical squares.
If the shorter side of rectangle is 4 cm, the longer side is: a) 8 cm b) 12 cm c) 16 cm d) 20 cm
How many squares are formed in total? a) 2 b) 3 c) 4 d) 6
What is the perimeter of each small square? a) 8 cm b) 12 cm c) 16 cm d) 20 cm
🔢 Insert NCERT image from p. 84 here – Consecutive number sums
5.1 Sums of Consecutive Numbers: Every odd number = sum of two consecutive numbers.
5.2 Parity of Expressions: For any 4 consecutive numbers, all +/– expressions yield even numbers.
5.3 Divisibility by 4: Even numbers → multiples of 4 (remainder 0) or remainder 2.
5.4 Divisibility Rules: If a divides M & N → divides M+N and M–N.
5.5 Remainders: Numbers leaving remainder r when divided by n: nk + r.
5.6 Digital Roots: Sum digits repeatedly until single digit. Multiples of 9 → digital root 9.
🔢 Insert NCERT image from p. 86 here – Divisibility shortcuts table
🧮 Multiple Choice Questions (20)
1. Which of the following is NOT a sum of consecutive natural numbers?
(a) 7
(b) 10
(c) 11
(d) 15
Explanation: 11 cannot be expressed as a sum of consecutive natural numbers. Powers of 2 generally cannot, except 2 itself. 7=3+4, 10=1+2+3+4, 15=7+8=4+5+6.
2. For any 4 consecutive numbers, all expressions with ‘+’ and ‘–’ yield:
(a) Odd numbers
(b) Even numbers
(c) Prime numbers
(d) Multiples of 3
Explanation: Changing a sign changes the value by an even number. Starting expression (e.g., a+b-c-d) gives even result, so all 8 possibilities yield even numbers.
3. The sum of two even numbers is divisible by 4 if:
(a) Both are multiples of 4
(b) Both leave remainder 2
(c) Either (a) or (b)
(d) None
Explanation: Even numbers are either multiples of 4 (4k) or 4k+2. Sum of two same type: 4k+4m=4(k+m) or (4k+2)+(4m+2)=4(k+m+1).
4. If a number is divisible by 8, it is also divisible by:
(a) 2
(b) 4
(c) Both (a) and (b)
(d) Only 2
Explanation: 8 = 2³, so any multiple of 8 contains factor 2²=4 and 2¹=2.
5. The remainder when 427 is divided by 9 is:
(a) 4
(b) 5
(c) 6
(d) 7
Explanation: Digit sum: 4+2+7=13 → 1+3=4. For divisibility by 9, remainder = digital root (unless digital root is 9, remainder 0).
🔍 Assertion & Reasoning (20)
1. Assertion: All odd numbers can be expressed as sums of two consecutive numbers. Reason: Odd numbers are of the form 2n+1, which equals n + (n+1).
(a) Both correct, Reason explains Assertion
(b) Both correct, Reason does not explain
(c) Assertion correct, Reason wrong
(d) Both wrong
Explanation: Assertion is true (e.g., 7=3+4, 9=4+5). Reason correctly shows odd number 2n+1 = n + (n+1), which are consecutive integers.
2. Assertion: For 4 consecutive numbers, all ‘+’/‘–’ expressions give even results. Reason: Changing a sign changes value by an even number.
(a) Both correct, Reason explains
(b) Both correct, no relation
(c) Assertion correct, Reason wrong
(d) Both wrong
Explanation: Assertion true (verified). Reason: Switching +b to -b changes total by 2b (even), so parity remains same.
✅ True/False (10)
1. Every even number can be expressed as a sum of consecutive numbers.
True
False
Explanation: False. Some even numbers like 2, 8, 32 (powers of 2) cannot be expressed as sums of consecutive natural numbers.
2. Digital root of a multiple of 9 is always 9.
True
False
Explanation: True. Multiples of 9 have digit sums divisible by 9, and repeated summing yields 9 (except 0 which gives 9 as digital root).
📝 Short Answer I (2 Marks × 15)
1. Express 15 as sums of consecutive numbers in two ways.
2. Show that for 4 consecutive numbers, a ± b ± c ± d is always even.
Answer: Let numbers be n, n+1, n+2, n+3. Changing a sign changes total by even number (e.g., +b to -b changes by 2b). Starting expression n+(n+1)-(n+2)-(n+3) = -4 (even). All other expressions have same parity.
📝 Short Answer II (3 Marks × 10)
1. Prove: If a number is divisible by both 9 and 4, it is divisible by 36.
Proof: Let N be divisible by 9 → contains factor 3². Divisible by 4 → contains factor 2². Thus N contains 2²×3²=36 as factor.
Alternatively: LCM(9,4)=36. If divisible by both, divisible by LCM.
2. Solve cryptarithm: UT × 3 = PUT
Solution: Try T=7 → 7×3=21, carry 2. U×3+2 ends with U. Try U=5 → 5×3+2=17, doesn't end with 5. Try U=1 → 1×3+2=5, doesn't end with 1. Try U=7 → 7×3+2=23, ends with 3? No.
Actually: 17×3=51 → U=1, T=7, P=5 ✓.
📚 Long Answer (5 Marks × 10)
1. Explore and write which numbers can be expressed as sums of consecutive numbers in more than one way. Provide reasoning.
Answer:
• Numbers with odd divisors >1 can be expressed in multiple ways.
• Example: 15 (divisors: 1,3,5,15) → 3 ways.
• General: If N has an odd divisor d>1, it can be written as sum of d consecutive numbers centered at N/d.
• Powers of 2 have only one representation (as single number).
• Formula: Number of ways = number of odd divisors of N minus 1.
• Example: 45 has odd divisors 1,3,5,9,15,45 → 6-1=5 ways.
2. Prove that the sum of three consecutive even numbers is divisible by 6.
Proof: Let three consecutive even numbers be 2n, 2n+2, 2n+4.
Sum = 2n + (2n+2) + (2n+4) = 6n + 6 = 6(n+1).
Clearly divisible by 6 for all integer n.
🧩 Case-Based Questions
Case 1: Sums of Consecutive Numbers
Anshu is exploring sums of consecutive numbers. He writes: 7=3+4, 10=1+2+3+4, 12=3+4+5, 15=7+8=4+5+6=1+2+3+4+5. He wonders: Can every natural number be written as a sum of consecutive numbers?
1. Which of the following numbers CANNOT be expressed as a sum of consecutive natural numbers?
(a) 9
(b) 11
(c) 16
(d) 25
Explanation: 11 cannot be expressed. Powers of 2 generally cannot (except 2 itself). 9=4+5, 16= cannot, 25=12+13.
2. Which number can be expressed as a sum of consecutive numbers in the most ways?
(a) 15
(b) 21
(c) 30
(d) 45
Explanation: 45 can be written in 5 ways: 22+23, 14+15+16, 7+8+9+10+11, 5+6+7+8+9+10, 1+2+3+4+5+6+7+8+9.
3. Which statement is TRUE?
(a) All odd numbers can be expressed as sum of 2 consecutive numbers
(b) All even numbers can be expressed as sum of consecutive numbers
(c) 0 can be expressed using only positive consecutive numbers
(d) Prime numbers cannot be expressed as sum of consecutive numbers
4. How many ways can 18 be expressed as sum of consecutive natural numbers?
(a) 1
(b) 2
(c) 3
(d) 4
Explanation: 18 = 5+6+7 = 3+4+5+6. Two ways only.
Case 2: Parity and 4 Consecutive Numbers
Take any 4 consecutive numbers, say 3,4,5,6. Place '+' and '–' signs between them in all 8 possible ways. All results are even numbers.
1. What is common about all results?
(a) All are positive
(b) All are negative
(c) All are even numbers
(d) All are multiples of 3
Explanation: Verified by evaluating all 8 expressions: 18,6,8,-4,10,-2,0,-12 → all even.
2. Why are all results even?
(a) Because 4 consecutive numbers contain 2 evens and 2 odds
(b) Because sum of 4 consecutive numbers is always even
(c) Because changing a sign changes value by an even number
(d) Both (a) and (c)
Explanation: Both reasons contribute. The parity remains same because sign changes alter by even amounts.
🔢 Insert NCERT image from p. 90 here – Digital root example
❓ "Figure It Out" Questions
1. The sum of four consecutive numbers is 34. What are these numbers?
Solution: Let numbers be n, n+1, n+2, n+3.
Sum = 4n + 6 = 34 → 4n = 28 → n=7.
Numbers: 7, 8, 9, 10.
2. "I hold some pebbles... When grouped by 3's, one remains... by 5's, one remains... by 7's, none remain. Less than 100. How many pebbles?"
Solution: N mod 3=1, mod 5=1, mod 7=0, N<100.
N=7k, try multiples of 7: 7,14,21,28,35,42,49,56,63,70,77,84,91,98.
Check odd (since mod 2=1): 7,21,35,49,63,77,91.
Check mod 5=1: 21,91.
Check mod 3=1: 91 ✓.
Answer: 91 pebbles.
