KENDRIYA VIDYALAYA STEEL PLANT - CLASS VIII MATHS SAMPLE PAPER (SET 1)
Time: 1.5 Hours | Total Marks: 40
Section – A (8 Questions x 1 Mark = 8 Marks)
1. How would the ancient Mesopotamians have written the base of their sexagesimal number system?
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Step 1: Identify the system mentioned in the sources as the "sexagesimal system".
Step 2: The sources explicitly state that this later became a "base-60 system".
Answer: c) 60
Step 2: The sources explicitly state that this later became a "base-60 system".
Answer: c) 60
2. In the Roman numeral system, which symbol is used to represent the landmark number 50?
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Step 1: Refer to the list of landmark numbers in the Roman system.
Step 2: The symbol 'L' is associated with the value 50.
Answer: b) L
Step 2: The symbol 'L' is associated with the value 50.
Answer: b) L
3. What is the sum of all interior angles in any quadrilateral?
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Step 1: Use the property derived from joining two triangles within a quadrilateral.
Step 2: Since each triangle is 180°, the total sum is 360°.
Answer: c) 360°
Step 2: Since each triangle is 180°, the total sum is 360°.
Answer: c) 360°
4. According to the distributive property, what is the expansion of a(b + c)?
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Step 1: Recall the distributive property of multiplication over addition.
Step 2: The multiplier 'a' is applied to both 'b' and 'c' individually.
Answer: c) ab + ac
Step 2: The multiplier 'a' is applied to both 'b' and 'c' individually.
Answer: c) ab + ac
5. A number is divisible by 9 if and only if:
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Step 1: Recall the algebraic shortcut for divisibility by 9.
Step 2: Verify that adding digits repeatedly reveals the remainder when divided by 9.
Answer: c) The sum of its digits is divisible by 9.
Step 2: Verify that adding digits repeatedly reveals the remainder when divided by 9.
Answer: c) The sum of its digits is divisible by 9.
6. What is the simplest form of the ratio 60 mm : 40 mm?
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Step 1: Find the HCF of 60 and 40, which is 20.
Step 2: Divide both terms by the HCF: 60/20 = 3 and 40/20 = 2.
Answer: b) 3 : 2
Step 2: Divide both terms by the HCF: 60/20 = 3 and 40/20 = 2.
Answer: b) 3 : 2
7. Every square is also a:
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Step 1: A square has 90° angles (Rectangle).
Step 2: A square has equal sides (Rhombus).
Step 3: A square has parallel opposite sides (Parallelogram).
Answer: d) All of the above
Step 2: A square has equal sides (Rhombus).
Step 3: A square has parallel opposite sides (Parallelogram).
Answer: d) All of the above
8. The digital root of the number 489710 is:
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Step 1: Sum the digits: 4+8+9+7+1+0 = 29.
Step 2: Repeat for the result: 2+9 = 11.
Step 3: Repeat again: 1+1 = 2.
Answer: c) 2
Step 2: Repeat for the result: 2+9 = 11.
Step 3: Repeat again: 1+1 = 2.
Answer: c) 2
Section – B (4 Questions x 2 Marks = 8 Marks)
9. To maintain the same sweetness, Kesang adds 10 spoons of sugar to 6 glasses of lemonade. How much sugar is needed for 18 glasses?
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Step 1: Model the proportion: 6 : 10 :: 18 : x.
Step 2: Identify the factor of change: 18 / 6 = 3.
Step 3: Apply the factor: 10 × 3 = 30.
Answer: 30 spoons
Step 2: Identify the factor of change: 18 / 6 = 3.
Step 3: Apply the factor: 10 × 3 = 30.
Answer: 30 spoons
10. Explain if the opposite sides of a rectangle are parallel using transversal properties.
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Step 1: In rectangle ABCD, AB is a transversal to AD and BC.
Step 2: The sum of internal angles A and B is 90° + 90° = 180°.
Step 3: Since the sum of internal angles on the same side is 180°, lines AD and BC are parallel.
Step 2: The sum of internal angles A and B is 90° + 90° = 180°.
Step 3: Since the sum of internal angles on the same side is 180°, lines AD and BC are parallel.
11. Expand $(a + 1)(b + 1)$ using the distributive property.
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Step 1: Treat $(a + 1)$ as a single term: $(a + 1)b + (a + 1)1$.
Step 2: Apply distributivity again: $ab + b + a + 1$.
Answer: ab + a + b + 1
Step 2: Apply distributivity again: $ab + b + a + 1$.
Answer: ab + a + b + 1
12. Find the missing number so the ratios are proportional: 18 : 24 :: 3 : ?
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Step 1: Find the factor of change for the first terms: 18 / 6 = 3.
Step 2: Divide the second term by the same factor: 24 / 6 = 4.
Answer: 4
Step 2: Divide the second term by the same factor: 24 / 6 = 4.
Answer: 4
Section – C (4 Questions x 3 Marks = 12 Marks)
13. Solve the cryptarithm: PQ × 8 = RS, where P, Q, R, S are distinct digits.
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Step 1: Recognize that a 2-digit number × 8 results in another 2-digit number.
Step 2: Test small values: 10 × 8 = 80 (Q=0, S=0; invalid as digits must be distinct).
Step 3: Test 12: 12 × 8 = 96. Here P=1, Q=2, R=9, S=6. All digits are distinct.
Answer: P=1, Q=2, R=9, S=6
Step 2: Test small values: 10 × 8 = 80 (Q=0, S=0; invalid as digits must be distinct).
Step 3: Test 12: 12 × 8 = 96. Here P=1, Q=2, R=9, S=6. All digits are distinct.
Answer: P=1, Q=2, R=9, S=6
14. Describe the properties of the diagonals of a Rhombus.
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Step 1: The diagonals of a rhombus bisect each other.
Step 2: They bisect the internal angles of the rhombus.
Step 3: Most uniquely, they intersect at right angles (90°).
Step 2: They bisect the internal angles of the rhombus.
Step 3: Most uniquely, they intersect at right angles (90°).
15. Prove that the sum of three consecutive even numbers is always divisible by 6.
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Step 1: Let the three consecutive even numbers be $2n, 2n+2,$ and $2n+4$.
Step 2: Sum them: $2n + (2n+2) + (2n+4) = 6n + 6$.
Step 3: Factor out 6: $6(n + 1)$. This proves it is always a multiple of 6.
Step 2: Sum them: $2n + (2n+2) + (2n+4) = 6n + 6$.
Step 3: Factor out 6: $6(n + 1)$. This proves it is always a multiple of 6.
16. If a car travels 90 km in 150 minutes, find the distance it covers in 4 hours.
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Step 1: Convert 4 hours to minutes: 4 × 60 = 240 minutes.
Step 2: Model the proportion: 150 : 90 :: 240 : x.
Step 3: Cross multiply: $150x = 240 × 90$.
Step 4: Solve for x: $x = 21600 / 150 = 144$ km.
Answer: 144 km
Step 2: Model the proportion: 150 : 90 :: 240 : x.
Step 3: Cross multiply: $150x = 240 × 90$.
Step 4: Solve for x: $x = 21600 / 150 = 144$ km.
Answer: 144 km
Section – D (3 Questions x 4 Marks = 12 Marks)
17. Find the number of circles in Step k of the pattern where Step 1 has 3 circles, Step 2 has 8, and Step 3 has 15.
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Step 1: Observe the values: Step 1 (3), Step 2 (8), Step 3 (15).
Step 2: Method 1: Notice $(k+1)^2 - 1$. For $k=1, 2^2-1=3$. For $k=2, 3^2-1=8$.
Step 3: Expand the expression: $k^2 + 2k + 1 - 1 = k^2 + 2k$.
Step 4: Verify with $k=3$: $3^2 + 2(3) = 9 + 6 = 15$.
Answer: k² + 2k
Step 2: Method 1: Notice $(k+1)^2 - 1$. For $k=1, 2^2-1=3$. For $k=2, 3^2-1=8$.
Step 3: Expand the expression: $k^2 + 2k + 1 - 1 = k^2 + 2k$.
Step 4: Verify with $k=3$: $3^2 + 2(3) = 9 + 6 = 15$.
Answer: k² + 2k
18. Using the Rule of Three (Trairasika), solve: If 120 students require 15 kg of rice, how much rice is needed for 80 students?
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Step 1: Define the terms: PramΔαΉa (120), Phala (15), IchchhΔ (80).
Step 2: Use the formula: IchchhΔphala = (Phala × IchchhΔ) / PramΔαΉa.
Step 3: Substitute: $(15 × 80) / 120$.
Step 4: Simplify: $1200 / 120 = 10$.
Answer: 10 kg
Step 2: Use the formula: IchchhΔphala = (Phala × IchchhΔ) / PramΔαΉa.
Step 3: Substitute: $(15 × 80) / 120$.
Step 4: Simplify: $1200 / 120 = 10$.
Answer: 10 kg
19. Prove that the diagonals of a square bisect each other at right angles.
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Step 1: Consider two triangles formed by the diagonals, $\triangle BOA$ and $\triangle BOC$.
Step 2: By SSS congruence, $\triangle BOA \cong \triangle BOC$ because sides are equal and diagonals bisect.
Step 3: Corresponding angles $\angle BOA$ and $\angle BOC$ are equal.
Step 4: Since they form a linear pair (180°), each must be 90°.
Step 2: By SSS congruence, $\triangle BOA \cong \triangle BOC$ because sides are equal and diagonals bisect.
Step 3: Corresponding angles $\angle BOA$ and $\angle BOC$ are equal.
Step 4: Since they form a linear pair (180°), each must be 90°.
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