Class 10 Maths Test Paper Chapter 1 Real Numbers | CBSE and GSEB Board | Detailed Solutions & Marking Scheme
VEDANT CLASSES
Class 10 – Mathematics Test
Chapter: Real Numbers
Time: 1 Hour | Maximum Marks: 30
Paper Type: Competency Based / Situation Based
Section A – Situation Based MCQs
(1 × 6 = 6 Marks)
1. Riya is arranging 24 red balls and 36 blue balls into identical packets so that each packet has the same number of balls of each colour. The maximum number of packets she can make is:
(a) 6 (b) 12 (c) 18 (d) 24
2. Two school bells ring after every 12 minutes and 15 minutes respectively. Both bells rang together at 8:00 AM. After how many minutes will they ring together again?
(a) 30 minutes (b) 45 minutes (c) 60 minutes (d) 90 minutes
3. A teacher asked students to write the prime factorisation of 72. Which student gave the correct answer?
(a) 2 × 2 × 2 × 3 × 3 (b) 2 × 3 × 12 (c) 3 × 3 × 8 (d) 2 × 6 × 6
4. A square park has an area of 50 m². A student tries to find its side length using √50. Which statement is correct about √50?
(a) It is a rational number (b) It is an irrational number (c) It is an integer (d) It is a natural number
5. A student finds HCF = 6 and LCM = 180 for two numbers. What is the product of the two numbers?
(a) 30 (b) 1080 (c) 186 (d) 174
6. A number is written as 2² × 3 × 5. How many distinct prime factors does this number have?
(a) 2 (b) 3 (c) 4 (d) 5
Section B – Situation Based Short Answer
(2 × 4 = 8 Marks)
7. A fruit seller has 84 apples and 126 oranges. He wants to pack them in identical boxes so that each box contains the same number of apples and oranges with none left.
Find the maximum number of boxes he can make.
8. Two traffic signals at a junction turn green every 30 seconds and 45 seconds respectively.
After how many seconds will they turn green together again?
9. A student claims that every composite number can be written as a product of prime numbers in only one way.
State the Fundamental Theorem of Arithmetic and explain whether the student is correct.
10. A student says that √7 is rational because it has a decimal value.
Explain why this statement is incorrect.
Section C – Situation Based Application Questions
(4 × 2 = 8 Marks)
11. A teacher asks two students to find the HCF and LCM of 48 and 72 using prime factorisation.
(a) Find the prime factorisation of both numbers
(b) Find the HCF
(c) Find the LCM
12. A sports academy has 90 footballs and 120 cricket balls. They want to distribute them into identical kits containing both types of balls.
Find the maximum number of kits possible.
Section D – Competency Based Reasoning Questions
(4 × 2 = 8 Marks)
13. A student assumes that √2 is a rational number and writes it in the form p/q.
Using the contradiction method, prove that √2 is irrational.
14. A mathematics club claims that the number 6ⁿ will always end with digit 6 for any natural number n.
Using prime factorisation reasoning, explain whether this statement is correct.
Here is your DETAILED ANSWER KEY with MARKING SCHEME (CBSE/GSEB style, step-wise, easy to copy):
VEDANT CLASSES
Class 10 – Mathematics Test
Chapter: Real Numbers
DETAILED ANSWER KEY & MARKING SCHEME
Section A – MCQs (1 × 6 = 6 Marks)
HCF(24, 36) = 12 → Maximum packets
Answer: (b) 12LCM(12, 15) = 60
Answer: (c) 60 minutes72 = 2 × 2 × 2 × 3 × 3
Answer: (a)√50 = 5√2 → Irrational
Answer: (b)Product = HCF × LCM = 6 × 180 = 1080
Answer: (b)Distinct primes = 2, 3, 5 → Total = 3
Answer: (b)
Section B – Short Answer (2 × 4 = 8 Marks)
7.
84 = 2 × 2 × 3 × 7
126 = 2 × 3 × 3 × 7
HCF = 2 × 3 × 7 = 42
Answer: Maximum number of boxes = 42
Marking Scheme:
Prime factorisation (1 mark)
Correct HCF (1 mark)
8.
LCM(30, 45)
30 = 2 × 3 × 5
45 = 3 × 3 × 5
LCM = 2 × 3 × 3 × 5 = 90
Answer: 90 seconds
Marking Scheme:
Prime factors (1 mark)
Correct LCM (1 mark)
9.
Fundamental Theorem of Arithmetic:
Every composite number can be expressed as a product of prime numbers in a unique way, apart from the order of factors.
Conclusion: Student is correct.
Marking Scheme:
Statement (1 mark)
Explanation (1 mark)
10.
√7 = 2.6457… (non-terminating, non-repeating)
Such decimals are irrational.
Conclusion: Statement is incorrect.
Marking Scheme:
Decimal explanation (1 mark)
Conclusion (1 mark)
Section C – Application (4 × 2 = 8 Marks)
11.
48 = 2⁴ × 3
72 = 2³ × 3²
HCF = 2³ × 3 = 24
LCM = 2⁴ × 3² = 144
Answer:
(a) Factorisation correct
(b) HCF = 24
(c) LCM = 144
Marking Scheme:
Factorisation (1 mark)
HCF (1 mark)
LCM (2 marks)
12.
90 = 2 × 3² × 5
120 = 2³ × 3 × 5
HCF = 2 × 3 × 5 = 30
Answer: Maximum kits = 30
Marking Scheme:
Factorisation (1 mark)
HCF (1 mark)
Section D – Reasoning (4 × 2 = 8 Marks)
13. Proof that √2 is irrational
Assume √2 = p/q (in lowest form)
2 = p²/q²
⇒ p² = 2q²
⇒ p² is even → p is even
Let p = 2k
Substitute:
(2k)² = 2q²
4k² = 2q²
q² = 2k² → q is even
Both p and q even → contradiction
Hence, √2 is irrational.
Marking Scheme:
Assumption (1 mark)
Contradiction steps (2 marks)
Conclusion (1 mark)
14.
6 = 2 × 3
6¹ = 6 → last digit 6
6² = 36 → last digit 6
6³ = 216 → last digit 6
Multiplication always retains last digit 6
Conclusion: Statement is correct.
Marking Scheme:
Verification (2 marks)
Conclusion (2 marks)
✅ TOTAL MARKS DISTRIBUTION SUMMARY
Section A: 6
Section B: 8
Section C: 8
Section D: 8
Total = 30 Marks
Comments
Post a Comment