Last Updated on April 2, 2025 by XAM CONTENT
Hello students, we are providing case study questions for class 10 maths. Case study questions are the new question format that is introduced in CBSE board. The resources for case study questions are very less. So, to help students we have created chapterwise case study questions for class 10 maths. In this article, you will find case study questions for CBSE Class 10 Maths Chapter 1 Real Numbers. It is a part of Case Study Questions for CBSE Class 10 Maths Series.
| Chapter | Real Numbers |
| Type of Questions | Case Study Questions |
| Nature of Questions | Competency Based Questions |
| Board | CBSE |
| Class | 10 |
| Subject | Maths |
| Unit | Unit 1 Number Systems |
| Useful for | Class 10 Studying Students |
| Answers provided | Yes |
| Difficulty level | Mentioned |
| Important Link | Class 10 Maths Chapterwise Case Study |
Case Study Questions on Real Numbers
Questions
Passage 1:
Teaching Mathematics through activities is a powerful approach that enhances students’ understanding and engagement. Keeping this in
mind, Ms. Mukta planned a prime number game for class 5 students. She announces the number 2 in her class and asked the first student to multiply it by a prime number and then pass it to second student. Second student also multiplied it by a prime number and passed it to third student. In this way by multiplying to a prime number, the last student got 173250.
Based on the above information, solve the following questions:
Q 1. What is the least prime number used by students?
Q 2. How many students are in the class?
Or
What is the highest prime number used by students?
Q 3. Which prime number has been used maximum times?
Answers
1. The prime factorisation of given number is
173250 = 2 × 3 × 3 × 5 × 5 × 5 × 7 × 11 = 2 × 32 × 53 × 7 × 11
Since, the first student multiply 2 by the prime number.
We see that in prime factorisation of given number, the smallest prime number other than 2 is 3.
2. The total number of prime numbers other than 2 is 7.
Hence, 7 students are in the class.
Or
The highest prime number used by students is 11.
3. Prime number 5 is a number which is used maximum number of times.
Passage 2:
Repeating and Terminating Decimals
Priya divided 7 by 8 and got 0.875. When she divided 1 by 3, the answer was 0.333… repeating endlessly. The teacher explained that rational numbers either have terminating or non-terminating repeating decimals, and this helps identify whether a number is rational.
Q1. Which of the following is a terminating decimal?
(a) 1/3
(b) 2/5
(c) 2/7
(d) 22/7
Difficulty Level: Easy
Answer: (b) 2/5
Q2. 1/3 is an example of:
(a) Irrational number
(b) Terminating decimal
(c) Non-terminating, repeating decimal
(d) Non-repeating decimal
Difficulty Level: Easy
Answer: (c) Non-terminating, repeating decimal
Q3. A rational number in decimal form is terminating only if the denominator (in lowest form) has:
(a) Only factors of 2 and/or 5
(b) Factor 3
(c) Any prime factor
(d) Factors greater than 7
Difficulty Level: Medium
Answer: (a) Only factors of 2 and/or 5
Q4. Which number has a non-terminating repeating decimal?
(a) 7/8
(b) 3/4
(c) 5/6
(d) 2/5
Difficulty Level: Medium
Answer: (c) 5/6
Passage 3:
Proving Irrationality
In a class debate, Arnav claimed that √5 is rational. His classmate proved otherwise using proof by contradiction, starting with the assumption that √5 = p/q and showing that both p and q must be divisible by 5—contradicting their co-prime condition.
Q1. What is the correct method to prove √5 is irrational?
(a) Use decimal expansion
(b) Use prime factorization
(c) Use proof by contradiction
(d) Use estimation
Difficulty Level: Hard
Answer: (c) Use proof by contradiction
Q2. In the proof of irrationality, if √5 = p/q, then:
(a) p and q must be even
(b) p and q must be divisible by 5
(c) p/q must be a whole number
(d) q must be a decimal
Difficulty Level: Hard
Answer: (b) p and q must be divisible by 5
Q3. Why is it a contradiction if both p and q are divisible by 5?
(a) They are not integers
(b) They are not co-prime
(c) They are both odd
(d) It makes √5 a perfect square
Difficulty Level: Medium
Answer: (b) They are not co-prime
Q4. Which of the following numbers can be proven irrational in a similar way?
(a) √16
(b) √9
(c) √2
(d) √25
Difficulty Level: Medium
Answer: (c) √2
Passage 4:
The Great Maths Duel
Aryan and Ved were in a friendly math competition. The question: “Prove √7 is irrational using contradiction.” Aryan assumed √7 = p/q (with p, q co-prime), squared both sides, and found that 7 divided both p and q — which contradicted the co-prime condition. Ved, meanwhile, tried to show √7 = 2.6457 and gave up. Aryan won the round!
Q1. What method did Aryan use to prove √7 is irrational?
(a) Estimation
(b) Decimal expansion
(c) Proof by contradiction
(d) Trial and error
Difficulty Level: Medium
Answer: (c) Proof by contradiction
Q2. If √7 = p/q is assumed and both p and q are divisible by 7, what does it contradict?
(a) That √7 is a perfect square
(b) That √7 is rational
(c) That p and q are co-prime
(d) That 7 is prime
Difficulty Level: Hard
Answer: (c) That p and q are co-prime
Q3. Why was Ved’s method incorrect?
(a) He did not try enough decimals
(b) Irrationality cannot be proven by decimal approximation
(c) He used the wrong formula
(d) He didn’t use a calculator
Difficulty Level: Medium
Answer: (b) Irrationality cannot be proven by decimal approximation
Q4. Which of the following is a valid irrational number?
(a) √7
(b) 3/4
(c) 22/7
(d) 0.666…
Difficulty Level: Easy
Answer: (a) √7
🚀 Boost Your Exam Prep: Get case study questions for all subjects (Class 6-12) now!
Also check
- Arithmetic Progressions Class 10 Case Study Questions Maths Chapter 5
- Quadratic Equations Class 10 Case Study Questions Maths Chapter 4
- Pair of Linear Equations in Two Variables Class 10 Case Study Questions Maths Chapter 3
- Polynomials Class 10 Case Study Questions Maths Chapter 2
- Real Numbers Class 10 Case Study Questions Maths Chapter 1
Topics from which case study questions may be asked
- Fundamental Theorem of Arithmetic
- Statement of the theorem after review
- Illustrations and motivation through examples
- Proofs of Irrationality
- √2
- √3
- √5
Case study questions based on above topics may be asked.
Understanding Real Numbers
Fundamental Theorem of Arithmetic: Every composite number can be uniquely expressed as a product of primes, except for the order in which these prime factors occur. This theorem is also known as Unique Factorisation Theorem.
For any Two Positive Integers $a$ and $b$ :
1. $\operatorname{HCF}(a, b)$ : Product of the smallest power of each common prime factor in the numbers.
2. LCM $(a, b)$ : Product of the greatest power of each common and uncommon prime factors in the numbers.
3. $\operatorname{HCF}(a, b) \times \operatorname{LCM}(a, b)=$ Product of numbers $(a \times b)$
A number is said to be a composite number, if it has at least one factor other than 1 and the number itself. e.g., 4, 6, 9, 24, … are composite numbers.
Frequently Asked Questions (FAQs) on Real Numbers Case Study
Q1: What is the Fundamental Theorem of Arithmetic?
A1: The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes, and this factorization is unique except for the order of the factors.
Q2: What is Euclid’s Division Lemma?
A2: A: Euclid’s Division Lemma states that for any two positive integers $a$ and $b$, there exist unique integers $q$ (quotient) and $r$ (remainder) such that:
$$
\begin{aligned}
& a=b q+r \\
& \text { where } 0 \leq r<b
\end{aligned}
$$
Q3: What is the difference between terminating and non-terminating decimals?
A3: Terminating Decimals: A decimal that comes to an end (e.g., 0.5, 0.25). These correspond to rational numbers with denominators having factors of only 2 or 5
Non-Terminating Decimals: A decimal that continues indefinitely.
Repeating (Recurring): The decimal has a repeating pattern (e.g., 0.333…).
Non-Repeating: The decimal does not repeat and is irrational (e.g., π, √2)
Q4: How can you determine if a rational number will have a terminating or non-terminating decimal expansion?
A4: A: A rational number $\frac{p}{q}$ (in its simplest form) will have a terminating decimal expansion if and only if the prime factorization of the denominator $q$ contains only the prime factors 2 or 5.
Q5: What is the LCM and HCF relationship of two numbers?
A5: For any two integers a and b:
HCF x LCM = a x b
Q6: What are co-prime numbers?
A6: Two numbers are co-prime if their HCF is 1.
Q7: Are there any online resources or tools available for practicing Real Numbers case study questions?
A7: We provide case study questions for CBSE Class 10 Maths on our website. Students can visit the website and practice sufficient case study questions and prepare for their exams. If you need more case study questions, then you can visit Physics Gurukul website. they are having a large collection of case study questions for all classes.
