Secondary School Examination — Mathematics
Real Numbers
Chapter 1 • Class X • Theory Examination
General Instructions
- This question paper contains four sections — Section A, B, C and D.
- Section A comprises 20 MCQ questions of 1 mark each. Total: 20 marks.
- Section B comprises 5 questions of 2 marks each. Total: 10 marks.
- Section C comprises 5 questions of 4 marks each. Total: 20 marks.
- Section D comprises 3 questions of 5 marks each and 2 questions of 5 marks (case study). Total: 30 marks.
- All questions are compulsory. There is no overall choice; however, internal choice is provided in some questions.
- Draw neat diagrams wherever required. Use of calculator is not permitted.
Section A — Multiple Choice Questions
Choose the most appropriate option (A, B, C or D) for each question.
1.
The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes in a:
(A) Non-unique way
(B) Unique way, regardless of order
(C) Way that depends on order
(D) Way with at least 3 prime factors
2.
The prime factorisation of 32760 is:
(A) 23 × 32 × 5 × 7 × 13
(B) 24 × 3 × 5 × 7 × 13
(C) 23 × 32 × 52 × 7
(D) 22 × 33 × 5 × 7 × 11
3.
For any two positive integers a and b, which relation is always true?
(A) HCF + LCM = a × b
(B) HCF × LCM = a + b
(C) HCF × LCM = a × b
(D) HCF ÷ LCM = a × b
4.
The HCF of 96 and 404 is:
(A) 8
(B) 4
(C) 2
(D) 12
5.
Which of the following is an irrational number?
(A) √4
(B) √9
(C) √7
(D) √25
6.
The LCM of 6, 72 and 120 is:
(A) 6
(B) 720
(C) 360
(D) 180
7.
Can 4n end with the digit 0 for any natural number n?
(A) Yes, when n is even
(B) Yes, when n is large enough
(C) No, never
(D) Only when n = 5
8.
The method used to prove √2 is irrational is:
(A) Proof by induction
(B) Direct proof
(C) Proof by contradiction
(D) Proof by exhaustion
9.
If HCF(306, 657) = 9, then LCM(306, 657) is:
(A) 11169
(B) 22338
(C) 2233
(D) 201042
10.
A rational number p/q (in lowest terms) has a terminating decimal if and only if:
(A) q is a prime number
(B) p < q
(C) q = 2n × 5m for some n, m ≥ 0
(D) p is even
11.
Which mathematician gave the first correct proof of the Fundamental Theorem of Arithmetic?
(A) Euclid
(B) Archimedes
(C) Isaac Newton
(D) Carl Friedrich Gauss
12.
If a prime p divides a2, then according to Theorem 1.2:
(A) p divides 2a
(B) p divides a
(C) p divides a + 1
(D) p2 divides a
13.
The decimal expansion of 7/40 is:
(A) Non-terminating repeating
(B) Terminating
(C) Non-terminating non-repeating
(D) Cannot be determined
14.
7 × 11 × 13 + 13 is a composite number because:
(A) It ends in an odd digit
(B) 13 is a common factor: 13(7 × 11 + 1)
(C) It is a product of three primes
(D) All its factors are odd
15.
Sonia and Ravi complete one round of a circular field in 18 min and 12 min respectively. They will meet at the starting point after:
(A) 6 minutes
(B) 216 minutes
(C) 36 minutes
(D) 24 minutes
16.
Which of the following decimal numbers represents an irrational number?
(A) 0.375
(B) 0.333…
(C) 0.101001000100001…
(D) 0.142857142857…
17.
In the proof that √2 is irrational, if √2 = a/b (coprime), then squaring gives 2b2 = a2. This means:
(A) a is odd
(B) b is even
(C) 2 divides a
(D) a = 2
18.
The HCF of 6, 72 and 120 is:
(A) 360
(B) 12
(C) 3
(D) 6
19.
Which of the following fractions has a non-terminating repeating decimal expansion?
(A) 3/125
(B) 7/40
(C) 5/12
(D) 11/32
20.
The sum of a rational number and an irrational number is always:
(A) Rational
(B) Irrational
(C) An integer
(D) A whole number
Section B — Short Answer Questions
Answer each question in 2–4 steps. Show all working clearly.
21.
Express 5005 as a product of its prime factors.
22.
Find the LCM and HCF of 26 and 91 by the prime factorisation method. Verify that LCM × HCF = product of the two numbers.
23.
Check whether 6n can end with the digit 0 for any natural number n. Justify your answer using the Fundamental Theorem of Arithmetic.
24.
Without performing the division, state whether 7/40 will have a terminating or non-terminating decimal expansion. Give reason.
25.
Explain why 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 is a composite number.
Section C — Long Answer Questions (I)
Show complete working and state theorems used where applicable.
26.
Prove that √3 is irrational.
27.
Find the HCF and LCM of 336 and 54 by prime factorisation. Hence verify HCF × LCM = product of the two numbers.
28.
Prove that 3√2 is irrational.
29.
Find the LCM and HCF of 12, 15 and 21 using the prime factorisation method. Also verify that:
LCM(12, 15, 21) ≠ HCF(12, 15, 21) × 12 × 15 × 21
30.
Show that 5 − √3 is irrational.
Section D — Long Answer Questions (II) & Case Studies
31.
(a) State and prove Theorem 1.2: “Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer.”
(b) Using the above theorem, prove that √2 is irrational.
OR
Prove that 1/√2 is irrational. Hence show that (5 + 3√2) is also irrational.
32.
An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?
OR
Three bells ring at intervals of 9 minutes, 12 minutes and 15 minutes. They start ringing together at 6:00 a.m. When will they ring together again? How many times will they ring together in the next 6 hours (excluding 6:00 a.m.)?
33.
Using the Fundamental Theorem of Arithmetic, find the HCF and LCM of 8, 9 and 25. Hence answer the following:
(i)What is HCF(8, 9, 25)?
(ii)What is LCM(8, 9, 25)?
(iii)Is HCF × LCM = 8 × 9 × 25? What does this tell us about the product rule for three numbers?
Case Study — I
34.
Read the passage carefully and answer the questions that follow:
A mathematics teacher is explaining the concept of prime factorisation to her class. She writes the number 2520 on the board and asks students to find its prime factorisation using a factor tree. She then explains that the prime factorisation of any natural number is the foundation for finding HCF and LCM. She reminds them that HCF is found by taking the smallest power of common prime factors, while LCM is found by taking the greatest power of all prime factors. She also tells them that the product rule HCF × LCM = a × b holds only for two numbers.
(i)Find the prime factorisation of 2520.
(ii)Find HCF(2520, 360).
(iii)Find LCM(2520, 360) using the product rule.
(iv)Why does the product rule HCF × LCM = a × b not extend to three numbers?
Case Study — II
35.
Read the passage carefully and answer the questions that follow:
Riya is studying rational and irrational numbers. She knows that every rational number p/q (in lowest terms, q ≠ 0) has a decimal expansion that either terminates or eventually repeats. She is trying to determine, without actual division, whether a given fraction will terminate. Her teacher tells her the secret: “Just factorise the denominator. If it has only 2s and 5s as prime factors, it terminates. Otherwise, it repeats.” Riya also learns that numbers like √2, √3, √5 are irrational because their decimal expansions never terminate and never repeat.
(i)Without dividing, determine whether 13/125 has a terminating or non-terminating decimal expansion. Give reason.
(ii)Without dividing, determine whether 11/98 terminates. Give reason.
(iii)Prove that √5 is irrational.
(iv)What type of decimal expansion does every irrational number have?
🔑 Official Answer Key
For teacher use only — Chapter 1: Real Numbers (80 Marks)
Section A — MCQ Answers (1 mark each)
Q1
(B)
Q2
(A)
Q3
(C)
Q4
(B)
Q5
(C)
Q6
(C)
Q7
(C)
Q8
(C)
Q9
(B)
Q10
(C)
Q11
(D)
Q12
(B)
Q13
(B)
Q14
(B)
Q15
(C)
Q16
(C)
Q17
(C)
Q18
(D)
Q19
(C)
Q20
(B)
Section B — Short Answers (2 marks each)
Q21 — Prime factorisation of 5005
Step 15005 ÷ 5 = 1001 → 5005 = 5 × 1001
Step 21001 ÷ 7 = 143 → 1001 = 7 × 143
Step 3143 ÷ 11 = 13
5005 = 5 × 7 × 11 × 13
Q22 — LCM and HCF of 26 and 91
Factor26 = 2 × 13 and 91 = 7 × 13
HCF(26, 91) = 13 LCM(26, 91) = 2 × 7 × 13 = 182
VerifyHCF × LCM = 13 × 182 = 2366 = 26 × 91 ✓
Q23 — Can 6n end with digit 0?
KeyFor a number to end in 0, it must be divisible by 10 = 2 × 5.
6n = (2 × 3)n = 2n × 3n
Concl.The prime factorisation of 6n contains only 2 and 3, never 5. By the uniqueness of FTA, 5 can never appear. Therefore 6n can never end with 0.
Q24 — Decimal expansion of 7/40
40 = 23 × 5
RuleSince 40 = 23 × 51 (only factors 2 and 5), the decimal expansion of 7/40 is terminating. (= 0.175)
Q25 — Why is 7! + 5 composite?
7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 = 5(7 × 6 × 4 × 3 × 2 × 1 + 1) = 5 × 1009
Concl.5 is a factor other than 1 and the number itself, so it is composite.
Section C — Long Answers I (4 marks each)
Q26 — Prove √3 is irrational
AssumeLet √3 = a/b where a, b are integers, b ≠ 0, and a, b are coprime.
SquareThen 3b2 = a2 → 3 divides a2 → 3 divides a (Theorem 1.2)
SubWrite a = 3c. Then 3b2 = 9c2 → b2 = 3c2 → 3 divides b.
Contr.Both a and b are divisible by 3 — contradicts them being coprime. Hence √3 is irrational.
Q27 — HCF and LCM of 336 and 54
336 = 24 × 3 × 7 54 = 2 × 33
HCF = 21 × 31 = 6 LCM = 24 × 33 × 7 = 3024
VerifyHCF × LCM = 6 × 3024 = 18144 = 336 × 54 ✓
Q28 — Prove 3√2 is irrational
AssumeLet 3√2 = a/b (coprime, b ≠ 0).
√2 = a / (3b)
Contr.Since a, 3, b are integers, a/3b is rational → √2 is rational. But √2 is irrational. Contradiction! Hence 3√2 is irrational.
Q29 — LCM and HCF of 12, 15, 21
12 = 22 × 3 15 = 3 × 5 21 = 3 × 7
HCF = 3 LCM = 22 × 3 × 5 × 7 = 420
VerifyHCF × LCM = 3 × 420 = 1260 ≠ 12 × 15 × 21 = 3780. This confirms the product rule does not hold for three numbers.
Q30 — Prove 5 − √3 is irrational
AssumeLet 5 − √3 = a/b (rational, coprime, b ≠ 0).
√3 = 5 − a/b = (5b − a)/b
Contr.RHS is rational (integers in numerator and denominator) → √3 is rational. But √3 is irrational. Contradiction! Hence 5 − √3 is irrational.
Section D — Long Answers II & Case Studies (5 marks each)
Q31 — Theorem 1.2 + Proof that √2 is irrational
Thm 1.2Let a = p1p2…pn. Then a2 = p12p22…pn2. If p | a2, by FTA uniqueness p must be one of p1,…,pn, so p | a. ✓
AssumeLet √2 = a/b (coprime). Then 2b2 = a2 → 2|a → a = 2c.
2b2 = 4c2 → b2 = 2c2 → 2|b
Contr.2 divides both a and b — contradicts coprime. √2 is irrational.
Q32 — Army contingent OR Three bells
MainMaximum columns = HCF(616, 32).
616 = 23 × 7 × 11 32 = 25
HCF = 23 = 8 columns
ORLCM(9,12,15) = 180 minutes. They ring together at 9:00 a.m. In 6 hours = 360 min: 360 ÷ 180 = 2 times.
Q33 — HCF and LCM of 8, 9, 25
8 = 23 9 = 32 25 = 52
(i)HCF = 1 (no common prime factors)
(ii)LCM = 23 × 32 × 52 = 1800
(iii)HCF × LCM = 1 × 1800 = 1800 ≠ 8 × 9 × 25 = 1800. (Coincidentally equal here, but the general rule does not hold for three numbers — the product formula is a special property of two numbers only.)
Q34 — Case Study I (Prime Factorisation of 2520)
(i)2520 = 23 × 32 × 5 × 7
(ii)360 = 23 × 32 × 5. Common factors: 23, 32, 5. HCF = 23 × 32 × 5 = 360
(iii)LCM = (2520 × 360) ÷ 360 = 2520
(iv)The formula HCF × LCM = a × b is derived from the properties of two numbers. For three numbers, the combined HCF and LCM interact differently — the simple product a × b × c is not equal to HCF × LCM in general.
Q35 — Case Study II (Decimals & Irrationals)
(i)125 = 53 (only 5s) → 13/125 terminates. (= 0.104)
(ii)98 = 2 × 72. Has factor 7 → 11/98 is non-terminating repeating.
(iii)Assume √5 = a/b (coprime). 5b2 = a2 → 5|a → a=5c. Then b2 = 5c2 → 5|b. Contradiction. √5 is irrational.
(iv)Irrational numbers have non-terminating, non-repeating decimal expansions.
Marking Scheme Summary
Section A
20 × 1
= 20 marks
Section B
5 × 2
= 10 marks
Section C
5 × 4
= 20 marks
Section D
5 × 6
= 30 marks
TOTAL = 80 MARKS
