Chapter 1: Real Numbers
Unique Previous Year Questions extracted from the CBSE Class 10 Standard Mathematics 2025 question paper series.
I
Multiple Choice Questions
1. HCF and LCM
1 Mark
If HCF(98, 28) = \(m\) and LCM(98, 28) = \(n\), then the value of
\(n-7m\) is:
Series 30/1 (Sets 1 & 3)
2. Rational Number Between Irrational Numbers
1 Mark
Which of the following is a rational number between
\(\sqrt{3}\) and \(\sqrt{5}\)?
Series 30/1 (Sets 1, 2 & 3)
3. Least Perfect Square Divisible by Given Numbers
1 Mark
The least number which is a perfect square and is divisible by each of
16, 20 and 50, is:
Series 30/2 (Sets 1, 2 & 3)
4. HCF of Three Numbers
1 Mark
The HCF of 40, 110 and 360 is:
Series 30/2 (Sets 1, 2 & 3)
5. Prime Factorisation
1 Mark
The sum of the exponents of prime factors in the prime factorisation
of 4004 is:
Series 30/2 (Sets 1, 2 & 3)
6. Prime Factorisation of 1080
1 Mark
If
\[
1080=2^p\times3^q\times5,
\]
then \((p-q)\) is equal to:
Series 30/2 (Sets 1, 2 & 3)
7. LCM of LCMs
1 Mark
If \(x\) is the LCM of 4, 6, 8 and \(y\) is the LCM of 3, 5, 7
and \(p\) is the LCM of \(x\) and \(y\), then which of the following
is true?
Series 30/3 (Sets 1, 2 & 3)
8. HCF and LCM of Algebraic Expressions
1 Mark
If \(x=ab^3\) and \(y=a^3b\), where \(a\) and \(b\) are prime
numbers, then [HCF \((x,y)\) · LCM \((x,y)\)] is equal to:
Series 30/4 (Sets 1, 2 & 3)
9. Rational and Irrational Numbers
1 Mark
\[
(1+\sqrt3)^2-(1-\sqrt3)^2
\]
is:
Series 30/4 (Sets 1, 2 & 3)
10. LCM with Unknown Exponents
1 Mark
Let
\[
x=a^2b^3c^n
\]
and
\[
y=a^3b^mc^2,
\]
where \(a,b,c\) are prime numbers. If LCM of \(x\) and \(y\) is
\[
a^3b^4c^3,
\]
then the value of \(m+n\) is:
Series 30/5 (Sets 1, 2 & 3)
11. Divisibility of Prime Factors
1 Mark
For any prime number \(p\), if \(p\) divides \(a^2\), where \(a\) is any
real number, then \(p\) also divides:
Series 30/5 (Sets 1, 2 & 3)
12. Classification of a Number
1 Mark
\(\sqrt{0.4}\) is a/an:
Series 30/6 (Sets 1, 2 & 3)
13. Assertion – Reason: HCF and LCM
1 Mark
Assertion (A):
For any two prime numbers \(p\) and \(q\), their HCF is 1 and LCM is
\(p+q\).
Reason (R):
For any two natural numbers, HCF × LCM = product of numbers.
II
Very Short Answer Type Questions
14. Least Common Multiple
2 Marks
Find the smallest number which is divisible by both 644 and 462.
Series 30/3 (Sets 1, 2 & 3)
15. HCF and LCM Using Ratio
2 Marks
Two numbers are in the ratio \(4:5\) and their HCF is 11.
Find the LCM of these numbers.
Series 30/3 (Sets 1, 2 & 3)
III
Short Answer Type Questions
16. Irrationality of \(\frac{1}{\sqrt5}\)
3 Marks
Prove that
\[
\frac{1}{\sqrt5}
\]
is an irrational number.
Series 30/1 (Sets 1 & 3)
17. Irrationality of \(\sqrt5\)
3 Marks
Prove that \(\sqrt5\) is an irrational number.
Series 30/1 (Sets 1 & 3)
18. Irrationality of \(\sqrt3\)
3 Marks
Prove that \(\sqrt3\) is an irrational number.
Series 30/2, 30/4 & 30/5
19. Irrationality of \(\sqrt2\)
3 Marks
Prove that \(\sqrt2\) is an irrational number.
Series 30/2, 30/4 & 30/5
20. Irrationality of an Algebraic Expression
3 Marks
Prove that
\[
5\sqrt3+\frac{2}{3}
\]
is an irrational number, given that \(\sqrt3\) is an irrational number.
Series 30/3 (Set 3)
21. Composite Number and Prime Number
3 Marks
Let \(p,q\) and \(r\) be three distinct prime numbers. Check whether
\[
p\cdot q\cdot r+q
\]
is a composite number or not.
Further, give an example for 3 distinct primes \(p,q,r\) such that:
(i) \(p\cdot q\cdot r+1\) is a composite number.
(ii) \(p\cdot q\cdot r+1\) is a prime number.
Series 30/6 (Sets 1, 2 & 3)
Further, give an example for 3 distinct primes \(p,q,r\) such that:
(i) \(p\cdot q\cdot r+1\) is a composite number.
(ii) \(p\cdot q\cdot r+1\) is a prime number.
22. Irrationality of \(\sqrt5\)
3 Marks
Prove that \(\sqrt5\) is an irrational number.
Series 30/6 (Sets 1, 2 & 3)