CHAPTER 1 : REAL NUMBERS
CBSE Class 10 Mathematics — PYQs 2026 | Basic & Standard
BASIC MATHEMATICS — 2026
SECTION A : MULTIPLE CHOICE QUESTIONS (1 Mark Each)
1.
If the HCF of two positive integers
a and
b is 1,
then their LCM is:
(A) a + b
(B) a
(C) b
(D) ab
[Sets: 430/1/1, 430/1/2, 430/1/3 | 1 Mark]
2.
If
p = 2³ × 3² × 5
and
q = 2² × 3³,
then the LCM of
p and
q is:
(A) 2³ × 3³
(B) 2² × 3²
(C) 2² × 3² × 5
(D) 2³ × 3³ × 5
[Sets: 430/3/1, 430/3/2, 430/3/3 | 1 Mark]
3.
The value of (HCF − LCM) for the two numbers 3 and 5 is:
(A) 2
(B) 4
(C) 14
(D) −14
[Sets: 430/2/1, 430/2/2, 430/2/3 | 1 Mark]
4.
The number
2ⁿ,
where
n is a natural number,
cannot end with the digit:
(A) 4
(B) 6
(C) 2
(D) 0
[Set: 430/2/1 | 1 Mark]
5.
3ⁿ,
where
n is a natural number,
cannot end with the digit:
(A) 3
(B) 5
(C) 7
(D) 9
[Sets: 430/3/1, 430/3/2, 430/3/3, 430/2/2 | 1 Mark]
6.
The number
3 + √2
is:
(A) Rational
(B) Irrational
(C) Integer
(D) Natural
[Set: 430/1/1 | 1 Mark]
7.
(√3 − 3)/√3
is:
(A) Rational
(B) Irrational
(C) Integer
(D) Natural
[Set: 430/1/2 | 1 Mark]
8.
(2 + √2)²
is:
(A) Rational
(B) Irrational
(C) Integer
(D) Natural
[Set: 430/1/3 | 1 Mark]
9.
A prime number has:
(A) exactly two prime factors
(B) exactly one prime factor
(C) at least one prime factor
(D) at least two prime factors
[Sets: 430/3/1, 430/3/2, 430/3/3 | 1 Mark]
SECTION B : ASSERTION–REASON QUESTIONS (1 Mark Each)
10.
Assertion (A):
For any two natural numbers
a and b,
the HCF of a and b
is a factor of the LCM of a and b.
Reason (R):
HCF of any two natural numbers divides both the numbers.
[Sets: 430/1/1, 430/1/2, 430/1/3 | 1 Mark]
11.
Assertion (A):
The prime numbers which divide 36 also divide 6.
Reason (R):
Any number which divides
p²
also divides p.
[Sets: 430/2/1, 430/2/2, 430/2/3 | 1 Mark]
SECTION C : SHORT ANSWER QUESTIONS (3 Marks Each)
12.
Prove that
√3 is an irrational number.
[Sets: 430/1/1, 430/1/2, 430/1/3 | 3 Marks]
13.
Prove that
√5 is an irrational number.
[Sets: 430/2/1, 430/2/2, 430/2/3 | 3 Marks]
14.
Prove that
√2 is an irrational number.
[Sets: 430/3/1, 430/3/2, 430/3/3 | 3 Marks]
15.
The factor tree of a number
x is shown below:
x → 2, y
y → 2, 210
210 → a, 70
70 → 2, 35
35 → 5, b
Find the values of
x, y, a, and
b.
Write
x as a product of prime factors.
[Sets: 430/1/1, 430/1/2, 430/1/3 | 3 Marks]
16.
State the Fundamental Theorem of Arithmetic and use it to find the LCM of 36 and 54.
[Sets: 430/2/1, 430/2/2, 430/2/3 | 3 Marks]
17.
Find which among the following numbers
a, b, and
c
are composite numbers:
a = 7 × 11 × 13 + 13
b = 6 × 5 × 4 + 4
c = 7 × 13 + 6
[Sets: 430/3/1, 430/3/2, 430/3/3 | 3 Marks]
STANDARD MATHEMATICS — 2026
SECTION A : MULTIPLE CHOICE QUESTIONS (1 Mark Each)
1.
The HCF of 960 and 432 is:
(A) 48
(B) 54
(C) 72
(D) 36
[Set 30/1/1 Q1, Set 30/1/2 Q3, Set 30/1/3 Q17 | 1 Mark]
2.
The LCM of 960 and 240 is:
(A) 960
(B) 240
(C) 60
(D) 15
[Set 30/2/1 Q1, Set 30/2/3 Q6 | 1 Mark]
3.
The natural number 2 is:
(A) a prime number
(B) a composite number
(C) prime as well as composite
(D) neither prime nor composite
[Set 30/1/1 Q2, Set 30/1/2 Q1, Set 30/1/3 Q18 | 1 Mark]
4.
The natural number 1 is:
(A) a prime number
(B) a composite number
(C) prime as well as composite
(D) neither prime nor composite
[Set 30/2/1 Q2, Set 30/2/2 Q4, Set 30/2/3 Q2 | 1 Mark]
5.
For any natural number
n,
6ⁿ ends with the digit:
(A) 0
(B) 6
(C) 3
(D) 2
[Set 30/1/1 Q3, Set 30/1/2 Q2, Set 30/1/3 Q1 | 1 Mark]
6.
For any natural number
n,
5ⁿ ends with the digit:
(A) 0
(B) 5
(C) 3
(D) 2
[Set 30/2/1 Q3, Set 30/2/2 Q6, Set 30/2/3 Q4 | 1 Mark]
7.
The value of
3 × 11 × 13 + 3
is:
(A) a prime number
(B) divisible by 13
(C) a composite number
(D) an odd number
[Set 30/3/1 Q4, Set 30/3/2 Q18, Set 30/3/3 Q11 | 1 Mark]
8.
There are two sections A and B of Grade X.
There are 28 students in Section A and 30 students in Section B.
What is the minimum number of books you will acquire for the class library
so that they can be distributed equally among students of Section A or Section B?
(A) 144
(B) 2
(C) 420
(D) 272
[Set 30/2/2 Q2 | 1 Mark]
SECTION B : ASSERTION–REASON QUESTIONS (1 Mark Each)
9.
Assertion (A):
H.C.F. (36m², 18m) = 18m, where m
is a prime number.
Reason (R):
H.C.F. of two numbers is always less than or equal to the smaller number.
[Set 30/5/1 Q20, Set 30/5/2 Q20, Set 30/5/3 Q19 | 1 Mark]
10.
Assertion (A):
(√3 + √5)
is an irrational number.
Reason (R):
Sum of any two irrational numbers is always irrational.
[Set 30/4/1 Q20, Set 30/4/2 Q19, Set 30/4/3 Q20 | 1 Mark]
SECTION C : VERY SHORT ANSWER QUESTIONS (2 Marks Each)
11.
Prove that
2 + 3√5
is an irrational number, given that
√5
is irrational.
[Set 30/4/1 Q24(a), Set 30/4/2 Q21(a), Set 30/4/3 Q25(a) | 2 Marks]
12.
Prove that
2 − 5√3
is an irrational number, given that
√3
is irrational.
[Set 30/5/1 Q21 | 2 Marks]
13.
Prove that
4 − 2√5
is an irrational number, given that
√5
is irrational.
[Set 30/5/2 Q23 | 2 Marks]
14.
Prove that
14 − 2√3
is an irrational number, given that
√3
is irrational.
[Set 30/5/3 Q25 | 2 Marks]
15.
If the HCF of 210 and 55 is expressed as
HCF = 210 × 5 + 55m
then find the value of
m.
[Set 30/4/1 Q24(b), Set 30/4/2 Q21(b), Set 30/4/3 Q25(b),
Set 30/5/1 Q21(b), Set 30/5/3 Q25(b) | 2 Marks]
16.
Find the length of the plank that can be used to measure the lengths
4 m 20 cm and 5 m 4 cm exactly, in the least time.
[Set 30/1/3 Q21, Set 30/3/1 Q21, Set 30/3/3 Q24 | 2 Marks]
SECTION D : SHORT ANSWER QUESTIONS (3 Marks Each)
17.
A trader has three different types of oils of volume
870 L, 812 L and 638 L.
Find the least number of containers of equal size required
to store all the oil without getting mixed.
[Set 30/4/2 Q26 | 3 Marks]
18.
Find the greatest number which divides 764 and 1198,
leaving remainders 8 and 10 respectively.
[Set 30/4/3 Q28 | 3 Marks]
19.
Find the greatest number less than 10,000 which is exactly divisible by
48, 60 and 65.
[Set 30/4/1 Q31 | 3 Marks]
20.
The dimensions of a window are
156 cm × 216 cm.
Arjun wants to put grill on the window creating complete squares
of maximum size.
Determine the side length of the square and hence find the number
of squares formed.
[Set 30/5/1 Q31, Set 30/5/3 Q29 | 3 Marks]
21.
Prove that
√3 is an irrational number.
[Set 30/1/1 Q26, Set 30/1/3 Q30, Set 30/2/2 Q31,
Set 30/3/1 Q30, Set 30/3/2 Q31 | 3 Marks]
22.
Prove that
√2 is an irrational number.
[Set 30/1/2 Q26, Set 30/3/3 Q26 | 3 Marks]
23.
Prove that
√5 is an irrational number.
[Set 30/2/1 Q26, Set 30/2/3 Q27 | 3 Marks]