Real Numbers

CBSE Class 10 — Maths — Previous Year Questions

70 questions139 total marksMCQ: 24VSA: 18Assertion-Reason: 3SA: 20LA: 5

Q1.MCQ1 mark(2020)Easy

The total number of factors of a prime number is:

Q2.MCQ1 mark(2020)Easy

The HCF and the LCM of

12

21

15

respectively are:

Q3.MCQ1 mark(2020)Easy

The sum of exponents of prime factors in the prime-factorisation of 196 is

Q4.MCQ1 mark(2020)Easy

Euclid's division Lemma states that for two positive integers

a

b

, there exists unique integer

q

r

a = b q + r

Q5.MCQ1 mark(2020)Easy

The HCF of 135 and 225 is

Q6.MCQ1 mark(2020)Easy

The exponent of 2 in the prime factorization of 144, is

Q7.VSA1 mark(2020)Easy

\frac{2 + 5}{3}

is ________ number.

Q8.VSA1 mark(2020)Easy

HCF (135, 225) = 45

LCM (135, 225)

Q9.VSA1 mark(2020)Easy

After how many decimal places will the decimal representation of the rational number

\frac{229}{2 ^{2} \times 5 ^{7}}

Q10.VSA1 mark(2020)Easy

The LCM of two numbers is

182

and their HCF is

13

. If one of the numbers is

26

, find the other.

Q11.Assertion-Reason1 mark(2023)Medium

Assertion (A): The perimeter of

△ A BC

is a rational number.
Reason (R): The sum of the squares of two rational numbers is always rational.

$Right triangle ABC with AB = 2 cm, BC = 3 cm$

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
(C) Assertion (A) is true but Reason (R) is false.
(D) Assertion (A) is false but Reason (R) is true.

Q12.MCQ1 mark(2023)Easy

If 'p' and 'q' are natural numbers and 'p' is the multiple of 'q', then what is the HCF of 'p' and 'q'?
(a)

pq

p

q

p + q

Q13.MCQ1 mark(2023)Easy

The ratio of HCF to LCM of the least composite number and the least prime number is:

Q14.Assertion-Reason1 mark(2023)Medium

**Assertion (A):** The number

5^{n}

cannot end with the digit

0

n

is a natural number.

**Reason (R):** Prime factorisation of

5

has only two factors,

1

5

Q15.MCQ1 mark(2023)Easy

p^{2} = \frac{32}{50}

p

Q16.MCQ1 mark(2024)Medium

If two positive integers

p

q

can be expressed as

p = 18 a^{2} b^{4}

q = 20 a^{3} b^{2}

a

b

are prime numbers, then LCM

(p, q)

Q17.MCQ1 mark(2024)Easy

(2520, 6600) = 40

(2520, 6600) = 252 \times k

, then the value of

k

Q18.MCQ1 mark(2024)Easy

The HCF of two numbers

65

104

13

65

104

40 x

, then the value of

x

Q19.MCQ1 mark(2024)Easy

The greatest number which divides 281 and 1249, leaving remainder 5 and 7 respectively, is:

Q20.MCQ1 mark(2024)Easy

The LCM of three numbers 28, 44, 132 is:

Q21.MCQ1 mark(2024)Easy

If the product of two co-prime numbers is 553, then their HCF is:

Q22.MCQ1 mark(2025)Easy

HCF (98, 28) = m

LCM (98, 28) = n

, then the value of

n - 7 m

Q23.MCQ1 mark(2025)Easy

(- 1)^{n} + (- 1)^{2 n} = 0

n

Q24.MCQ1 mark(2025)Easy

Which of the following is a rational number between

3

5

Q25.MCQ1 mark(2025)Medium

The least number which is a perfect square and is divisible by each of 16, 20 and 50, is:

Q26.MCQ1 mark(2025)Easy

The sum of the exponents of prime factors in the prime factorisation of 4004 is:

Q27.MCQ1 mark(2025)Easy

The HCF of 40, 110 and 360 is:

Q28.MCQ1 mark(2025)Medium

x

is the LCM of 4, 6, 8 and

y

is the LCM of 3, 5, 7 and

p

x

y

, then which of the following is true?

Q29.MCQ1 mark(2025)Medium

x = a b^{3}

y = a^{3} b

a

b

are prime numbers, then

HCF (x, y) - LCM (x, y)

Q30.MCQ1 mark(2025)Easy

(1 + 3)^{2} - (1 - 3)^{2}

Q31.Assertion-Reason1 mark(2025)Medium

4^{n}

ends with digit 0 for some natural number

n

.
Reason (R): For a number '

x

' having 2 and 5 as its prime factors,

x^{n}

always ends with digit 0 for every natural number

n

Q32.VSA2 marks(2020)Medium

5 + 27

is an irrational number, where

7

is given to be an irrational number.

Q33.VSA2 marks(2020)Medium

1 2^{n}

can end with the digit

0

for any natural number

n

Q34.VSA2 marks(2023)Medium

Find the greatest number which divides 85 and 72 leaving remainders 1 and 2 respectively.

Q35.VSA2 marks(2023)Medium

2 + 3

is an irrational number, given that

3

is an irrational number.

Q36.VSA2 marks(2023)Easy

Two numbers are in the ratio

2 : 3

and their LCM is 180. What is the HCF of these numbers?

Q37.VSA2 marks(2023)Easy

Using prime factorisation, find HCF and LCM of

96

120

Q38.VSA2 marks(2023)Medium

6^{n}

can not end with digit

0

for any natural number

n

Q39.VSA2 marks(2023)Easy

Find the HCF and LCM of

72

120

Q40.VSA2 marks(2024)Medium

5 - 23

is an irrational number. It is given that

3

is an irrational number.

Q41.VSA2 marks(2024)Easy

Show that the number

5 \times 11 \times 17 + 3 \times 11

is a composite number.

Q42.VSA2 marks(2024)Easy

(15)^{n}

n

being a natural number, end with the digit

0

Q43.VSA2 marks(2024)Easy

Three bells toll at intervals of

9

12

15

minutes respectively. If they start tolling together, after what time will they next toll together?

Q44.VSA2 marks(2025)Easy

Find the smallest number which is divisible by both 644 and 462.

Q45.VSA2 marks(2025)Easy

Two numbers are in the ratio

4 : 5

and their HCF is 11. Find the LCM of these numbers.

Q46.SA3 marks(2020)Medium

5

is an irrational number.

Q47.SA3 marks(2020)Medium

Use Euclid Division Lemma to show that the square of any positive integer is either of the form

3 q

3 q + 1

for some integer

q

Q48.SA3 marks(2023)Medium

5

is an irrational number.

Q49.SA3 marks(2023)Easy

Find by prime factorisation the LCM of the numbers 18180 and 7575. Also, find the HCF of the two numbers.

Q50.SA3 marks(2023)Easy

Three bells ring at intervals of 6, 12 and 18 minutes. If all the three bells rang at 6 a.m., when will they ring together again?

Q51.SA3 marks(2023)Medium

5

is an irrational number.

Q52.SA3 marks(2023)Medium

3

is an irrational number.

Q53.SA3 marks(2023)Medium

The traffic lights at three different road crossings change after every

48

72

108

seconds respectively. If they change simultaneously at

7

a.m., at what time will they change together next?

Q54.SA3 marks(2023)Easy

Find the HCF and LCM of

26

65

117

, using prime factorisation.

Q55.SA3 marks(2023)Medium

2

is an irrational number.

Q56.SA3 marks(2024)Medium

In a teachers' workshop, the number of teachers teaching French, Hindi and English are 48, 80 and 144 respectively. Find the minimum number of rooms required if in each room the same number of teachers are seated and all of them are of the same subject.

Q57.SA3 marks(2024)Medium

5

is an irrational number.

Q58.SA3 marks(2024)Medium

\frac{2 - 3}{5}

is an irrational number, given that

3

is an irrational number.

Q59.SA3 marks(2024)Medium

3

is an irrational number.

Q60.SA3 marks(2024)Medium

(2 + 3)^{2}

is an irrational number, given that

6

is an irrational number.

Q61.SA3 marks(2025)Medium

\frac{1}{5}

is an irrational number.

Q62.SA3 marks(2025)Medium

3

is an irrational number.

Q63.SA3 marks(2025)Medium

(53 + \frac{2}{3})

is an irrational number given that

3

is an irrational number.

Q64.SA3 marks(2025)Medium

2

is an irrational number.

Q65.SA3 marks(2025)Medium

x

y

be two distinct prime numbers and

p = x^{2} y^{2}

q = x y^{4}

r = x^{3} y^{2}

. Find the HCF and LCM of

p

q

r

. Further check if

HCF (p, q, r) \times LCM (p, q, r) = p \times q \times r

Q66.LA4 marks(2020)Hard

Show that the square of any positive integer cannot be of the form

(5 q + 2)

(5 q + 3)

for any integer

q

Q67.LA4 marks(2020)Hard

Prove that one of every three consecutive positive integers is divisible by

3

Q68.LA4 marks(2020)Medium

5

is an irrational number.

Q69.LA4 marks(2020)Medium

(12)^{n}

cannot end with digit 0 or 5 for any natural number

n

Q70.LA4 marks(2020)Medium

(2 + 5)

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