chap 2 pyq

✦ 10 Years Archive (2015-2026) ✦

Chapter 1: Real Numbers

Complete Board Exam PYQs | 55 Authentic Questions

🔹 1 Mark Questions (Objective/MCQ)

Year: 2026 | Set-1 | Code: 30/2/1 | Marks: 1

Q: The LCM of $960$ and $240$ is :
(A) $960$    (B) $240$    (C) $60$    (D) $15$

✔️ Answer: (A) 960

Year: 2025 | Set-1 | Code: 30/1/1 | Marks: 1

Q: If two positive integers $a$ and $b$ are written as $a = x^3 \times y^2$ and $b = x \times y^3$, where $x, y$ are prime numbers, then $\text{HCF}(a, b)$ is :
(A) $x \times y$    (B) $x \times y^2$    (C) $x^3 \times y^3$    (D) $x^2 \times y^2$

✔️ Answer: (B) $x \times y^2$

Year: 2025 | Set-1 | Code: 30/1/1 | Marks: 1

Q: If two positive integers $a$ and $b$ are written as $a = x^3 \times y^2$ and $b = x \times y^3$, where $x, y$ are prime numbers, then $\text{LCM}(a, b)$ is :
(A) $x \times y^2$    (B) $x^3 \times y^3$    (C) $x^2 \times y^2$    (D) $x^4 \times y^5$

✔️ Answer: (B) $x^3 \times y^3$

Year: 2020 | Set-1 | Code: 30/1/1 | Marks: 1

Q: The sum of exponents of prime factors in the prime-factorisation of $196$ is :
(A) $3$    (B) $4$    (C) $5$    (D) $2$

✔️ Answer: (B) 4

Year: 2023 | Set-3 | Code: 30/2/3 | Marks: 1

Q: If $\text{HCF}(2520, 6600) = 40$ and $\text{LCM}(2520, 6600) = 252 \times k$, then the value of $k$ is :
(A) $1650$    (B) $1600$    (C) $165$    (D) $1625$

✔️ Answer: (C) 165

Year: 2023 | Set-1 | Code: 30/3/1 | Marks: 1

Q: Two positive numbers are in the ratio $2 : 3$. If their LCM is $180$, then their HCF is :
(A) $30$    (B) $60$    (C) $90$    (D) $15$

✔️ Answer: (A) 30

Year: 2026 | Set-1 | Code: 30/2/1 | Marks: 1

Q: Let $p$ be a prime number and $k$ be a positive integer. If $p$ divides $k^2$, then which of the following is definitely true?
(A) $p$ divides $k$    (B) $p$ divides $\sqrt{k}$    (C) $p^2$ divides $k$    (D) None of these

✔️ Answer: (A) p divides k

Year: 2024 | Set-1 | Code: 30/1/1 | Marks: 1

Q: If $p$ is a prime number, then $\sqrt{p}$ is always :
(A) a rational number    (B) an irrational number    (C) an integer    (D) a composite number

✔️ Answer: (B) an irrational number

Year: 2020 | Set-1 | Code: 30/1/1 | Marks: 1

Q: The decimal expansion of $\frac{23}{2^2 \times 5^3}$ terminates after how many decimal places?
(A) $1$    (B) $2$    (C) $3$    (D) $4$

✔️ Answer: (C) 3

Year: 2019 | Set-3 | Code: 30/3/3 | Marks: 1

Q: The decimal expansion of the rational number $\frac{14587}{1250}$ terminates after how many decimal places?
(A) $1$    (B) $2$    (C) $3$    (D) $4$

✔️ Answer: (D) 4

Year: 2018 | Set-1 | Code: 30/1/1 | Marks: 1

Q: If $p_1$ and $p_2$ are two odd prime numbers such that $p_1 > p_2$, then $p_1^2 - p_2^2$ is always :
(A) an even number    (B) an odd number    (C) an odd prime number    (D) a prime number

✔️ Answer: (A) an even number

Year: 2017 | Set-1 | Code: 30/1/1 | Marks: 1

Q: If the HCF of $65$ and $117$ is expressible in the form $65m - 117$, then the value of $m$ is :
(A) $1$    (B) $2$    (C) $3$    (D) $4$

✔️ Answer: (B) 2

Year: 2016 | Set-1 | Code: 30/1 | Marks: 1

Q: The largest number which divides $70$ and $125$, leaving remainders $5$ and $8$ respectively, is :
(A) $13$    (B) $65$    (C) $875$    (D) $1750$

✔️ Answer: (A) 13

Year: 2015 | Set-1 | Code: 30/1 | Marks: 1

Q: If two positive integers $p$ and $q$ are written as $p = a \times b^2$ and $q = a^3 \times b$, where $a, b$ are prime numbers, then $\text{HCF}(p, q)$ is :
(A) $a \times b$    (B) $a^2 \times b^2$    (C) $a^3 \times b^2$    (D) $a^2 \times b$

✔️ Answer: (A) $a \times b$

Year: 2015 | Set-1 | Code: 30/1 | Marks: 1

Q: If two positive integers $p$ and $q$ are written as $p = a \times b^2$ and $q = a^3 \times b$, where $a, b$ are prime numbers, then $\text{LCM}(p, q)$ is :
(A) $a \times b$    (B) $a^2 \times b^2$    (C) $a^3 \times b^2$    (D) $a \times b^2$

✔️ Answer: (C) $a^3 \times b^2$

Year: 2020 | Set-1 | Code: 30/1/1 | Marks: 1

Q: The LCM of the smallest two-digit composite number and the smallest composite number is :
(A) $12$    (B) $4$    (C) $20$    (D) $44$

✔️ Answer: (C) 20

Year: 2018 | Set-1 | Code: 30/1/1 | Marks: 1

Q: The HCF of the smallest prime number and the smallest composite number is :
(A) $1$    (B) $2$    (C) $4$    (D) $3$

✔️ Answer: (B) 2

Year: 2016 | Set-1 | Code: 30/1 | Marks: 1

Q: The product of a non-zero rational and an irrational number is always :
(A) rational    (B) irrational    (C) integer    (D) natural number

✔️ Answer: (B) irrational

Year: 2021 | Set-1 | Code: 30/1/1 | Marks: 1

Q: The exponent of $2$ in the prime factorisation of $144$ is :
(A) $4$    (B) $5$    (C) $3$    (D) $6$

✔️ Answer: (A) 4

Year: 2017 | Set-1 | Code: 30/1/1 | Marks: 1

Q: The HCF of $135$ and $225$ is :
(A) $15$    (B) $30$    (C) $45$    (D) $75$

✔️ Answer: (C) 45

Year: 2022 | Set-1 | Code: 30/3/1 | Marks: 1

Q: If $a = 2^2 \times 3^3 \times 5^4$ and $b = 2^3 \times 3^2 \times 5$, then $\text{HCF}(a, b)$ is equal to :
(A) $180$    (B) $360$    (C) $540$    (D) $90$

✔️ Answer: (A) 180

Year: 2020 | Set-1 | Code: 30/1/1 | Marks: 1

Q: The LCM of $12, 15$ and $21$ is :
(A) $420$    (B) $210$    (C) $140$    (D) $35$

✔️ Answer: (A) 420

Year: 2021 | Set-2 | Code: 30/1/2 | Marks: 1

Q: The product of two numbers is $1600$ and their HCF is $5$. The LCM of the numbers is :
(A) $8000$    (B) $1605$    (C) $320$    (D) $315$

✔️ Answer: (C) 320

Year: 2022 | Set-1 | Code: 30/3/1 | Marks: 1

Q: If $p$ is a prime number, then HCF of $p$ and $p + 1$ is :
(A) $p$    (B) $1$    (C) $p + 1$    (D) $p(p+1)$

✔️ Answer: (B) 1

Year: 2023 | Set-2 | Code: 30/3/2 | Marks: 1

Q: The HCF of two co-prime numbers $a$ and $b$ is :
(A) $a \times b$    (B) $1$    (C) $a + b$    (D) $0$

✔️ Answer: (B) 1

Year: 2019 | Set-1 | Code: 30/1/1 | Marks: 1

Q: If $\text{LCM}(x, 18) = 36$ and $\text{HCF}(x, 18) = 2$, then the value of $x$ is :
(A) $2$    (B) $3$    (C) $4$    (D) $1$

✔️ Answer: (C) 4

Year: 2015 | Set-1 | Code: 30/1 | Marks: 1

Q: The number $\pi$ ($3.14159\dots$) is a/an :
(A) rational number    (B) irrational number    (C) integer    (D) natural number

✔️ Answer: (B) irrational number

Year: 2018 | Set-1 | Code: 30/1/1 | Marks: 1

Q: The number $3.\overline{27}$ is :
(A) an integer    (B) a rational number    (C) an irrational number    (D) a natural number

✔️ Answer: (B) a rational number

Year: 2025 | Set-1 | Code: 30/1/1 | Marks: 1

Q: The LCM of three numbers $2^3 \times 3$, $2 \times 3 \times 5$, and $3 \times 5 \times 7$ is :
(A) $3780$    (B) $840$    (C) $120$    (D) $420$

✔️ Answer: (B) 840

Year: 2024 | Set-1 | Code: 30/1/1 | Marks: 1

Q: If $\text{HCF}(a, b) = 1$, then the numbers $a$ and $b$ are called :
(A) composite    (B) prime    (C) co-prime    (D) even

✔️ Answer: (C) co-prime

✍️ 2 Marks Questions (Short Answer-I)

Year: 2024 | Set-1 | Code: 30/1/1 | Marks: 2

Q: Prove that $5 - \sqrt{3}$ is an irrational number.

Year: 2020 | Set-1 | Code: 30/1/1 | Marks: 2

Q: Prove that $3 + 2\sqrt{5}$ is an irrational number, given that $\sqrt{5}$ is irrational.

Year: 2019 | Set-1 | Code: 30/1/1 | Marks: 2

Q: Find the HCF and LCM of $306$ and $657$.

✔️ Answer: HCF = 9, LCM = 22338

Year: 2020 | Set-1 | Code: 30/1/1 | Marks: 2

Q: Express $5005$ as a product of its prime factors.

✔️ Answer: 5005 = 5 × 7 × 11 × 13

Year: 2018 | Set-1 | Code: 30/1/1 | Marks: 2

Q: Find HCF of $96$ and $404$ by prime factorisation method. Hence, find their LCM.

✔️ Answer: HCF = 4, LCM = 9696

Year: 2017 | Set-1 | Code: 30/1/1 | Marks: 2

Q: If HCF of $144$ and $180$ is expressed in the form $13m - 16$, find the value of $m$.

✔️ Answer: m = 4

Year: 2016 | Set-1 | Code: 30/1 | Marks: 2

Q: Check whether $6^n$ can end with the digit $0$ for any natural number $n$.

✔️ Answer: Does not end with 0

Year: 2015 | Set-1 | Code: 30/1 | Marks: 2

Q: Check whether $4^n$ can end with the digit $0$ for any natural number $n$.

✔️ Answer: Does not end with 0

Year: 2019 | Set-3 | Code: 30/5/3 | Marks: 2

Q: Find the HCF of $612$ and $1314$ using prime factorisation.

✔️ Answer: HCF = 18

Year: 2020 | Set-1 | Code: 30/1/1 | Marks: 2

Q: Explain why $7 \times 11 \times 13 + 13$ is a composite number.

✔️ Answer: Composite because it has factors other than 1 and itself

Year: 2021 | Set-1 | Code: 30/1/1 | Marks: 2

Q: Find LCM and HCF of $120$ and $144$ using the Fundamental Theorem of Arithmetic.

✔️ Answer: HCF = 24, LCM = 720

Year: 2025 | Set-1 | Code: 30/1/1 | Marks: 2

Q: Given that $\sqrt{2}$ is irrational, prove that $(5 + 3\sqrt{2})$ is an irrational number.

📝 3 Marks Questions (Short Answer-II)

Year: 2026 | Set-1 | Code: 30/2/1 | Marks: 3

Q: Prove that $\sqrt{3}$ is an irrational number.

Year: 2024 | Set-1 | Code: 30/1/1 | Marks: 3

Q: Prove that $\sqrt{5}$ is an irrational number.

Year: 2023 | Set-1 | Code: 30/1/1 | Marks: 3

Q: Prove that $\sqrt{2}$ is an irrational number.

Year: 2022 | Set-1 | Code: 30/3/1 | Marks: 3

Q: Prove that $\sqrt{7}$ is an irrational number.

Year: 2015 | Set-1 | Code: 30/1 | Marks: 3

Q: Two tankers contain $850$ litres and $680$ litres of petrol respectively. Find the maximum capacity of a container which can measure the petrol of either tanker in exact number of times.

✔️ Answer: 170 litres

Year: 2017 | Set-1 | Code: 30/1/1 | Marks: 3

Q: A sweet shopkeeper prepares $396$ gulab jamuns and $342$ rasgullas. He packs them into containers. Each container consists of either gulab jamuns or rasgullas but must have equal number of pieces. Find the minimum number of containers required.

✔️ Answer: 41 containers

Year: 2016 | Set-1 | Code: 30/1 | Marks: 3

Q: In a seminar, the number of participants in Hindi, English, and Mathematics are $60, 84$, and $108$ respectively. Find the minimum number of rooms required if in each room the same number of participants are to be seated and all of them being in the same subject.

✔️ Answer: 21 rooms

Year: 2018 | Set-1 | Code: 30/1/1 | Marks: 3

Q: Find the largest number which divides $318$ and $739$ leaving remainders $3$ and $4$ respectively.

✔️ Answer: 15

📊 4 Marks Questions (Case Study/Passage)

Year: 2023 | Set-1 | Code: 30/3/1 | Marks: 4

Q: Three bells toll together at intervals of $9, 12$, and $15$ minutes respectively. If they toll together now, answer the following:
(i) After how many hours will they toll together next?
(ii) How many times will they toll together in $36$ hours (excluding the start)?

✔️ Answer: (i) 3 hours, (ii) 12 times

Year: 2024 | Set-1 | Code: 30/1/1 | Marks: 4

Q: To enhance the reading habits of Class X students, a school manages a library. There are two sections - Section A and Section B. Section A has $32$ students and Section B has $36$ students. Answer the following:
(i) Find the minimum number of books required for their class library so that they can be distributed equally among students of Section A or Section B.
(ii) If the product of two positive integers is equal to the product of their HCF and LCM, find $\text{HCF}(32, 36)$.

✔️ Answer: (i) 288 books, (ii) 4

Year: 2025 | Set-1 | Code: 30/1/1 | Marks: 4

Q: A circular track is around a sports field. Sonia takes $18$ minutes to drive one round of the field, while Ravi takes $12$ minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. Answer the following:
(i) After how many minutes will they meet again at the starting point?
(ii) If another cyclist, Amit, takes $15$ minutes to complete one round, after how many minutes will all three meet together at the starting point?

✔️ Answer: (i) 36 minutes, (ii) 180 minutes

Year: 2020 | Set-1 | Code: 30/1/1 | Marks: 4

Q: 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. Answer the following:
(i) What is the maximum number of columns in which they can march?
(ii) What is the fundamental theorem of arithmetic statement applied here?

✔️ Answer: (i) 8 columns, (ii) Prime Factorisation used for HCF

🏆 5 Marks Questions (Long Answer)

Year: 2019 | Set-1 | Code: 30/1/1 | Marks: 5

Q: Prove that the square of any positive integer cannot be of the form $5m + 2$ or $5m + 3$ for any integer $m$.