Real Numbers: Class 10 Important Questions with Answers

HOTS (higher-order) practice

H1. Show that $n^2-1$ is divisible by 8 when $n$ is an odd positive integer.

H2. The HCF of two numbers is 23 and their LCM is 1449. If one number is 161, find the other.

Case-based questions (4 marks each)

20 MCQs (1 mark each)

1. The prime factorisation of 156 is
   (a) $2\cdot 3\cdot 13$ (b) $2^2\cdot 3\cdot 13$ (c) $2^2\cdot 39$ (d) $4\cdot 3\cdot 13$
2. HCF of 12 and 18 is
   (a) 2 (b) 3 (c) 6 (d) 36
3. LCM of 15 and 20 is
   (a) 5 (b) 30 (c) 60 (d) 300
4. If HCF(a, b) = 4 and ab = 96, then LCM is
   (a) 24 (b) 48 (c) 384 (d) 12
5. Which number can end with the digit 0 for some n?
   (a) $4^n$ (b) $6^n$ (c) $15^n$ (d) $20^n$
6. The HCF of two co-prime numbers is
   (a) 0 (b) 1 (c) their product (d) the smaller one
7. Which of these is irrational?
   (a) $\sqrt{16}$ (b) $\sqrt{0.09}$ (c) $\sqrt{12}$ (d) $\frac{22}{7}$
8. If $a=2^3\cdot 3$ and $b=2\cdot 3^2\cdot 5$, HCF(a, b) is
   (a) 6 (b) 12 (c) 30 (d) 360
9. For the $a,b$ of Q8, LCM(a, b) is
   (a) 180 (b) 360 (c) 720 (d) 120
10. $\sqrt{2}\times \sqrt{8}$ is
    (a) irrational (b) 4 (c) $2\sqrt{2}$ (d) $\sqrt{10}$
11. The least number divisible by 2, 3, 4, 5 and 6 is
    (a) 30 (b) 60 (c) 120 (d) 720
12. If 3 divides $p^2$ ($p$ an integer), then
    (a) 9 divides p (b) 3 divides p (c) p is prime (d) p = 3
13. $2+\sqrt{5}$ is
    (a) rational (b) irrational (c) an integer (d) a natural number
14. HCF × LCM of 25 and 40 is
    (a) 200 (b) 1000 (c) 40 (d) 5
15. The largest number dividing 70 and 125 leaving remainders 5 and 8 is
    (a) 13 (b) 65 (c) 875 (d) 1750
16. $7\times 11\times 13+13$ is
    (a) prime (b) composite (c) neither (d) irrational
17. $12^n$ ends with the digit 0 for
    (a) n = 1 (b) n = 5 (c) every even n (d) no natural n
18. The product of a non-zero rational and an irrational number is
    (a) always rational (b) always irrational (c) an integer (d) zero
19. Two numbers are $2x$ and $3x$ with LCM 72. Their HCF is
    (a) 6 (b) 12 (c) 24 (d) 36
20. The exponent of 2 in the prime factorisation of 144 is
    (a) 2 (b) 3 (c) 4 (d) 6

Answer key

1 (b) · 2 (c) · 3 (c) · 4 (a) · 5 (d) · 6 (b) · 7 (c) · 8 (a) · 9 (b) · 10 (b) · 11 (b) · 12 (b) · 13 (b) · 14 (b) · 15 (a) · 16 (b) · 17 (d) · 18 (b) · 19 (b) · 20 (c)

Hints: Q4 LCM = 96/4 = 24. Q14 HCF × LCM = 25 × 40 = 1000. Q19 LCM = 6x = 72 ⇒ x = 12. Q20 144 = 2⁴ × 3².

Assertion–Reason (10)

(a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true but R is not the correct explanation of A. (c) A is true but R is false. (d) A is false but R is true.

1. A: HCF(18, 24) × LCM(18, 24) = 432. R: 18 and 24 are both even numbers.
2. A: $\sqrt{9}$ is a rational number. R: 9 is a perfect square, so $\sqrt{9}=3$.
3. A: $6^n$ can never end with 0. R: A number ending in 0 has both 2 and 5 as prime factors.
4. A: HCF(7, 11) = 1. R: 7 and 11 are both odd numbers.
5. A: $5+\sqrt{3}$ is irrational. R: The sum of a rational and an irrational number is irrational.
6. A: For 12, 15 and 21, HCF × LCM ≠ 12 × 15 × 21. R: HCF × LCM = product holds only for two numbers.
7. A: 1 is a prime number. R: A prime number has exactly two distinct factors.
8. A: If LCM(a, 18) = 36 and HCF(a, 18) = 2, then a = 4. R: HCF × LCM = product of two numbers.
9. A: $\sqrt{2}\times \sqrt{2}$ is rational. R: The product of two irrational numbers is always irrational.
10. A: 3825 = $3^2\times 5^2\times 17$ is its only prime factorisation. R: The Fundamental Theorem of Arithmetic says the factorisation is unique apart from order.

Key: 1 (b) · 2 (a) · 3 (a) · 4 (b) · 5 (a) · 6 (a) · 7 (d) · 8 (a) · 9 (c) · 10 (a)

Very short answer (1 mark each)

1. Write the HCF of the smallest prime and the smallest composite number. Ans. HCF(2, 4) = 2
2. If $p$ and $q$ are distinct primes, find LCM(p, q). Ans. pq
3. Express 98 as a product of primes. Ans. $2\times 7^2$
4. Is $0.\overline{3}$ rational or irrational? Ans. Rational (= 1/3)
5. Give one irrational number between 2 and 3. Ans. $\sqrt{5}$ (or $\sqrt{7}$, $\sqrt{8}$)
6. If HCF(a, b) = 1, what is LCM(a, b)? Ans. ab
7. Find the sum of the exponents in $360=2^3\cdot 3^2\cdot 5$. Ans. 6
8. If LCM(26, 169) = 338, find HCF(26, 169). Ans. 26 × 169 ÷ 338 = 13

Short answer practice (2 marks each)

1. Find the HCF and LCM of 404 and 96, and verify HCF × LCM = product.

2. Can $(12{)}^n$ end with the digit 0? Justify.

3. Prove that $3\sqrt{2}$ is irrational, given that $\sqrt{2}$ is irrational.

4. Three bells toll at intervals of 9, 12 and 15 minutes. If they toll together at 8 a.m., when will they next toll together?

Short answer practice (3 marks each)

5. Prove that $\sqrt{7}$ is irrational.

6. Find the largest number that divides 2053 and 967 and leaves remainders 5 and 7 respectively.

7. A shopkeeper has 120 litres of petrol, 180 litres of diesel and 240 litres of kerosene. He wants to sell them in cans of equal size, each can holding only one fuel. Find the greatest capacity of a can and the least number of cans needed.

8. Prove that $4-5\sqrt{2}$ is irrational, given that $\sqrt{2}$ is irrational.

Long answer (5 marks)

9. (i) Prove that $\sqrt{3}$ is irrational. (ii) Hence show that $2\sqrt{3}-1$ is irrational. (iii) Is $(2\sqrt{3}-1)(2\sqrt{3}+1)$ rational? Explain.